4. Drafting THE PRIMAL PROBLEM Therefore, the primal problem leads the manufacturer to find an optimal production plan that maximizes the sales with available resources. Let us consider maximum profit obtained as Z Manufacturer: Maximization of Profit (Z) with the available resources (A,B)
5. Drafting THE PRIMAL PROBLEM OBJECTIVE FUNCTION Max Z = 40X1 + 35X2 SUBJECT TO THE CONSTRAINTS 2X1 + 3X2 ≤60 Raw Material Constraint 4X1 + 3X2 ≤96 Labour Constraint X1,X2 ≥0 … non negative constraint
6. Optimal Solution TO THE PRIMAL PROBLEM M: Will produce 18 units of X1 type and 8 units of X2 type to attain maximum profit of Rs. 1000 Maximization of Profit 18 units of X1 and 8 units X2 profit = 18x40 + 8x35 = Rs.1000
7. Dual OF THE PRIMAL PROBLEM Let’s assume the manufacturer (M) rents his facilities to R for a week. R wants to negotiate the unit rental pricefor resources with M to minimize the total price in obtaining them. M will not rent his resources for anything below the market price. The market price and the “product-resource” conversion ratio (how many units of resource(s) are required for a product) are open information on market.
8. Dual OF THE PRIMAL PROBLEM The rental rates should be at least as attractive as producing the products. Setup constraints keeping in mind that renting is at least as favourable as the others.
9. Dual OF THE PRIMAL PROBLEM R: Minimize the value of Z to the least possible, i.e. acceptable to manufacturer M Let’s assume the manufacturer (M) rents his facilities to R for a week. M’s assets are in the form of: 60 kgs of Raw material and 96 hrs of labour Considering, Y1 = Rate of rent (per kg) of raw materials Y2 = Rate of rent (per hr) of labour. Z = 60Y1 + 96Y2 Were Z = rent receivable from the R by manufacturer M.
10. Formulation OF THE DUAL PROBLEM Now, to produce 1 unit of type X1, 2 kg of raw material and 4 hrs of labour is required. Also, the rent must be at least equal to the profit obtainable from type X1i.e. 40. 2Y1 + 4Y2 ≥40 Similarly, to produce 1 unit of type X2, 3 kg of raw material and 3 hrs of labour is required. Also, the rent must be at least equal to the profit obtainable from type X2i.e. 35. 3Y1 + 3Y2 ≥35
11. Formulation OF THE DUAL PROBLEM The objective of R is to minimize the cost of rent to the least amount acceptable to the manufacturer M. OBJECTIVE FUNCTION Min Z = 60Y1 + 96Y2 SUBJECT TO THE CONSTRAINTS 2Y1 + 4Y2 ≥40 3Y1 + 3Y2 ≥35 Y1,Y2 ≥0
12. Optimal Solution TO THE DUAL PROBLEM Minimization of Rent 10/3 units of Y1 and 25/3 units Y2 Min. Rent = 1000
13. Interpretations OF THE SOLUTION The value of the objective function for both the dual and primal solution is the same. This implies that, the minimum rental that the Manufacturer M earns from R is equal to the maximum profit that it can earn from using the available resources. An addition of 1 kg of material causes a 10/3 unit rise in profit. Similarly, an addition of 1 labour hour causes a rise in the profit by 25/3 units.
16. They are assigned to the services of the two resources: Material and labour hours.
17. Are imputed from the profit obtained from the utilization of the resources.
18. Have no relationship with the original cost of the two. Total imputed cost for A: Known, Shadow prices Y1 = 10/3 and Y2 = 25/3 2 kgs of Y1 @ 10/3 per kg= 20/3 4 hrs of Y2 @ 25/3 per hr= 100/3 Imputed Cost for A = 40 HENCE, Imputed cost = profit obtained Total imputed cost for B: Known, Shadow prices Y1 = 10/3 and Y2 = 25/3 3 kgs of Y1 @ 10/3 per kg= 30/3 3 hrs of Y2 @ 25/3 per hr= 75/3 Imputed Cost for B = 35 HENCE, Imputed cost = profit obtained
19. Interpretations OF THE SOLUTION Maximum Marginal Price The Shadow Price of the optimal dual solution represents the maximum marginal price. This implies that if there were a market for rentals of raw materials and labour hours, the manufacturer M would take them if the price of raw material is less than 10/3 and the price of labour is less than 25/3 units.
20. Interpretations OF THE SOLUTION Complementary Slackness Condition When a resource is not fully utilized (say the labour hours is underutilized to 90 hrs), then the complementary slackness condition requires that yi*=0 , were yi*= optimal dual solution, which means the manufacturer is not willing to pay a paisa more to get an additional amount of that resource from the supplier.
21. Interpretations OF THE SOLUTION Complementary Slackness Condition When the supplier ask too much for a resource, (say he is asking 10 per kg for raw material) the complementary slackness condition requires that X1*=0 (i.e. primal solution of X1 be 0.) Which means that the manufacturer is no longer willing to produce any amount of the product X1.
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