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 3.00 2.00 1.50 B 1.00 2.50 2.00 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 $28 for product 1, $30 for product 2, and $25 for product 3. (a) Formulate a linear programming model for maximizing total profit contribution. If the constant is "1" it must be entered in the box. If required, round your answers to two decimal places. Let Pi = units of product i produced Max $ P1 + $ P2 + $ P3 s.t. P1 + P2 + P3 - Select your answer ≤, ≥, = P1 + P2 + P3 - Select your answer ≤, ≥, = P1 + P2 + P3 - Select your answer ≤, ≥, = 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? Product 1 Product 2 Product 3 Amount to Produce Profit $ (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 $400 for product 1, $550 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 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 provided in part (c) into account. Management also stated that we should not consider making more than 175 units of product 1, 150 units of product 2, or 140 units of product 3. What are the new objective function and additional equation constraints? If the constant is "1" it must be entered in the box. Let Yi is one if any quantity of product i is produced and zero otherwise. Max $ P1 + $ P2 + $ P3 - $ Y1 - $ Y2 - $ Y3 s.t. P1 - Select your answer ≤, ≥, = Y1 P2 - Select your answer ≤, ≥, = Y2 P3 - Select your answer ≤, ≥, = Y3 P1, P2, P3 ≥ 0 (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 contribution? Compare this profit contribution to that obtained in part (c). If required, round your answers to nearest whole number. If your answer is zero enter “0”. Product 1 Product 2 Product 3 Amount to Produce Updated Profit $

Answer 1

Here, the given information is that we have 450 in department A, 350 in department B, and 50 in department C and the profit contributions per unit are $28 for product 1, $30 for product 2, and $25 for product 3.

Also, we are given the per product unit labor hour requirement for the process by each of the department A, B, and C.

So, let a, b and c be the number of units produced for each of the products 1, 2 and 3 respectively.

We have to maximize the total profit contribution of (28a + 30b + 25c) subject to the following conditions:

Solution for part (a) and (b):

The first condition is that the quantity produced will be an integer and non-negative.

Hence, a, b and c >=0

Also, we have a labor-hour availability constraint

Hence, For Department A: 3a + 2b + 1.5c <=450

Similarly, for department B: a + 2.5b + 2c <=350

Department C: 0.2a + 0.25b + 0.25c <=50

Solving this, we get

Labor-hours consumed will be: Department A: 450, Department B: 348, Department C: 49.8

Production plan: a= 84 (Product 1), b=0 (Product 2) and c=132 (Product 3) units and the total profit contribution will be $5652

Solution for part (c):

Now, taking into account the setup costs of $400 for product 1, $500 for product 2 and $600 for product 3,

Concept: Setup cost for any product will be incurred if we produce even 1 unit of that product.

Here, Product 2 is not being manufactured at all (b=0)

Hence, Hart Manufacturing will only be incurring setup costs for products 1 and 3.

Total setup costs = $400 + $600 = $1000

Net profit contribution = $5652 - $1000 = $4652

Solution for part (d) and (e):

We introduce a binary variable Make/No Make (x, y, and z for products 1, 2 and 3 respectively) which will decide whether we are producing that particular product or not and if we are producing even a single unit of that product, only then its fixed costs which are the setup costs will be included in the linear model calculation.

The conditions remain the same as those mentioned in part (a). Additional variables are x, y and z will be inculcated.

The total profit contribution that has to be maximized is [(28a + 30b + 25c) - (400x + 500y + 600z)]

An additional constraint is x, y and z should be binary {1 is to produce and 0 is no production of that product}

Solving this, we get Labor-hours consumed will be: Department A: 450, Department B: 348, Department C: 49.8

Production plan: a= 84 (Product 1), b=0 (Product 2) and c=132 (Product 3) units and the total profit contribution will be $4652

The profit contribution is the same as that of the part (c) which is also $4652

Another way to verify the answer is to see if the resources (Labour-hour) available are consumed to the fullest.

For both the scenarios, labor-hours consumed are equal and to the fullest possible such that from the remaining labor-hours unutilized, no new unit of any product could be produced.

All the best!

Thank you!