OPERATIONAL RESEARCH ASSIGNMENT............(CA FINAL COST AND OR)

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

LEARNING CURVE What is Learning curve? Learning curve is essentially a measure of the experience gained in production of an article by an organization. As more and more units are produced, workers involved in production become more efficient than before. Each subsequent unit takes fewer man-hours to produce. The Learning curve exists during a worker's startup or familiarization period on a particular job. After the limits of experimental learning are reached, productivity tends to stabilize and no further improvement is possible. -

learning curve measures the efficiency of labour (not machine) It is effective only for new jobs Once the job is familiarized, concept of learning curve is not effective st If learning curve is 80%, then if 1 unit is produced in 10 hours, two units will be produced in 16 hrs, i.e as output doubles average time per unit will be 80 % of original time. When 2 units are produced average time per unit is 8hrs. If 4 units are produced, average time per unit is (80% of 8 hrs) 6.4 hrs per unit. Thus total time for producing 4 units is 25.6 hrs.If 8 units are produced, average time per unit is (80% of 6.4 hrs) 5.12 hrs. Total time of producing 8 units is 40.96 hrs.

The nature of learning curve phenomenon can be described as a constant percentage reduction in the average direct labour input time required per unit as the cumulative output doubles (increases) LC model is the mathematical (graphical) portrayal of the decreasing rate of increase in the costs of labour whilst experience is being gained in a new task

Y

Cumulative average time

Q

0

N o. O f units

X

L e a rn in g C u rv e According to the graph, as production increases, average time per unit of output decreases. But this reduction in average labour time per unit gradually diminishes. When total production is large enough the time per unit becomes quite stable. The learning curve thus tapers off and gradually ceases to influence the time taken to perform the task. The learning curve ratio can be calculated with the help of the following formula: Learning curve ratio = Average labour cost of first 2N units Average labour cost of first N units

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Q

Enlist the applications of learning curve ? Following are the areas where the effects of learning curve would be useful to decision making: a. Setting up of standard costssince in the learning phase average time per unit is known due to learning curve , it helps in setting standards in this phase. Once the job is familiarized standard cost can be set without the application of learning curve model as time per unit will remain same. b. Pricing decisions- Since learning curve permits better cost predictions, it should be employed in cost predictions c.

Capital budgeting- Most important aspect of capital budgeting is cash flows. Learning curve suggests that unit costs are likely to begin high and reduce afterwards. It permits improved estimates of production levels that can be attained and thus has implications on cash flows.

d. Training- once it is known that gradual experience on job will reduce average time per unit o production, thereby average cost per unit, it is better to provide intensive training to workers to save the initial increased cost and time. Training will lead to greater scope of learning and larger reduction in inputs in successive attempts. e. Marketing- the study of learning curve helps in sales projections and planning for advertisement, delivery schedules to coincide with expected production schedules. f.

Direct costs- Learning curve applies to an industry where there is high labour turnover or when products and process are subject to frequent changes. A knowledge of learning curve helpps in direct labour cost budget, as the labour hours or cost is reduced for repeat orders.

g. work scheduling - Learning curve increases a firms ability to predict their required labour input and make it possible to forecast labour needs.

Q

Enumerate limitations of Learning curve?

N11

1. All activities are not subject to learning effect. It measures only efficiency of labour and not any other element of production (eg machine). It applies only to new labour or new operations. 2. It involves careful and tedious analysis of data ana valid data for computation of learning effect is not easily available. 3. Learning curve is affected by factors other than adaptation of workers of a new jo.Such other factors may be in the form of motivation of employees, improvement of facilities, arrangements and equipment. 4. Any alterations in the existing circumstances renders learning curve obsolete. 5. According to learning curve effect average time per unit decreases as production increases initially, but it is very unlikely that there will be a regular consistent rate of decrease. Thus cost predictions based on conventional Learning curve are unreliable. 6. Learning curve assumes that production will continue without any interruptions. If for any reason work is interrupted the curve will change its shape.

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Q

Explain distinctive features of learning curve theory in manufacturing envoirment

N10

The production quantity of a given item doubled the cost of that item decrease at a fixed rate. This phenomenon is the basic premise on which the theory of learning curve has been formulated. The distinctive features of a learning curve are: 1. Better tooling methods are developed and used 2. More productive equipments are designed and used to make the product. 3. Design bugs are detected and corrected. 4. Better design engineering reduces material and labour costs. 5. Early teething problems are overcome. As production progresses management is prompted to achieve better planning and better management. 6. Rejections and rework tend to diminish over time. 7. As quantity produced increases the Cost per unit decreases, since each unit entails: (i) Lesser labour (Ii) Greater productivity of material and labour (III) Fewer delays and lesser time losses.

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Q1

The direct labour hours to assemble the first unit of a new equipment is Rs 400. Assuming that this type of rd th assembly will experience a learning effect of 90%. Compute the average direct labour or 3 and 4 units and th th th th also for the 5 to 8 units (For p = 90%, take b = -1520). Also calculate the average labour for 6 and 7 units.

Q2

A first batch of 25 transistors took a total of 20 direct labour hours. It s proposed to assemble another 40 units. What will be the average labour per unit in this lot?. Assume that there is 85% learning rate.

Q3

A company has found that the average direct labour just after completion of X units was 26.4 hours. The average at the end of first unit was 52 hours. If there is a learning curve effect of 85%, what has been the total output to the date?

Q4

A company which has developed a new machine has observed that the time taken to manufacture the first machine is 600 hours. Calculate the time which the company will take to manufacture the second machine if the actual learning curve rate is (i) 80% and (ii) 90%. Explain which of the two learning rates will show faster learning. N08

Q5

A firm has received an order to make and supply 8 units of a standard product, which involves intricate labour operations. The first unit was made in ten hours. It is understood that this type of operations is subject to 80% learning effect. The workers are paid a wage rate of Rs 12 per hour. 1. What is the total time and labour cost required to execute the above order? 2. If repeat order of 24 units is also received from the same customer, What is the labour cost necessary for the second order?

Q6

A company has made 6,000 units of a product. The labour to make each 1000 unit is as follows – Unit No. (000’s) 1 2 3 4 5 6 Labour Content (person- hour) 385 344 325 310 301 292 1. Estimate the Learning rate for this product using 1000 units of the product as the unit of production. 2. Predict the labour requirements to make the next 2,000 units.

Q7

Illustrate the use of Learning curve for calculating the expected average unit cost of making: (a) 4 machines (b) 8 machines Using the data given below: Direct labour needed to make the first machine = 1000 hours Learning curve ratio = 90% Direct labour cost = Rs 15 per labour hour Direct material cost per unit = Rs 1,50,000

Q8

XYZ Co. has observed that a 90% learning curve ratioapplies to all labour related costs each time a new product enters production. It is anticipated that 320 units will be manufactured during 2002. Direct labour cost for the first lot of 10 units amount to 1000 hours@ Rs 8 per hour. Variable overhead cost is assigned to product @ Rs 2 per direct labour hour. Determine: (i) (ii) (iii)

Total labour and labour related costs to manufacture 320 units of output. Average cost of firstv(a) 40 (b) 80 (c) 100 units produced Incremental cost of (a) units 41-80 and (b) units 101-200

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Q9

Q10

A customer has asked your company to prepare a bid on supplying 800 units of a new product. Production will be in batches of 100 units. You estimate that the cost for the first batch of 100 units will average Rs 100 a unit. You also expect that a 90% learning curve will apply to the cumulative labour cost on this contract. Required: (a) Prepare an estimate of the labour costs of fulfilling this contract. (b) Estimate the incremental labour cost of extending the production run to produce an additional 800 units. (c) Estimate the incremental labour cost of extending the production run from 800 units to 900 units. XYZ and company has given the following data: Learning curve ratio 80% No of units 1 2 3 4

Average hours 100 80 ? 64

Total hours 100 160 ? 256

Marginal hours 100 60 ? ?

You are required to fill in the blanks. b

Q11

The learning curve model is Y=ax where ‘’Y’’ is the average time per unit for X units, ‘’a’’ is the time for the first unit, ‘’x is the cumulative number of units and ‘’b’’ is the learning coefficient. Taking ‘’b’’ = (Log 0.8 ‚ Log 2) = 0.322 for a learning rate is 80% and ‘’a’’ = 10 hours, calculate- (a) Average time for 20 units, (b) Total time for 30 units, and (c) Time for units 31 to 40. Given that Log 2 = 0.3010 Antilog of 0.5811 = 3.812 Log 3 = 0.4771 Antilog of 0.5244 = 3.345 Log 4 = 0.6021 Antilog of 0.4841 = 3.049.

Q12

Boeing is an aircraft manufacturer that has introduced a new airplane model. It expects the plane to cost Rs 1500 lakhs to manufacture (not including development cost) but their products have historically exhibited in 85% experience curve, Assuming the same experience rate will apply to this product, compute the cost of computing the each of the first five planes, and yhe average cost per plane to produce the first 25 planes RTP

Q13

Havoc company developed and manufactured a new machine. The manufacture of the first machine took 800 direct labour hours. The direct wage rate is Rs 20 per hour. The company experiences a learning curve effect of 80% (index is -0.3219). The first piece was used as a demonstration piece and was not intended for sale. On the basis of demonstration , the company obtained an order for the manufacture of 20 machines. The direct Material cost is Rs 16,000 per machine. The variable overhead rate is Rs 25 per direct labour hour. The fixed overheads on absorbtion costing amounted to Rs 40 per direct labour hour. The selling price is to include a profit margin of 20% on selling price. Subsequently , after the delivery of the 20 machines, the company receives a repeat order for supply of 30 machines. I. II.

Calculate the Selling price per machine of the first lot of 20 machines. What reduction in Selling price can the company allow in respect of the repeat order

5

RTP

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Q14

A company has 10 direct workers, who work for 25 days a month of 8 hours per day. The estimated downtime is 25% of the total available time. The company received an order for a new product. The first unit of the new product requires 40 direct labour hours to manufacture the product. The company expects 80% (index is – 0,322) learning curve for this type of work. The company uses standard absorbtion costing and the cost data are as under: Direct Material Rs 60 per unit Direct labour Rs 6 per direct labour hour Variable overheads Rs 1 per direct labour hour Fixed overheads Rs 7,500 per month Required: (a) Calculate the cost per unit of the first order of 30 units. (b) If the company receives a repeat order for 20 units, what price will be quoted to yield a profit of 25% on selling price? N02

Q15

An electronics firm which has developed a new type of fire alarm system has been asked to quote for a prospective contract. The customer requires separate price quotations for each of the following possible ordersOrder Number of fire alarm systems

First 100

Second 60

Third 40

The Firm estimates the following cost per unit for the first orderDirect Materials Direct Labour: Department A (Highly automatic) Department B (skilled Labour) Variable overheads Fixed overheads Department A Department B

Rs 500 20 hours @ Rs 10 per hour 40 hours @ Rs 15 per hour 20% of direct labour Rs 8 per hour Rs 5 per hour

Determine a price per unit for each of the 3 orders, assuming the Firm uses a mark-up of 25% on total costs and allows for an 80% Learning curve. Extract from 80% Learning curve table M05. X Y

1.0 100,0

1.3 91.7

1.4 89.5

1.5 87.6

1.6 86.1

1.7 84.4

1.8 83.0

1.9 81.0

2.0 80.0

X represents the cumulative total volume produced to date expressed as a multiple of the initial order Y is the Learning curve factor, for a given X value, expressed as a percentage of the cost of the initial order. Q16

A company has just completed the manufacture of 40 units of a new product. The manufacturing costs areDirect materials 2,00,000 Direct labour hours: 8000 hours @ Rs 20 per hour 1,60,000 Variable overheads 80,000 Special tools (re-usable) 10,000 Fixed overheads apportioned 1,00,000 Total 5,50,000 The Company’s policy is to add a profit of 12% on selling price. The company received another order for 120 units of this product for which the company quoted, based on its policy on absorbtion cost basis, a price of rs 15,625 per unit. The customer struck the order to Rs 11,000 per unit. The Company is short of work and so is keen to take up more orders but it is reluctant to accept this order price because it is against the policy to accept any price before its cost. The company experiences a Learning curve of 90%. 1. Compute the gain or Loss arising from acceptance of the order of Rs 11,000 per unit. 2. Advise whether the company should accept this order for 120 units or not. RTP

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Q17

The Gifts company makes momentos for offering chief guests and other dignitaries at functions. A customer wants 4 identical pieces of hand-crafted gifts for 4 dignitaries invited to its function. For this product, the gifts company estimates the following costs for the first unit of the product Rs/unit Direct variable cost (excluding labour) 2,000 Direct labour (20 hours @ Rs 50 per hour) 1,000 90% Learning curve ratio is applicable and one labourer works foe one customer’s order. i. What is the price per piece to be quoted for this customer if the targeted contribution is Rs 1,500 per unit? ii. I 4 different labourers made the 4 products simultaneously to ensure faster delivery to the customer, can the price at (i) above be quoted? Why? N09

Q18

PQ Ltd. Makes and sells a labour intensive product. Its labour force has a learning rate of 80%, applicable only to direct labour and not to variable overhead. The cost per unit of the first product is as follows: Direct materials 10,000 Direct labours 8,000 (@ Rs 4 per hour) Variable overheads 2,000 Total variable cost 20,000 PQ Ltd has received an order from X Ltd for 4 units of the product. Another customer, Y Ltd is also interested in purchasing 4 units of the product. PQ Ltd has the capacity to fulfill both the orders. Y Ltd presently purchases this product in the market for Rs 17,200 and is willing to pay this price per unit of PQ’s product. But X Ltd lets PQ choose one of the following options: i.

A price of Rs 16,500 per unit for the 4 units it proposes to take from PQ Or ii. Supply X Ltd’s idle labour force to PQ for only 4 units of production, with PQ having to pay only Re 1 per labour hour to X Ltd workers. X Ltd’s workers will be withdrawn after the first 4 units are produced. In this case, PQ need not use its labour for producing X Ltd’s requirement. X Ltd assures PQ that its labour force also has a learning rate of 80%. In this option, X Ltd offers to buy the product from PQ at only Rs 14,000 per unit. X and Y shall not know of each others offer. If both orders came before any work started, what is the best option that PQ may choose? Present suitable calculations in favour of your argument. J09 Q19

M Ltd manufactures a special product purely carried out by manual labour. It has a capacity of 20,000 units. It estimates the following cost structure: Direct Material Rs 30/ unit Direct labour (1 hour / unit) Rs 20 per unit Variable overhead Rs 10 per unit Fixed overheads at maximum capacity are Rs 1,50,000 It is estimated that at the current level of efficiency, each unit requires one hour for the first 5,000 units. Subsequently it is possible to achieve 80% learning rate. The market can absorb the first 5,000 units at Rs 100 per unit. What should be the minimum selling price acceptable for an order of 15,000 units for a prospective client? M08

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Q20

A company with two production departments has set the following standard for the forthcoming year:-

Direct labour hours available per period Standard wage rate per hour Expected Learning curve Standard variable overheads per hour Standard fixed overheads per hour Direct labour hours required for first unit in lot of 100 units

Department S 6,000 Rs 6 80% Rs 9 Rs 12 18

Department W 4,000 Rs 5 70% Rs 5 Rs 8 9

The direct materials are introduced in Department S. The company is able to negotiate the following prices for purchase of direct materials during the year. Level of output (units)

Price of direct materials per Unit of output 100 Rs 72.00 200 Rs 64.80 800 Rs 54.00 Overtime, if required is paid at time and a half. The overhead rates as given above does not include overtime premium. It is policy of the company to add profit margin as under in quoting the prices: Department %age on total labour and overhead cost S 25% W 15% Subcontracted work 5% on subcontract price The company has received a special order. Special tooling costs of the order amounts to rs 1200. If this order is for 200 units or less, it will be executed in the period which has a workload of 3840 direct labour hours in Department S and 2100 direct labour hours in Department W. For the work which is done in Department W, a subcontract, an associate company quotes price of Rs 50 per unit. Required: (1) If the company decided to get the work executed entirely within the company, what price, on cost plus basis, should be quoted for the order, if it consists of 100, 200 units (2) Assuming that the initial order placed by the customer is for 200 units. What lowest price should be quoted for a repeat order of 600 units? Assume that this order will be executed when there are no capacity constraints. (3) State the output level at which the company should close down Department W to get the work executed through subcontractors

8

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Q21

SV Ltd. Which has a fairly full order book is approached by a customer with the offer of a contract for a model that is a variant, in terms of dimensions and materials used, of one of its existing products. Though the customer expects to pay a normal price for the model he wants SV Ltd to take account of an 80% Learning curve in its price calculation; this level has been shown to be reasonable in SV Ltd’s industry for relevant work. The prospective contract is for a total of 464 units made up of an initial order of 160 units, two subsequent orders of 80 units each and three subsequent orders of 48 units each. SV Ltd estimates the following costs for the initial order. Direct materials: P- 8 meter At Rs 3.50 per meter Q-12 kg At Rs 1.00 per kg Direct wages Departments Hours Rs per hour 1 100 2.00 2 320 3.00 3 160 1.00 Variable overheads: 20% of Direct wages. Fixed overheads Department Recovery rate per Hour (Rs) Department 1 Rs 2 per hour 2 Rs 1 per hour 3 Rs 2 per hour The nature of work in 3 production departments is as follows Department 1 uses highly automatic machines. Although the operators on these machines need to be fairly skilled, their efficiency only affects the quality of the work but can have little impact on the quantity of his departments output which is largely machine controlled. Department 2 and 3 the skill of operators is a major determinant of the volume of output. You are required to calculate the cost per unit for: a. The initial order of 160 units b. The second, third and fourth orders, if given successively but without guarantee of further orders and c. The whole contract of six orders if given from the start but on the same basis of production and delivery. Note:- An 80% learning curve on ordinary graph paper would show the following relation ship between the X axis (volume) and Y axis ( Cumulative average price of elements subjects to the learning curve) X Y%

1.0 100.00

1.1 96.0

1.2 93.3

1.3 91.7

1.4 89.5

1.5 87.6

1.6 86.1

1.7 84.4

1.8 83.0

1.9 81.5

2.0 80.0

X Y%

2.1 78.9

2.2 77.8

2.3 76.8

2.4 76.0

2.5 74.9

2.6 74.0

2.7 73.2

2.8 72.3

2.9 71.5

3.0 70.7

3.1 70.0

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LINEAR PROGRAMMING

Q

Define linear programming

M95

Linear programming deals with the situations in which activities compete for limited resources, along with restrictions and constraints of markets, to attain the given objective in a situation in which the constraints (restrictions) and objective can be expressed as linear mathematical functions. Thus Linear programming is a mathematical technique for optimum allocation of scarce resources, which may be in the form of limited raw materials, labour hours, machine time, capital, market demand, to achieve the specified objective which may be of cost minimization or profit maximization. In order to apply Linear programming , there are certain requirements to be met, these are a. There should be a defined objective which can be measured in quantitative terms. It is denoted by function Z. It should be expressed as linear function of decision variables. b. The activities should be distinctly identifiable and measurable in quantitative terms. c. The resources should be limited, distinctively identifiable and measurable in quantitative terms d. The objective function and constraint equations should be linear in nature. Q

Enlist the assumptions under Linear programming Linear programming model is based on following assumptions: a. Proportionality- It exists in the objective function and constraint inequalities - Proportionate change in output will lead to proportionate change in Profits Eg if profit of one unit is Rs 4, then total profit is 4x1 , where x1 is the number of units of product. If 10 units are produced profit is Rs 40 and for 20 units profit is Rs 80. -

Proportionate change in output will lead to proportionate change in allocation and consumption of resources Eg if one unit takes 5 hours of labour, 10 units consume 50 hours and 20 units consume 100 hours.

b. Additivity- sum total of each activity conducted separately defines objective function and constraint inequalities. Eg total profit in objective function is determined by the sum of profit contributed by each of the products separately Total resource used is sum total of resources used by each activity separately. c.

