Question

The company "Dual-Best" producing paper has you solve an LP problem, the objective being measured in dollars of profit. You solve the primal LP problem, arriving at a final and optimal dictionary that is non-degenerate. And you also solved the dual problem. Currently they are getting 1000000 kilograms of pulp ingredients per day from the contractor "Pulp-good" at the price $1 per kilogram. In your dual optimal solution T*, the component, say, yi that corresponds to the constraint of pulp < 1000000 kg, is found to be yi 10. In below (a) are independent questions. (a) 4 marks Suppose a neighbouring company "Neighbor" started a new business and needs some small amount of pulp, say, 10 kg per day, and wants the company "Dual- Best" to resell the pulp to them, at the price $10 per kilogram. Would you advise the company "Dual-Best" to agree on reselling its pulp to the company "Neighbor"? Justify your decision. Explain your answer clearly using relevant theorems.

Answer 1

Reselling 10 kg of pulp to "Neighbor" at $10 per kilogram is advisable as it matches the shadow price in the dual optimal solution, suggesting an additional profit without disrupting the current optimal production scenario.

Step-by-step explanation:

The company "Dual-Best" asked whether they should resell pulp at a higher price to "Neighbor". Given that the dual optimal solution indicates a shadow price of $10 for the pulp, as represented by the variable yi, and the company is currently buying pulp for $1 per kilogram, reselling at $10 per kilogram indicates a direct marginal benefit equal to the shadow price. This suggests that the company could make additional profit per kilogram resold without impacting their optimal production scenario, as long as the quantity resold does not significantly affect their operations.

Under the assumptions of the Linear Programming model, as long as the primal is non-degenerate, small changes in resource availability or prices will not change the optimal solution significantly. This supports the argument that reselling a small amount of pulp will not disrupt the current optimal solution. Therefore, reselling the 10 kg of pulp to "Neighbor" at the given price is advisable as it aligns with increasing profitability without affecting the company's optimal production.