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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Hart Manufacturing makes three products. Each product requires manufacturing operations in three departments: A, B, and C. The labor-hour requirements, by department, are as follows.

Department	Product 1	Product 2	Product 3
A	1.50	3.00	2.00
B	2.00	1.00	2.50
C	0.25	0.25	0.25

During the next production period, the labor-hours available are 450 in department A, 350 in department B, and 50 in department C. The profit contributions per unit are $27 for product 1, $30 for product 2, and $32 for product 3.

(a)

Formulate a linear programming model for maximizing total profit contribution. (Let

P_i =

units of product i produced, for

i = 1, 2, 3.)

Max ?

Capitalization of variable(s) is important. Check that your upper and lower case match the question.

s.t.

Department A ?

Department B ?

Department C ?

P₁, P₂, P₃ ≥ 0

(b)

Solve the linear program formulated in part (a). How much of each product should be produced, and what is the projected total profit contribution (in dollars)?

(P₁, P₂, P₃) = 60,80,60 correct

with profit $ 5940 correct

(c)

After evaluating the solution obtained in part (b), one of the production supervisors noted that production setup costs had not been taken into account. She noted that setup costs are $410 for product 1, $520 for product 2, and $600 for product 3. If the solution developed in part (b) is to be used, what is the total profit contribution (in dollars) after taking into account the setup costs?

$ ?

(d)

Management realized that the optimal product mix, taking setup costs into account, might be different from the one recommended in part (b). Formulate a mixed-integer linear program that takes setup costs into account. Management also stated that we should not consider making more than 160 units of product 1, 185 units of product 2, or 200 units of product 3. (Let P_i =units of product i produced and y_ibe the 0-1 variable that is one if any quantity of product i is produced and zero otherwise, for i = 1, 2, 3.)

What is the objective function of the mixed-integer linear program?

Max ?

In addition to the constraints from part (a), what other constraints should be added to the mixed-integer linear program?

s.t.

units of Product 1 produced ?

units of Product 2 produced ?

units of Product 3 produced ?

P₁, P₂, P₃ ≥ 0; y₁, y₂, y₃ = 0, 1

(e) Solve the mixed-integer linear program formulated in part (d). How much of each product should be produced, and what is the projected total profit (in dollars) contribution?

(P₁, P₂, P₃, y₁, y₂, y₃) = ( )? with profit $ ?

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