Continuity- activities are continuous in time i.e after attaining optimum combination of the alternative activities in attaining the given objective, such activities will be carried on in next period.(fractional production is possible, fractional output will be carried forward as WIP and completed in next period)

d. Certainity- The coefficients of objective function and constraints are known with certainity i.e profit per unit and consumption of resource per unit is known and constant

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e. Non negative values – The optimum solution variables cannot be negative i.e the production of any product cannot be negative in optimum solution

Q

Enumerate the applications of Linear programming

M07, M03

Linear programming techniques has following applications a. Industrial Applications - Product mix - Blending applications - Production planning - Trim applications b. Administrative Applications - Personnel assignment c.

Product distribution - Transportation applications

d. Marketing Applications - Advertising Mix problems e. Financial Applications - Investment portfolio f.

Operational scheduling Applications - Flight scheduling applications

g. Agriculture applications - Fertilizer allocation h Q

Diet ascertainment

Write short notes on limitation of linear programming

M98, M04, N07

1. Linear programming can ascertain the optimal solution to the problem only if constraints inequalities and objective function can be expressed as linear mathematical functions. Practically such linear relation does not always exist. 2. It is assumed that all constraints to production are identified and measured in quantitative terms, whereas all constraints may not be quantified. So optimum solution ascertained by considering quantifiable constraints may render the objective ineffective. 3. In LP coefficients of the objective function must be known and certain and should remain constant during the given period, whereas practically in the open market any variable may change giing uncertainity to such coefficients 4. In LPP fractional values are permissible in optimum solution. Rounding off fractional values may not provide optimal solution. 5. Constraints are not always fixed. Some constraints can be removed by incurring additional costs.

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6. Simplex method is difficult to understand and involve tedious calculations, making the planning process expensive 7. Graphical method is effective only in case of two variables

Q

Write short notes on Slack, Surplus and artificial variable Slack variable To ascertain the optimal solution by simplex method all constraint inequalities are to be converted into equalities. To convert ‘’Less than’’ inequalities into equalities, slack variable is added to equations. Slack variable represents the idle or unused resource. The coefficient of such slack variable in constraint is Positive (+1), it cannot be negative and in the objective function coefficient of slack variable is 0. It helps in finding the solution by locating the identity matrix Surplus variable To ascertain the optimal solution by simplex method all constraint inequalities are to be converted into equalities. To convert ‘’More than’’ inequalities into equalities, surplus variable is reduced from equations. It represents the excess amount of resources utilized. Since identity matrix cannot be located by these variables, so these variables does not help in finding solution independently. Coefficient of surplus variable in constraint is -1 and in objective function is 0. Artificial variable If the constraint is ‘’more than’’ or equality constraint, then solution cannot be obtained as identity matrix cannot be located. To find solution to the LPP, in case of such constraints artificial variables are introduced in the constraints. It is a fictitious varialble with no economic meaning. Coefficient of Artificial variable in constraint is +1 and in objective function it is –M in case of maximization and +M in case of minimization. Such variable will not appear in final solution. If the optimal table consist of Artificial variable, then LP has no feasible solution.

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Q1

A firm is engaged in producing two products, A and B. Each unit of product A requires 2 kg of raw material and 4 labour hours for processing, whereas each unit of product B requires :3 kg of raw material and 3 hours of labour of the same type: Every week, the firm has an availability of 60 kg of raw material and 96 labour hours. One unit of product A sold yields Rs 40 and one unit of product B sold gives Rs 35 as profit. ' Formulate this problem as a linear programming problem to determine as to how many units of each of the products should be produced per week so that the firm can eam the maximum profit. Assume that there is no marketing constraint so that all that is produced can be sold.

Q2

A manufacturer can produce two products, A and B, during a given time period. Each of these products requires. four different manufacturing operations: grinding, turning, assembly and testing. The manufacturing requirements in hours per unit of product are given below for A and B.

Grinding Turning Assembly Testing

A 1 3 6 5

B 2 1 3 4

The available capacities of these operation in hours for the given time period are: grinding 30; turning 60; assembly 200; testing 200. The contribution to profit is Rs. 2 for each unit of A and Rs. 3 for each unit of B. The firm can sell all that it produces at the prevailing market price. Determine the optimum amount of A and B to produce during the given time period. Q3

A toy manufacturer produces two types of dolls, A and B. Each doll of type B requires twice as much time as required by a doll of type A. If all dolls were of type A, the company could make 2000 dolls per day. The supply of plastic is sufficient to produce 1500 dolls per day and each type requires equal amount of it. Doll B requires a fancy dress and only 600 fancy dresses are available per day. If company makes a profit of Rs. 3 and Rs. 5 per doll, respectively, on doll A and B, how many of each should be produced per day in order to maximize profit? Formulate this as a linear programming problem and solve by graphical method.

Q4

A box manufacturer makes small and large boxes from a large piece of cardboard. The large boxes require 4 sq. ft. per box, while the small boxes require 3 sq. ft. per box. The manufacturer is required to make at least three large boxes and at least twice as many small boxes as large boxes. If 60 sq. ft. of cardboard is in stock, and if the profits on the small and large boxes are Rs. 2 and Rs. 3 per box, respectively, how many of each should be made in order to maximize the total profit? (Use graphic method).

Q5

Using Graphic method, find the maximum value of Z = 7x + 10y Subject to X + y ≥ 30,000 Y ≤ 12,000 X ≥ 6,000 X ≥y

Q6

Maximize Subject to

x,y≥ 0

Z = 2x1 + x2 – x3 x1 +x2 ≤1 X1 – 2x2 – x3 ≥ -2

x1, x2, x3 ≥ 0

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2nd yr 156

2nd yr 195

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Q7

A firm produces three products A, B, and C, each of which passes through three departments: Fabrication, Finishing and Packaging. Each unit of product A requires 3,4 and 2; a unit of product B requires 5, 4 and 4, while each unit of product C requires 2, 4 and 5 hours respectively in the three departments. Every day, 60. hours are available in the fabrication department, 72 hours in the finishing department and 100 hours in packaging department. The unit contribution of product A is Rs 5, of product B is Rs 10., and of product C is Rs 8. Required: (a) Formulate the problem as an LPP and determine the number of units of each of the products, that should be made each day to maximise the total contribution. Also determine if any capacity would remain unutilized. (b) If the optimal solution obtained does not require the production of some product, explain as to why such product would not be produced. In this context, indicate the quantity (quantities) of other product/s that would be foregone for producing such product. (c) What would be the effect on the solution of each of the following: (i) obtaining an order for 6 units of product"A, which has to be met. (ii) an increase of 20 percent capacity in the fabrication department.

Q8

A firm uses three machines in the manufacture of three products. Each unit of product A requires 3 hours on machine I, 2 hours on machine II and one hour on machine III. Each unit of product B requires 4 hours on machine I, one hour on machine II and 3 hours on machine III, while each unit of product C requires 2 hours on each of the three machines. The contribution margin of the three products is Rs 30, Rs 40 and Rs 35 per unit respectively. The machine hours available on three machines are 90, 54 and 93 respectively. (i) Formulate the above problem as a linear programming problem. (ii) Obtain optimal solution to the problem by using the simplex method. Which of the three products shall not by produced by the firm? Why? (iii) Calculate the percentage of capacity utilisation in the optimal solution. (iv) What are the shadow prices of the machine hours? (v) Is the optimal solution degenerate?

Q9

Minimize Subject to

Z = 40x1 + 24 x2

Total cost

20x1 + 50x2 ≥ 4800 80x1 + 50 x2 ≥ 7200 X1, x2 ≥ 0 Q10

Q11

Ashok Chemicals Company manufactures two chemicals A and B which are sold to the manufacturers of soaps and detergents. On the basis of the next month's demand, the management has decided that the total production for chemicals A and B should be at least 350 kilograms. Moreover, a major customer's order for 125 kilograms of product A must also be supplied. Product A requires 2 hours of processing time per kilogram and product B requires one hour of processing time per kilogram. For the coming month, 600 hours of processing time are available. The company wants to meet the above requirements at minimum total production cost. The production costs are Rs 2 per kilogram for product A and Rs 3 per kilogram for product B. Ashok Chemicals Company wants to determine its optimal product mix and the total minimum cost relevant thereto. (i) Formulate the above as a linear programming problem. (ii) Solve the problem with the simplex method. (iii) Does the problem have multiple optimal solutions? Why? Minimize Subject to

Z = - 4x1 + 3x2 X1 – 2x2 ≥ -4 2x1 + 3x2 ≥ 13 X1 – x2 ≥ 4

x1, x2 ≥ 0

14

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Q12

Maximise Subject to

Z = 8x1 – 4x2 4x1 + 5x2 ≤ 20 -x1 + 3x2 ≥ - 23

Q13

Minimize Subject to

x1 ≥ 0, x2 unrestricted in sign

Z = x1 + x2 + x3 X1 – 3x2 + 4x3 = 5 X1 – 2x2 ≤3 2x2 – x3 ≤4

Q14

Maximize Subject to

x1, x2 ≥ 0, x3 unrestricted in signs

Z = 4x1 + 5x2 – 3x3 X1 + x2 + x3 = 10 2x1 + 3x2 + x3 ≤ 30 X1 – x2 ≥1

Q15

Q16

x1, x2, x3 ≥ 0

The standard weight of a special purpose brick is 5 kg and it contains two basic ingredients Bl and B2. Bl costs Rs. 5 per kg and B2 costs Rs. 8 per kg. Strength considerations dictate that the brick contains not more than 4 kg of Bl and minimum of 2 kg of B2. Since the demand for the product is likely to be related to the price of the brick, find the minimum cost of brick satisfying the above conditions. Maximize Subject to

Z = x1 + 3x2 – 2x3 -X1 – 2x2 – 2x3 = -6 -x1 – x2 + x3 ≤ -2 X1, x2, x3 ≥ 0

Q17

Q18

Change the following LPP suitably so that an initial solution to it mav be obtained by applying the simplex method Maximize Z = 4x1 + 2x2 + 10x3 Subject to X1 + 3x2 – x3 = 11 3x1 + 2x2 + 4x3 = 54 2x1 + 5x2 + x3 = 39 x1, x2, x3 ≥ 0 Maximize Subject to

Z = 3x1 + 2x2

-x1 + 2x2 3x1 + 2x2 X1 – x2 Q19

Q20

≤4 ≤ 14 ≤3

x1, x2 ≥ 0

A company produces three products. P1, P2 and P3 from two raw materials A and 8, and labour L. One unit of product P1 requires one unit of A, 3 units of 8 and 2 units of L. One unit of product P2 requires 2 units of A and 8 each, and 3 units of L, while one unit of P3 needs 2 units of A, 6 units of 8 and 4 units of L. The company has a daily availability of 8 units of A, 12 units of 8 and 12 units of L. It is further known that the unit contribution margin for the products is Rs 3, 2 and 5 respectively for P1, P2 and P3. Formulate this problem as a linear programming problem, and then solve it to determine the optimum product mix. Is the solution obtained by you unique? Identify an alternate optimum solution, if any. Maximize Subject to

Z = 8x1 + 16x2 X1 + x2 X2 3x1 + 6x2

15

≤ 200 ≤ 125 ≤ 900

x1, x2 ≥ 0

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q21

Maximize Subject to

Z = 20x1 + 30x2 2x1 + x2 4x1 – x2 X1

Q22

Maximize Subject to

Maximize Subject to

Maximize Subject to

≤8 ≥ 36

≥ 16 ≥ 15

Maximize Subject to

Maximize Subject to

Maximize Subject to

Maximize Subject to

X1, x2 ≥ 0

≤ 18 ≤8 ≤ 30

X1, x2 ≥ 0

≤3 ≤4

x,y≥ 0

Z = 5x1 + 2x2 4x1 + 2x2 ≤ 16 3x1 + x2 ≤ 9 3x1 – x2 ≤ 9

Q29

≥ 24 ≥ -3

Z = 80x + 100y 20x + 30y 60x + 40y

Q28

X1, x2, x3 ≥ 0

Z = 28x1 + 30 x2 6x1 + 3x2 3x1 + x2 4x1 + 5x2

Q27

≤5 ≤2 ≤3

Z = x1 + x2 3x1 + 2x2 -x1 + 3x2

Q26

X1, x2 ≥ 0

Z = 5x1 + 6x2 + x3 9x1 + 3x2 – 2x3 4x1 + 2x2 – x3 X1 – 4x2 + x3

Q25

X1, x2 ≥ 0

Z = 10x1 + 20x2 2x1 + 4x2 X1 + 5x2

Q24

x1, x2 ≥ 0

Z = x1 + 2x2 X1 + x2 4x1 + 3x2

Q23

≤ 40 ≤ 20 ≥ 30

X1, x2 ≥ 0

A company makes two kinds of leather belts, A and B. Belt A is a high quality bell, and B is of lower quality. The respective profits are Rs. 4 and Rs. 3 per belt. Each belt of type A requires twice as much time as a belt of type B, and, if all belts were of type B, the company could make 1000 per day. The supply of leather is sufficient for only 800 belts per day (both A and B combined). Belt A requires a fancy buckle, and only 400 per day are available. There are only 700 buckles a day available for belt B. What should be daily production of each type of belt? Formulate the problem as a linear programming problem.

16

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q30

A company produces 2 types of leather belts - Type A and B. Contribution per belt is Rs. 4 for type A and Rs. 3 for type B. The time requirements of one belt of type A and type B are is the ratio 2 : 3. Time available is sufficient to produce 500 belts of type A. The leather is sufficient for only 400 belts. Belt A required a fancy buckle and only 200 fancy buckles are available. (i) Formulate above as a linear programming problem. nd (ii) Solve it by simplex method and comment on this optimum solution. 2 yr 202

Q31

A company makes two kinds of leather belts. Belt A is a high quality belt, and belt B is of lower quality. The respective profits are Rs 0.40 and Rs 0.30 per belt. Each belt of type A requires twice as much time as belt of type B, and if all belts were of type B, the company could make 1000 per day. The supply of leather is sufficient for only 800 belts per day (both A and B combined). Belt A requires a fancy buckle, and only 400 per day are available. There are only 700 buckles a day available for belt B. What should be the daily production of each type of belts? Form the linear programming problem and hence solve it by simplex method

Q32

A retired person wants to invest upto an amount of Rs 30,000 in fixed income securities. His broker recommends investing in two bonds : Bond A yielding 7% and Bond B yielding 10%. After some consider ion, he decides to invest at most Rs 12,000 in Bond B and at least Rs 6,000 in Bond A. He also wants t e amount invested in Bond A to be at least equal to the amount invested in Bond B. What should the broker recommend if the investor wants to maximise his retum on investment? Solve graphically.

Q33

An agriculturist has a 125-acre farm. He produces radish, muttar and potato. Whatever he raises is fully sold in the market. He gets Rs 5 for radish per kg, Rs 4 for muttar per kg and Rs 5 for potato per kg. The average per acre yield is 1500 kg of radish, 1800 kg of muttar and 1200 kg of potato. To produce each 100 kg of radish and muttar and 80 kg of patato, a sum of Rs 12.50 has to be used for manure. Labour required for each acre to raise the crop is 6 man-days for radish and potato each and 5 man-days for muttar. A total of 500 man-days of labour at a rate of Rs 40 per man-day are available. Formulate this as a Linear programming model to maximize the agriculturist's total profit.

