Question

Quality Air Conditioning manufactures three home air conditioners: an economy model, a standard model, and a deluxe model. The profit per unit are $63, $95, and $135, respectively. The production requirements per unit are as follows.

       Number       Number of           Manufacturing
   of Fans Cooling Coils Time (hours)
Economy   1           1               8
Standard   1           2               12
Deluxe   1           4               14

For the coming production period, the company has 200 fan motors, 320 cooling coils, 2400 hours of manufacturing time available. How many economy models (E), standard models (S), and deluxe models (D) should the company produce in order to maximise profit? The linear programming model for the problem is as follows:

Max     63E + 95S + 135D

s.t.   1E + 1S + 1D <= 200 Fan motors
   1E + 2S + 4D <= 320 Cooling coils
   8E + 12S + 14D <= 2400 Manufacturing time

   E, S, D >= 0

The Computer Solution Excel Sensitivity Report is shown below in Table 1.

Table 1: Microsoft Excel 16.0 Sensitivity Report

Variable Cells                      
         Final   Reduced   Objective   Allowable   Allowable  
   Variable        Value   Cost   Coefficient   Increase   Decrease  
   E      80   0   63   12   15.5  
   S      120   0   95   31   8  
   D      0   -24   135   24   1E+30  
                              

Constraints                      
         Final   Shadow   Constraint   Allowable   Allowable  
   Variable       Value   Price   R.H. Side   Increase   Decrease  
   E      200   31   200   120   40  
   S      320   32   320   80   120  
   D      320   0   2400   1E+30   2080  

Use the information in Table 1 above answer the following questions from a-g.  
  

(a)   What is the optimal solution and what is objective function value?
(b)   Which Constraints are binding?
(c)   Which constraints show extra capacity? How much?   
(d)   If the profit for deluxe model were increased to $150 per unit, would the optimal solution change? Explain showing brief calculations.
(e)   Identify and calculate range of optimality for each objective function coefficient.

(f)   Identify the range of feasibility for the right-hand-side values.   

(g)   If the number of fan motors available for production increased by 100, will the dual value for that constraint change? Explain showing brief calculations.   

Answer 1

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.

e. 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)

f. 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.