Question

Maximize 8C + 10R + 7T         Total profit

Subject to

        7C + 10R + 5T ? 2000       Production budget constraint

          2C + 3R + 2T ? 660         Labor hours constraint

                              C ? 200         Maximum demand for clocks constraint

                              R ? 300         Maximum demand for radios constraint

                              T ? 150         Maximum demand for toasters constraint

             And C, R, T ? 0             Non-negativity constraints

Where C = number of clocks to be produced

          R = number of radios to be produced

          T = number of toasters to be produced

The QM for Windows output for this problem is given below.

Solution List:

Variable           Status Value

C Basic 178.57

R NONBasic 0

T Basic 150

slack 1 NONBasic        0

slack 2 Basic 2.86

slack 3 Basic 21.43

slack 4 Basic    300

slack 5 NONBasic        0

Optimal Value (Z)        2478.57

Linear Programming Results:

            C          R          T                     RHS     Dual

Maximize         8          10        7                                 

Constraint 1     7          10        5          <=        2000    1.14

Constraint 2     2          3          2          <=        660      0

Constraint 3     1          0          0          <=        200      0

Constraint 4     0          1          0          <=        300      0

Constraint 5     0          0          1          <=        150      1.29

Solution           178.57 0          150                  2478.57

Ranging Results:

Variable           Value   Reduced Cost   Original Val      Lower Bound   Upper Bound

C 178.57 0 8 7 9.8

R 0 1.43 10 -Infinity 11.43

T 150 0 7 5.71 Infinity

            Dual Value       Slack/Surplus   Original Val      Lower Bound   Upper Bound

Constraint 1     1.14     0          2000 750 2010

Constraint 2     0          2.86     660 657.14 Infinity

Constraint 3     0          21.43   200 178.57 Infinity

Constraint 4     0          300      300 0 Infinity

Constraint 5     1.29     0          150 120 155

. (a) What are the ranges of optimality for the profit of a clock, a radio and a toaster?

(b) Find the dual prices of the five constraints and interpret their meanings. Determine the ranges in which each of these dual prices is valid.

(c) If the profit contribution of a clock changes from $8 to $9, what will be the optimal solution? What will be the new total profit? (Note: Answer this question by using the ranging results given above).

(d) Which resource should be obtained in larger quantity to increase the profit most? (Note: Answer this question using the ranging results given above.).

Answer 1

The problem solved on tora with the output and answers is given below:-

a)

The different range of optimality are:-

Clock:- 7 to 9.8

Radio:- - Infinity to 11.43

Toaster:- 5.71 to Infinity

b) The dual price of the constraints are:-

Constraint 1     1.14

Constraint 2     0

Constraint 3     0

Constraint 4     0

Constraint 5     1.29

The dual price shows the improvement or the increase/decrease in value of the objective function if the constraint is allowed to be relaxed by one unit. So, if the RHS of Constraint 1 is increased to 2001 then the maximum value of the objective function would increase by 1.14 units to 2479.71.Similarly for Constraint 5 if the RHS of Constraint 5 is increased to 151 then the value of the objective function would increase by 1.29 units.

c) As, per the above result if the profit contribution of a clock changes from $8 to $9, then the optimal solution would remain the same because the value of 9 is within the range of optimality for clocks ( between 7 to 9.8) but the new total profit would change as per the new profit equation 9C + 10R + 7T, where C= 178.71, R = 0, T = 150

So, Z = 9* 178.71 + 10* 0 + 7* 150 = 1608.4+0+1050 = 2658.4

d) Since the Dual price for toasters is higher than that of radios and clocks it must be obtained in larger quantity to get higher profits.