Sensitivity Problem Solved and Assignment 2 (1)

Solved Problem: Problem #1 The computerized solution of a given problem is given below:

Sensitivity Analysis: Objective function:

Variable

Final Value

Reduced Cost

Objective Coefficient

Allowable Increase

Allowable Decrease

M B R D

25 425 150 0

0 0 0 -45

90 84 70 60

1E+30 6 17 45

6 34 1E+30 1E+30

Sensitivity Analysis: Right hand side:

Constraints

Final Value

Shadow Price

Constraint R.H. Side

Allowable Increase

Allowable Decrease

1 2 3 4

5000 1775 600 150

3 0 60 -17

5000 1800 600 150

850 1E+30 3.571428571 50

50 25 85 150

Answer the following questions: a) Should manager increase the capacity of resource 1 by 100 units if the cost of such a change is Tk. 200? Why or why not? b) What will be the change in objective value if resource 1 is increased by 1000 units? c) If it cost Tk. 300 to increase 1 unit of resource 1 and Tk. 5000 to increase 1 unit of resource 3, which resource will get the priority, considering enough money is available? Why or why not? d) The Price of 1 unit of product D is Tk. 90. What should the product sell for in order to be included in the solution? e) Which constraints are fully utilized? What is the utilization (in percent) of resource 2? f) Whether there will be a change or not in the optimum solution, if the profit of 1 unit of product M is Tk. 80? Tk. 110?

Solution: a) The allowable increase of resource 1 is up to 850 from its current level and shadow price (Unit worth of resource) for per unit increase is 3. So increasing resource 1 by 100 units will contribute 3X100 = Tk. 300 to the profit at a cost of Tk. 200 only. So management should increase the capacity of resource 1 by 100 units. b) The allowable increase of resource 1 is up to 850 from its current level. So increasing this resource by 1000 units will make the resource abundant. So contribution to profit for such a change will be 850X3= Tk. 2550. c) If fund is available, then we should go for increasing the resource that will contribute the most to the total profit. Resource 1: 850X3= Tk. 2550 Resource 2: 3.571428571X60= Tk. 214.29. Here, resource 1 will be the choice. If fund is not sufficient, then we should go for increasing the resource that will contribute the most on per taka basis. (Contribution/Cost) Resource 1: 3/300 = .01 Resource 2: 60/1200 = .012. Here, resource 2 will be the choice. d) The product should sell for Tk.(90 + 45) = Tk. 135 in order to be included in the optimum solution. Product D is a non basic variable in the optimum solution. Its reduce cost -45.00 indicates that it has less contribution to the profit by Tk.45.00. So selling price should be increased by Tk. 45.00.

e) Resource 1, 3 and 4 are fully utilized. (These 3 resources have shadow price other than zero) The utilization of resource 2 is 1775/1800 = 98.61%.

f) The profit range of M within which current solution will remain optimum is: Upper limit: Infinity to Lower limit: 90-6 = 84. So, when profit will change to Tk. 110, current solution will remain optimum. But when profit will change to Tk. 80, current solution will change (since it is outside the range).

Problem#2 Tucker Inc. produces high-quality suits and sport coats for men. Each suit requires 1.2 hours of cutting time and 0.7 hours of sewing time, uses 6 yards of material, and provides a profit contribution of $190. Each sport coat requires 0.8 hours of cutting time and 0.6 hours of sewing time, uses 4 yards of material, and provides a profit contribution of $150. For the coming week, 200 hours of cutting time, 180 hours of sewing time, and 1200 yards of fabric are available. Additional cutting and sewing time can be obtained by scheduling overtime for these operations. Each hour of overtime for the cutting operation increases the hourly cost by $15, and each hour of overtime for the sewing operation increases the hourly cost by $10. A maximum of 100 hours of overtime can be scheduled. Marketing requirements specify a minimum production of 100 suits and 75 sport coats. Let

The computer solution is shown in the following Figure. a. What is the optimal solution, and what is the total profit? What is the plan for the use of overtime? b. A price increase for suits is being considered that would result in a profit contribution of $210 per suit. If this price increase is undertaken, how will the optimal solution change? c. Discuss the need for additional material during the coming week. If a rush order for material can be placed at the usual price plus an extra $8 per yard for handling, would you recommend the company consider placing a rush order for material? What is the maximum price Tucker would be willing to pay for an additional yard of material? How many additional yards of material should Tucker consider ordering? d. Suppose the minimum production requirement for suits is lowered to 75. Would this change help or hurt profit? Explain.

