OR

Hand out No1 Formulation of Linear Programming Problem 1. The Holiday Meal Turkey Ranch is considering buying two different brands of turkey food and blending then to provide a good low cost diet for its turkeys. Each feed containd, in varying proportions, some or all of the three nutritional ingredients essential for flattening turkeys. Each pound of brand I purchased for example contains 5 ounces of ingredient A4 ounces of gradient B and ½ ounce of gradient C. Each pound of brand II contains 10 ounces of ingredient A, 3 ounces of ingredient B but no ingredient C. The brand I feed costs the rnch 2 cents a pound, while the brand II feed costs 3 cents a pound. The owner of the ranch would like to use LP to determine the lowest cost dient meets the minimum intake requirement for each nutritional ingredient as shoen below Ingredient

composition of each pound of feed (ounce) feed (o unce) Brand I feed Brand II feed A 5 10 B 4 3 C ½ 0 Cost per pound 2 cents 3 cents

Minimum monthly requirement per turkey 90 48 3/2

Formulate the LPP. 2. A post office requires different number of full time employees on different days of the week. The number of full time employees required each day is given below. Union rules state that each full time employee must work 5 consecutive days and then receive 2 days off. Th post office wants to meet its daily requirement using only full time employees. Its objective is to minimize the number of full time employees that must be hired. Days Monday Tuesday Wednesday Thursday Friday Saturday Sunday Minimum 17 13 15 19 14 16 11 requirement 3. In the above example, suppose that each full time employee works 8 hours per day, thus Monday’s requirement is 8*17=136 hours. The post office can meet its daily labour requirement by both full time and part time employees. During each week a full time employee can work 8 hrs a dya and five consecutive days and a part time employee can work 4 hrs a day and five consecutive days. A full time employee costs post office $ 15 per hrs while the part time employee with reduced fringe benefits cost the post office only $ 10 per hrs. Union requirements limit part time labour to 25% of the weekly labour requirement. Formulate LPP. 4. A cargo plane has three compartments for storing cargo: front, centre and rear. These compartments have the following limits on both weight and space: Compartment Weight capacity (tonnes) Space capacity (cubic metres) Front 10 6800 Centre 16 8700 Rear 8 5300 Furthermore, the weight of the cargo in the respective compartments must be the same proportion of that compartment's weight capacity to maintain the balance of the plane. The following four cargoes are available for shipment on the next flight:

Cargo Weight (tonnes) Volume (cubic metres/tonne) Profit (£/tonne) C1

18

480

310

C2 15 650 380 C3 23 580 350 C4 12 390 285 Any proportion of these cargoes can be accepted. The objective is to determine how much (if any) of each cargo C1, C2, C3 and C4 should be accepted and how to distribute each among the compartments so that the total profit for the flight is maximised. Formulate the above problem as a linear program 5. A canning company operates two canning plants. The growers are willing to supply fresh fruits in the following amounts: • • •

S1: 200 tonnes at £11/tonne S2: 310 tonnes at £10/tonne S3: 420 tonnes at £9/tonne

Shipping costs in £ per tonne are: To: Plant A Plant B From: S1 S2 S3

3 2 6

3.5 2.5 4

Plant capacities and labour costs are: Capacity Labour cost

Plant A 460 tonnes £26/tonne

Plant B 560 tonnes £21/tonne

The canned fruits are sold at £50/tonne to the distributors. The company can sell at this price all they can produce. The objective is to find the best mixture of the quantities supplied by the three growers to the two plants so that the company maximises its profits. •

Formulate the problem as a linear program.

Hand out no. 3 Assignment problem 1. Five employees of a company are to be assigned five jobs which can be done by any of them. Because of the different number of years with the firm, the workers get different wages per hour. These are Rs. 5 / hour for A,B and C, and Rs. 3 per hour for D and E each. The amount of time taken by each employee to do a job is given in the following table. Determine the assignment pattern that minimize the total time taken and minimize the total cost of getting five units of work done. Employee A B C D E 1 7 9 3 3 2 2 6 1 6 6 5 3 3 4 9 10 7 4 1 5 2 2 4 5 6 6 9 4 2 2. A dispatcher of the police department has received four requests for police assistance. Currently, six patrol cars are available for six assignment and the estimated response time in minute are shown in the time that follows Which patrol units should respond? Incident Patrol unit 1 2 3 4 5 6 I 6 5 3 4 5 8 II 5 6 2 3 5 6 III 4 4 7 6 5 5 IV 3 7 9 8 4 7 What will be average response time? 3. Fowle Marketing Research has received requests for marketing research from three clients. The company faces the task of assigning a project leader to each client. Folwe’s management realizes that the time required to complete each study will depend upon the experience and ability of the project leader assigned. The three projects have approximately the same priority. The company wants to assign project leaders to minimize the total number of days required to complete all three projects. if a project leader is to be assigned to on client only, what assignment should be made? Project Leader Client 1 2 3 Terry 10 15 9 Carle 9 18 5 McClymonds 6 14 3 4. The management of Salisbury Discount Inc. is facing the problem about the locating the various department within the store in the newly leased store. The shop manager has four locations that have not been assigned a department and is considering five departments that might occupy the four locations. The departments under consideration are shoe, toy, auto parts, house wares and video. After a careful study of the layout of the remainder of store, the store manager has made estimates of the expected annual profit for each department in each location. The estimates are as below

