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The following is a linear programming model for analyzing the product mix of Maxine's Hat Company, which produces three hat types: Maximize: Z = $7X1 + $8X, +S6X Subject to: 2X2 + 4X2 + 2X, 5120 (machine A time) 5X, +3X2 + 2X, $ 400 (machine B time) 2x, + 2X2 + 4%, $110 (machine C time) X, 20. X, 20. and X: 20 The POM for Windows printout below shows the optimal solution to the problem. (a) Results Screen X X RHS Shadow Price Maximize 7 6 Machine A 2 2 120 0.5 Machine B 0.20 0.05 0.10 400 Machine C 0.15 0.10 0.10 110 3 Solution 50 5 0 390 (6) Ranging screen SE Slot 0 0 Variable Value Reduced Cost X1 50 0 X2 5 X3 7 Constraint Shadow Price Slack/Surplus Machine A .5 0 Machine B 0 135 Machine C 3 Original Val 7 8 6 Original Val 120 400 110 Lower Bound 4.6667 7 - Infinity Lower Bound 110 285 80 Upper Bound 8 14 13 Upper Bound 220 Infinity 120 Consider each of the following statements independently, and state whether it is true or false. Explain each answer. a. If the price of hat 3 were increased to $12.25, it would be part of the optimal product mix. (Hint: Hat 3 is represented by X, and its optimal value currently is 0, which means that it is not to be produced (and not part of the optimal product mix).) O A. True, because the price of Hat 3 increased more than 50% of its current price. B. Unknown, because the optimal value of Hat 3 should be recalculated with the new selling price. OC. False, because the optimal value of Hat 3 would remain at its current value until its price increases above $13.00. D. True, because the reduced cost of Hat 3 is $7.00. b. The capacity of machines can be reduced to 320 hours without affecting profits. O A. False, because the new limit is lower than its previous value and the constraint would become more restrictive. B. True, because the new RHS value is within the range in which the shadow price of 0 would remain valid. OC. False, because the new limit is lower than its previous value and its shadow price would change. OD. True, because the new limit is lower than its previous value. c. If machine A has a capacity of 140 hours, the production output would remain unchanged. O A. True, because the new RHS value is within the range in which the shadow price of 0.5 would remain valid. OB. False, because constraint A is a binding constraint C. False, because the new limit is higher than its previous value and its shadow price would change. OD. True, because the new limit is higher than its previous value and the constraint would become less restrictive