1. Duality (a) A farmer can plant three types of crops: corn, soy beans and wheat. The farmer is maximizing prot and is constrained by the land, fertilizer and working-hours availability. Additionally, an existing contract requires that she produces at least 1000 units of corn. The linear programming problem for the farmer is given below: max 30 + 68 + 4W s. t. (land) 120 + 103 + 15W 5 18000 (fertilizer) 150 + 153 + 12W 3 18000 (hers) 40 + 38' + 2W 3 9000 (contract) 0 Z 1000 C, S, W 2 0. Suppose the following information is given about the optimal solution: Variable Value Dual C S W 1000 0 200 0 0 -0.8 Constraint Value Dual land 14000 0 fertilizer 18000 0.4 hours 4600 0 contract 1000 -3 ii. iii. iv. . Formulate the dual problem and verify that you are in fact given the optimal solution (explain how you check it). Suppose that due to logistics difculties, the farmer's availability of fertilizer is reduced by 1000 units. The farmer has an option to purchase additional 1000 units on the market for $300. Should she do it? Suppose that the contract could be expanded to require 1200 units of corn for additional $500 premium. Is it a good offer? Suppose that a few of the farmer's workers have quit, reducing the available worker hours to 8000. What will happen to the optimal solution. What if the available hours are reduced to 4000