Q34

A firm assembles and sells two different types of outboard motors, A and B, using four resources. The production process can be described as follows: Resources Capacity per month Motor unit shop resource 400 Type A units or 250 Type B units or any linear combination of the two Type A gear and drive 175 Type A units shop resource Type B gear and drive 225 Type B units shop resource Final assembly resource 200 Type A units or 350 Type B units or any linear combination of the two Type A units bring in a profit of Rs. 90 each and Type B units Rs. 60 each. Formulate the above as a linear programming problem to maximize profit and solve the same by graphic method.

Q35

A firm buys casting of P and Q type of parts and sells them as finished products after machining, boring and polishing. The purchasing costs for casting are Rs. 3 and Rs. 4 each for parts P and Q respectively and selling costs are Rs. 8 and Rs. 10 respectively. The per. hour capacity of machines used for machining, boring and polishing for the two products is given below:

Machine Boring Polishing

P 30 30 45

Capacity per hour or 0r or or

Q 50 45 30

The’ running cost for machining, boring and polishing are Rs. 30, Rs. 22.5 and Rs. 22.5 per hour respectively. Formulate the linear programming problem to find out the product mix to maximize the profit. DO NOT SOLVE.

17

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q36

A city hospital has the following minimal daily requirement for nurses: Period 1 2 3 4 5 6

Clock time (24 hours day)

Minimal number of nurses required

6 am – 10 am 10am - 2 pm 2pm - 6 pm 6pm - 10 pm 10pm - 2 am 2am - 6 am

2 7 15 8 20 6

Nurses report to the hospital at the beginning of each period and work for S consecutive hours\. The hospital wants to determine the minimal number of nurses to be employed so that there will be sufficient number of nurses available for each period. Formulate this as a Linear Programming Problem by setting up appropriate constraints and objective function. Do not solve. Q37

A person is interested in investing Rs 50,00,000 in a mix of investments. The investment choices and expected rates of returns on each one of them are: Investment Mutual Fund A Mutual Fund B Money Market fund Government Bonds Share Y Share X

Expected rate of return 0.12 0.09 0.08 0.085 0.16 0.18

The investor wants at least 35 per cent of his investment in government bonds. Because of the higher perceived risk of the two shares, he has specified that the combined investment in these not to exceed Rs. 80,000. The investor also has specified that at least 20 per cent of the investment should be in the money market fund and that the amount of money invested in shares should not exceed the amount invested in mutual funds. His final investment condition is that the amount invested in mutual fund A should be no more than the amount invested in mutual fund B. The problem is to decide the amount of money to invest in each alternative so as to obtain the highest annual. total return. Formulate the problem as a linear programming Q38

A company manufacturing television sets and radios has four major departments: chassis, cabinet, assembly and final testing. Monthly capacities are as follows:

Chassis Cabiinet Assemble Testing

Television capacity 1500 1,000 2,000 3,000

or or 0r or

Radio capacity 4,500 8,000 4,000 9,000

The contribution of television is Rs. 150 each and the contribution of radio is Rs. 25 each. Assuming that the company can sell any quantity of either product, determine the optimal combination of output.

18

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q39

A pharmaceutical company has 100 kg of A, 180 kg of B and 120 kg of C available per month. They can use these materials to make three basic pharmaceutical products, viz., 5 -10 - 5, 5 - 5 - 10 and 20 - 5 - 10, where the numbers in each represent the percentage by weight of A, B' and C respectively in each of the products. The cost of these ingredients arc given below: Ingredient A B C Inert ingredients

Cost per kg (Rs.) 80 20 50 20

Selling prices of these products are Rs. 40.50, Rs. 43 and Rs. 45 per kg respectively. There is a capacity restriction of the company for the product 5 - 10 - 5 so as they cannot produce more than 30 kg per month. Determine how much of each of the products they should produce in order to maximize their monthly profit. Q40

An advertising agency wishes to reach two types of audience!). Customers with annual incomes greater than Rs. 40.000 (target audience A) and customers with annual incomes of less than Rs. 40,000 (target audience B). The total advertising budget is Rs. 2.00,000. One programme of TV advertising costs Rs. 50,000, one programme of radio advertising costs Rs. 20,000. For contract reasons, at least 3 programmes ought to be on TV and the number of radio programmes must not exceed 5. Survey indicates that a single TV programme reaches 4,50,000 customers in target audience A and 50,000 in target audience B. One radio programme reaches 20,000 in target audience' A and 80,000 in target audience B. Determine the media mix to maximize the total reach.

Q41

The daily Florist company is planning to make up floral arrangements for the upcoming festival. The company has available the following supply of flowers at the costs shown: Type Red roses Garnenias Carnations White roses Yellow roses

Number available 800 456 4,000 920 422

Cost per flower(Rs.) 0.20 0.25 0.15 0.20 0.22

These flowers can be used in any of the four popular arrangements whose make up and selling prices are as follows: Arrangement

Requirements

Selling price

Economy

4 red roses 2 garnenias 8 carnations

Rs.6

May time

8 white roses 5 garnenias 10 carnations 4 yellow roses

Rs 8

Spring colour

9 red roses 10 carnations 9 white roses 6 yellow roses

Rs 10

Deluxe rose

12 red roses 12 white roses 12 yellow roses

Rs 12

How many units of each arrangement should be made up in order to maximize profits, assuming all arrangements can be sold.

19

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q42

Hifashion Furnishers manufacture two models of dining table P and Q. The market standing of the fir~ is such that they are able to sell at profit margins of Rs. 300 and Rs. 400 respectively of as many numbers of P and Q as they make. The dining tables are manufactured in two stages, viz, carpentry followed by paint shop whose capacities per month are as follows:

Carpentry:

70 of model P only or 120 of model Q only . or Some intermediate mix

Paint Shop:

90 of model P only or. 60 of model Q only or Some intermediate mix

Within these manufacturing capacity constraints, find how many of each model should be manufactured per month so that the profit of the firm is maximized. FORMULATE the problem and DO NOT SOLVE.

Q43

A local travel agent is planning a charter trip to a major sea port. The eight day/seven night package includes the fare for round trip, surface transportation, board and lodging and selected tour options. The charter trip is restricted to 200 persons and the past experience indicates that there will not be any problem for getting 200 clients. The problem for the travel agent is to determine the number of Deluxe, Standard and Economy tour packages to offer for this charter. These three plans differ according to seating and service for the flight, quality of accommodations, meal plans and tour options. The following table summarizes the estimated prices for the three packages and the corresponding expenses for the travel agent. The travel agent has hired an aircraft for the flat fee of Rs 2,00,000 for the entire trip. In planning the trip, the following considerations must be taken into account: (i) At least 10% of the packages must be of the Deluxe type. (ii) At least 35% but not more than 70% must be of the Standard type. (iii) At least 30% must be of the Economy type. (iv) The maximum number of Deluxe packages available in any aircraft is restricted to 60. (v) The Hotel desires that at least 120 of the tourists should be on the Deluxe and Standard packages taken together. Tour Plan Deluxe Standard Economy

Price 10,000 7,000 6,500

Hotel Costs 3,000 2,200 1,900

Meal and other expenses 4,750 2,500 2,200

The travel agent wishes to determine the number of packages to offer in each type so as to maximize the total profit. (a) Formulate this as a linear programming problem. (b) Restate the above LPP in terms of two decision variables, taking advantage of the fact that 200 packages will be sold. (c) Find the optimal solution using graphical method for the restated problem and interpret your results.

20

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q44

Let us assume that you have inherited Rs 100,000 from your father-in-law that can be invested in a combination of only two stock portfolios, with the maximum investment allowed in either portfolio set at Rs 75,000. The first portfolio has an average retum of 10%, whereas the second has 20%. In terms of risk factors associated with these portfolios, the first has a risk rating of 4 (on a scale from 0 to 10), and the second has 9. Since you want to maximise your retum, you will not accept an average rate of retum below 12% or a risk factor above 6. Hence, you then face the important question. How much should you invest in each portfolio? Formulate this as a linear programming problem and solve it by graphic method.

Q45

A furniture manufacturer produces two types of desks: Standard and Executive. These desks are sold to an office furniture wholesaler, and for all practical purposes, there is an unlimited market for any mix of these desks, at least within the manufacturer's production capacity. Each desk has to go through four basic operations: cutting of the lumber, joining of the pieces, pre-finishing, and final finish. Each unit of the Standard desk produced takes 48 minutes of cutting time, 2 hours of joining, 40 minutes of pre-finishing, and 5 hours and 20 minutes of final finishing time. Each unit of the Executive desk required 72 minutes of cutting, 3 hours of joining, 2 hours of pre-finishing and 4 hours of final finishing time. The daily capacity for each operation amounts to 16 hours of cutting, 30 hours of joining, 16 hours of pre-finishing and 64 hours of final finishing time. The profit per unit produced is Rs 40 for the Standard desk and Rs 50 for the Executive desk. Determine the product mix that maximizes total revenue.

Q46

Clarified solutions is in the process of drawing up a Capital Budget for the next three years. It has funds to the tune of Rs. 100000 which can be allocated across the projects A, B, C, D and E. The net cash flows associated with an investment of Re.1 in each project are provided in the following table:

From inv. A From inv. B From inv. C From inv. D From inv. E

0 i.e present

Cash Flow at Time 1year from now 2year from now

-Re. 1 Re. 0 -Re. 1 -Re. 1 Re. 0

+Re. 0.5 -Re. 1 +Rs. 1.2 Re. 0 Re. 0

+Re. 1 +Re. 0.5 Re. 0 Re. 0 -Re. 1

3year from now Re. 0 +Re. 1 Re. 0 +Rs. 1.9 +Rs. 1.5

To ensure that the firm remains reasonably diversified, the firm will not commit an investment exceeding Rs. 75000 in any project. The firm cannot borrow funds; therefore the cash available for investment at any time is limited to cash on hand. The firm will earn interest.at 8% per annum by parking the uninvested funds in money market instruments. Assume that the returns from investments can be immediately reinvested. For example, the positive cash flow received from project C at time 1 can immediately re reinvested in project B. Required: Formulate an LP that wiIl Maximize cash on hand 3 years from now

21

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q47

A leading Firm of Chartered Accountants Is attempting to determine a best investment portfolio and is considering six alternative Investment proposals. The following table Indicates point estimates for the price per share, the annual growth rates in the price per share, the amount of dividend per share and a measure of the risk associated with each investment

Particulars Current price per share (Rs) Projected annual growth Projected annual dividend per share (Rs) Projected risk in return

A 80 0.08 4.00 0.05

Shares under consideration B C D E 100 160 120 150 0.07 0.10 0.12 0.09 4.50 7.50 5.50 5.75 0.03 0.10 0.20 0.06

F 200 0.15 0.00 0.08

The total amount available for investment is Rs.25 Lakhs and the following conditions are to be satisfied. The maximum rupee amount to be invested in alternative F is Rs.2,50,000 Not more than Rs.5,00,000 should be invested in alternatives A and B combined. Total weighted risk should not be greater than 0.10 where Total weighted risk = (Amount invested in Alternative j) x (Risk of Alternative j) Total Amount invested in all alternatives For the sake of diversity, at least 100 shares of each stock should be purchased. At least 10% of the total investment should be in alternatives A and B combined. Dividends for the year should be at least Rs.10,000 Rupees Return per share is defined as the price per share one year hence less current price per share plus dividend per share. If the objective is to maximize total rupee return, formulate (but do not solve) the LPP determining the optimum number of shares to be purchased in each of the shares under consideration. Assume that the time horizon for the investment is one year. N91 Q48

Kali has 2 plants. Orders from 4 customers have been received. The number of units ordered by each customer and the shipping cost of each plant are shown in the following table. Customer

Units ordered

Shipping cost Plant 1

Plant 2

A 500 Rs 15 p.u Rs 40 p.u B 300 Rs 20 p.u Rs 30 p.u C 1000 Rs 30 p.u Rs 25 p.u D 200 Rs 35 p.u Rs 20 p.u Each unit of the product must be machined and assembled. These cost, together with the capabilities at each plant are shown below – Hours / unit Cost (Rs / hr) Hours available Plant No. 1

Machining Assembling

0.10 0.20

40 30

120 260

Plant No.2

Machining

0.11 0.22

40 30

140 250

Formulate LP problem to minimize cost.

RTP

22

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854 Q49

The management accountant of Atul enterprises Ltd has suggested that a Linear Programming model might be used for selecting the best mix of 5 possible products A, B, C, D and E. The following information is available

Particulars Selling Price Costs:

Materials Direct Labour ** Fixed overheads Total Costs Net Profit **

A 48

B 42

C 38

15 18 9

14 16 8 38 4

16 16 3 25 13

42 6

Per unit of Product (Rs) D 31 15 4 2 21 10

E 27 16 4 2 22 5

Based on 50% of direct labour cost a. Expected Maximum unit demand per week for each product at the prices indicated A 1,500

B 1,200

C 900

D 600

E 600

b. Cost of materials includes a special component, which is in short supply; it costs Rs 3 p.a unit. Only 5800 units will be available to the company during the week. The number of unit of the special component needed for a unit of each product is: A 1

B 1

C 3

D 4

E 5

c. Labour is paid at the rate of Rs 1.50 per hpur and only 20,000 hours will be available in a week. d. The company has ruled that expenditure on materials must not exceed a sum of Rs 30,000 e. All other resources are freely available in sufficient quantities for planned needs. Formulate a Linear Programming model stating clearly the criterion you use

23

RTP

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q50

The most recent audited summarized Balance Sheet of Best Yield Financial Services is given below Liabilities Equity share capital Reserves and Surplus

Rs in Lakhs 65 110

Term loan from IFCI Public deposits Bank Borrowings Other current Liabilities

80 150 147 50

Assets Fixed Assets - Assets on lease (Cost 550 lakhs) - Other fixed assets Investment in wholly owned subsidiaries Currents Assets - Stock on hire - Recievables - Other current assets - Misc expenditure not written off

602

Rs in Lakhs 375 50 20 80 30 35 12 602

The Company intends to enhance its investment in the lease portfolio by another Rs.1000 Lakhs. For this purpose, it would like to raise a mix of debt and equity in such a way that the overall cost of raising additional funds is minimized. The following constraints apply to the way the funds can be mobilized , Total debt divided by net owned funds cannot exceed 10. Amount borrowed from financial institutions cannot exceed 25% of the Net Worth. Maximum amount of bank borrowings cannot exceed three times the Net Owned Funds The Company would like to keep the Public Deposit limited to 40% of the total debt. The post tax costs of the different sources of finance are as follows Equity 25% Term Loans 8.5% Public Deposits 7% Bank borrowings 10% . Formulate the above as a LPP. Q51

RTP

Welltype Manufacturing Company produces three types of typewriters: Manual typewriters, Electronic typewriters, and Deluxe Electronic typewriters. All the three models are required to be machined first and then assembled. The time required for the various models are as follows: Types Manual Typewriter Electronic Typewriter Deluxe Electronic Typewriter

Machine Time 15 hours 12 hours 14 hours

Assembly time 4 hours 3 hours 5 hours

The total available machine time and assembly time are 3,000 hours and 1,200 hours respectively. The data regarding the selling price and variable costs for the three types are: Type Manual Electronic Deluxe Electronic Selling Price (Rs.) 4,100 7,500 14,600 Labour, Material and other Variable Costs (Rs.). 2,500 4,500 9,000 The Company sells all the three types on credit basis, but will collect the amounts on the first on next month. The Labour, Material and other Variable Expenses will have to be paid in cash. The Company has taken a loan of Rs.40,000 from a Co-Operative Bank and will have to repay it to the bank on April 200X. The TNC Bank from whom the Company has borrowed Rs.60,000 has expressed its approval to renew the loan. st

The Balance Sheet of the Company as on 31 March 200X is as follows: Liabilities Rs Assets Equity share capital 1,50,000 Land Capital Reserve 15,000 Building General reserve 1,10,000 Plant & Machinery Profit and Loss A/c 25,000 Furniture & Fixtures Long term loan 1,00,000 Vehicles Loan from TNC bank 60,000 Inventory Loan from TNC bank 40,000 Receivables Cash 5,00,000

24

Rs 90,000 70,000 1,00,000 15,000 30,000 5,000 50,000 1,40,000 5,00,000

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

The Company will have to pay Rs.10,000 towards salary for top management executives and other fixed overheads for the month. Interest on long-term loans is to be paid every month at 24% per annum. Interest on loans from TNC and Co-operative Banks may be taken to be Rs. 1,200 for the month. Also the Company has promised to deliver 2 Manual Typewriters and 8 Deluxe Electronic Typewriters to one of its valued customers next month. Make sure that the level of operations in the Company is subject to the availability of cash next month. The Company will also be able to sell all their types of typewriters in the market. The Company desires to know as to how many units of each type writer must be manufactured in the factory next month so as to maximise the profits of the Company. Formulate as a LPP. M93 Q52

A Company must produce two products over a production period of three months. The Company can pay for materials and labour from two sources: Company Funds and Borrowed Funds. The firm faces three decisions: (1) (2) (3)

How many units of Product 1 should it produce? How many units of Product 2 should it produce? How much money should it borrow to support the production of the products 1 and 2?