Department

location 1 2 3 4 Shoe 10 6 12 8 Toy 15 18 5 11 Auto parts 17 10 13 16 House wares 14 12 13 10 Video 14 16 6 12 5. In the modification of the plant layout of a factory, four new, machines M 1, M2, M3, and M4 are to be installed in a machine shop. There are five vacant places A,B,C,D, and E available. Because of the limited space Machine B can not placed at C and Machine 3 can not be placed at A. The cost of locating the machine i to place j is shown here. Find the optimal assignment schedule. A B C D E M1 9 11 15 10 11 M2 12 9 10 9 M3 11 14 11 7 M4 14 8 12 7 8 6. Three employees of a company are to be assigned to three jobs which can be done by any of them. Because of the different number of years spent in the organization, John, Jim and Sally get different wages per hour, $6, $8, and $10, respectively. The amount of time in hours taken by each employee to do the job is given in table below. Determine the cost of best assignment. Job John Jim Sally A 5 3 6 B 4 5 8 C 6 7 1 7. Scott and Associates Inc., is an accounting firm that has three new clients. Project leaders will be assigned to the three clients. Based on the different backgrounds and experiences of the leaders, the various leader-client assignments differ in terms of projected completion times. The possible assignments and the estimated completion times in days are Client Project leader 1 2 3 Jackson 10 16 32 Ellis 14 22 40 Smith 22 24 34 8. A job shop has purchased three new machines of different types. There are four available locations in the shop where a machine could be installed. Some of these locations are more desirable than others for particular machines because of their proximity to work centers which would have a heavy work flow to and fro from the machines. The estimated cost per unit time of materials handling involving each of the machines is given below for the respective locations. Machines Location 1 2 3 4 1 13 10 12 11 2 15 13 20 3 5 7 10 6 What is the optimal assignment? 9. A department head has four subordinates and four tasks to be performed. The subordinates differ in efficiency and the tasks differ in their intrinsic difficulty. His estimates of the time that each man would take to perform each task I given below Tasks Subordinates I II III IV A 8 26 17 11 B 13 28 4 26 C 38 19 18 15 D 19 26 24 10

How should the tasks be allotted to subordinates so as to maximize the total man-hours. 10. A project work that consists of four major jobs for which an equal number of contractors have

submitted tenders. The tender amount quoted in lakhs of rupees is given in the matrix. Job Contractor A B C D 1 10 24 30 15 2 16 22 28 12 3 12 20 32 10 4 9 26 34 16 find the assignment which minimizes the total cost of project, when each contractor has to be assigned only one job. Hand out no. 4 Transportation Problem 1.A company has four factories situated in four different locations in the country and four sales agencies located in four other locations. The cost of production (rupees per unit), the sales price (rupees per unit) and shipping costs (rupees per unit) in the cells of the matrix, and monthly capacities and monthly requirements are given in Table 1. Find the monthly production and distribution schedule, which will maximizeprofit.

2. Solve the following transportation problem for total minimum distances distances Markets Plants New York Chicago Topeka Supply Seattle 2.5 1.7 1.8 350 San Diego 2.5 1.8 1.4 600 Demand 325 300 275 3. A product is produced at three plants and shipped to three warehouses. The transportation cost per unit are shown in the following table Plant Ware house Plant capacity W1 W2 W3 P1 20 16 24 300 P2 10 10 8 500 P3 12 18 10 100 Warehouse Demand 200 400 300 Formulate it as linear programming model for minimizing transportation cost. Solve the above problem using any of the transportation method to get the optimal transportation schedule. 4. Sound electronics, Inc. produces a battery tape recorder at plants located in Martinsville, North Carolina; Plymouth, New York; and Franklin, Missouri. The unit transportation cost for the shipments from the three plants to the distribution centers in Chicago, Dallas, and New York are as follows:

To From Chicago Dallas New York Martinsville 1.45 1.60 1.40 Plymouth 1.10 2.25 0.60 Franklin 1.20 1.20 1.80 After considering transportation costs, management has decided that under no circumstances will it use the Plymouth-Dallas route. The plant capacities and distributor orders for the next month are as follows: Capacity Orders Plant (units) Distributor (units) Martinsville 400 Chicago 400 Plymouth 600 Dallas 400 Franklin 300 New York 400 Because of different wage scales at the three plants, the unit production cost varies from plant to plant. Assuming the costs are $29.50 per unit at Martinsville, $31.20 per unit at Plymouth, and$ 30.35 per unit at Franklin, find the production distribution plan that minimizes production and transportation costs. 5. The following table shows the necessary information on the availability of supply to each warehouse, the requirement of each market and unit transportation cost in Rs. from each warehouse to each destination. Markets Warehouses P Q R S SUPPLY A 6 3 5 4 22 B 5 9 2 7 15 C 5 7 8 6 8 Demand 7 1 1 9 45 2 7 The shipping clerk has worked out the following schedule from experience. 12 units from A to Q, 1 unit from A to R, 9 units from A to S , 15 units from B to R , 7 units from C to P and one unit from C to R. Check if the clerk has the optimal schedule if no find the optimal schedule. 6. A baking firm can produce a specialty bread in either of its two plants, as follows Plant Production capacity loaves Production cost Cents/loaf A 2500 23 B 2100 25 Four restaurant chains are willing to purchase this bread, their demands and the prices that they are willing to pay are as follows Chain Maximum demand loaves Price offered Cents/loaf 1 1800 39 2 2300 37 3 550 40 4 1750 36 The cost in cents of shipping the loaf from a plant to a restaurant chain is given in the following table Plant Chain 1 Chain 2 Chain 3 Chain 4 Plant A 6 8 11 9 Plant B 12 6 8 5 Determine a delivery schedule for the baking firm. 7. A company has three production units A,B,C which supply to the four retail chains W,X,Y,Z . the cost of manufacturing one unit at the three different units is as follows A: Rs.20 B: Rs. 25 C: Rs.22

