Maximize 120P1 + 150P2 + 180P3 + 105P4 Subject to Total profit Production budget
ID: 3146795 • Letter: M
Question
Maximize 120P1 + 150P2 + 180P3 + 105P4 Subject to Total profit Production budget constraint Labor hours constraint Minimum quantity needed for Product 1 constraint Minimum quantity needed for Product 2 constraint Non-negativity constraints 4P1 + 12P2 10P3 +8P4 s 4P1 +3P2+2P3 +3P4 12000 4000 P3 2 200 P42 100 And P1, P2, P3, P420 The QM for Windows output for this problem is given below. Linear Programming Results: Status Variable Basic Value 456.25 Pl P2 P3 P4 slack 1 slack 2 surplus 3 NONBasic 0 Basic Basic NONBasic 0 NONBasic 0 Basic 937.5 100 737.5 surplus 4 NONBasic 0 Optimal Value (Z) 234000 Original problem w/answers: Dual P3 180 10 P4 105 P1 RHS P2 150 12 Maximize 120 Constraint 1 4 Constraint 2 4 Constraint 3 0 Constraint 4 0 15 15 12000 4000 200 100 -60Explanation / Answer
a) The optimal solution to the problem is P1= 465.25; P2= 0, P3= 937.5 and P4= 100, i.e P1 ,P3 and P4 are Basic Variables and P2 is non Basic variable.
The optimal value is 234000 (in $)
i.e if we use the product mix as the optimal solution values then the Profit is maximized. On all other combinations the profit value will be less than 234000
b) The slack values are 737.5 which is the amount in excess of 12000 that the optimal solution requires i.e. 465.25+0+937.5+100=1502.75 the other three slacks are 0 which is the amount of resource 1 and 3 left. That is to produce 465.25 of product P1 ,937.5 of product P3 and 100 units of P4 one would use all of resource1- 12000 , all of resource 2 -4000 and all of resources3- 100