Question

The Weigelt Corporation has three branch plants with excess production capacity. Fortunately, the corporation has a new product ready to begin production, and all three plants have this capability, so some of the excess capacity can be used in this way. This product can be made in three sizes--large, medium, and small--that yield a net unit profit of $420, $360, and $300, respectively. Plants 1, 2, and 3 have the excess capacity to produce 750, 900, and 450 units per day of this product, respectively, regardless of the size or combination of sizes involved.

The amount of available in-process storage space also imposes a limitation on the production rates of the new product. Plants 1, 2, and 3 have 13,000, 12,000, and 5,000 square feet, respectively, of in-process storage space available for a day's production of this product. Each unit of the large, medium, and small sizes produced per day requires 20, 15, and 12 square feet, respectively.

Sales forecasts indicate that if available, 900, 1,200, and 750 units of the large, medium, and small sizes, respectively, would be sold per day.

At each plant, some employees will need to be laid off unless most of the plant’s excess production capacity can be used to produce the new product. To avoid layoffs if possible, management has decided that the plants should use the same percentage of their excess capacity to produce the new product.

Management wishes to know how much of each of the sizes should be produced by each of the plants to maximize profit.

1. Formulate the dual of the above problem and solve it.

2. Weigelt.lp

   /* objective function */

   Max: 420 L1 + 360 M1 + 300 S1 + 420 L2 + 360 M2 + 300 S2 + 420 L3 + 360 M3 + 300 S3;

   /* Constraints */

   /* capacity */

   L1 + M1 + S1 <= 750;

   L2 + M2 + S2 <= 900;

   L3 + M3 + S3 <= 450;

   /* square footage */

   20 L1 + 15 M1 + 12 S1 <= 13000;

   20 L2 + 15 M2 + 12 S2 <= 12000;

   20 L3 + 15 M3 + 12 S3 <= 5000;

   /* sales */

   L1 + L2 + L3 <= 900;

   M1 + M2 + M3 <= 1200;

   S1 + S2 + S3 <= 750;

   /* same percentage of capacity */

   900 L1 + 900 M1 + 900 S1 - 750 L2 - 750 M2 - 750 S2 = 0;

   450 L1 + 450 M1 + 450 S1 - 750 L3 - 750 M3 - 750 S3 = 0;

   * Formulate the dual of the above problem and solve it. I just need help with the dual formulation of this LP solution.

Answer 1