The retail chain are willing to pay for ont unit as follows:W: Rs.45 X: Rs. 55 Y: Rs.53 and Z: Rs.57 The capacity of the manufacturing units is 50,60 and 40 respectively and the demand of the retail chains is 20,70 , 50 and 10 respectively. The cost in Rs. Of shipping one unit of commodity from different manufacturing units to various retail stores is as follows: Retail units Unit W A 5 B 8 C 15 Hand out No 5 1 .Min 40 X1+ 24 X2 Subject to 20 X1+ 50X2 ≥4800 80 X1 + 50X2≥7200

X Y Z 15 7 6 7 9 1 8 9 8 Find the optimal schedule that will maximize the total profit. Solution to minimization problem and dual problem, dual simplex method

Xi’s are non negative

2. Min 10 X1+ 20 X2 Subject to 3 X1+ 2X2 ≥18 X1 + 3X2≥8 2X1-X2 ≤6 Xi’s are non negative 3 Maximize 2 X1+ 4 X2 Subject to 2 X1+ X2 ≤ 18 3 X1 + 2X2≥30 X1+ 2 X2 = 26 Xi’s are non negative 4. Minimize Z = 50x1 + 60x2 + 55x3 20x1 + 25x2 + 30x3 ≥ 6000 30x1 + 25x2 + 20x3 ≥ 7500 And x1, x2 and x3 ≥ 0

Hand out no. 6 Game theory (Two persons zero sum games) 1. You are the head coach of the Alphas (Team A), and are attempting to come up with a strategy to deal with your rivals, the Betas (Team B). Team A is on offense, and Team B is on defense. You have five preferred plays, but are not sure which to select. You know, however, that Team B usually employs one of three defensive plays. Over the years, you have diligently recorded the yardage gained by your team for each combination of plays used, and have come up with the following table. Team

A

B 1

2

3

1

0

-1

5

2

7

5

10

3

15

-4

-5

4

5

0

10

5

-5

-10

10

2. . Determine optimal strategies for both the players and the value of the game from the following pay off table Player A a1 a2 a3 a4 b1 4 4 4 4 Player B b2 3 4 2 4 B3 1 3 9 2 3. The defense lawyer and district attorney in a criminal case could conceivably use a game theory approach. Solve this game using the analytical approach. Attorney A1 A2 A3

L1 70 75 85

L2 45 55 60

Lawyer L3 75 83 70

L4 80 95 25

4. Solve the game: A b1 -2 -1 1

A1 A2 A3

B b2 -4 3 2

5. Two break fast food manufacturers, ABC, XYZ are competing for an increased market share. The pay off matrix, shown in the following table describes the increase in market share for ABC and decrease in market share of XYZ. Determine the optimal strategies for both the manufacturers and the value of the game. XYZ ABC Give coupons decrease maintain increase price present strategy advertising Give coupons 2 Decrease price 6 Maintain -3 present strategy

-2

4

1

1 2

12 0

3 6

Increase advertising

-3

7

1

2

6. Solve the following game Player B Player A B1 B2 A1 2 4 A2 5 6 A3 6 7 A4 4 2

B3 3 3 9 8

B4 8 7 8 4

B5 4 8 7 3

7. The north American Air Defence Command (NORAD) and the Strategic Air Command (SAC) are participating in a large war-game exercise. Each Will consider the other an opponent. The designated aggressor SAC has developed four strategie for “ penetrating” the air defence network. NORAD has developed four “counter strategies”. Table below shows a pay off matrix which represents the number of targets “destroyed” by SAC. Determine the pure strategies for each player. NORAD SAC N1 N2 N3 N4 S1 65 100 65 60 S2 68 27 29 27 S3 75 80 60 41 S4 102 40 77 60 8. Solve the following game by simplex method. 5  5 8 

4 8 5

7  4 6 

9. The pay off matrix is given below. Solve the game using graph method. Player B [ b1 b2 ]

 A1   4 A    2  − 2 Player A  A   6 3    A4   − 4  A5   5   Hand out 7

Player B

5 1 2 − 2  − 8 − 3

[ b1

Player A

b2

b3

b4 ]

 a1   8 5 − 7 9  a     2   − 6 6 4 − 2

Title : Sensitivity Analysis/ post optimality Analysis

Special Cases Infeasible solution: If at the optimal iteration any of the artificial variable is present Unbounded solution: if the entering variable is having coefficients zero and negative Alternate solutions: if the number of zeros in the zj-cj row are more than the number of variables 1. Consider the following linear programming problem, Maximize z = X1+9X2+X3 (profit function) Subject to constraints X1+2X2 +3X3 ≤ 9 ( Machine hours) 4X1+X2+2X3 ≤ 15 (labour hours) X1, X2, X3 non negative After solving the above problem, the following table is arrived at. C 1 9 1 0 0 Basis Cb X1 X2 X3 S1 S2 B X2 9 ½ 1 3/2 1/2 0 9/2 X5 0 2 0 -1 -1 1 6 Zj-Cj 7/2 0 25/2 9/2 0 Z=81/2 Answer the following questions. i. The availability of machine hours is reduced by 2 hours will the current solution change? ii. If the product I (X1) is to be produced, what change is required in the profit contribution? iii. What is the percent utilization of two resources? iv. If a new product proposal is as follows what do you suggest about introducing that product. Consumption of machine hours=2units Consumption of labour hours=3 units Profit contribution=5units/unit. 2. In the process of solving the given LPP the following table is obtained. Answer the following questions. Maximize Z= 15x1+ 6 x2+9x3+2x4 profit function Subject to constraints