FIGURE THE SOLUTION FOR THE TUCKER INC. PROBLEM

Sensitivity Analysis: Objective function: Variable Name S SC D1 D2

Value 100

Cost 0

Objecti ve Coeffici ent 190

150 40 0

0 0 -10

150 -15 -10

Final

Reduc ed

Allowabl e

1E+30 15 10

Decrease 1E+30 23.333333 33 172.5 1E+30

Allowabl e

Allowable

Increase 35

Allowable

Sensitivity Analysis: Right hand side Sensitivity

1 2

Value 200 160

Price 15 0

Constra int R.H. Side 200 180

3 4 5 6

1200 40 100 150

34.5 0 -35 0

1200 100 100 75

Constraints

Final

Shado w

Increase 40 1E+30 133.3333 333 1E+30 50 75

Decrease 60 20 200 60 100 1E+30

ANSWER a. The optimal solution calls for the production of 100 suits and 150 sport coats. Forty hours of cutting overtime should be scheduled, and no hours of sewing overtime should be scheduled. The total profit is $40,900. (190X100+150X150-40X15). b. The objective coefficient range for suits shows and upper limit of $225. Thus, the optimal solution will not change. But, the value of the optimal solution will increase by ($210$190)100 = $2000. Thus, the total profit becomes $42,990. c. The slack for the material coefficient is 0. Because this is a binding constraint, Tucker should consider ordering additional material. The dual price of $34.50 is the maximum extra cost per yard that should be paid. Because the additional handling cost is only $8 per yard, Tucker should order additional material. Note that the dual price of $34.50 is valid up to 1333.33 -1200 = 133.33 additional yards. d. The dual price of -$35 for the minimum suit requirement constraint tells us that lowering the minimum requirement by 25 suits will improve profit by $35(25) = $875.

Problem#3

The computer solution is shown in the following Figure. a. What is the optimal solution, and what is the value of the objective function? b. Which constraints are binding? c. Which constraint shows extra capacity? How much? d. If the profit for the deluxe model were increased to $150 per unit, would the optimal solution change? e. Identify the range of optimality for each objective function coefficient. f. Suppose the profit for the economy model is increased by $6 per unit, the profit for the standard model is decreased by $2 per unit, and the profit for the deluxe model is increased by $4 per unit. What will the new optimal solution be? g. Identify the range of feasibility for the right-hand-side values.

h. If the number of fan motors available for production is increased by 100, will the dual value for that constraint change? Explain.

FIGURE: THE SOLUTION FOR THE QUALITY AIR CONDITIONING PROBLEM

Sensitivity Analysis: Objective function: Varia ble Name E S D

Fin al Val ue 80 120 0

Reduc ed Cost 0 0 -24

Objecti ve Coeffici ent 63 95 135

Allowa ble Increas e 12 31 24

Allowa ble Decrea se 15.5 8 1E+30

Sensitivity Analysis: Right hand side Sensitivity Constraints

1 2 3

Fin al Val ue 200 320 208 0

Shado w Price 31 32

Constra int R.H. Side 200 320

Allowa ble Increas e 80 80

Allowa ble Decrea se 40 120

0

2400

1E+30

320

Solution: a. E = 80, S = 120, D = 0, Profit = $16,440. b. Fan motors and cooling coils c. Labor hours; 320 hours available. d. Objective function coefficient range of optimality: No lower limit to 159 (135+24). Since $150 is in this range, the optimal solution would not change. e. Range of optimality: E 47.5 to 75 S 87 to 126 D No lower limit to 159. f.

g. Range of feasibility Constraint 1 160 to 180 Constraint 2 200 to 400 Constraint 3 2080 to No Upper Limit h. Yes, fan motors = 200 + 100 = 300 is outside the range of feasibility. The dual price will change.

