Solve the below problem:
Q1. A clothing company makes coats and slacks. The two resources required are wool cloth and labor. The company has 200 square yards of wool and 300 hours of labor available. Each coat requires 5 square yards of wool and 10 hours of labor, whereas each pair of slacks requires 6 square yards of wool and 4 hours of labor. The profit for a coat is $25, and the profit for a pair of slacks is $18. The company wants to determine the number of coats and pairs of slacks to make so that profit will be maximized.
(a) Formulate a linear programming model for this problem by defining the decision variables, objective function and all the constraints. What do they represent?
(b) Find the optimal solution of this model by hand using the corner points graphical method.
Q2. Solve the following linear programming model graphically. In addition, write the problem in standard form and do a constraint analysis for the optimal solution. Show all details.
Maximize Z = 20x + 10y
Subject to
x + 2y < 20
x + y < 15
x > 8
x, y ≥ 0
Q3. Charm Mining Company owns two mines, each of which produces three grades of ore- high, medium, and low. The company has a contract to supply a smelting company with at least 140 tons of high-grade ore, 120 tons of medium-grade ore, and 160 tons of low-grade ore. Each mine produces a certain amount of each type of ore during each hour that it operates. Mine 1 produces 7.2 tons of high-grade ore, 3.5 tons of medium-grade ore, and 4.6 tons of low grade ore per hour. Mine 2 produces 3.2, 4.4, and 9.8 tons, respectively, of high-, medium-, and low-grade ore per hour. It costs the company $350 per hour to operate mine 1, and it costs $250 per hour to operate mine 2. The company wants to determine the number of hours it needs to operate each mine so that its contractual obligations can be met at the lowest cost.
Formulate a linear programming model for the above situation by determining
(a) The decision variables
(b) Determine the objective function. What does it represent?
(c) Determine all the constraints. Briefly describe what each constraint represents.
Q4. Determine whether the following linear programming problem is infeasible, unbounded, or has multiple optimal solutions. Draw a graph and explain your conclusion.
Maximize Z = 8x + 6y
Subject to
-3x + 4y < 60
2x + y > 40
2x + 3y > 60
x, y ≥ 0