2x1+ x2+5x3+6x4 ≤ 20 resource 1 3x1+ x2+3x3+x4 ≤ 24 resource 2 7x1 + x4 ≤ 70 resource 3 x1, x2,x3,x4 ≥ 0 C 15 6 9 2 0 0 0 B B Cb X1 X2 X3 X4 S1 S2 S3 X2 6 0 1 9 -32 3 -2 0 12 X1 15 1 0 -2 57/3 -1 1 0 4 S3 0 0 0 14 -132 7 -7 1 42 Zj-Cj 0 0 15 91 3 3 0 Z=132 i. What is the percent utilization of the three resources? ii. If product 3 is to be produced what change do you suggest in the profit contribution of product 3? iii. What is the range of profit contribution of product 1 over which the current solution remains optimal? iv. If the availability of the resource 2 is increased to 30, how does this change affect the current solution? 3. HiDec produces two models of electronic gadgets that use resistors, capacitors and chips. The following table summarizes the data of the situation Resource unit resource requirement Maximum availability Model 1 model 2 units Resistors 2 3 1200 Capacitors 2 1 1000 Chips 0 4 800 Unit profit $ 3 4 In getting the solution to this problem the following simplex table is found. C 3 4 0 0 0 Basis Cb X1 X2 X3 S1 S2 b X1 3 1 0 -1/4 ¾ 0 450 X5 0 0 0 -2 2 1 400 X2 4 0 1 ½ -1/2 0 100 Zj-Cj 0 0 5/4 -1/2 0 Z=1750 Determine the status of each resource. In terms of optimal profit, determine the worth of one resistor, one capacitor and one chip. If the available number of resistors is increased to 1300 units, find the new optimum solution. If the available number of chips is reduced to 350 units, determine the optimal solution. 4. Consider the following linear programming problem Maximize Z= 4X1 +6 X2 +2 X3 Profit function St. X1 +X2+ X3 ≤ 12 Man power constraint X1 +4X2+9 X3 ≤ 9 Raw material constraint And X1,X2 ,X3≥ 0 Following is the optimal solution for the above LPP. C 4 6 2 0 Basis Cb X1 X2 X3 S1 X1 4 1 0 -1 4/3 X2 6 0 1 2 -1/3 Zj-Cj 0 0 6 10/3

0 S2 1/3 1/3 2/3

B 1 2 Z=16

1) 2) 3) 4)

Find the range of the manpower and raw material over which the current solution remains optimal. Find the range of the profit contribution over which the current solution remains optimal. If the product three is to be produced what change in the profit contribution do you suggest? A new product consuming the same set of resources is conceived. It takes the 3 units of manpower and 2 units of the raw material. Which fetches Rs.5 per unit of the product what do you suggest to the company? 5. Solve the following problem and answer the following questions: Maximize Z= -X1+3X2-2X3 Subject to constraints 3X1-X2+2X3≤ 7 -2X1+4X2≤ 12 -4X1+3X2+8X3≤ 10 X1, X2, and X3are non negative The following able was obtained in the process of getting the optimal solution -1 3 -2 0 0 0 X1 X2 X3 S1 S2 b X1 1 0 4/5 2/5 1/10 0 4 X2 0 1 2/4 1/5 3/10 0 5 S3 0 0 10 1 -1/2 1 11 Zj-Cj 0 0 12/5 1/5 4/5 0 Z=11 i. Is the solution an optimal solution? If yes, why? ii. Is the solution unique? Why? iii. What is the percentage utilization of the resources? iv. If product 3 is to be produced what change do you suggest in the profit contribution of the product? v. What is the unit worth of each resource? vi. What is the range of the availability of resource 1 over which the current solution remains optimal? vii. What is the range of the profit contribution of product 2 over which the current solution remains optimal? viii. If a new product is proposed that consumes two units of resource 1, 3 units of resource 2 and 2 units of resource 3 and the profit contribution is estimated to be Rs. 5. What is your suggestion about introducing this new product? 6. Maximize Z=5x1+10x2+8x3 profit function Subject to 3x1+5x2+2x3≤ 60 Fabrication hours 4x1+4x2+4x3≤ 72 Finishing hours 2x1+4x2+5x3≤ 100 Packaging hours x1, x2, x3 ≥0 C * * * * * * Basic X1 X2 X3 S1 S2 S3 bi X2 10 1/3 * * 1/3 -1/6 * 8 X3 8 2/3 * * -1/3 5/12 * 10 S3 0 -8/3 * * 1/3 -17/12 * 18 Zj-Cj * * * * * * Z=* i. Replace the * by appropriate numbers ii. Is the solution optimal? Why? Basis

Cb -1 3 0

iii. iv. v. vi.