In making these decisions, the firm wishes to maximise the profit contribution on the following conditions. Since the Company's products are enjoying a seller's market, it can sell as many units as it can produce. The Company would therefore like to produce as many units as possible subject to production capacity and financial constraints. The capacity constraints, to ether with cost and rice data, are given below. Products Selling price Production cost Required hours per unit in department (Rs per unit) (Rs per unit) A B C 1 14 10 2 11 8 Available hours per production period of 3 months

0.5 0.3 500

0.3 0.4 400

0.2 0.1 200

The available Company funds during the production period will be Rs.3 Lakhs. A bank will give loans upto Rs.2 Lakhs per production period at an interest rate of 20% provided the Company's Acid Test ratio is at least 1 to 1 while the loan is outstanding. Acid-test ratio is given by Surplus cash on hand after oroduction + Accounts Receivable Bank borrowings + Interest accrued thereon Also make sure that the needed funds are made available for meeting the production costs. . Formulate the above as a LPP. Q53

N92

The Southern Zone of Corporation Bank is trying to determine Its overall loan policy for the region for the coming month. Its customers can be classified into four groups: industrial, agricultural, personal loans without solidarity and personal loans with solidarity. The total amount available for loans in the coming month is estimated at Rs. 5 Crores. The interest rates charged and the percentage of bad debts are given in following table Customer type Industrial Agricultural Personal loans without solidarity Personal loans with solidarity

Interest rate charged p.a (%) 9 10 13 14

Risk level (% of bad debts) 0.2 0.5 1.0 2.0

There are a number of restrictions on loan policy, which the region must observe, due to the regulations set by the Reserve Bank of India and .the national policy of Corporation Bank. These can be summarised as follows:

25

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Personal loans must not exceed 40 per cent of the value of total loans. Personal loans with solidarity must not exceed 20 per cent of the total personal loans. Total agricultural loans must not exceed Rs. 10 Lakhs and Industrial loans should not be less than Rs. 2 Crores. The average risk factor must not exceed 0.8 per cent. (a) (b)

Formulate a Linear Programming Model for the ,bank's problem. State clearly in your answer how did you derive your objective function and constraints. If you were to solve this problem by the simplex method, explain (but do not solve) how you would deal with this situation where you have inequalities of both types, (i.e., "greater than" and "less than").

Q54

A Mutual Fund Company has Rs.20 Lakhs available for investment in Government Bonds, Blue Chip Stocks, Speculative Stocks and Short-Term Bank Deposits. 'The annual expected return and risk factor are given below Types of investments Annual expected return (%) Risk factor (0 to 100) Government Bonds 14 12 Blue chip stock 19 24 Speculative stock 23 48 Short term deposits 12 6 The Company is required to keep at least Rs.2 Lakhs in short-term deposits and not to exceed an average risk factor of 42. Speculative stocks must be at most 20 % of the total amount invested. How should the Company invest the funds so as to maximise its total expected annual return? Formulate as a LPP

Q55

The portfolio manager of Morgan Stanley has been asked to invest Rs 10,00,000 of a large pension fund. The investment research department has identified six mutual funds with varying investment strategies, resulting in different potential returns and associated risks, as given below Fund 1

2

3

Price 45 76 110 Expected return (%) 30 20 15 Risk category high high high The management has specified the following guidelines (a) (b) (c) (d) (e)

4

5

6

17 12 medium

23 10 medium

22 7 low

The total amount invested in high risk funds must be between 50% and 75% of the portfolio The total amount invested in medium risk funds must be between 20% and 30% of the portfolio The total amount invested in low risk funds must be at least 5% of the portfolio The amount invested in the high risk funds 1,2 and 3 should be in the ratio 1:2:3 respectively The amount invested in the medium risk funds 4 and 5 should be 1:2

With these guidelines, what portfolio should you recommend so as to maximize the expected rate of return? Formulate the LP problem. Q56

Write the Dual to the following LPP’s a. Maximise Subject to

Z = 20x1 + 15x2 + 18x3 + 10x4 4x1 – 3x2 + 10x3 + 4x4 ≤ 60 X1 + x2 + x3 = 27 -x2 + 4x3 +7x4 ≥ 35

26

x1,x2,x3 ≥ 0, x4 : unrestricted in sign

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

b. Maximise Subject to

Z = 4x1 + 3x2 3x1 – x2 X1 + x2 - 4x1 + x2

c.

Maximize subject to

≤2 ≥1 ≥3

x1, x2 ≥ 0

Z = 4x1 + X2 + 2x3 2x1 + x2 + 3x3 ≤ 10 x1-x2+x3 =4

d. Maximize subject to

xl,x2,x3 ≥0

Z = 3x1 + 5x2 + 7x3 x1 + x2 + 3x3 ≤ 10 4x1 – x2 + 2x3 ≥ 15

e

Minimise Subject to

x1,x2 ≥ 0, x3 : unrestricted in sign

Z = x1 + 2x2 + x3 X1 – x2 – x3 ≤-1 6x1 + 3x2 + 2x3 = 12

f

Minimize subject to

Z = 5x1 + 10x2 + 15x3 + 12x4 2x1 + 3x2 + 7x3 + x4 X1 + x2 + x4 -x1 + 4x2 + 5x3

G

Maximize:

xl,x2,x3 ≥0

≥ 50 = 45 ≤ 30

x1,x2,x3 ≥ 0, x4 : unrestricted in sign

100X1 + 90X2 + 40X3 + 60 X4 Subject to, 6X1 + 4X2 + 8X3 + 4 X4 ≤140 10X1 + 10X2 + 2X3 + 6X4 ≤ 120 10X1 + 12x2 + 6X3 + 2X4 ≤ 50 Xlo X2, X3.X4 2: 0;

(Only formulation is required. Please do not solve.)

Q57

A factory produces three different products, A, B, and C. The profits per unit of these products are Rs. 3, Rs. 4 and Rs. 6 respectively. The products are processed. in three operations, viz., X, Y and Z and the time (in hours) required in each operation for each unit is given below:

Operations

X Y Z

A 4 5 1

Products B C 1 6 3 1 2 3

The factory has 3 machines for operation X, 2 machines for operation Y and only one machine for operation Z. The factory works 25 days in a month, at the rate of 16 hours a day in two shifts. The effective working of all the processes is only 80% due to power cuts or breakdown of machines. (i) Formulate the problem mathematically. (ii) Use simplex method to find how many units of each product should be produced monthly in order to maximize profit? (iii) Write the dual to the above problem and determine the optimum values of the dual variables. (iv) What are the shadow prices of the resources?

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PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q58

Noah’s Boats make three different kinds of boats. All can be made profitably by the Company, but the monthly production is constrained by the limited amount of labour, wood and screws available each month. The Director should choose the combination of boats that maximizes his revenue In view of the Information given Input Labour (hours) Wood (board feet) Screws (Kg) Selling price (Rs)

Rowboat 12 22 2 4,000

Canoe 7 18 4 2,000

Kayak 9 16 3 5,000

Monthly availability 1260 hours 19,008 board feet 396 Kg

1 2 3

Formulate the above as a linear programming problem, Solve it by the Simplex method, From the optimal table of the solved LPP, answer the following questions How many boats of each type will be produced and what will be the resulting revenue? Which, If any, of the resources are not fully utillsed? If so, how much of spare capacity Is left? How much wood will be used to make all of the boats given in the optimal solution? State the dual of the formulated LPP, Q59

Store More Company has three departments - Assembly, Painting and Packing with the capability of making three types of almirah. An almirah of Type I requires one hour of assembly, 40 minutes of painting and 20 minutes of packing time respectively. Similarly, Type II requires 80 minutes, 20 minutes and one hour respectively. The Type III requires 40 minutes each of assembly, painting and packing time. The total time available at assembly, painting and packing departments are 600 hours, 400 hours and 800 hours respectively. Determine the number of each type of almirah that should be produced in order to maximize profits. The unit for types I, II, and III are Rs.40, 80 and 60 respectively Suppose the Manager is thinking of renting the productive capacities of the three departments to another manufacturer - Steel Racks Company. Steel Racks Is Interested In minimizing the rental charges. On the other hand the Store More Company would like to know the worth of a productive hour to them, in each of the departments determine the rental rates. Formulate the problem as a LPP. Explain clearly M90

Q60

Given below are the objective function, the constraints and the final simplex tableau for a linear programming product mix problem: Maximize Subject to

Z = 2x1 + 5x2 + 8x3 6x1 + 8x2 + 4x3 ≤ 96 2x1 + x2 + 2x3 ≤ 40 5x1 + 3x2 + 2x3 ≤ 60

x1,x2,x3 ≥ 0

Final simplex tableau ____________________________________________________________________________________ Cj Product 2 5 8 0 0 0 Quantity Mix X1 x2 x3 s1 s2 s3 ____________________________________________________________________________________ 5

x2

1/3

1

0

1/6

-1/3

0

8/3

8

x3

5/6

0

1

-1/12

2/3

0

56/3

0 s3 7/3 0 0 -1/3 -1/3 1 44/3 ____________________________________________________________________________________ Zj 25/3 5 8 1/6 11/3 0 162 -2/3 Cj – Zj -19/3 0 0 -1/6 -11/3 0 ____________________________________________________________________________________

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PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

(i) (ii) (iii) (iv) (v) (vi) (vii) Q61

Write the optimum product mix and the profit contribution shown by the above solution. Is this solution feasible? Why? give brief reason. Does the problem have any alternative optimum solution? If so, find the other solution. What are the shadow prices of three departments. If the company wishes to expand the production capacity, which of the three depart ments should be given priority. If the company produces 6 units of Xl, how many units of X2 andx3 shall have to be reduced, if any? If a customer is prepared to pay higher prices for product Xl, how much should the price be increased so that the company's profit remains unchanged? _____________________________________________________________________________ X1 X2 S1 S2 A1 A2 Basis

Cj

15

25

0

0

-M

-M

A1

-M

7

6

-1

0

1

0

20

S2

0

8

5

0

1

0

0

30

A2 -M 3 -2 0 0 9 1 18 _____________________________________________________________________________ Zj

-10M

-4M

M

0

-M

-M

-38M

Cj-Zj 15+10M 25+4M -M 0 0 0 ____________________________________________________________________________ write down the original primal problem represented by the above tableau. Find out the optimal solution of this problem. Is it a unique solution? Why? Write the dual of the problem and determine the optimal solution of the dual Q62

The Simplex table for a maximization problem of linear programming problem is given below: ________________________________________________________________________________ Cj Product 4 5 0 0 Quantity Mix X1 x2 s1 s2 ________________________________________________________________________________ 5

x2

1

1

1

0

10

0 s2 1 0 -1 1 3 _________________________________________________________________________________ Zj 5 5 5 0 50 Cj – Zj -1 0 -5 0 Answer the following questions, giving brief reasons 1. 2. 3. 4.

5. 6. 7.

Is the above solution optimal? Are there more than one optimal solution? Is this solution degenerate and is the solution feasible? If sl is the slack in Machine A (in hours/week) and s2 is the slack in Machine B (in hours / week), which of these machines is being used to the full capacity when producing according to this solution? A customer would like to have one unit of product xl and is willing to pay in excess of the normal price to get it. How much should the price be increased in order to ensure no reduction in profit? Machine B (associated with slack s2 in hours/week) has to be shut down for repairs for 2 hours. next week. What will be the effect on profits? How many units of the products xl and x2 are being produced as per this solution and what is the total profit?

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PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q63

The costs and selling prices per unit of two 'products manufacturing by a company are as under: Product

A (Rs)

B(Rs)

Selling Price 500 450 Variable costs Direct Materials @ 25 per kg 100 100 Direct Labour @ 20 per kg 80 40 Painting @ Rs. 30 per hour 30 190 Variable Overheads: 190 175 Fixed costs @ Rs. 17.50/Direct labour hour 70 35 Total costs 470 410 Profit . 30 40 In any month the maximum availability of input is limited to the following: Direct Materials 480 kgs Direct Labour hours 400 hours Painting hours 200 hours Required: (i) Formulate a linear programme to determine the production plan which maximjzes the profits by using graphical approach. . (ii) State the optimal product mix and the monthly profit derived from your solution in (i) above. (iii) If the company can sell the painting time as Rs. 40 per hour as a separate service, show what the modification will be required in the formulation of the linear programming problem. You are required to re-formulate the problem but not to solve. N08 Q64

An oil refinery can blend three grades of crude oil to produce quality A and 4 quality B petrol. Two possible blending processes are available. For each production run, the older process uses 5 units of crude Q, 7 units of crude P and 2 units of crude R and produces 9 units of A and 7 units of 8. The newer process uses 3 units of crude Q, 9 units of crude P and 4 units of crude R to produce 5 units of A and 9 units of B. .. . Because of prior .contract commitments, the refinery must produce at least 500 units of A and at least 300 units of B for the next month. It has 1,500 units of crude Q, 1,900 units of crude P and 1,000 units of crude R. For' each unit of A, refinery receives Rs. 60 while for each unit of B it receives Rs. 90. Formulate the problem as linear programming model so as to maximise the 'revenue. N09

Q65

A firm produces three products A,B & C. Its uses two types of raw materials I & II of which 5000 and 7500 units respectively are available. The raw material requirements per unit of product are given below: Raw Material I II

Requirement per unit of product A B C 3 4 5 5 3 5

The labour time for each unit of product A is twice that of product B and three times that of product c. The entire labour force of the firm can produce the equivalent of 3000 units of A. The minimum demand of the three products is 600, 650 and 500 units respectively. Also, the ratios of the number of units must be equal to 2:3:4. Assuming the profits per unit of A,B & C as Rs. 50, 50 and 80 respectively. . Formulate the problem as linear programming model in order to determine the number of units of each product which will maximize the profit N97

30

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q66

A refinery makes three grades of petrol A, B, C from 3 crude oils d, e, f. Crude oil f can be used in any grade but the others must satisfy the following specifications: Grade

Selling price per liter

Specification

A

18.0

Not less than 50% crude d Not more than 25% crude e

B

16.5

Not less than 25% crude d Not more than 50% crude e

C

15.5

No specifications

There are capacity limitations on the amounts of the 3 crude elements that can be used Crude Capacity (Kl) Price per liter d 500 19.5 e 500 14.5 f 360 15.1 It is desired to obtain maximum profit. Formulate this as a LPP Q67

XYZ company has three departments - Assembly. painting and packing with the capability of making three types of almirahs. An almiras of Type 1 requires one hour of assembly. 40 minutes of painting and 20 minutes of packing time respectively. Similarly. almirah of type II needs 80 minutes of assembly, 20 minutes of painting and one hour of packing time respectively. The last type requires 40 minutes each of assembly, painting and packing time. The total times available at assembly, painting, and packirig departments are 600 hours, 400 hours, and 800 hours respectively. Determine the number of each type of almirahs that should be produced in order to maximize the profit. The unit profits for types I, II and III are Rs. 40, 80 and 60 respectively. Suppose the manager is thinking of renting the productive capacities of the three departments to another almirah manufacturer - Steel Racks Company. Steel Racks is. interested in minimizing the rental charges. On the other hand the Z company would like to know the worth of a productive hour to them, in each of the departments to determine the rental rates. Formulate the problem as a LPP. Explain clearly. May/90

Q68

Three grades of Coal A, B and C contains phosphorus and ash as impurities. In a particular industrial process, fuel upto 100 ton (maximum) is required, which could contain ash not more than 3% and phosphorus not more than 0.03%. It is desired to maximize the profit while satisfying these conditions. There is an unlimited supply of each grade. The percentage of impurities and profits of each grade are as follows N05 Coal Phosphorus(%) Ash (%) Profit (Rs per ton) A 0.02% 3.0% Rs 12.00 B 0.04% 2.0% Rs 15.00 C 0.03% 5.0% Rs 14.00 Formulate the LP model to solve it using Simplex Method to determine optimal product mix and profit.

