Question

A company is attempting to decide the mix of products which it should produce next week. It has six products, each with a unit profit (unit profit =selling price –variable cost per unit) and a unit production time as shown below:

Product	Profit ($/unit)	Production time (hours/unit)
1	10	1
2	22	2
3	35	1.7
4	19	2.4
5	55	4.5
6	115	9.5

The company has 720 hours available next week. The company has the following additional linear constraints:

• If any of the above six products are produced a fixed cost of $1000 is incurred.

• If product 3 is produced, 40 hours are needed for production setup.

Formulate an integer programming model to find next week's production schedule to maximize profit with linear objective function and all linear constraints.

Additional constraint 1 : Product 1 must be produced if any of Product 2, 3 and 4 is produced.

Additional constraint 2 : Product 3 must be produced if all of Product 4, 5 and 6 are produced.

**Clearly State the Decision Variables. Objective Function, and Constraints**

Answer 1

Let Pi= 1 when product i produced in the next week

Xi= number of product i produced next week

Objective function =

Max Z= 10X1 + 22X2 + 35X3 + 19X4 + 55X5 + 115X6 – 1000(P1 + P2 + P3 + P4 + P5 + P6)

Subjected to=

Xi ≤ 720Pi ( For total production hour)

1X1 + 2(X2-Y2) + 3Y2 + 1.7X3 + 2.4X4 + 4.5X5 + 9.5X6 ≤ 720 – 40P3 ( for product 3 set up)

P1 ≤ P2,P3, P4 ( condition of the production of product 1)

P3≤ P4+P5+ P6 ( condition of the production of product 3)