Is the solution unique? Why? What is the percent utilization of the three resources? What is the unit worth of the three resources? What is the range of availability of the fabrication hours over which the current solution remains optimal? vii. What is the range of the profit contribution of product 2 for which the current solution remains optimal? viii. What change do you suggest in the profit contribution of product one if it is to be produced? ix. If a new product is introduced which consumes 2,3 and 4 units of resources 1,2 and 3 respectively and fetches profit of 6 units per unit then what would you suggest? 7. Max Z = 6X1 + 4X2 Subject to constraints 2X1 + 3X2 ≤ 100 ………… Raw material 4X1+ 2X2 ≤ 120 …………. Time X1, X2 ≥ 0 ...………. Non-negativity C Basis CB X2 4 X1 6 Zj-Cj

6 X1 0 1 0

4 X2 1 0 0

0 S1 1/2 -1/4 1/2

0 S2 -1/4 3/8 5/4

B 20 20 Z=200

Answer the following questions I. Is the solution optimal? II. Is the solution unique? III. If the availability of the raw material is reduced to 90 units, will the current solution be optimal? IV. If over time is allowed, because of which the availability is increased to 150 units, will the current solution be optimal? What is the unit worth of the resources? V. If a new product is introduced whose one unit consumed 2 units of raw material and one unit of the time and gets 5 units as profit should you advice to go for the product? 8. The following linear programming model for analysing the product mix of Maxine’s Hat company which produces three hat styles: Maximize Z= 7X1+5X2 +2X3 3X1+5X2 +X3 ≤ 150 7X1+5X2 +2X3≤ 100 7X1+5X2 +2X3≤160 X1¸X2 , X3≥0 The print out below shows the optimal solution to the problem. Consider each of the following statements independently and state whether true or false. Explain each answer. Basic x1 x2 x3 sx4 sx5 sx6 Solution ------------------------------------------------------------------------x2 0.00 1.00 -0.06 0.31 -0.19 0.00 28.13 x1 1.00 0.00 0.44 -0.19 0.31 0.00 3.12 sx6 0.00 0.00 0.69 -0.44 0.06 1.00 100.62 -------------------------------------------------------------------------Zj-Cj 0.00 0.00 0.75 0.25 1.25 0.00 162.50

i) ii) iii)

If the price of the hat 3 were increased to 2.50 it would be part of the optimal solution. The capacity of machine C can be reduced to 65 hours without affecting the profits If machine A had a capacity of 170 hours the production out remain unchanged.

9. After a few iterations in an attempt to solve an LPP given here below: Maximize Z= 3X1+ 2X2+5X3 Subject to X1+ 2X2+X3≤430 operation 1 3X1 +2X3≤460 operation 2 X1+ 4X2 ≤420 operation 3 Xi ≥ 0 i= 1,2,3. The following simplex table emerged: C * * * * * * B CB X1 X2 X3 S1 S2 S3 b X2 * -1/4 1 0 ½ -1/4 0 * X3 * 3/2 0 1 0 ½ 0 * S3 * 2 0 0 -2 1 1 * Zj-Cj Fill in the blanks. Write the dual of the problem. Identify the solution to the dual. Find the range of values over which the profit contribution of product 2 could vary without changing the optimality? What are the shadow prices of the three operations? What are the capacity utilization of three operations 10. The product planning manager at Westlake Electronics wants to determine the optimal television product mix for the next quarter. The production capacities for the firm’s three manufacturing facilities are Facility fabrication hrs Assembly hrs 1 10000 50000 2 15000 60000 3 5000 35000 total 30000 145000 Westlake can produce three different types of TVs: Portable (20 inches), regular(27 inches), and home theatre(40 inches). the gross profit and production requirements by type of TV are Television gross profit fabrication hr/unit assembly hr/unit Portable $75 3 9 Regular $125 4 12 Home Theatre $200 7 16 Basic x1 x2 x3 x4 x5 x6 x7 --------------------------------------------------------------------------1) x4 0.75 0.00 0.00 1.00 0.00 0.00 1.75 2) x5 0.00 0.75 0.00 0.00 1.00 0.00 0.00 3) x6 0.00 0.00 0.75 0.00 0.00 1.00 0.00 4) sx13 0.00 0.00 0.00 0.00 0.00 0.00 -5.00 5) sx14 0.00 0.00 0.00 0.00 0.00 0.00 0.00 6) sx15 0.00 0.00 0.00 0.00 0.00 0.00 0.00 --------------------------------------------------------------------------z 18.75 18.75 18.75 0.00 0.00 0.00 18.75 --------------------------------------------------------------------------Basic x8 x9 sx10 sx11 sx12 sx13 sx14 --------------------------------------------------------------------------1) x4 0.00 0.00 0.25 0.00 0.00 0.00 0.00 2) x5 1.75 0.00 0.00 0.25 0.00 0.00 0.00 3) x6 0.00 1.75 0.00 0.00 0.25 0.00 0.00 4) sx13 0.00 0.00 -3.00 0.00 0.00 1.00 0.00 5) sx14 -5.00 0.00 0.00 -3.00 0.00 0.00 1.00

6) sx15 0.00 -5.00 0.00 0.00 -3.00 0.00 0.00 -----------------------------------------------------------------------------z 18.75 18.75 31.25 31.25 31.25 0.00 0.00 ¦ Basic sx15 Solution ¦----------------------------------¦ 1) x4 0.00 2500.00 ¦ 2) x5 0.00 3750.00 ¦ 3) x6 0.00 1250.00 ¦ 4) sx13 0.00 20000.00 ¦ 5) sx14 0.00 15000.00 ¦ 6) sx15 1.00 20000.00 ¦----------------------------------¦ z 0.00 937499.94

What product mix optimises product mix? What is the value of the objective function? What is the value of an additional hour of fabrication time at facility 1? What should be the impact on the optimal solution of the company if the company were to produce at least 1500 portable and 500 home theatre TVs? What should be the impact on the optimal solution if the production capacities for all three plants were increased by 10%? 11. Maximise 8X1+5X2+10X3 subject to 2X1+3X2+1X3≤400 1X1+1X3≤150 2X1+4X3≤200 X2≤50 Xi≥0 C Basis S1 S2 X1 X2 zj-cj

Cb 0 0 8 5

5 x1 0 0 1 0 0

8 x2 0 0 0 1 0

10 x3 -3 -1 2 0 6

0 s1 1 0 0 0 0

0 s2 0 1 0 0 0

Analyze the table with all aspects.

