Linear Programming: Graphical Methods for Minimization

1.<s>, The Charm City Clothiers Inc. 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 $15. The company wants to determine the number of coats and pairs of slacks to make so that profit will be maximized.</s>

a. Formulate a linear programming model for this problem.
b. Solve this model by hand using the corner points graphical method.

2. Solve the following linear programming model graphically. In addition, write the problem in standard form and do a constraint analysis for the optimal solution.

Maximize Z = 7x + 10y
Subject to
x + 2y < 20
x + y < 15
x > 8
x, y ? 0

3. Copperfield 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 10 tons of high-grade ore, 10 tons of medium-grade ore, and 20 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 5 tons of high-grade ore, 2.5 tons of medium-grade ore, and 4 tons of low grade ore per hour. Mine 2 produces 2, 2, and 10 tons, respectively, of high-, medium-, and low-grade ore per hour. It costs Copperfield $250 per hour to mine each ton of ore from mine 1, and it costs $180 per hour to mine each ton of ore from 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 this problem.
Note: Do NOT solve the model after formulating.

4. Solve the following linear programming model graphically and explain the solution result.

Maximize Z = 100x1 + 80x2
Subject to
-3x1 + 4x2 < 60
2x1 + 3x2 > 60
2x1 + x2 > 40
x1, x2 ? 0

1. The Charm City Clothiers Inc. 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 $15. 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.
b. Solve this model by hand using the corner points graphical method.

Click to expand...

Let c=the number of coats and s=the number of slacks.

But each coat needs 5 yards of wool, so \(\displaystyle 5c\leq 200\)

plus 10 hours of labor each: \(\displaystyle 10c\leq 300\)

The slacks can be made from what remains:

\(\displaystyle 6s\leq 200-5c\)

\(\displaystyle 4s\leq 300-10c\)

The constraint to maximize is the profit: \(\displaystyle P=25c+15s\)

Graph them and check the points.

Make sure I set this up right.