31

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q69

A gear manufacturing company makes two types of gears – A and B. Both gears are processed on 3 machines, Hobbing M/c, Shaping M/c and Grinding M/c. The time required by each gear and total time available per week on each M/c is as follows: M07 compiler 363 Gear (A) Gear (B) Available Machine (Hours) (Hours) (Hours) Hobbing M/c 3 3 36 Shaping M/c 5 2 60 Grinding M/c 2 6 60 Other data: Selling price (Rs.) 820 960 Variable cost (Rs.) 780 900 Determine the optimum production plan and the maximum contribution for the next week by simplex method. The initial table is given below: _________________________________________________________________________________ Cj 40 60 0 0 0 Qty _________________________________________________________________________________ Cj Variable x1 x2 x3 x4 x5 0 x3 36 3 3 1 0 0 0 x4 60 5 2 0 1 0 0 x5 60 2 6 0 0 1 __________________________________________________________________________________

Q70

The following matrix gives the unit cost of transporting a product from production plants P1 P2 & P3 to destinations D1, D2 and D3. Plants P1, P2 and P3 have a maximum production of 65, 24 and 111 units respectively and destinations 01, O2 and 03 must receive at least 60, 65 and 75 units respectively. __________________________________________________________________________________ To D1 D2 D3 Supply From __________________________________________________________________________________ P1 400 600 800 65 P2 1000 1200 1400 24 P3 500 900 700 111 Demand 60 65 75 200 ___________________________________________________________________________________ You are required to formulate the above as LPP

N08

32

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q71

Transport Ltd. provides tourist vehicles of 3 types- 20-seater vans, 8-seater big cars & 5-seater small cars. These seating capacities are excluding the drivers. The company has 4 vehicles of the 20-seater van type, 10 vehicles of the eight-seater big car types 20 vehicles of the 5-seater small car types. These vehicles have to be used to transport employees of their client company from their residences to their offices and back. All the residences are in the same housing colony. The offices are at two different places, one is the Head Office and the other is the Branch. Each vehicle plies only one round trip per day, if residence to office in the morning and office to residence in the evening. Each day, 180 officials need to be transported in Route I (from residence to Head Office & back) and 40 officials need to be transported in Route II (from Residence to Branch office & back). The cost per round trip for each type of vehicle along each route is given below. _______________________________________________________________________________________ Rs per round trip per customer 20 seater vans 8 seater big cars 5 seater small cars ______________________________________________________________________________________ Route 1: Residence:

Head office and back

30

50

60

Route II: Residence: Branch office and back 25 40 40 _______________________________________________________________________________________ You are required to formulate the information as a linear programming problem, with the objective of minimizing the total cost of hiring vehicles for the client company, subject to the constraints mentioned above. ********

33

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

TRANSPORTATION

The transportation problem is concerned with the allocation of items between suppliers (called origins) and consumers (called destinations) so that the total cost of the allocation is minimised. The problem can be solved using either linear programming methods or the special transportation algorithm. Transportation problems are basically allocation models. The objective is to minimise the cost of transportation of homogeneous commodity from different supply points to different demand points. Application A few applications of the transportation method are mentioned below: Minimise shipping costs from factories to warehouses (or from warehouses to retail outlets) Determine lowest cost location for new factory, warehouse, office or other facility. Find minimum cost production schedule that satisfies firm's demand and production limitations (called 'production smoothing) Conditions to use transportation algorithm To use the transportation algorithm the following conditions must be satisfied: The cost per item for each combination of origin and destination must be specified. The supply of items at each origin must be known. The requirement of items at each destination must be known. The total supply must equal the total demand. Q

Explain the steps to solve transportation problem. Step 1 Find Initial Basic Solution (for balanced minimization problem) using any of the following 3 methods Points to be noted before ascertaining initial basic solution Transportation problem should be a balanced problem i.e. demand should be equal to supply, or availability should be equal to requirements. If the problem is not balanced, add a dummy row or column If the transportation problem is a maximization problem, convert the profit matrix into Loss matrix. This is done by subtracting each entry in the table from the largest No. in the table. If it is a restrictive transportation problem with prohibited routes, then assign M as a maximum cost to the respective cell. If it is a committed transportation problem i.e. a minimum or maximum no. of units are to be transferred from one factory to a specified depot, then reduce the demand and supply of respective depot and factory. But after attaining initial basic solution allocate that minimum or maximum units to a respective cell

34

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

1. Northwest Comer Method Allocation always starts from upper left hand corner (Le. North side first row-first column). Assignment is made in such a way that the resources available are exhausted or demand is fully satisfied. If the resources available are fully exhausted then we move down to second next row and continue the process till the whole demand is exhausted. If the first allocation completely satisfies the demand then we move to the next column of the same row, and continue the allocation process till all availability and requirements are met. The procedure is repeated till all row availability and column requirements are met.

2. Least cost method. Allocation always starts from the cell whose transportation cost per unit is least. The lowest cell is filled as much as possible in .view of the availability and destination requirement of its column. Then we move to the next lowest cell and so on continue the procedure in view of the remaining availability of demand and supply. The procedure is repeated till all row availability and column requirements are met. In case of tie for the lowest cell during allocation, choice may be made for a row or column by which maximum requirement is exhausted. 3. Vogel’s Approximation method Find the difference between two least cost cells in every row and column. Identify the' row or' column with the highest of the difference. It is in this row or column where allocation should be made. In the row or column selected in step 2, identify the least cost cell. It is in this cell allocation should be made. If there is a tie amongst the largest differences, the choice may be made for a row or column which has least cost. In case there is a tie in cost cell also, choice may be made for a row or column by which maximum requirement is exhausted. Hatch that row or column containing this cell whose totals have been exhausted so that this row or column is ignored in further consideration. Re-compute the row & column differences for the reduced TLP 'go ahead to Step 3. Continue the above steps till all allocations are made.

35

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Step 2 Ascertain if initial basic solution is feasible or not

Initial basic solution is feasible if number of occupied cells = m + n – 1, where m = no. of rows and n = No. of columns If no. of occupied cells ≠ m + n – 1 , then initial basic solution is degenerate To remove such degeneracy a very small quantity Є (epsilon) is allocated to one or more least cost independent unoccupied cell ( such least cost cell should be independent means that a closed path cannot be made from this cell.). If the least cost cell is not independent then next least cost independent cell is chosen (Є can be added to unoccupied dummy cell ) Degeneracy may also arise at later stage. Thus after every improved solution degeneracy is to be checked (Є may have to be used in improved solution also)

Step 3 Check if initial basic feasible solution is optimal or not It can be done by Modified distribution method (MODI) or Algebra method. let the row wise costs are Ui &column wise costs are Vj, where i = 1,2,3….m & j = 1,2,3……n Assign 0 value arbitrarily to a row or column variable Uj or Vj. (assign 0 to U1) Taking each cost cell (Uj+Vj) = Cij , calculate individually all values of Ui & Vj Calculate opportunity cost for each unallocated cell i.e. Ui + Vj – Cij = Δij If all Δij values are 0 or –ve, solution is optimal, if any Δij value is +ve then solution is not optimal. If any Δij value is 0 it means alternate optimal solution also exist If solution is not optimal, the cell with largest +ve opportunity cost Δij should be selected. Form this cell a closed loop starting and ending at this cell should be made and reallocate the solution

36

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q1

.

Plant A B C

Transportation cost (Rs/unit) Distribution centers X Y 50 30 15 27 25 25

Z 20 40 45

Demand

10,000

25,000

15,000

Availabilty (units) 20,000 18,000 12,000

Suggest optimal solution for the following transportation problem and indicate the total minimum transportation cost (no degeneracy, optimal sol) If the company wants atleast 5,000 units to be transported from plant B to distribution center Z, what will be the transportation schedule and effective cost? (restriction, no degeneracy, not optimal) Q2

A product is manufactured by four factories A, B, C and D. The Unit production costs are Rs.2, Rs.3, Re.1 and Rs.5 respectively. Their daily production capacities are 50, 70, 30 and 50 units respectively. These factories supply the product to four P, Q, R and S. The demand made by these stores are 25, 35, 105 and 20 Units transportation cost in rupees from each factory to each store is given in the following table; Stores P Q R S A 2 4 6 11 B 10 8 7 5 C 13 3 9 12 D 4 6 8 3 Determine the extent of deliveries from each of the factories to each of the stores so that the total cost (production and transportation together ) is minimum. (May 2002) (only table imp, non degeneracy, optimal)

Q3

A compressed Natural Gas (CNG) company has three plants producing gas and four outlets. The cost of transporting gas from different production plants to the outlets, production capacity of each plant and requirement at different outlets is shown in the following cost-matrix table: outlets Plants A B C D capacity of production X 4 6 8 6 700 Y 3 5 2 5 400 Z 3 9 6 5 600 Requirement 400 450 350 500 1700 Determine a transportation schedule so that the cost is minimized. The cost in the cost-matrix is given in thousand of rupees. (Nov 2001) (Simple, non deg, optimal, multiple optimal)

37

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q4

Solve the following problem using transportation method, obtaining the initial feasible solution by VAM. The cell entries in the table are unit costs To 1 2 3 4 5 supply From 1 80 69 103 64 61 12 2 47 100 72 65 40 16 3 16 103 87 36 94 20 4 86 15 57 19 25 8 5 27 20 72 94 19 8 Demand 16 14 18 6 10 (Degeneracy, non optimal, balanced, difficult path)

Q5

A company has 4 terminals U,V,W and X. At the start of a particular day 10,4,6 and 5 trailers respectively are available at these terminals. During the previous night 13,10,6 and 6 trailers respectively were loaded at plants A,B,C,D. The company dispatcher has come up with the costs between the terminals and plants as follows Plants A B C D U 20 36 10 28 Terminals V 40 20 45 20 W 75 35 45 50 X 30 35 40 25 Find the allocation of loaded trailers from plants to terminals in order to minimize transportation cost.

Q6

Garg and Garg a leading firm has 3 auditors. Each auditor can work upto 160 hours during the next month, during which time 3 projects must be completed. Project 1 will take 130 hours, Project 2 will take 140 hours and project 3 will take 130 hrs. The amount per hour billed for assigning the auditor a project is: Project (amount in rupees) 1 2 3 Auditor A 1200 1500 1900 B 1400 1300 1200 C 1600 1400 1500 Formulate this as a transportation problem and find the optimal solution. Also find out the maximum total billings during the next month

Q7

Consider the following transportation cost table. The costs are given in rupees, supply and demand are in units. Determine an optimal solution Destination 1 2 3 4 5 Supply Source I 40 36 26 38 30 160 II 38 28 34 34 198 280 III 36 38 24 28 30 240 Demand 160 160 200 120 240 n88 (unbalanced, degeneracy, non optimal, difficult path, two e)

38

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q8

A company has two factories at A, B, and C which supply warehouses at D, E, F and G. Monthly factory capacities are 160, 150, and 190 units respectively. Monthly warehouse requirements are 80, 90, 110 and 160 units respectively. Unit shipping costs (in rupees) are as follows: To D E F G From A 42 48 38 37 B 40 49 52 51 C 39 38 40 43 Determine the optimum distribution for this company to minimize shipping costs. (degen, unbal, min, )

Q9

The cost per unit of transporting goods from factories X, Y, Z to destinations A, Band C and the quantities demanded and supplied are tabulated below. As the company is working out the optimium logistics, the company has announced a fall in the oil prices. The revised unit costs are exactly half the costs given in the table. You are required to evaluate the minimum transportation costs. Destination A B C Supply Factories X 15 9 6 10 Y 21 12 6 10 Z 6 18 9 10 Demand 10 10 10 30 J09 (Min,bal,deg,2e)

Q10

The initial allocation of a transportation problem along with the unit cost of transportation from each origin to the destination is given below. You are required to arrive at the minimum transportation cost by the Vogel’s approximation method and check for optimality (Hint take u1 = 0 at Row 1 for initial cell valuation) Requirement 11 9

10

7 9 Availability

8

2

12

2

6

4

8

6

9

12

9

6

10

6

3

7

7

8

3

5

11

4

8

8

2

6 8

8

2

4

18

40

(degeneracy, optimal)

39

M07

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q11

Alpha Company has 3 plants and 3 warehouses. The cost of sending a unit from different plants to the warehouses, production at different plants and demand at different warehouses are shown in the following matrix: Warehouses A B C Production Plant X 8 16 16 152 Y 32 48 32 164 Z 16 32 48 154 Demand 144 204 82 Determine the transportation schedule so that cost is minimized. Assume that cost in the cost matrix is given in thousands of rupees. M01 (Min, unbal,tie,no deg, optimal,multi)

Q12

Solve the following transportation problem for minimum cost: Destination/origin A B C 1 7 4 3 2 3 2 7 3 4 4 3 4 9 7 5 Availability 12 8 35 (Min,unbal,deg,multi)

Q13

D 4 5 7 3 25

Requirement 15 25 20 40 n85

Home building construction company is interested in taking loans from banks for its projects- P,Q,R,S,T. The rates of interest and the lending capacity differ from bank to bank. All these projects are to be completed. The relevant details are provided below . Source bank P 20 16 15 200

Private bank Nationalized bank Co-operative bank Amount required (in 000’s)

Interest rate in % for projects Q R S T 18 18 17 17 16 16 15 16 15 15 13 14 150 200 125 75

Max credit (in 000’s) Any amount 400 250

Assuming the role of a consultant, advise the company as to how it should take the loans so that the total interest payable is least. Find out the alternate optimal solutions, if any. N90 …(very multi, min, bal, no deg) Q14

Solve the following transportation problem and state whether the solution derived by you is unique. Godown Factory 1 Factory 2 Factory 3 Factory 4 Demand N89, big M,min, multi

1 7 9 11 9 60

2 5 11 10 10 20

3 7 6 6 9 40

4 7 11 2 6 20

40

5 5 2 9 40

6 3 5 8 12 40

Stock available 60 20 90 50

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q15

Given the following transportation problem: Market A B C Supply Warehouse 1 10 12 7 180 2 14 11 6 100 3 9 5 13 160 4 11 7 9 120 Demand 240 200 220 It is known that currently nothing can be sent from warehouse 1 to market A and from warehouse 3 to market C. Solve the problem and determine the least cost transportation schedule. Is the optimal solution obtained unique, if not what are the other optimal solutions ,(min.bal,non deg,multi, M)

Q16

Timely and Co, a manufacturer must produce a product in sufficient quantity to meet contractual sales in the next four months. The production capacity and unit cost of production vary from month to month. The production produced in one month may be held for sale in later months but an estimated storage costs of Re 1 per unit per month. No storage cost is incurred for goods sold in the same month in which they are produced. There is no opening inventory and none is desired at the end of four months. The necessary details are given below: Month Contracted sales Maximum production Unit cost of production 1 20 40 14 2 30 50 16 3 50 30 15 4 40 50 17 How much should the manufacturer produce each month to minimize total cost? RTP , M,min, Consider the following data for the transportation problem:

Q17

Factory

Destination Supply to be exhausted 1 2 3 A 5 1 7 10 B 6 4 6 80 C 3 2 5 15 Demand 75 20 50 Since there is not enough supply, some of the demands at the three destinations may not be satisfied. For the unsatisfied demands, let the penalty costs be rupees 1, 2 and 3 for destinations (1), (2) and (3) respectively. M98 Compiler 331, Q18

Following is the profit matrix based on four factories and three sales depots of the company: Sales depos S1 S2 S3 Availability F1 6 6 1 10 Factories F2 -2 -2 -4 150 F3 3 2 2 50 F4 8 5 3 100 Requirement 80 120 15 Determine the most profitable distribution schedule and the corresponding profit, assuming no profit in case of surplus production. Compiler 320 (max,unbal,nondeg,non optimal)

41

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q19

A Company has four factories F1, F2, F3 and F4, manufacturing the same product. Production and raw material costs differ from factory to factory and are given in the first two rows of the following table. The Transportation costs from the factories to sales depots S1, S2 and S3 are given in the next three rows of the table. The production capacity of each factory is given in the last row. The last two columns in the table given the sales price and the total requirement at each depot: Item per Factory Sales price Requirement unit Per unit F1 F2 F3 F4 Production cost 15 18 14 13 Raw material cost 10 9 12 9 80 Transportation cost 3 9 5 4 34 120 1 7 4 5 32 50 5 8 3 6 31 Production capacity 10 150 50 100 Determine the most profitable production and distribution schedule and the corresponding profit. The surplus should be taken to yield zero profit. N00, max,unbal,multi compiler 311,

Q20

Q21

A company produces a small component for all industrial products and distributes it to five wholesalers at a fixed prices of Rs.2.50 per unit. Sales forecasts indicate that monthly deliveries will be 3,000, 3,000, 10,000, 5,000 and 4,000 units to wholesalers 1,2,3,4 and 5 respectively. The monthly production capabilities are 5,000, 10,000, 12,500 at plants 1, 2 and 3 respectively. The direct costs of production of each unit are Rs.1.00 and Rs.0.80 at plants 1, 2 and 3 respectively. The transportation costs of shipping a unit from a plant to a wholesaler are given below: 1 2 3 4 5 1 0.05 0.07 0.10 0.15 0.15 Plant 2 0.08 0.06 0.09 0.12 0.14 3 0.10 0.09 0.08 0.10 0.15 Find how many components each plant supplies to each wholesaler in order to maximize profit.