0 s3 -1 -0.5 0.5 0 4

0 s4 -3 0 0 1 5

b 50 50 100 50 1050

Hand out no. 8 Network analysis 1. Jackson Community Hospital has a small automated hematology lab. The following procedure is used in the laboratory operation. ________________________________________________________________________ Description activity time minutes ___________________________________________________________________ Blood sample taken 1-2 4 Lab collects sample 2-3 2 Emergency sample given priority 2-4 1 Initial reports on samples 3-4 5 Inadequate samples rejected 3-6 6 Secondary reports on samples 4-5 15 Reports on emergency samples sent down 5-6 5 All other reports issued 6-7 9 Design the appropriate network. Calculate slack for each event and also the floats for each activity. 2. Consider the following activity sequence for a project: ______________________________________________ Activity Immediately Duration Preceding Days ______________________________________________ A 2 B 3 C 2 D B 4 E A,B 3 F B 2 G F,C 5 H G 4 I C,F 3 J I,D 2 K J 1 L E 6

______________________________________________ Construct a network and calculate the activity time estimates. 3. An architect has been awarded a contract to prepare plans for an urban renewal project. The project consists of the following activities and their estimated times. Activity Description Immediate predecessors time(days) A Prepare preliminary sketches 2 B Outline specifications 1 C Prepare drawings A 3 D Write specifications A,B 2 E Run off prints C,D 1 F Have Specification B,D 3 G Assemble E,F 1 Indicate the critical path. Calculate the activity floats.

PERT and crashing 1. Consider the time estimates in weeks for activities as presented below: Activity immediate optimistic most likely pessimistic Predecessor A 4 5 12 B 1 1.5 5 C A 2 3 4 D A 3 4 11 E A 2 3 4 F C 1.5 2 2.5 G D 1.5 3 4.5 H B,E 2.5 3.5 7.5 I H 1.5 2 2.5 J F, G, I 1 2 3 Draw the network. Calculate the expected duration and its variance. What is the probability that the project will be completed in 15 weeks? 2. Office Automation Inc. has developed a proposal for introducing a new computerized office system that will improve word processing and interoffice communication for a particular company. Contained in the proposal is a list of activities that must be accomplished to complete the new office system project. Use the following information about the activities. Immediate Time (weeks) Cost($000s) Activity predecessor normal crash Normal crash A 10 8 30 70 B A 8 6 120 150 C B 10 7 100 160 D A 7 6 40 50 E D 10 8 50 75 F C,E 3 3 60 60 Develop a project network Develop activity schedule Find the optimal project cost and the duration associated with it.

3. The basic cost time data for the jobs in a project are as given below: Job

Normal Days 3 6 2 4 2 7 4 3

Crash Cost of crashing Cost Days Cost per day A 140 2 210 70 B 215 5 275 60 C 160 1 240 80 D 130 3 180 50 E 170 1 250 80 F 165 4 285 40 G 210 3 290 80 H 110 2 160 50 Total 1300 1890 The activity dependencies are as followsA,B,.C are starting activities. Activity D,E,F can start when A is completed. G can start after B,D is completed. Activity H can start after C and E are completed. Activities G,F and H are final activities. Draw the network and indicate the critical path. If indirect cost per day is Rs.120, what is the optimal project duration? 4. Consider the data of the project Duration (weeks) Activity Predecessors a m b A 3 5 8 B 6 7 9 C A 4 5 9 D B 3 5 8 E A 4 6 9 F C,D 5 8 11 G C,D,E 3 6 9 H F 1 2 9 What is the probability of completing the project on or before 30 weeks? 5. A prerequisite for the funding of a government project is the construction of a CPM network. The data on the time and cost of various activities are as follows Time in weeks cost in dollars ______________________________________________________ Activity normal crash normal crash 1-2 16 14 1500 2000 1-3 25 20 2000 2500 2-4 10 7 2500 4000 3-4 32 26 1000 1600 3-5 40 35 1750 2250 4-5 16 12 4000 6000 4-6 12 8 3000 4200 5-6 9 6 1500 3000 Determine the critical path , The crash critical path , the minimum project completion time with the least increase in the costs over the normal cost. 6. A marketing firm is studying the possibility of introducing a new product. In order to facilitate the product development, the firm wants to build a PERT chart based on the following activities: ________________________________________________________________________ Time (weeks)

