CS 600 Test II

November 6, 1997

1. Find an initial basic feasible solution for the following LP problem, and give the initial dictionary for the problem. (33 points)

Max z = 3x₁ + x₂
Subject to
$-x_{1} + x_{2} \leq 1$
$-x_{1} - x_{2} \leq -2$
$x_{1} + 2x_{2} \leq 4$

2. Find an initial feasible tree solution for the transshipment problem with the following network. Use node 4 as w. (33 points)

A figure is missing here.

3. A certain corporation has three branch plants with excess capacity. All three plants have the capability for producing a certain product, and management has decided to use some of the excess production capacity in this way. This product can be made in three sizes - large, medium, and small. The net unit profit for these products are as follows: For the first 10 units of the large size, the unit profit would be $12. It would be reduced to $2 for any additional units of the large size. For the first 15 units of the medium size, the unit profit is estimated at $7. It would be $6 for each of the next 15 units and $4 for any additional units. For the first 8 units of the small size, the unit profit would be $10. The unit profit would be $9 for each of the next 5 units and $7 for any additional units.

Plants 1, 2, and 3 have the excess manpower and equipment capacity to produce 1,500, 1,800, and 900 units per day of this product, respectively, regardless of the size or combination of sizes involved. However, the amount of available in-process storage space also imposes a limitation on the production rates. Plants 1, 2, and 3 have 26,000, 24,000, and 10,000 square feet 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 40, 30, and 24 square feet, respectively.

Sales forecasts indicate that 1,800, 2,400, and 1,500 units of the large, medium, and small sizes, respectively can be sold per day.

To maintain a uniform work load among the plants and to retain some flexibility, management has decided that the additional production assigned to each plant must use the same percentage of the excess manpower and equipment capacity.

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

Formulate the linear programming model for this problem. (34 points)