(May 2000)

A company has 3 factories and 4 customers. It furnishes the following schedule of profit per unit on transportation of goods to customers in rupees. You are required to solve the transportation problem to maximize the profit. Determine the resultant optimal profit Factory/customer A B C D Supply P 40 25 22 33 100 Q 44 35 30 30 30 R 38 38 28 30 70 Demand 40 20 60 30 m03,

42

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q22

ABC enterprises is having 3 plants manufacturing dry-cells, located at different locations.Production cost differ from plant to plant. There are 5 sale offices of the company located in different regions of the country. The sales prices can differ from region to region. The shipping cost from each plant to each sales office and other data are given by following table: Production data table Production cost per unit Maximum capacity in No. Of units Plant No 20 150 1 22 200 2 18 125 3

Plant 1 Plant 2 Plant 3

Sales office1 1 9 4

Shipping cost sales office 2 1 7 5

sales office 3 5 8 3

sales office 4 9 3 2

Demand and sales prices Sales office1 sales office 2 sales office 3 sales office 4 Demand 80 100 75 45 Sales price 30 32 31 34 Find the production and distribution schedule most profitable to the company.

sales office 5 4 6 7

sales office 5 125 29 N98

Q23

A company has 3 warehouses W1, W2, W3. It is required to deliver a product from these warehouses to 3 customers A,B and C. The warehouses have the following units in stock. Warehouse W1 W2 W3 No. of units 65 42 43 Customer requirements Customer A B C No. of units 70 30 50 The table below shows the cost of transporting one unit from warehouse to the customer: Warehouse W1 W2 W3 A 5 7 8 Customer B 4 4 6 C 6 7 7 Find the optimal transportation route. M99

Q24

The manufacturer of jeans is interested in developing an advertisement campaign that will reach four different: age groups. Advertising campaigns can be conducted through TV, Radio and Magazines. The following table gives the estimated cost in paise per exposure for each age group according to the medium employed. In addition, maximum exposure levels possible in each of the media, namely TV, Radio and Magazines are 40, 30 and 20 million respectively. Also the minimum desired exposures within each age group, namely 13-18, 19-25, 26-35 and 36 and older are 30, 25, 15 and 10 millions. The objective is to maximize the cost of attaining the minimum exposure level in each group Age group 13-18 19-25 26-35 36 and older TV 12 7 10 10 Radio 10 9 12 10 Magazines 14 12 9 12 (a) Formulate the above as a transportation problem, and find the optimum solution. (b) Solve this problem if the policy is to provide atleast 4 million exposures through TV in the 13 - 18 age group, and atleast 8 million exposures through TV in the age group 19 - 25. M91

43

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q25

A Company wishes to determine an investment strategy for each of the next four years. Five investment types have been selected, investment capital has been allocated for each of the coming four years, and maximum investment levels have been established for each investment type. An assumption is that amounts invested b:: any year will remain invested until the end of the planning horizon of four years. The following table summarizes the data for this problem. The values in the body of the table represent net return on investment of one rupee upto the end of the planning horizon. For example, a rupee invested in investment Type B at the beginning of Year 1 will grow to Rs.l.90 by the end of the fourth year, yielding a net return of Re.0.90. Investment made at beginning of Year 1 2 3 4 Max Rupee Invest (000’s)

Net Return Data from Investment type A B C D E 0.80 0.90 0.60 0.75 1.00 0.55 0.65 0.40 0.60 0.50 0.30 0.25 0.30 0.50 0.20 0.15 0.12 0.25 0.35 0.10 750 600 500 800 1000

Rupees available (000’s) 500 600 750 800

The objective in this problem is to determine the amount to be invested at the beginning of each year in an investment type, so as to maximize the net rupee return for the four year period. Solve the above transportation problem and get an optimal solution. Also calculate the net return on investment for the 4 year planning period. M93

Q26

The following matrix is a minimization problem for transportation costs. The unit transportation costs are given at the right hand corner of the cells and the Δij values are encircled

D1 F1 F2 F3

D2 3

4

4

9

6

7

8

Demand 300

6

5

2

400

Supply

500

300

2

4 0

D3

300

Find the optimum solutions and minimum cost

200

1000

M11

44

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q27

A company has 3 plants located at A, B and C. The production of these plants is absorbed by 4 distribution centres located at X, Y , W and Z. The transportation cost per unit has been shown in small cells in the following table: Distribution Centers

X

Y

W

Z

Supply . (units)

Factories 6

9

13

7

B

6

10

11

5

6000

C

4

7

14

8

6000

4000

4000

4500

5000

A

6000

18000 Demand (units)

17500

Find the optimal solution to the transportation problem by applying Vogel’s approximation method. N10 Q28

Goods manufactured at 3 plants A, B and C are required to be transported to sales outlets X, Y and Z.The unit costs of transporting the goods from the plants to the outlets are given below: Sales outlets / Plants A B C Total Demand X 3 9 6 20 Y 4 4 6 40 Z 8 3 5 60 Total supply 40 50 30 120 You are required to: (i) Compute the initial allocation by North West corner rule (ii) Compute the initial allocation by VAM and check whether it is optional. (iii) State your analysis on the optionality of allocation under NWC rule and VAM M08

Q29

Solve the following transportation problem: Plant/ Market M1 M2 P1 10 5 P2 6 4 P3 9 12 Demand 110 80 Show initial solution by i. North west corner rule ii. Least cost rule iii. VAM

M3 7 8 10 190

45

M4 8 5 7 120

Supply 150 125 225 500

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q30

Determine the optimal solution to the problem under VAM To market _____________________________________________ M1 M2 M3 M4 _____________________________________________ From P1 6 4 9 1 Plant P2 20 6 11 3 P3 7 1 0 14 P4 7 1 12 6 ____________________________________________ Demand 90 30 50 30 ____________________________________________ Dummy, degeneracy, non optimal…difficult [path

46

Supply 40 40 50 90

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Assignment The assignment problem, is a special case of the linear programming problem. In general items, it is concerned with a one-to-one assignment. One person to one machine, one machine to one job, etc. The cost (or profit) of each person-machine, machine-job, or other assignment is known; the objective is to minimise the total cost or to maximise the total profit of the resource - job assignment. Hungarian Assignment Method (HAM) Step 1 Locate the smallest cost element in each row of the cost table. Now subtract this smallest element from each element in that row. As a result, there shall be at least one zero in each row of this new table, called the Reduced Cost Table. Step 2 In the reduced cost table obtained, consider each column and locate the smallest element in it. Subtract the smallest value from every other entry in the column. As a consequence of this action, there would be at least one zero in each of the rows and columns of the second reduced cost table. Step 3 Draw the minimum number of horizontal and vertical lines (not the diagonal ones) that are required to cover all the 'zero' elements. If the number of lines drawn is equal to n (the number of rows/columns) the solution is optimal, and proceed to step 6. If the number of lines drawn is smaller than n, go to step 4. Step 4 Select the smallest uncovered (by the lines) cost element. Subtract this element from all uncovered elements including itself and add this element to each value located at the intersection of any two lines. The cost elements through which only one line passes remain unaltered. Step 5 Repeat steps 3 and 4 until an optimal solution is obtained. Step 6 Given the optimal solution, make the job assignments as indicated by the 'zero' elements. This is done as follows: (a) Locate a row which contains only one 'zero' element. Assign the job corresponding to this element to its corresponding person. Cross out the zero's, if any, in the column corresponding to the element, which is indicative of the fact that the particular job and p«rson are no more available. (b) Repeat (a) for each of such rows which contain only one zero. Similarly, perform the same operation in respect of each column containing only one 'zero' element, crossing out the zero(s), if any, in the row in which the element lies. (c)

If there is no row or column with only a single 'zero' element left, then select a row/column arbitrarily and choose one of the jobs (or persons) and make the assignment. Now cross the remaining zeros in the column and row in respect of which the assignment is made.

(d)

Repeat steps (a) through (c) until all assignments are made.

(e)

Determine the total cost with reference to the original cost table.

47

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q1

A project consist of four (4) major jobs, for which four (4) contractors have submitted tenders. The tender amounts, in thousands of rupees, are given below. Jobs Contractors

A

B

C

D

1

120

100

80

90

2

80

90

110

70

3

110

140

120

100

4

90

90

80

90

Find the assignment, which minimizes the total cost of the project. Each contractor has to be assigned one job. Compiler 349 M01 Q2

A Production supervisor is considering, how he should assign five jobs that are to be performed, to five mechanists working under him. He wants to assign the jobs to the mechanists in such a manner that the aggregate cost to perform the jobs is the least. He has following information about the wages paid to the mechanists for performing these jobs: Jobs Mechanist

1

2

3

4

5

A

10

3

3

2

8

B

9

7

8

2

7

C

7

5

6

2

4

D

3

5

8

2

4

E

9

10

9

6

10

Assign the jobs to the mechanists so that the aggregate cost is the least. Q3

N01

compiler 347

A Marketing Manager has 4 subordinates and 4 tasks. The subordinates differ in efficiency. The tasks also differ in their intrinsic difficulty. His estimates of the time each subordinate would take to perform each task is given in the matrix below. How should the task be allocated one to one man so that the total man-hours are minimized ? Tasks I II III IV

Subordinates

1 2 3 4

16 26 76 38

52 56 38 52

34 8 36 48

48

22 52 30 20

Compiler 361

N04

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q4

A BPO company is taking bids for 4 routes in the city to ply pick-up and drop cabs. Four companies have made bids as detailed below: Min prohibited routes Compiler 362 Bids for Routes (Rs.) Company/Routes R1 R2 R3 R4 C1 4,000 5,000 − − C2 − 4,000 − 4,000 C3 3,000 − 2,000 − C4 − − 4,000 5,000 Each bidder can be assigned only one route. Determine the minimum cost that the BPO should incur. N06

Q5

To stimulate interest and provide an atmosphere for intellectual discussion, a finance faculty in a management school decides to hold special seminars on four contemporary topics: leasing, portfolio management, private mutual funds, swaps and options. Such seminars should be held once in a week in the afternoons. However, scheduling these seminars (one for each topic, and not more than one seminar per afternoon) has to be done carefully so that the number of students unable to attend is kept to a minimum. A careful study indicates that the number of students who cannot attend a particular seminar on a specific day is as follows: Leasing Portfolio Management Private Mutual Fund Swaps & Options Monday 50 40 60 20 Tuesday 40 30 40 30 Wednesday 60 20 30 20 Thursday 30 30 20 30 Friday 10 20 10 30 Find an optimal schedule of the seminars. Also find out the total number of students who will be missing at least one seminar. Unbal min Compiler 357 M99

Q6

Solve the following unbalanced assignment problem of minimizing total time for doing all the jobs. M85 Min unbal pad 16.16 Jobs I II III IV V Operators

Q7

1

6

2

5

2

6

2

2

5

8

7

7

3

7

8

6

9

8

4

6

2

3

4

5

5

9

3

8

9

7

6

4

7

4

6

8

Five workers are available to work with the machine and the respective costs (in Rs) associated with each worker-machine assignment is given below. A sixth machine is available to replace one of the existing machines and the associated costs are also stated – unbal min prohib routes Machines Workers M1 M2 M3 M4 M5 M6 A 12 3 6 5 8 B 4 11 5 3 C 8 2 10 9 7 5 D 7 8 6 12 10 E 5 8 9 4 6 RTP

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PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q8

Five swimmers are eligible to compete in a relay team which is to consist of four swimmers swimming four different swimming styles; back stroke, breast stroke, free style and butterfly. The time taken for the five swimmers - Anand, Balu, Chandru, Deepak and Eswar - to cover a distance of 100 meters in various swimming styles are given below in minutes : seconds.' Anand swims the back stroke in 1 : 09, the breast stroke in 1 : 15 and has never competed in the free style or butterfly. Balu is a free style specialist averaging 1 : 01 for the 100 meters but can also swim the breast stroke in 1 : 16 and butterfly in 1 : 20 Chandru swims all styles - backstroke 1 :10 breaststroke 1 : 12 free style 1 : 05 and butterfly,1 : 20 Deepak swims only the butterfly 1 : 11 while Easwar swims the back stroke 1 : 20, the breast stroke 1 : 16, the free style 1 : 06 and the butterfly 1 : 10. Which swimmers should be assigned to which swimming style? Who will not be in the Team? unbal min prohib routé N91

Q9

The cost of transporting different products to different products to different warehouses of a company is given. ‘’x’’ in the matrix denotes that the particular product is not required in the particular warehouse. Make optimal assignments of the products and also find the cost of such assignment. Pad 16.13 min unbal prohib route Warehouses I II III IV V Product A 28 19 ‘’x’’ 21 22 B 32 25 22 13 18 C 21 18 21 ‘’x’’ 17 D 22 29 21 17 28

Q10

Solve the assignment problem represented by the following effective matrix a b c d e f ___________________________________ A 9 22 58 11 19 27

Q11

B

43

78

72

50

63

48

C

41

28

91

37

45

33

D

74

42

27

49

39

32

E

26

11

57

22

25

18

F

3

56

53

31

17

28

min multiple opti sol

N99

ABC airline operating 7 days a week has given the following time-table. Crews must have minimum layover of 5 hours between flights. Obtain the pairing flights that minimize the layover time away from home. For any given pairing the crew will be based at the city that results in the smaller layover. M00 Hyderabad-Delhi ________________________________ Flight No. Departure. Arrival A1 6 AM 8 AM A2 8 AM 10 AM A3 2 PM 4 PM A4 8 PM 10 PM

Delhi-Hyderabad ________________________________ Flight No. Departure. Arrival B1 8 AM 10 AM B2 9 AM 11 AM B3 2 PM 4 PM B4 7 PM 9 PM Compiler 353

50

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q12

WELLDONE Company has taken the third floor of a multistoried building for rent with a view to locate one of their zonal offices. There are five. main rooms in this floor to be assigned to five managers. Each room has its own advantages and disadvantages. Some have windows, some are closer to the washrooms or to the canteen or secretarial pool. The rooms are of all different sizes and shapes. Each of the five managers was asked to rank their room preferences amongst the rooms 301, 302, 303, 304 and 305. Their preferences were recorded in a table as indicated below:

M1 302 303 304

M2 302 304 305 301

Manager M3 303 301 304 305 302

M4 302 305 304 303

M5 301 302 304

Most of the manager did not list all the 5 rooms sice they were not satisfied with some of these Rooms and they have left off these from the list. Assuming that their preferences can be qualified by numbers, find out as to which manager would be assigned to which room so that their total preference is minimum. N90, minimize ranking Q13

The XYZ co. has 5 jobs A,B,C,D and E to be done and 5 men L,M,N,O and P to do these jobs. The number of hours each man would take to accomplish each job is given by take to accomplish each job is given by the following table. Work out the optimum assignment and the total minimum time taken. Men

Jobs

bal,min,multi

L

M

N

O

P

A

16

13

17

19

20

B

14

12

13

16

17

C

14

11

12

17

18

D

5

5

8

8

11

E

5

3

8

8

10

Q14

Solve the following assignment problem and obtain the minimum cost at which all the jobs can be performed. Min unbal multi Job (cost in ’00 Rs) Worker 1 2 3 4 5 A 25 18 32 20 21 B 34 25 21 12 17 C 20 17 20 32 16 D 20 28 20 16 27

Q15

A solicitor’s firm employ’s typists on hourly piece rate basis for their daily work. There are 5 typists and their charges and speed are different. According to an earlier understanding , only one job is given to one typist and the typist is paid for a full hour even when he works for a fraction of an hour. Find the least cost allocation for the following data: Typist A B C D E

Rate/hour (Rs) No. of pages typed per hour 5 12 6 14 3 8 4 10 4 11

51

Job P Q R S T

No. of pages 199 175 145 298 178 min

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q16

A hospital has to pay Nurses for 40 hours a week. One Nurse is assigned to one patient. The cost per hour for each of the Nurses is given below: (a) Find the Nurse- Patient combination to minimize cost to the hospital. (b) How much does each nurse earn per week? Patient W Patient X Patient Y Nurse K 10 10 30 Nurse L 30 10 20 Nurse M 20 30 20 Suppose that a new patient Z is admitted, and that a new nurse N is appointed. The new patient is charged Rs 40 per hour by each of the existing nurses. The new nurse charges Rs 50 per hour irrespective of the patient. (c) (d)

What would be your revised calculations? Comment on the new solution.

M10

Q17

A city corporation has decided to carry out road repairs on main four arteries of the city. The government has agreed to make a special grant of Rs 50 lakhs towards the cost with a condition that the repairs must be done at the lowest cost and quickest time. If conditions worsens. then a supplementary token grant will also be considered favorably. The corporation has floated tenders and 5 contractors have sent in their bids. In order to expedite work. one road will be awarded to only one contractor Cost of Repairs (Rs lakhs) Contractors /Road R1 R2 R3 R4 C1 9 14 19 15 C2 7 17 20 19 C3 9 18 21 18 C4 10 12 18 19 C5 10 15 21 16 (i) Find the best way of assigning the repair work to the contractors and the costs. (ii) If it is necessary to seek supplementary grants, then what should be amount sought? (iii) Which of the five contractors will be unsuccessful in his bid? (iv) If C1 unable to accept any work, find best assignment

Q18

A company plans to assign 5 salesman to 5 districts in which it operates. Estimates of sales revenue in thousands of rupees for each salesman in different districts are given in the following table. In your opinion, what should be the placement of the salesman if the objective is to maximize the expected sales revenue? District D1 D2 D3 D4 Salesman S1 40 46 48 36 48 S2 48 32 36 29 44 S3 49 35 41 38 45 S4 30 46 49 44 44 S5 37 41 48 43 47 bal max A firm produces 4 products. There are 4 operators who are capable of producing any of these 4 products. The firm records 8 hours a day and allows 30 minutes for lunch. The processing time in minutes and profit for each of the products are given below: Products A B C D Operator 1 15 9 10 6 2 10 6 9 6 3 25 15 15 9 4 15 9 10 10 Profit (Rs) per unit 8 6 5 4 Find the optimal assignment of products to operators bal max,

Q19

52

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q20

A company has four zones open and four marketing managers available for assignment. The zones are not equal in sales potentials. It is estimated that a typical marketing manager operating in each zone would bring in the following Annual sales: Zones Rs. East 2,40,000 West 1,92,000 North 1,44,000 South 1,20,000 The four marketing manages are also different in ability. It is estimated that working under the same conditions, their yearly sales would be proportionately as under: Manager M : 8 Manager N : 7 Manager O : 5 Manager P : 4 Required: If the criterion is maximum expected total sales, find the optimum assignment and the maximum sales. N07 Compiler 365

Q21

The Marketing director of a multi unit company, Mr Ramesh Arora is faced with the problem of assigning 5 senior marketing managers to 6 zones. The efficiency table of these managers is given belowZones I II III IV V VI Managers A 75 93 89 84 80 32 B 83 87 71 78 76 87 C 77 74 85 86 80 93 D 95 98 88 93 85 84 E 92 93 81 91 71 78 As an advisor to the company, recommend which zone should be managed by a junior manager because of non availability of one more senior marketing manager, so as to maximize the efficiency of the company. Max,unbal

Q22

Solve the following assignment problem to maximize the production. Machines A B C D Operators 1 10 5 7 8 2 11 4 9 10 3 8 4 9 7 4 7 5 6 4 5 8 9 7 5

Q23

max unbal

An airline has drawn up a new flight schedule involving 5 flights. To assist in allocating 5 pilots to the flights, it has asked them to state their preference scorers by giving each flight a number out of 10. The higher the number, greater the preference is. Certain of these flights are unsuitable to some pilots owing to domestic reasons. These have been marked with a ‘’x’’. max,bal,prohib Flight Number F1 F2 F3 F4 F5 Pilots A 8 2 x 5 4 B 10 9 2 8 4 C 5 4 9 6 x D 3 6 2 8 7 E 5 6 10 4 3 What should be the allocation of pilots to flights in order to meet as many preferences as possible.

