Home furnishings for department stores. Planning is underway for the production

Question

Home furnishings for department stores. Planning is underway for the production of items in the "flower" fabric pattern during the next production period.

Bedspread

Curtains

Dust Ruffle

Fabric required (yds)

   7

  4

9

Time required (hrs)

1.5

  2

.5

Packaging material

   3

  2

1

Profit

12

10

8

Inventory of the flower fabric is 3000 yards. Five hundred hours of production time have been scheduled. Four hundred units of packaging material are available. Each of these values can be adjusted through overtime or extra purchases.

Home furnishings would like to achieve a profit of $3200, avoid purchasing more fabric or packaging material, and use all of the hours scheduled. Give the goal programming model. Set up the LP Model.

Bedspread

Curtains

Dust Ruffle

Fabric required (yds)

   7

  4

9

Time required (hrs)

1.5

  2

.5

Packaging material

   3

  2

1

Profit

12

10

8

Explanation / Answer

The Goal is to use all of the hours scheduled without purchasing more fabric and or packaging material.

The objective is to maximize the total profit. Decision variables are the number of Bedspreads, Curtains, and Dust Ruffles to be made using the given resources. Let us denote the number of bedspreads as B, number of Curtains as C. and D represents the number of Dust ruffles to be made out of the materials and hours scheduled.

Formulation of Integer Linear Programming problem is as follows:

Maximize Z = 12B + 10C + 8D subject to the following constraints:

7B + 4C + 9D <=(less than equal) 3000      Availability of fabric

1.5B + 2C + .5D <=(less than equal) 500 Hours available

3B + 2C + 1D <= (less than equal) 400   Units of packaging material

B, C, and D >= 0 (non-negative numbers)

Solution using excel solver is as follows:

The question is having the following contradicting statements:

"Each of these values can be adjusted through overtime or extra purchases."

"avoid purchasing more fabric or packaging material, and use all of the hours scheduled."

In case, hours scheduled needs to be used, profit per hour used in case of Bedspread is 12/1.5, in case of curtain is 10/2 and in case of dust ruffle is 8/.5

400 units of Dust ruffles results in profit of $3,200 but this require additional 600 yards of fabric, packaging material is fully utilized whereas 300 hours not utilized.

320 units of curtains results in profit of $3,200 but requires additional hours as well as additional packaging materials and fabric is not utilized fully.

267 units of bedspreads required to achieve the target profit of $3,200 which results in additional packaging material and under utilization of fabric and hours.

Solutions for target profit of $3,200 and Hours fully utilized are as follows (using excel solver)

Minimization of fabric is the objective in above case

Minimization of packaging materials is the objective in above case.

Decision Variables B C D Values 0 43 314 SumProdt Objective 12 10 8 2942 RHS Fabric contraint 7 4 9 2998 3000 Hours constraint 1.5 2 0.5 243 500 Packaging constraint 3 2 1 400 400

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