1) A.DECISION VARIABLE X1 = No. of units of model 101 to produce/month | X2= No. of units of model 102 to produce/month B.OBJECTIVE FUNCTIONTo maximize profit in dollars. Z= 3000 X1 + 5000 X2 C. SUBJECTED TO (CONSTRAINTS):Engine – X1+2X2 <= 4000 hoursMetal Stamping – 2X1 + 2X2 <= 6000 hoursModel 101 assembly - 2X1 <=5000 hoursModel 102 assembly – 3X2<= 4500 hoursX1, X2 >= 0Hours here refer to no. of machine hours3) OPTIMUM RANGE Range Objective Coeff. X1 (2500 to 5000) | 3000 | X2 (3000 to 6000) | 5000 |
{If the current unit contribution moves outside the range then our decisions would change} | 4) FEASIBILITY RANGE Engine (3500 to 4500) | 4000 | Metal stamp(5000 to 6500) | 6000 | Model 101(4000 to IE+30) | 5000 | Model 102(3000 to IE+30) | 4500 | |

{If the current capacity moves outside the range, then the constraint's binding status will change}

5)
No, Merton cannot continue to produce with the old optimal mix. They would now have to produce 1000 units of X1 and 1500 units of X2. This will be because of model 101 assembly now functioning only at 40% of its original capacity thus leading to loss of capacity of metal stamping too making it function at only 83.3% of its capacity. Solving by excel, optimal range for obj coeff. For X1 is 2500 -3000, since the contribution changes below 2500, therefore it will have an impact on the decision changing the number of units in the optimal mix.

6) Monthly availability of machine hours came down by 400 hours in metal stamping ie. 5600 hours department. This would not influence the binding status because this decrease is within the feasibility range of 5000 to 6500 hours therefore it continues to function as a binding status. Thus new optimum mix would be 1600 X1 and 1200 X2. This would result in a profit of only 10 lac 80thousand.

7) Canonical form-
Decision variables –
Y1 - No. of hours for engine assembly
Y2 – No. of hours for metal stamping
Y3 – No. of hours for model 101 assembly
Y4 – No. of hours for model 102 assembly

Objective function- To minimize the no. of hours Z= 4000 Y1 +6000 Y2+ 5000 Y3 + 4500 Y4

Constraints: 1 Y1 + 2 Y2 +2 Y3 + 0 Y4 >= 3000 units
2 Y2 + 2Y2 + 0 Y3+ 3 Y4 >= 5000 units
Y1, Y2, Y3, Y4 >= 0

8) No, there is no need to produce 103 model trucks for the product mix of 2000 model 101 and 1000 model 102 trucks. For it to be worthwhile to produce the model 103 trucks, since shadow price is -550, the contribution should be greater than 550$.