# Question

The MFG Corporation is planning to produce and market three different products. Let x1, x2, and x3 denote the number of units of the three respective products to be produced. The preliminary estimates of their potential profitability are as follows.

For the first 15 units produced of Product 1, the unit profit would be approximately $360. The unit profit would be only $30 for any additional units of Product 1. For the first 20 units produced of Product 2, the unit profit is estimated at $240. The unit profit would be $120 for each of the next 20 units and $90 for any additional units. For the first 20 units of Product 3, the unit profit would be $450. The unit profit would be $300 for each of the next 10 units and $180 for any additional units.

Certain limitations on the use of needed resources impose the following constraints on the production of the three products:

Management wants to know what values of x1, x2 and x3 should be chosen to maximize the total profit.

(a) Plot the profit graph for each of the three products.

(b) Use separable programming to formulate a linear programming model for this problem.

(c) Solve the model. What is the resulting recommendation to management about the values of x1, x2, and x3 to use?

(d) Now suppose that there is an additional constraint that the profit from products 1 and 2 must total at least $12,000. Use the technique presented in the “Extensions” subsection of Sec. 13.8 to add this constraint to the model formulated in part (b).

(e) Repeat part (c) for the model formulated in part (d).

For the first 15 units produced of Product 1, the unit profit would be approximately $360. The unit profit would be only $30 for any additional units of Product 1. For the first 20 units produced of Product 2, the unit profit is estimated at $240. The unit profit would be $120 for each of the next 20 units and $90 for any additional units. For the first 20 units of Product 3, the unit profit would be $450. The unit profit would be $300 for each of the next 10 units and $180 for any additional units.

Certain limitations on the use of needed resources impose the following constraints on the production of the three products:

Management wants to know what values of x1, x2 and x3 should be chosen to maximize the total profit.

(a) Plot the profit graph for each of the three products.

(b) Use separable programming to formulate a linear programming model for this problem.

(c) Solve the model. What is the resulting recommendation to management about the values of x1, x2, and x3 to use?

(d) Now suppose that there is an additional constraint that the profit from products 1 and 2 must total at least $12,000. Use the technique presented in the “Extensions” subsection of Sec. 13.8 to add this constraint to the model formulated in part (b).

(e) Repeat part (c) for the model formulated in part (d).

## Answer to relevant Questions

The Dorwyn Company has two new products that will compete with the two new products for the Wyndor Glass Co. (described in Sec. 3.1). Using units of hundreds of dollars for the objective function, the linear programming ...For each of the following cases, prove that the key property of separable programming given in Sec. 13.8 must hold. (a) The special case of separable programming where all the gi(x) are linear functions. (b) The general case ...Reconsider the linearly constrained convex programming model given in Prob. 13.6-13. Starting from the initial trial solution (x1, x2, x3) = (0, 0, 0), apply two iterations of the Frank- Wolfe algorithm. Reconsider the linearly constrained convex programming model given in Prob. 13.9-8. (a) If SUMT were to be applied to this problem, what would be the unconstrained function P(x; r) to be maximized at each iteration? Reconsider the convex programming model with an equality constraint given in Prob. 13.6-11. (a) If SUMT were to be applied to this model, what would be the unconstrained function P(x; r) to be minimized at each iteration?Post your question

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