Assignment: Main Text: Anderson sweeney Management science, Page no 131 and onward Problem no: 14, 15 Problem # 3 The Porsche Club of America sponsors driver education events that provide high performance driving instruction on actual race tracks. Because safety is a primary consideration at such events, many owners elect to install roll bars in their cars. Deegan Industries manufactures two types of roll bars for Porsches. Model DRB is bolted to the car using existing holes in the car’s frame. Model DRW is a heavier roll bar that must be welded to the car’s frame. Model DRB requires 20 pounds of a special high alloy steel, 40 minutes of manufacturing time, and 60 minutes of assembly time. Model DRW requires 25 pounds of the special high alloy steel, 100 minutes of manufacturing time, and 40 minutes of assembly time. Deegan’s steel supplier indicated that at most 40,000 pounds of the high-alloy steel will be available next quarter. In addition, Deegan estimates that 2000 hours of manufacturing time and 1600 hours of assembly time will be available next quarter. The profit contributions are $200 per unit for model DRB and $280 per unit for model DRW. The linear programming model for this problem is as follows:

The computer solution is shown in the following Figure. a. What are the optimal solution and the total profit contribution? b. Another supplier offered to provide Deegan Industries with an additional 500 pounds of the steel alloy at $2 per pound. Should Deegan purchase the additional pounds of the steel alloy? Explain. c. Deegan is considering using overtime to increase the available assembly time. What would you advise Deegan to do regarding this option? Explain. d. Because of increased competition, Deegan is considering reducing the price of model DRB such that the new contribution to profit is $175 per unit. How would this change in price affect the optimal solution? Explain. e. If the available manufacturing time is increased by 500 hours, will the dual value for the manufacturing time constraint change? Explain.

FIGURE THE SOLUTION FOR THE DEEGAN INDUSTRIES PROBLEM

Sensitivity Analysis: Objective function: Variable Name DRB DRW

Final Valu e 1000 800

Reduc ed Cost 0 0

Objecti ve Coeffici ent 200 280

Allowabl e

Allowabl e

Increase 24 220

Decrease 88 30

Sensitivity Analysis: Right hand side Sensitivity Constrai nts

1 2 3 Problem #4

Final Valu e 4000 0 1200 00 9200 0

Shado w Price

Constra int R.H. Side

8.8

Allowabl e

Allowable Decrease

40000

Increase 909.0909 091

0.6

120000

40000

5714.285714

0

96000

1E+30

4000

10000

A large sporting goods store is placing an order for bicycles with its supplier. Four models can be ordered: the adult Open Trail, the adult Cityscape, the girl's Sea Sprite, and the boy's Trail Blazer. It is assumed that every bike ordered will be sold, and their profits, respectively, are 30, 25, 22, and 20. The LP model should maximize profit. There are several conditions that the store needs to worry about. One of these is space to hold the inventory. An adult’s bike needs two feet, but a child's bike needs only one foot. The store has 500 feet of space. There are 1200 hours of assembly time available. The child's bike need 4 hours of assembly time; the Open Trail needs 5 hours and the Cityscape needs 6 hours. The store would like to place an order for at least 275 bikes. a. Formulate a model for this problem. b. Solve your model with any computer package available to you. c. How many of each kind of bike should be ordered and what will the profit be? d. What would the profit be if the store had 100 more feet of storage space? e. If the profit on the Cityscape increases to $35, will any of the Cityscape bikes be ordered? f. Over what range of assembly hours is the dual price applicable?

g. h.

If we require 5 more bikes in inventory, what will happen to the value of the optimal solution? Which resource should the company work to increase, inventory space or assembly time?