53

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q24

A methods engineer wants to assign 4 new methods to 3 work centers. The assignment of the new methods will increase production, details of which are given below. Determine the optimal assignment, if only one method can be assigned to each work center. Increase in production(units) in work centers A B C Methods 1 10 7 8 2 8 9 7 3 7 12 6 4 10 10 8 pad 16.5 unbal max multiple

Q25

The captain of the playwell cricket team has to allot 5 middle order batting positions to five batsmen. The average run scored by each batsmen at these positions are as follows M92 Batting positions I II III IV V P 40 40 35 25 50 Batsman Q 42 30 16 25 27 R 50 48 40 60 50 S 20 19 20 18 25 T 58 60 59 55 53 1. Find the assignment of batsmen to positions, which would give the maximum number of runs. 2. What will be the total runs scored if batsmen T wants only the III position 3. If another bats men U with the following average runs in the batting positions as given below: Batting position I II III IV V Average Runs 45 52 38 50 49 Is added to the team, should he be included to play in the team?, If so who will be replaced by him?

Q26

A manufacturing company has four zones East, West, North and South and four sales engineers A,B,C, and D respectively for assignment. The zones are not equally rich in sales potential. Therefore, it is estimated that a particular engineer operating in a particular zone will bring the following sales East – Rs 4,20,000, West – Rs 3,36,000,

North – Rs 2,94,000, South Rs 4,62,000

The sales engineers have different sales ability, working under the same conditions, their yearly sales are [proportional to 14, 9, 11 and 8 respectively. The criteria for maximum expected total sales is to be met by assigning the best sales engineer to the richest zone, the next best to the second richest zone and so on. Find the optimum assignment and the maximum sales. M98 Q 27

five star hotel has 4 banquet halls that can be used for all functions including weddings. The halls are all about the same size but the facilities in each hall differs. During the heavy marriage season, 4 parties approached the executive director to reserve a hall for wedding to be celebrated on the same day. These marriage parties were told that the first choice among these four halls would cost Rs 10,000 for the day. They were also required to indicate the second third and fourth preferences and the price that they would be willing to pay. Marriage party A and D indicated that they woudn’t be interested in halls H3 and H4. Other relevant particulars are given belowRevenue per hall(Rs) H1 H2 H3 H4 Marriage party A 10,000 9,000 B 8,000 10,000 8,000 5,000 C 7,000 10,000 6,000 8,000 D 10,000 8,000 Decide on the allocation that will maximize the revenue to the hotel.

54

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q28

A car hiring company has one car at each of the 5 depots A,B,C,D and E. A customer in each of the 5 towns V, W, X, Y and Z requires a car. The distance in Kms. Between depots (origin) and the towns (destinations) are given in the following table: Depots A B C D E V 3 5 10 15 8 W 4 7 15 18 8 X 8 12 20 20 12 Z 10 10 15 25 10 Find out as to which car should be assigned to which customer so that the total distance travelled is a minimum. How much is the total travelled distance?

Q29

The cost matrix giving selling costs per unit of a product by salesman A,B,C and D in Regions R1, R2, R3 and R4 is given below: Assign one salesman to one region to minimize selling cost. If the selling price of the product id Rs 200 per unit and variable cost excluding the selling cost given in the table is Rs 100 per unit, find the assignment that would maximize the contribution. What other conclusions can you make from the above? A B C D R1 R2 R3 R4

Q30

4 20 36 52

12 28 44 60

16 32 48 64

8 24 40 56

N08

A firm is contemplating the introduction of three products 1,2 and 3 in the 3 plants A,B and C. Only a single product is decided to be introduced in each of the three plants. The unit cost of production is given in the following matrix: A 8 10 7

1 2 3

B 12 6 6

C M 4 5

(M indicates impossible assignment)

A. How should the products be assigned so that the total unit cost is minimum? B. If the quantity of different products to be produced is as follows then which assignments shall minimize the aggregate production cost? C. It is expected that the selling prices of the products produced by different plants would be different. Assuming that the the quantities given at (b) above would be produced and sold, how should the products be assigned to maximize profit?

1 2 3

A 15 18 12

B 18 16 10

C 10 8

55

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q31

A company has 4 territories and 4 salesmen available for assignment. Te territories are not equally rich in their sales potential. The annual total sales potential in each territory is- Territory I – Rs 60,000, Territory II – Rs 50,000, Territory III – Rs 40,000, Territory IV – Rs 30,000. Four salesman differ in ability. Estimation of their yearly sales proportion under the same working conditions areSalesman A B C D Proportion 7 5 5 4 To maximize expected total sales, the intitutive answer is to assign the best salesman to the richest territory the next best salesman to the second richest and so on. Verify this answer by assignment technique RTP

56

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

SIMULATION

Q

What is simulation and what are the steps in simulation? SIMULATION means to assume the mere appearance of without the reality. Thus the appearance is true but not real, which implies that simulation is imitation of reality. Simulation is the representation of a system by a model which will react to change in a similar way to that which is being simulated. Simulation may be defined as "a quantitative procedure which describes a process by developing a model of that process and later conducting a series of organized trial and error experiments to predict the behaviours of the process over a period. In other words, simulation portrays how the real process would react to certain change." Steps Involved in Simulation Process The following are the steps involved generally in a Simulation Process. 1. 2. 3. 4. 5. 6. 7. 8.

Q

Identification of the problem or system which is intended to be s!mulated. Formulation of the model intended to be used. Testing of the model by comparing its behaviour with the behaviour of the actual problem environment Identification and collection of the data required for testing of the model. Running of the simulation. Analysing the results of simulation in case necessary, the solution which is being evaluated may be changed Re-running of the simulation to test the new solution Validation of the simulation, Le., the inferences drawn which will be appropriate for running the simulation valid

Enumerate the advantages and disadvantages of Simulation Advantages Simulation can be used to investigate the behaviour of problems which are too complex to be modelled mathematically. The technique can also be used when the variables in the problem e.g., arrival time, service time do not follow the standard distributions required for the mathematical models, i.e., Poisson distribution, Normal exponential distribution. The basic principles of the simulation technique are fairly simple and it is, therefore, more attractive to people who are not expert in quantitative techniques. Simulation does not interfere with the real world system but only with table model and, therefore it results in saving of cost. It is a micro analysis of big and complicated system by breaking into each sub-system and studying the interface of the various sub-systems. Time will be saved in simulation e.g.,the effects of ordering, advertising or other policies over many months or years can be obtained by computer simulation in a short time. Simulation allows us to study the interactive effect of individual components or variables in order to determine which are important.

57

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Disadvantages Simulation is not an optimizing technique. It simply allows us to select the best of the alternative systems examined. Reliable results are possible only if the simulation is continued for a long period. A computer is essential to cope with the amount of calculation in simulation modeling. To develop a simulation model means consumption of voluminous data and it may be very costly. Each simulation model is unique and its solution cannot be applied to other problems however similar they may be. The simulation model does not produce answers by itself. Managers must generate all of the conditions and constraints for solutions they want to examine. Simulation methods generally are not as efficient as the analytical methods Q

What is Monte carlo Simulation and what are the steps in Monte carlo simulation? Also enlist its usage in inventory control. Monte Carlo simulation Is a special technique of simulation which Involves the selection of random observations within the simulation model. It is used to solve problems that depend upon probability, where physical experimentation is virtually unpractical and it is practically impossible to create mathematical formula. It is used to examine Inventory, queing, scheduling and forecasting problems. Steps: 1.

Determination and measure of effectiveness which may be either of (I) Maximisatlon of ROI (II). Minimlsation of inventory holding cost, stock etc.

2.

Identification of those variables which Influences the measures of effectiveness significantly.

3

Determination and cumulative probability distribution of each variable selected under steps

4.

Plotting these cumulative probability distribution with probability on the vertical axis and values of variables on horizontal axis.

5.

Obtaining a set of random numbers from tables

6.

Now consider each random number as decimal value of the cumulative probability distribution and enter the cumulative distribution plot from the vertical axis.

7.

Now project this point horizontally, until! it intersects cumulative probabilitydistribution curve. Then project the point of intersection down into the vertical axis.

8.

Then record the value generated above in the formula derived.

9.

Solve and record the value. This value is the objective for that simulated value.

10.

Steps 6 to 9 are repeated until sample size is large enough to the satisfaction of the decision maker.

Monte carlo Simulation has following applications in Inventory controla. b. c. d.

Determination of Reorder level and Reorder quantity. Computation of stock out costs and impact on profit. Analysis of value of storage facilities for avoding stockouts and impact on profits Analysis of demand distribution during lead time.

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Q1

A car rental agency has collected the following Data on the demand for five- seater vehicles over the past 50 days Daily Demand No. of days

4 4

5 10

6 16

7 14

8 6

The agency has only 5 cars currently. i. Use the following 5 random numbers to generate 5 days of demand for the rental agency. Random Numbers: 15,48,71,56,90 ii. What is the average numbers of cars rented per day for 5 days ? iii. How many rentals will be lost over the 5 days ? M11 Q2

ABC Co-operative Bank receives and disburses different amount of cash in each month. The bank has an opening cash balance of Rs 15 crores in the first month. Pattern of receipts and disbursements from past data is as follows: Monthly cash receipts Monthly Cash disbursements Rs. In crores Probability Rs. In crores Probability 30 0.20 33 0.15 42 0.40 60 0.20 36 0.25 39 0.40 99 0.15 57 0.25 Simulate the Cash position over a period of 12 months. Required: 1. Calculate the probability that the ABC Cooperative Bank will fall short in payments. M10 2. Calculate the average monthly shortfall. 3. If ABC Bank can get an overdraft facility of Rs.45 Crores from other nationalized banks, what Is the probability that they will fall short in Monthly Payments? Use the following sequence (row-wise) of paired Random Numbers. 1778 4316 7435 3123 7244 4692 5158 6808 9358 5478 9654 0977

Q3

With a view to improving the quantity of customer services, a Bank is interested In making an assessment of the Waiting time of its Customers coming to of Its branches located in residential area. This branch has only one telling counter. The arrival rate of the Customers and the Service Rate of Teller are given below:

Time between two consecutive arrivals of customers 3 minutes 4 minutes 5 minutes 6 minutes 7 minutes

Probability

Service time by the teller

0.17

3 minutes 4 minutes 5 minutes 6 minutes 7 minutes

0.25 0.25 0.20 0.13

Probability 0.10 0.30 0.40 0.15 0.05

You are required to Simulate 10 arrivals of Customers in the system starting from 11 AM and show the waiting time of the Customers and idle time of the teller. Use the following random numbers taking the first two random numbers in two digits each for first trial and. so on. 11, 56, 23, 72, 94, 83, 83, 02, 97, 99, 83, 10, 93, 34, 33, 53, 49, 94, 37 and 97. M10

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Q4

At a small store of readymade garments, there is one clerk at the counter who is to check bills, receive payments the packed garments into fancy bags. The arrival of customer at the store is random and service time varies from one minute to 6 minutes the frequency distribution for which is given below: Time between arrivals (minutes) 1 2 3 4 5 6

Frequency 5 20 35 25 10 5

Service time (in minutes) 1 2 3 4 5 6

Frequency 1 2 4 2 1 0

The store starts work at 11 a.m. and closes at 12 noon for lunch and the Customers are served on the "first came ft served basis". Using Monte Carlo Simulation technique, find average length of waiting line, average waiting time, average service time and total time spent by a customer in system. You are given the following set of random numbers, first twenty for arrivals and last twenty for service: N09 64 07 30 11 Q5

04 08 75 79

02 59 38 61

70 53 24 77

03 01 57 10

60 62 09 16

16 36 12 55

18 27 18 52

36 97 65 59

38 86 25 63

A single counter ticket booking centre employs one booking clerk. A passenger on arrival Immediately goes to the booking counter for being served If the counter is free. If, on the other hand, the counter is engaged, the passenger will have to wait. The passengers are served on first come first served basis. The time of arrival and the time of service varies from one minute to six minutes. The distribution of arrival and services time is as under: Arrival / service time 1 2 3 4 5 6

Arrival (Probability) 0.05 0.20 0.35 0.25 0.10 0.05

Service (Probability) 0.10 0.20 0.40 0.20 0.10 -

Simulate the arrival and service of 10 passengers starting from 9 A.M by using the following random numbers in pairs respectively for arrival and services. Random numbers 60, 09, 16, 12, 08, 18, 36, 65, 38, 25, 07, 11, 08, 79, 59, 61, 53, 77, 03, 10. Determine the total duration of Idle time of Booking Clerk and Waiting time of passengers N08 Q6

A Publishing house has bought out a new monthly magazine, which sells at Rs. 37.5 per copy. The cost of producing it is Rs. 30 per copy. A Newsstand estimates the sales pattern of the magazine as follows: Demand Copies Probability 0 < 300 0.18 300 < 600 0.32 600 < 900 0.25 900 < 1200 0.15 1200 < 1500 0.06 1500 < 1800 0.04 The newsstand has contracted for 750 copies of the magazine per month from the publisher. The unsold copies are returnable to the publisher who will take them back at cost less Rs. 4 per copy for handling charges. The newsstand manager wants to simulate of the demand and profitability. The of following random number may be used for simulation: 27, 15, 56, 17, 98, 71, 51, 32, 62, 83, 96, 69.

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You are required to(i) Allocate random numbers to the demand patter forecast by the newsstand. (ii) Simulate twelve months sales and calculate the monthly and annual profit/loss. (iii) Calculate the loss on lost sales. Q7

A company trading in motor vehicle spares wishes to determine the level of stock it should carry for the item in its range. Demand is not certain and replenishment of stock takes 3 days. For one item X, the following information is obtained: Demand (unit per day) Probability 1 .1 2 .2 3 .3 4 .3 5 .1 Each time an order is placed, the company incurs an ordering cost of Rs. 20 per order. The company also incurs carrying cost of Rs. 2.50 per unit per day. The inventory carrying cost is calculated on the basis of average stock. The manager of the company wishes to compare two options for his inventory decision. (A) Order 12 units when the inventory at the beginning of the day plus order outstanding is less than 12 units. (B) Order 10 units when the inventory at the beginning of the day plus order outstanding is less than 10 units. Currently (on first day) the company has a stock of 17 units. The sequence of random number to be used is 08, 91, 25, 18,40, 27, 85, 75, 32, 52 using first number for day one. You are required to carry out a simulation run over a period of 10 days, recommended which option the manager should chose

Q8

A bakery shop keeps stock of a popular brand of cake. Previous experience indicates the daily demand as given here: Daily demand : 0 10 20 30 40 50 Probability : 0.01 0.20 0.15 0.50 0.12 0.02 Consider the following sequence of random numbers; Random. No. 48, 78, 19, 51, 56, 77, 15, 14, 68, 09 Using this sequence, simulate the demand for the next 10 days. Find out the stock situation if the owner of the bakery decides to make 30 cakes every day. Also, estimate the daily average demand for the cakes on the basis of simulated data N99

Q9

A book-store wishes to carry Systems Analysis and Design in stock. Demand is probabilistic and replenishment of stock takes 2 days (i.e., if an order is placed in March 1, it will be delivered at the end of the day on March 3). The probabilities of demand are given below: Demand (daily) : 0 1 2 3 4 Probability : 0.05 0.10 0.30 0.45 0.10 Each time an order is placed, the store incurs an ordering cost of Rs.10 per order. The store also incurs a carrying cost of Rs.0.50 per book per day. The inventory carrying cost is calculated on the basis of stock at the end of each day. The manger of the book-store wishes to compare two options for his inventory decision: A. Order 5 books, when the inventory at the beginning of the day plus orders outstanding is less than 8 books. B. Order 8 books, when the inventory at the beginning of the day plus orders outstanding is less than 8 books. Currently (beginning of the 1st day) the store has stock of 8 books plus 6 books plus 6 books ordered 2 days ago and expected to arrive next day. Using Monte-Carlo simulation for 10 cycles, recommend which option the manager should choose? The two digits random numbers are given below: 89, 34, 78, 63, 81, 39, 16, 13, 73 (M00)

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Q10

A retailer deals in a perishable commodity. The daily demand and supply are variables. The data for the past 500 days show the following demand and supply: Supply Availability (Kg) 10 20 30 40 50

No. of days 40 50 190 150 70

Demand (Kg) 10 20 30 40 50

Demand No. of days 50 110 200 100 40

The retailer buys the commodity at Rs.20 per kg and sells it at Rs.30 per kg. Any commodity remains at the end of the day, has no saleable value. Moreover, the loss (unearned profit) on any unsatisfied demand is Rs.8 per kg. Given the following pair of random numbers, simulate 6 days sales, demand and profit. (31, 18); (63, 84); (15, 79); (07, 32) (43, 75); (81, 27) The first random number in the pair is for supply and the second random number is for demand viz. in the first pair (31, 18), use 31 to simulate supply and 18 to simulate demand. N00 Q11

An investment company wants to study the investment projects based on market demand, profit and the investment required, which are independent of each other. Following probability distributions are estimated for each of these three factors: Annual Demand (Units in thousands): 25 30 35 40 45 50 55 Probability 0.05 0.10 0.20 0.30 0.20 0.10 0.05 Profit Probability

3.00 0.10

5.00 0.20

7.00 0.40

Investment required (in 000’s of Rs): Probability

2,750 0.25

3,000 0.50

3,500 0.25

9.00 0.20

10.00 0.10

Using simulation process, repeat the trial 10 times, compute the investment on each trail taking these factors into trail. What is the most likely ret Use the following random numbers : urn ? (30, 12, 16); (59, 09, 69); (63, 94, 26); (27, 08, 74); (64, 60, 61);

(28, 28, 72);

(31, 23, 57);

(54, 85, 20);

(64, 68, 18); (32, 31, 87). In the bracket above, the first random number is for annual demand, the second one is for profit and the last one is for the investment required. M01 Q12

A Car Manufacturing Company manufactures 40 cars per day. The sale of cars depends upon demand which has the following distribution: Sales of cars Probability 37 0.10 38 0.15 39 0.20 40 0.35 41 0.15 42 0.05 The production cost and sale price of each car are Rs.4 lakh and Rs.5 lakh respectively. Any unsold car is to be disposed off at a loss of Rs.2 lakh per car. There is a penalty of Re.1 lakh per car, if the demand is not met. Using the following random numbers, estimate total profit/loss for the company for the next ten days: 9, 98, 64, 98, 94, 01, 78, 10, 15, 19 If the company decides to produce 39 cars per day, what will be its impact on profitability?

