Problem 8-08 (Algorithmic) Let S represent the amount of steel produced (in tons). Steel production is related to the amount of labor used (L) and the amount of capital used (C) by the following function: S = 3040.700.30 In this formula L represents the units of labor input and C the units of capital input. Each unit of labor costs $150, and each unit of capital costs $200. a. Formulate an optimization problem that will determine how much labor and capital are needed in order to produce 70,000 tons of steel at minimum cost. If the constant is "1" it must be entered in the box; if your answer is zero, enter "0". Min L s.t. L, C b. Solve the optimization problem you formulated in part (a). Hint: Use the Multistart option as described in Appendix 8.1. Add lower and upper bound constraints of 0 and 5000 for both L and C before solving. Round your answers for L and C to three decimal places. Round your answer for optimal solution to one decimal place. L = and C= for an optimal solution of $ Problem 8-10 (Algorithmic) Heller Manufacturing has two production facilities that manufacture baseball gloves. Production costs at the two facilities differ because of varying labor rates, local property taxes, type of equipment, capacity, and so on. The Dayton plant has weekly costs that can be expressed as a function of the number of gloves produced: TCD(X) = x2 - X + 4 where X is the weekly production volume in thousands of units and TCD(X) is the cost in thousands of dollars. The Hamilton plant's weekly production costs are given by TCH(Y) = y2 + 2Y + 5 where Y is the weekly production volume in thousands of units and TCH(Y) is the cost in thousands of dollars. Heller Manufacturing would like to produce 8000 gloves per week at the lowest possible cost. a. Formulate a mathematical model that can be used to determine the optimal number of gloves to produce each week at each facility. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. If the constant is "1" it must be entered in the box. x2 + X + + y2 + Y + s.t. X + Y X, Y Z 0 b. Use Excel Solver or LINGO to find the solution to your mathematical model to determine the optimal number of gloves to produce at each facility. If required, round your answers to two decimal places. The optimal solution is X = and Y = for an optimal objective value of $