Question

Performance Cooling Inc. manufactures air conditioners for home use and they currently have three models serving this market: (1) the essential model, (2) the comfort model, and (3) the majestic model. The units sell for reasonable prices and bring in a profit of $63, $95, and $135 respectively. Currently, the resources required for production of each unit is given in the table below. Fans Cooling Coils Assembly Time (hr) 1 8 1 2 12 Essential Comfort Majestic In Stock/Available 1 4 14 200 320 2400 The manager has developed the following linear programming model to capture this problem, where: E represents the number of essential model units that the company should produce C represents the number of comfort model units that the company should produce M represents the number of majestic model units that the company should produce Max Z = 63E + 95C +135M Subject to: 1E + 1C + 1M 5200 1E + 20 + 4M S 320 8E + 12 + 14M S 2400 E,C,M20 The solved sensitivity report is given below: Variable Cells Cell Name $B$2 Dec Var E $C$2 Dec Varc $D$2 Dec Var M Final Reduced Objective Allowable Allowable Value Cost Coefficient Increase Decrease 80 63 12 15.5 120 0 95 31 0 135 24 1E30 Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease SE$5 Fans 200 31 200 80 40 $E$6 Cooling Coils 320 32 320 80 120 $E$7 Assembly Time 2080 0 2400 1E+30 320 Answer the questions on the following page using the information presented in the problem and the sensitivity report above. Answer each question independently and refer back to the original formulation for each question. Page 2 of 7 ADM2302 x Final Spring/Summer 2020 (a) What is the optimal solution and how much can Performance Cooling Inc. expect in profit? (2 points) (b) Which resource has extra capacity? How much extra capacity does it have? (2 points) (c) If the profit for the majestic model increased to $150 per unit, would the optimal solution change? Explain. (3 points) (d) If 20 fans happen to be damaged, how would it affect the total profit for Performance Cooling Inc.? (3 points) (e) One cell is blanked out in the sensitivity report darkened black). What value would go in this cell? Explain how you were able to calculate this value without running it in Excel again. (5 points) (1) Suppose the profit for the essential model increased by $6 per unit, the profit for the comfort model decreased by $2 per unit, and the profit for the majestic model increased by S4 per unit. Would this change the optimal production schedule? What can Performance Cooling Inc. expect in terms of profit? (5 points)

Answer 1

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Answers::

(A). The optimal values are represented by final values in the variable cells table given above. Therefore,

Optimal number of Essential model = 80

Optimal number of comfort model = 120

Optimal number of majestic model = 0

Given, objective function = 63E + 95C + 135M

Therefore,

Maximum Profit = 63×80 + 95×120 + 135×0

Maximum Profit = $ 16,440

(B). From the constraints table given in the question, it can be observed that for assembly time, the final value is less than constraint R.H. side value which demonstrates that it has some extra Capacity.

Amount of extra Capacity for assembly time = 2400 - 2080 = 320

(C). No, the optimal solution will not change.

The objective coefficient for Majestic model is 135 and its allowable increase is 24 which means the the optimal solution will not change even if the coefficient of majestic model is increased to (135+24) = 159 and for any value of coefficient of majestic model above 159, the optimal solution changes. Therefore, if objective coefficient of majestic model is increased to 150, the optimal solution won't change.

(D). The shadow price corresponding to fans is 31 which demonstrates that a unit change in the constraint R.H. side of fans will change the value of optimal solution by $ 31.

Therefore, if 20 fans are damaged, then constraint R.H. side decreases by 20. Therefore the profit will reduce by (20 × $31) = $ 620

(E). It can be seen that the reduced cost for majestic model is blank. It can be observed that unlike essential and comfort models, Majestic model has a final value of zero which means the reduced cost of Majestic model will not be zero and it will take the negative value of kts allowable increase. Therefore, reduced cost for majestic model will be -24.

(F) In this, we first calculate the percentage of each model by dividing the amount decreased or increased with its corresponding allowable decrease or increase.

Therefore,

Percentage for essential model = 100 × (6/12) = 50%

Percentage for comfort model = 100 × (2/8) = 25%

Percentage for majestic model = 100 × (4/24) = 16.67%

Therefore, total percentage = 50% + 25% + 16.67% = 91.67%

Since the total percentage is less than 100%, therefore the optimal solution will not change.

Therefore, new objective function becomes:

Z = 69E + 93C + 139M

New Maximum profit = 69×80 + 93×120 + 139×0

New Maximum profit = 16,680