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M02

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q13

Param and Karam are workers on a two-station assembly line. The distribution of activity time at their stations is as follows: Time (in sec) Time frequency for param Time frequency for karam 10 4 4 20 6 5 30 10 6 40 20 7 50 40 10 60 11 8 70 5 6 80 4 4 (i) Simulate operation of the line for eight times. Use the random numbers given below:

14 01 95 40 (ii)

Operation 1 61 82 00 03

Operation 2 36 76 55 25

97 41 13 34

Assuming Karam must wait until Param completes the first item before starting work. Will he have to wait to process any of the other eight items? Explain your answer, based upon your simulation

Q14

A company manufactures around 200 mopeds. Depending upon the, availability of raw materials and other conditions, the daily production has been varying from 196 mopeds to 204 mopeds, whose probability distribution is as given. below: ' Production/day: Probability 196 1.00 197 0.95 198 0.86 199 0.74 200 0.60 201 0.40 202 0.25 203 0.14 204 0.06 The finished mopeds are transported in a specially designed three storied lorry that can accommodate only 200 mopeds. Use the following 15 random numbers 82, 89,78,24, 53, 61, 18, 45, 04, 23,50, 77, 27, 54,10 to simulate the process to find out: (i) What will be the average number of mopeds waiting in the factory? (ii) What will be the average number of empty space on the lorry?

Q15

Dr. STRONG is a dentist' who schedules all her patients for 30 minutes appointments. Some of the patients take more or less than 30 minutes depending on the type of dental work to be done. The following summary shows the various categories of work, their probabilities and the time needed to complete the work: Category Time Required Probability Filling 45 0.40 Crown 60 0.15 Cleaning 15 0.15 Extraction 45 0.10 Check up 15 0.20 Simulate the dentist's clinic for four hours and determine the average waiting time for the patients as well as the idleness of the doctor. Assume that all the patients show up at the clinic at exactly their scheduled arrival time starting at 8.00 A.M. Use the following random numbers handling the above problem: 40 82 11 34 25 66 17. 79 N90

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Q16

The occurrence of rain in a city on a day is dependent upon whether or not it rained on the previous day. If it rained on the previous day, the.rain distribution is given by : Event Probability

No rain 0.50

1 cm. Rain 0.25

2 cm. Rain 0.15

3 cm. Rain 0.05

4cm.Rain 0.03

5cm.Rain 0.02

If it did not rain the previous day, the rain distribution is given by : Event Probability

No rain 0.75

1 cm. Rain 0.15

2 cm. Rain 0.06

3 cm. Rain 0.04

Simulate the city's weather for 10 days and determine by simulation the total days without rain as well as the total rainfall during the period. Use the following random numbers: 67 63 39 55 29 78 70 06 78 76 for simulation. Assume that for the first day of the simulation it had not rained the day before. N93 Q17

The output of a production line is checked by an inspector for one or more of three different types of defects, called defects A, 8 and C. If defect A occurs, the item is scrapped. If defect B or C occurs, the item must be reworked. The time required to rework a B defect is 15 minutes and the time required to rework a C defect is 30 minutes: The probabilities of an A, B and C defects are 0.15, 0.20 and 0.10 respectively. For ten items coming off the assembly line, determine the number of items without any defect, the number scrapped, the total minutes of rework time & total no of good units. Use the following random numbers: RN for defect A RN for defect B RN for defect C

Q18

48 47 82

55 36 95

91 57 18

40 04 96

93 79 20

01 06 04

83 10 56

63 13 11

47 57 52

52 09 03

M04

A book store wishes to carry 'Ramayana' in stock. Demand is probabilistic and replenishment of stock takes 2 days (i.e. if an order is placed on March 1, it will be delivered at the end of the day on March 3). The probabilities of demand are given below Demand (daily) Probability

0 0.05

1 0.10

2 0.30

3 0.45

4 0.10

Each time an order is placed, the store incurs an ordering cost of Rs10 per order. The store also incurs a carrying cost of Rs 0.50 per book per day. The stock out cost is Rs 5 per unit. The inventory carrying cost is calculated on the basis of stock at the end of each day. The manager of the book store wishes to compare two options for his inventory decision. A.

Order 5 books when the inventory at the beginning of the day plus orders outstanding is less than 8 books. B. Order 14 books when the inventory at the beginning of the day plus orders outstanding is less than 8. Currently (beginning of 1 st day) the store has a stock of 8 books plus 6 books ordered two days ago and expected to arrive next day. Using Monte Carlo Simulation for 10 cycles, recommend which options the manager should choose. The two digit random nos. are : 89, 34, 78, 63, 61, 81, 93, 16, 13, 73

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PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q19

A dealer in colour TVs wants to use a scientific method for reducing his investment in stocks. The daily demand for a T.V. is random and varies from day to day in an unpredictable pattern. From the past sales records, the dealer has been able to establish a probability distribution of the demand as given below: Daily Demand (units) 2 3 4 5 6 7 8 9 10 Probability 0.06 0.14 0.18 0.17 0.16 0.12 0.08 0.06 0.03 The dealer also known from his past experience that the lead time is almost fixed at 5 days. The dealer would like to study the implications of a possible inventory policy of ordering 30 units, whenever the inventory at the end of the day is 20 units. The inventory on hand is 30 units & use the following sequence of random numbers to simulate the demand for next 25 days. Random Numbers: 3, 38,17,32,69,24,61,30,3,48,88,71,27,80,33,90,78,55, 87,16,34, 45, 59, 20, 59.

Q20

The management of ABC company is considering the question of marketing a new product. The fixed cost required in the project is Rs 4,000 3 factors are uncertain viz, the selling price, variable cost and the annual sales volume. The product has a life of only one year. The management has the data on these 3 factors as under: Selling price Rs 3 4 5

Probability 0.2 0.5 0.3

Variable cost Rs 1 2 3

Probability 0.3 0.6 0.1

Sales volume (units) 2,000 3,000 5,000

Probability 0.3 0.3 0.4

Considering the following sequence of thirty random numbers: 81, 32, 60. 04, 46, 31, 67, 25, 24, 10, 40, 02, 39, 68, 08, 59, 66, 90, 12, 64, 79, 31, 86, 68, 82, 89, 25, 11, 98, 16 Using the sequence (first 3 random numbers for the first trial etc) simulate the average profit for the above project on the basis of 10 trials n94 Q21

The Ever alert-Ltd.: which has a satisfactory preventive maintenance system in its plant, has installed a new Hot Air Generator based on electricity instead of fuel oil for drying its finished products. The Hot Air Generator requires periodicity shutdown maintenance. If.the shut down is scheduled yearly, the cost of maintenance will be as under: Maintenance cost (Rs) Probability 15,000 0.3 20,000 0.4 25,000 0.3 The costs are expected to be almost linear i.e. if the shutdown is scheduled twice a year the maintenance cost will be double. There is no previous experience regarding the time taken between break downs. Costs associated with break down will vary depending upon the periodicity of maintenance. The probability distribution of break down cost is estimated as under: Breakdown costs Rs per annum Shut down once a year Shut down twice a year 75,000 0.2 0.5 80,000 0.5 0.3 1,00,000 0.3 0.2 Simulate the total costs maintenance and breakdown cost and recommend whether shutdown overhauling should be restored to once a year or twice a year?

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Q22

A doctor who has introduced an appointments system for daily consultations has derived the following information regarding patient punctually Minutes early 3 6% 2 29% 1 41% On time Minutes late

12% 1

7% 5% The doctor times his consultations over a period, and derives the following frequency distribution: Minutes 12 10% 13 15% 14 28% 15 34% 16 13% The doctor would like to issue appointments at 15 minutes interval and wishes to have an idea of his idle time, the patient a waiting time, and whether he can complete his appointments on schedule. Simulate sixteen consultations and derive the required information Given the following series of random numbers: 17 14 50 40 83 13 94 08 49 98 79 51 43 74 92 24 09 21 40 12 46 91 09 05 95 44 52 79 91 53 15 16

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PERT/ CPM Q

Enumerate basic Assumptions of PERT/ CPM 1. A project can be sub-divided into a set of predictable independent activities 2. The precedence relationships of project activities can be completely represented by a noncyclical network graph in which each activity connects directly into its immediate successors. 3. activity times may be estimated either as single-point estimates or as three-point PERT estimates and are independent of each other 4. In PERT/CPM model, activity duration is assumed to follow the beta distribution. The standard deviation of the distribution is assumed to be 1/6th of its range; the mean is approximated by 1/6( to + 4tm + tp) and the variances in length of a project is assumed to be equal to the sum of the variances of activities on the critical path.

5. The CPM Model also has its special assumptions. One is; the duration of an activity is linearly (and inversely) related to the cost of resources applied to the activity.

Q

Distinguish between PERT and CPM The PERT and CPM models are similar in terms of their basic structure, rationale and mode of analysis. However, there are certain distinctions between PERT and CPM networks which are enumerated below: 1.

CPM is activity oriented i.e. CPM network is built on the basis of activities. Also results of various calculations are considered in terms of activities of the project. On the other hand, PERT is event oriented.

2.

CPM is a deterministic model i.e. it does not take into account the uncertainties involved in the estimation of time for execution of a job or an activity. It completely ignores the probabilistic element of the problem. PERT, however, is a probabilistic model. It uses three estimates of the activity time; optimistic, pessimistic and most likely; with a view to take into account time uncertainty. Thus, the expected duration for each activity is probabilistic and expected duration indicates that there is fifty per probability of getting the job done within that time.

3.

CPM places dual emphasis on time and cost and evaluates the trade-off between project cost and project time. By deploying additional resources, it allows the critical path project manager to manipulate project duration within certain limits so that project duration can be shortened at an optimal cost. On the other hand, PERT is primarily concerned with time. It helps the manager to schedule and coordinate various activities so that the project can be completed on scheduled time.

4.

CPM is commonly used for those projects which are repetitive in nature and where one has prior experience of handling similar projects. PERT is generally used for those projects where time required to complete various activities are not known as prior. Thus PERT is widely used for Planning and scheduling research and development projects. .

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Basic steps involved in drawing a CPM/PERT network are : (i) (ii) (iii) (iv) (v) Q

Breaking up of the entire project into smaller systems known as tasks. For each task ascertain the activities and events to be performed. For each activity determine the preceding and succeeding activities. For each activity determine or estimate the time and other resources needed. Draw a network depicting the assembly of tasks into a project.

Explain the following in the context of a network: a. b.

N10

M08, M09

Critical path Dummy activity

A Dummy Activity is a hypothetical activity which consumes no resources and time. Such an activity is represented by dotted lines and is inserted in the network to clarify activity pattern under the following situations: (i) It is created to make activities with common starting and finishing events distinguishable easily. (II) To Identify and maintain the proper precedence relationship between activities that are not connected by any events. (III) To bring all "loose ends' to a single initial and a single terminal event In each network using dummies, If required. Critical Path: The longest time duration path is known as critical path & represented by thick line or double lines. All activities lying in this critical path are called critical activities. Any delay in their execution will lead to a delay in the completion of the entire project. . Sometimes there may be more than one critical paths as they have same highest time duration. In that case, the critical path having highest variance is called principle critical path. Q

Define proiect. State some of its characteristics.

N91

A project can be defined as a set of activities or jobs that are performed in a certain sequence determined logically or technologically and it has to be completed within (I) a specified time, (ii) a specified cost and (iii) meeting the performance standards. A project is a new work for which organisation has no preliminary experience. Examples of a project from fairly diverse fields could be cited. Some of them are given below: 1. Introducing a new product in the market. . 2. Construction of a new bridge over a river or construction of a 25 -- storied building. 3. Executing a large and complex order on jobbing production. 4. Sending a space craft to the mars. All these projects are characterized by the following set of common implications, although they pertain to widely different fields. (i) The large-scale characteristic : These projects are generally unusually large and complex. Thousands of suppliers, workers and other categories of persons are involved and their efforts have to be .coordinate for completion of the project. (ii) The non-recurring characteristic: These projects are generally of a one time nature. Neither in the past, nor in the future they are likely to be undertaken substantially in the same form. (iii) Uncertain and critical dates : Duration of the various activities involved in such projects are usually uncertain. Further in such type of projects, many critical dates exist by which operations must be completed in order to complete the entire project on schedule. (iv) Completion dead line : The fourth distinct feature of these projects is that there is dead line for the completion of the entire project. In case of any delay in the completion of the project, some penalty is levied for such delay beyond the dead line.

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PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

Q

Define various types of floats. The four types of floats used in network analysis are briefly explained below: (i) Total float:

The total float of an activity represents the amount of time by which an activity can be delayed, without delaying the project completion date. In other words, it refers to the amount of free time associated with an activity which can be used before, during or after the performance of this activity. Total float is the positive difference between the earliest finish time and the latest finish time, or the positive difference between the earliest start time and the latest start time of an activity depending upon which way it is defined.

(ii) Interfering float:

Utilization of the float of an activity may affect the float times of the other activities in the network. Interfering float is that part of the total float which causes a reduction in the float of the successor activities. It is the difference between the latest finish time of the activity in question and the earliest starting time of the following activity or zero, whichever is larger. It indicates the portion of the float of an activity which cannot be consumed without affecting adversely the float of the subsequent activities.

(iii) Free Float:

Free float is that portion of the total float within which an activity can be manipulated without affecting the float of subsequent activities. It is computed for an activity by subtracting the head event slack from its total float. The head event slack is the difference between its latest and earliest event timings that is its (LE).

(iv) Independent Float: It is that portion of the total float within which an activity can be delayed for start without affecting floats of the preceding activities. It is computed by subtracting the tail event slack from the free float. If the result is negative, it is taken as zero. Last column of table above gives independent floats of various activities of the network Q

Explain the terms Resource Smoothing and Resource Leveling.

M02, N02

Resource smoothing :

M99

It is a technique used for smoothening peak resource requirements during different periods of a project net work. The total product duration is maintained at the minimum level. The constraint is on the project duration time. It helps to estimate the total resource requirements for various projects. In resource smoothing, time scaled diagram of various activities of a project and their floats along with their resource requirements are used. The period of maximum demand for resources are identified and non critical activities during these periods are staggered by rescheduling them according to their floats for balancing the resource requirements. Resource Leveling

M00

It is a net work technique used for Reducing the requirement of a particular resource due to its scarcity. It utilizes the large floats available on non critical activities and cuts down the demand on resources. The maximum demand of a resource should not exceed the available limit at any point of time. Non critical activities are rescheduled by utilizing their floats.

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Q1

Draw a network for the following a.

b. Activity

Preceding activity

A B C D

c.

Activity

A,B A

g.

i.

Activity A B C D E F

Preceding activity A,B B,C B

Activity

Preceding activity

Activities

d.

B C B,C

A B C D E F

f.

1 – 3,

1 – 4,

2 – 5,

70

B A,B

Activity A B C D E F G H

Preceding activity C A,B B B,C E,F,G

Activity A B C D E F

Preceding activity A A,B A,B,C

Activity A B C D E F G

h.

A A,B B,C

1 – 2,

Preceding Activity

A B C D

Preceding activity

A B C D E F

e.

Activity

3 – 6,

3 – 7,

Preceding activity B B B E A,D,C 4 – 7,

5 – 8,

6 – 8,

7 – 9,

8 – 9.

PRAVINN MAHAJAN CLASSES 9871255244, 8800684854

J. Activity A B C D E F J

Preceding Activity ---A,B B,C A,B C

Activity H I J K L M N

Preceding Activity D,E,F D G G H,J K I,L

Q2

A project has 14 activities A through M. The relationships which obtain among these activities are given here: construct the network and number them. A is the first operation B and C can be performed in parallel and are immediate successors to A D,E and F follow B G follows E H follows D, but it cannot be started until E is complete I and J succeed G F and J precede K H and I precede L M succeeds L and K The Last operation N succeeds M and C.

Q3

Construct the network diagram comprising activities B, C,……..Q and N Such that the following constraints are satisfied: B < E,F; C < G,L; E,G < H; L,H < I; L < M; H < N; H < J; I,J < P; P < Q. The notation X < Y means that the activity X must be finished before Y can begin.

Q4

A project consists of a series of tasks labeled A, B,……..H, I with the following relationship ( W < X,Y means X,Y can start until W is completed; X, Y < W means W cannot start until both X and Y are completed). With this notation construct the network diagram having the following constraints: A