A company has three branch plants with excess production capacity and wants to use it to produce a new product. 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.
this is how i have solved the problem so far. kind of stuck on how to determine how the plants will use the excess capacity to produce the new products(to minimise layoffs)
Z = 420X1 + 360X2 + 300X3 + 420X4 + 360X5 + 300X6 + 420X7 + 360X8 + 300X9
20X1 + 15X2 + 12X3 ≤ 13,000
20X4 + 15X5 + 12X6 ≤ 12,000
20X7 + 15X8 + 12X9 ≤ 5,000
X1 + X2 + X3 ≤ 750
X4 + X5 + X6 ≤ 900
X7 + X8 + X9 ≤ 450
X1 + X2 + X3 ≤ 900
X4 + X5 + X6 ≤ 1200
X7 + X8 + X9 ≤ 750
thanks
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.
this is how i have solved the problem so far. kind of stuck on how to determine how the plants will use the excess capacity to produce the new products(to minimise layoffs)
Z = 420X1 + 360X2 + 300X3 + 420X4 + 360X5 + 300X6 + 420X7 + 360X8 + 300X9
20X1 + 15X2 + 12X3 ≤ 13,000
20X4 + 15X5 + 12X6 ≤ 12,000
20X7 + 15X8 + 12X9 ≤ 5,000
X1 + X2 + X3 ≤ 750
X4 + X5 + X6 ≤ 900
X7 + X8 + X9 ≤ 450
X1 + X2 + X3 ≤ 900
X4 + X5 + X6 ≤ 1200
X7 + X8 + X9 ≤ 750
thanks