_________________________________ Description Activity a m b ________________________________________________________________________ Market survey 1-2 2 3.5 5 Product test I 1-3 3.5 5 6.5 Market Analysis 2-4 4 4.5 7 Product test II 3-5 3 5 6.5 Small market test 4-5 2.5 4 5.5 Promotion campaign 4-6 4.5 5 5.5 FDA approval 5-6 1 3 4 Product test III 6-7 2 3 4 Citywide distribution 4-7 6 7.5 10 Test final analysis 7-8 3 4 4.5 General Distribution 8-9 8 11 16 Determine the critical path. What is the probability that the entire project would be completed by the end of 30 weeks? 7. The following table gives the activities in a construction project and other relevant information: ________________________________________________________________________ Activity Immediate Time (days) Direct cost (Rs) predecessors Normal Crash Normal Crash ________________________________________________________________________ A 4 3 60 90 B 6 4 150 250 C 2 1 38 60 D A 5 3 150 250 E C 2 2 100 100 F A 7 5 115 175 G B,D,E 4 2 100 240 Indirect cost varies as follows Days 15 14 13 12 11 10 9 8 7 6 Cost 600 500 400 250 175 100 75 50 35 25 Draw an arrow diagram for the project. Determine the project duration which will return the minimum total project cost. 8. The list of activities for erecting a canteen in the factory is given below with other relevant details . Job A must precede all others while job E ust follow all others: apart from this ,jobs can run concurrently ________________________________________________________________________ Code Job description Normal crash Duration Cost Duration Crash Days Rs Days Rs ________________________________________________________________________ A Lay foundations and build walls 5 3000 4 4000 B Tile roofing 6 1200 2 2000 C Install electricity 4 1000 3 1800 D Install plumbing 5 1200 3 2000 E Connect services to finish 3 1600 3 1600 Draw network and identify critical path. Crash the network fully to find out minimum duration. If indirect costs are Rs. 300/day , determine the time cost trade off for the project. 9. A project is represented by the network shown below and has the following data Task: A B C D E F G H

I

Optimistic Pessimistic Most likely

5 10 8

18 22 20

26 40 33

16 20 18

15 25 20

6 12 9

7 12 10

7 9 8

3 5 4

Determine expected times and their variances. The probability of an event occurring at the expected completion date if original scheduled time of completing the project is 41.5 weeks.

F

3 B 1

I

6 E

A

2

7

H D 5

G

C 4

Hand out no. 9 Waiting line models 1. The manger at a grocery store in the retirement community of Sunnyville is interested in providing good service to the senior citizens who shop in his store. Presently the store has a separate check out counter for the senior citizens. On an average, 30 senior citizens per hour arrive at the counter according to the poison distribution and are served at an average rate of 35 customers per hour, with exponential service time. Find i. Utilization of the check out clerk. ii. What is the probability of having more than three customers in the system? iii. Time spent in the system. 2. Speedy Oil Provides a single channel automobile oil change and lubrication service. New arrival occurs at the rate of 2.5 cars per hour and the mean service rate is 5 cars per hour. Assume that arrivals follow a Poisson distribution and the service time follows an Exponential distribution. What is the average number of cars in the system? What is the average time the car waits for the oil and lubrication service to begin? What is the average time the car spends in the system? What is the probability that the arrival has to wait for the service on the arrival? 3.Vehicles pass through a toll gate at a rate of 90 per hour. The average time to pass through the gate is 36 seconds . The arrival rate and the service rate follow poisson distribution . There is a complaint that the vehicle waits for long duration. The authorities are willing to install one more gate to reduce the average time to pass through the toll gate to 30 seconds if the idle time of the toll gate is less than 10% and the average queue length at the gate is more than 5 vehicles. Check whether the installation of the second gate is justified. 4. Arrivals at telephone booth are considered to be Poison with an average time of 10 minutes between one arrival and the next. The length of the phone call is assumed to be exponentially distributed with mean three minute. Assuming that the telephone is only one What is the probability that a person arriving at the booth will have to wait? What is the average length of the queue that forms from time to time? What is the probability that the booth will have five persons waiting for the call? 5. The Taj Service station has a central store where service mechanics arrive to take spare parts for the jobs they work upon. The mechanics wait in queue if necessary and are served on first come first served basis. The store is manned by one attendant, who can attend 8 mechanics per hour, on an average. The arrival rate of the mechanics averages 6 per hour. Assuming that the pattern of mechanics’ arrival is Poisson distributed and the serving time is exponentially distributed. Determine the parameters of this queue. The information about the cost to the company is as follows The mechanics are paid Rs. 8/hr and the store keeper is paid Rs.5/hour If one more attendant is hired for the storeroom, the service rate can be increased form 8 per hour to 12 per hour . If an additional attendant is hired for the storeroom, the service rate can be increased form 12 per hour to 16 per hour. 6. A TV repairman finds that the time spent on his job has an exponential distribution with a mean of 30 minutes. If the repairs set on FIFO basis and if the arrival of sets with an average of 10 per 8 hour day, what is the repairman’s expected idle time each day? Also obtain average number of sets in the system.

7. A computer maintenance contract is to be signed by your office. At an average three computers go off road due to various defects. The cost of computer being unavailable is Rs. 8000 per month. Two companies have bid for the contract. Alfa computers have quoted at Rs 3000 per month whereas Beta Bytes has quoted at Rs 5000 per month of the contract. Enquiries reveal that Alfa computers has an average repair capability of 5 computers per month and Beta bytes can repair six computers per month on an average. Who should be given the contract? Note The performance measures of the queue systems are judged using the following: 1.The probability that the system has n customers in the system including the customers waiting for service as well as customers in the service facility is given by Pn= probability that there are n customers in the system (queue +service) Pn = ρ n(1-ρ ) If there are no customers we have P0= (1-ρ ). When there are no customers the server is idle. This is also the probability that the server is idle. The probability that the arriving customer has to wait is also the same as 1-P0 2. Expected number of customers in the systems (queue + service) at any time is denoted by Ls= ρ /(1-ρ ) = λ /(µ -λ ) 3. The expected number of customers waiting in the queue at any time is given by Lq= Ls-(λ /µ ) = ρ 2/(1-ρ ) = λ 2/µ (µ -λ ) 4. The expected number of customers in the queue given that there is a queue formed is a conditional. It is given by Average length of a non empty queue=µ / (µ -λ ) 5. The average time spent by the customer in the queue is given by Wq =(1/λ ) Lq =ρ / (µ -λ ) 6.The average number of customers in the system; queue and service facility is given by Ws=(1/λ ) Ls = 1/(µ -λ )

7. The probability that a customer has to spend more than t units of time in the system Ws(t) = e-t/ws 8.The probability that a customer spends more than t units of time in the queue,

Wq (t)= ρ e-t/ws.

