Question

12. a-d and 13. a-d

12. Quality Air Conditioning manufactures three home air conditioners: an cconomy model a standard model, and a deluxe model. The profits per unit are S63-995, and $135, np tively. The production requirements per unit are as follows Number of Fans Number of Cooling Cods Manufacturing Time hours Standard Deluxe 12 14 For the coming production period, the company has 200 fan motons, 320 cooling coils and 2400 hours of manufacturing time available. How many ecconomy models (E), sts dard models (S), and deluxe models (D) should the company produce in oeder to maximie profit? The linear programming model for the problem is as follows Max 63E955+135D 1S+Ds 200 Fan motors E+25 4Ds 320 Cooling coils E+12514D 2400 Manfacturing time FIGURE 3.17 THE SOLUTION FOR THE QUALITY AIR CONDTIONING PROBLEM Optimal objective Value 16440.00000 Reduced Cost 80.00000 120.00000 0.00000 -24.00000 Dual Value 2.00000 abjective Variable Coeffielent 63.00000 135.00000 12.00000 24.0000O Allowable 15.50000 Intinice Allovable 200.00000 20.00000 2400.0b000 80.00000 40.00000 Infinite 20.00000 The computer solution is shown in Figure 3.17. h Which constraints are binding? d. If the profit for the deluxe model were increased to $150 per unit, would the ogtima What is the optimal solution, and what is the value of the otjective function? Which constraint shows extra capacity? How much? solution change? Use the informatioe in Figure 317 so answer this question 13. Refer to the computer solution of Problem 12 in Figure 3.17 a. Identify the range of optimality for each objective function ooefficient h Suppose the profit for the economy model is increased by $6 per unit, the proit for the standard model is decreased by $2 per unit, and the profit for the deluse model is increased by 34 per unit. What will the new optimal solution be? e. Identify the range of feasibility for the right-hand-side valucs d. If the number of fan motors available for prodaction is increased by 100, wil the dua valac for that constraint change? Explain.

Answer 1

12.

a. The optimal solution from the figure (see first line and first table in the figure) is 80 units of Economy, 120 units of Standard and 0 units of Deluxe and the value of the objective function i.e. maximum profit is 16440$.

b. Binding constraints are those whose change also change the optimal results, i.e. the constraints are fulfilled in the optimal solution and if the values are changed then the results of optimal solution also change.

In the figure (see second table), constraints 1 & 2 have a slack or surplus value of 0, indicating that these constraints are fulfilled. Therefore, these are the binding constraints.

Constraint 1: 1E + 1S + 1D < 200 and Constraint 2: 1E + 2S + 4D < 320 are the binding constraints.

Also, the dual value of only those constraints are positive which are binding.

c. From figure (see second table) constraint 3 has a slack or surplus value of 320. Therefore constraint 3 has extra capacity of 320.

Constraint 3: 8E + 12S + 14D < 2400 i.e. constraint for manufacturing time has extra capacity of 320 meaning, 320 hours of manufacturing time is unused.

d. No. From the third table, it is seen that the allowable increase in coefficient of variable D (i.e. profit of variable D) is 24. which means that results wont change up to the profit value of D = 135+24 = 159. Therefore, for profit of 150 the results will not change.

13.

a. The range of optimality for each objective co-efficient is obtained by increasing/ decreasing the coefficient value by the allowable increase and allowable decrease

Therefore the range is:

63-15.5 < E < 63+12 : 47.5 < E < 75

95-8 < S < 95+31: 87 < S < 126

135 - Infinite < D < 135 + 24: 0 < D < 159 (as the allowable decrease is infinite, but in the question 12 it is mentioned that these values should be greater than 0)

b.

The new coefficients of E, S & D are 69, 93 and 139

The results wont change as the new values are within the range of optimality. Therefore, the optimal solution shall remain same as before i.e. 80 units of Economy, 120 units of Standard and 0 units of Deluxe and the value of the objective function i.e. maximum profit is 16440$.

c.

The range of feasibility for right hand side (RHS) values are obtained by increasing/ decreasing the RHS value by the allowable increase and allowable decrease. This range indicates the range within the dual values will not change.

Therefore the range is:

200-40 < Constraint 1 < 200+80 : 160 < Constraint 1 < 280

320-120 < Constraint 2 < 320+80: 200 < Constraint 2 < 400

2400-320 < Constraint 3 < 2400 + Infinite: 2080 < Constraint 3 < Infinite

d. Yes. As the value is being increased by more than the range of feasibility, the dual value of the constraint will change. Dual value the change in objective function per unit to change in unit value of constraint. The dual values are valid for range of feasibility of RHS values.