A tailor makes wool tweed sport coats and wool slacks. He is able to get a shipment of 150 square yards of wool cloth from Scotland each month to make coats and slacks, and he has 200 hours of his own labor to make them each month. A coat requires 3 square yards of wool and 10 hours to make, and a pair of slacks requires 5 square yards of wool and 4 hours to make. The tailor earns $50 in profit from each coat he makes and $40 from each pair of slacks. He wants to know how many coats and pairs of slacks to produce to maximize profit. a. Formulate an integer linear programming model for this problem.

b. Determine the integer solution to this problem by using the computer. Compare this solution with the solution without integer restrictions and indicate whether the rounded-down solution would have been optimal.

Explanation:

a) The linear programming model takes into account material and labor. It seeks to maximize profit.

Let c and s represent the numbers of coats and slacks produced in a month.

  Maximize profit = 50c +40s subject to ...

  3c + 5s ≤ 150 . . . . . limit on yards of wool

  10c + 4s ≤ 200 . . .  limit on hours of labor

  c ≥ 0, s ≥ 0 . . . . . . . quantities produced cannot be negative

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b) Many on-line solvers, even with an "integer" restriction, produce a rational number as a solution. We did find one that was able to accept a restriction that values be integers. Its solution is ...

  (coats, slacks) = (10, 24) . . . . first attachment

The non-integer solutions are ...

  (coats, slacks) = (200/19, 450/19) ≈ (10.526, 23.684) . . . . second attachment

A graphical solution lets one find integer grid points that are close to the optimum non-integer solution.

  (coats, slacks) = (10.526, 23.684) = (10, 24) . . . third attachment

Rounding down the non-integer solution would give (10, 23), which is less than the optimal solution.

answer; closing the loop;

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