Hand out No. 10 Replacement Problem. 1. A firm is using a machine whose purchase price is Rs.13000. The installation charges amount to Rs.3600 and the machine has a scrape value of only Rs.1600 because the firm has a monopoly of this type of work. The maintenance cost in various years is given in the following table: Year 1 2 3 4 5 6 7 8 9 mi 250 750 1000 1500 2100 2900 4000 4800 6000 The firm wants to determine after how many years should the machine be replaced on economic considerations, assuming that the machine replacement can be done only at the end of the year. 2. The cost of a pressing machine is Rs.50000. The maintenance of the machine is Maintenance Cost = (n+1) 600+ 50. The resale value of the press is 20000 irrespective of the period for which it is used. Find the best time to replace the machine. 3. An engineering company is offered a material handling equipment A. It is priced at Rs.60000 including cost of installation, and cost of operation and maintenance are estimated to be Rs.10000 for each of the first five years, increasing every year by Rs.3000 in the sixth year and subsequent year. The company expects a return of ten percent per annum on its investment. What is the optimal replacement period assuming that money value changes at 10% per annum? 4. The cost of machine is Rs.6000. The maintenance of n th year is given by 250(n-1), n=1,2,3….Suppose that the discount rate is 5% per annum . After how many years will it be economical to replace the machine by a new one? 5.A large establishment has an installation with 1000 bulbs of a certain type. From the past experience it has been observed that the failure rate of these bulbs is as below: End of the week: 1 2 3 4 5 Probability of Failure to date : 0.1 0.25 0.50 0.70 1.00 The cost of replacing an individual bulb is Rs.3 while if the entire group of bulbs is replaced, the cost would be Rs.1 per bulb. It is decided to replace all bulbs simultaneously at fixed intervals of time and also to replace the individual bulbs that fail in between. 6. The manufacturer has to decide between two alternative machines M1 and M2, about which pertinent information is given below M1 M2 Cost 5000 Maintenance Cost 800 /year for first 5 years Increasing by 200 every year thereafter Scrap value Nil Cost of capital (to be used as discounted factor) Determine i) optimal replacement period for M1, M2 ii)Which of the two is better alternative?

2500 1200/year for first 6 years increasing by 200 thereafter. thereafter. nil 10% pa

7. The cost of the new machine is Rs.65000 . the maintenance cost of the nth year is Cn= 1000(2n-1); n=1,2,3,4…. Suppose that the discount rate 10% pa. After how many years will it be economical to replace the machine by a new one?

8. An electronic equipment contains 500 resistors. When any resistor fails, it is replaced. The cost of replacing a resistor individually is Rs.20. If all the resistors are replaced at the same time, the cost per resistor is Rs.5. The percentage surviving, S(i) at the end of month I is given in the table below. Suggest a optimal replacement plan Percent survival rate Month ( i) 0 1 2 3 4 5 S(i) 100 90 75 55 30 0 9. The firm is considering replacement of an equipment whose first cost is Rs. 4000 and the scrape value is negligible at the end of any year . based on experience it has been found that the maintenance cost is zero during the first year and it is Rs. 1000 for the second year . it increases by Rs.300 every year there after. When should the equipment be replaced if the rate of change of money is 10% per annum. 10. A firm is considering replacement of a machine, whose cost price is Rs.12,200; and the scrape value is Rs.200 only. The maintenance costs are found as follows: Year 1 2 3 4 5 6 7 8 Maintenance cost 200 500 800 1200 1800 2500 3200 4000 When should the machine be replaced? 11. The failure rate of 1000 street bulbs in a colony are summarized as under End of month 1 2 3 4 5 6 Probability of 0.05 0.20 0.40 0.65 0.85 1.00 Failure to date The cost of replacing an individual bulb is Rs.60 If all the bulbs are replaced simultaneously it would cost Rs.25 per bulb. Find out the optimal replacement policy. If group replacement policy is optimal, then find at what equal intervals should all bulbs be replace? 12 department sells 1000 water beds per year. At a 100% mark up ,the beds sell for $80 each. It costs $200 to place an order with the supplier. The annual holding cost per dollar value of items held in inventory is $0.25. Find the economic water bed order size. How often should orders be placed?

Hand out no. 11 Inventory management 1. A manufacturer has to supply his customer with 600 units of his product per year. Shortages are not allowed and the storage cost amounts to Rs.0.60 per unit per year. The set up cost per run is Rs.80 .Find the optimal run size and the minimum average yearly cost.

2.The annual consumptions of an item is 2000items. The ordering cost is Rs.100 per order. The carrying cost is Rs. 0.80 per unit per year. Assuming working days as 200 lead-time 20 days, and safety stock 100units. Calculate EOQ, number of orders per year, reorder level and total annual cost. 3 Find the optimal order quantity for a product for which the price breaks are as follows: Quantity unit cost (Rs.) 0≤ q1