Question

You are in the business of producing and selling snow shovels, and you need to determine how many shovels should be produced during each of the next four quarters to meet the following demands: 11,000 shovels in quarter 1; 48,000 shovels in quarter 2; 64,000 shovels in quarter 3; and 15,000 shovels in quarter 4.

Due to labor limitations, at most 65,000 shovels can be produced in any one quarter at a cost of $5/shovel. Additionally, a fixed cost of $30,000 must be paid for any quarter in which shovels are produced. You may assume that any shovels produced during a quarter can be used to satisfy demand for that quarter. At the end of the quarter, a holding cost of $0.50 per shovel in inventory is incurred. Currently, you have no shovels in inventory.

Required:
Formulate an integer-linear program to determine a production schedule that minimizes the sum of production and inventory costs over the next four quarters.

Answer 1

Quarter Production

Q1 11000

Q2 62000

Q3 65000

Q4 0

This will generate lower production and inventory cost as it savesthe fixed cost of 30,000 if we produce in the fourth quarter.

Step-by-step explanation:

First, we construct the formula for the relevant cost:

Holding Cost: $0.50 per shovel

$0.50 x 2 x (Q2-48,000) + $0.50 x (Q1-11,000) = Holding Cost Q2

$0.50 x 1 x (Q3-64,000) = Holding Cost Q3

First, the restrictions:

P1 P2 P3 P4 are Integer

P1 < 65,000

P2 < 65,000

P3 < 65,000

P4 < 65,000

Then, we have the inventory formulas:

I1 = P1 - S1

I2 = P2 + I1 -S2

I3 = P3 + I2 - S3

I4 = P4 + I3 - S4

The holding cost

H1 = I1 x 0.50

H2 = I2 x 0.50

H3 = I3 x 0.50

H4 = I4 x 0.50

The fixed cost

if P1> 0 then FC1 = 30,000

if P2> 0 then FC2 = 30,000

if P3> 0 then FC3 = 30,000

if P4> 0 then FC4 = 30,000

And last,the total cost:

FC1 + H1 +FC2 + H2 +FC3 + H3 +FC4 + H4 = Total Cost

This is the formula we want to minimize

We place this into excel solver and get the answer: