Q2. The final tableau for a maximisation LP with only resource (3) constraints i
ID: 423366 • Letter: Q
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Q2. The final tableau for a maximisation LP with only resource (3) constraints is Basis xiT 2 3 8152 s3 RHS Ratio 16 1 0-0.021 010 3-63/10 0 0.05 2 0 6 198 0 0 0.1 -28 1 40 z37 0 0 30 0 900 T2 S3 (i) Write down the optimal solution and the total profit (ii) Identify the shadow prices for this problem. (ii) Identify the reduced costs for this problem (iv) Determine how much the profit will change if the amount of the resource in constraint 2 is increased by 4. Assume this will not affect the optimal basis. (v) Determine what level of profitability r1 must achieve before it is optimal to start making ri. Assume this will not affect the optimal basis. (vi) Find the new optimal solution to the problem if 20 more units of the resource in constraint 3 become available.Explanation / Answer
(i) Optimal solution can be read from the final tableau RHS
x2 = 10
x3 = 6
s3 = 40
Total profit = 900
(ii) Shadow prices are the values corresponding to slack variables (s1, s2, s3) in the bottom row
Shadow price for constraint 1 = 1
Shadow price for constraint 2 = 30
Shadow price for constraint 3 = 0
(iii) Reduced costs are the values corresponding to decision variables (x1, x2, x3) in the bottom row
Reduced cost for x1 = 37
Reduced cost for x2 = 0
Reduced cost for x3 = 0
(iv) Shadow price of constraint 2 is 30. Therefore, by increasing the resource 2 by 4, the total profit will increase by = 4*30 = 120
(v) Profitability of x1 must increase equal to its reduced cost, i.e. 37 before it is optimal to start making x1.
(vi) Shadow price of constraint 3 is 0. Therefore, 20 more units of resource in constraint 3 are not any useful. Optimal solution remains unchanged.
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