Question

1 2 12 1 14 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, 995, and $135 respectively. Currently, the resources required for production of each unit is given in the table below. Fans Cooling Coils Assembly Time (hr) Essential 8 Comfort Majestic In Stock/Available 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 Crepresents 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 + 10 + 1M S 200 1E + 2C + 4M S 320 8E + 120 + 14M S 2400 E,C,M20 The solved sensitivity report is given below: Variable Cells Final Reduced Objective Allowable Allowable Cell Name Cost Coefficient Increase Decrease $B$2 Dec Var E 0 63 12 15.5 $C$2 Dec Var C 95 31 $D$2 Dec Var M 135 1E430 Value 80 120 0 0 24 Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $E$5 Fans 200 31 200 80 40 SE$6 Cooling Coils 320 32 320 80 120 $E$7 Assembly Time 2080 0 2400 15:30 320 Answer the questions on the following page using the information presented in the problem and the sensitivity report above. Answer cach question independently and refer back to the original formulation for each question.

Answer 1

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Answers::

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

Optimal number of Essential models = 80

Optimal number of comfort models = 120

Optimal number of majestic models = 0

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

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

Maximum Profit = $ 16,440

(B). From the constraints table, it can be seen that for assembly time, the final value is less than constraint R.H. side value which means that it has some slack or extra Capacity.

Amount of extra Capacity = 2400 - 2080 = 320

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

The objective coefficient for Majestic models is 135 with an allowable increase of 24 which means the the optimal solution will not change even if coefficient of majestic models 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 means that a unit change in the constraint R.H. side of fans will change the optimal solution value by $ 31.

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

(E). The reduced cost for majestic model is blanked. Now, it can be seen 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 allowable increase. Therefore, reduced cost for majestic model = -24

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

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%

Now, 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