Answered Essay: EZ-Windows, Inc., manufactures replacement windows for the home remodeling business. In January, the company produced 15,000 windows and ended the month with 9,000 windows in inventory.

EZ-Windows, Inc., manufactures replacement windows for the home remodeling business. In January, the company produced 15,000 windows and ended the month with 9,000 windows in inventory. EZ-Windows’ management team would like to develop a production schedule for the next three months. A smooth production schedule is obviously desirable because it maintains the current workforce and provides a similar month-to-month operation. However, given the sales forecasts, the production capacities, and the storage capabilities as shown, the management team does not think a smooth production schedule with the same production quantity each month is possible.

Sales forecast Production capacity Storage capacity February 15,000 14,000 6,000 March 16,500 14,000 6,000 April 20,000 18,000 6,000

The company’s cost accounting department estimates that increasing production by one window from one month to the next will increase total costs by $1.00 for each unit increase in the production level. In addition, decreasing production by one unit from one month to the next will increase total costs by $0.65 for each unit decrease in the production level. Ignoring production and inventory carrying costs, formulate and solve a linear programming model that will minimize the cost of changing production levels while still satisfying the monthly sales forecasts. If required, round your answers to two decimal places.

Let:

F = number of windows manufactured in February

M = number of windows manufactured in March

A = number of windows manufactured in April

Im = increase in production level necessary during month m

Dm = decrease in production level necessary during month m

sm = ending inventory in month m

Min I1 + I2 + I3 + D1 + D2 + D3
s.t.
(1) F – s1 = February Demand
(2) s1 + M – s2 = March Demand
(3) s2 + A – s3 = April Demand
(4) F – I1 + D1 = Change in February Production
(5) M – F – I2 + D2 = Change in March Production
(6) A – M – I3 + D3 = Change in April Production
(7) F ≤ February Production Capacity
(8) M ≤ March Production Capacity
(9) A ≤ April Production Capacity
(10) s1 ≤ February Storage Capacity
(11) s2 ≤ March Storage Capacity
(12) s3 ≤ April Storage Capacity

If required, round your answers to the nearest dollar.

Cost: $

If required, round your answers to the nearest whole number.

February March April
Production Level
Increase in Production
Decrease in Production
Ending Inventory

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Expert Answer

 

Let F = number of windows manufactured in February
M = number of windows manufactured in March
A = number of windows manufactured in April
Im = increase in production level necessary during month m
Dm = decrease in production level necessary during month m
sm = ending inventory in month m

Min 1I1 + 1I2 + 1I3 + 0.65D1 + 0.65D2 + 0.65D3

9000 + F – s1 = 15,000February Demand

(1) F1 – s1 = 6000
(2) s1 + M – s2 = 16,500 March Demand
(3) s2 + A – s3 = 20,000 April Demand
F – 15,000 = I1 – D1 Change in February Production
(4) F – I1 + D1 = 15,000
M – F = I2 – D2 Change in March Production
(5) M – F – I2 + D2 = 0
A – M = I3 – D3 Change in April Production
(6) A – M – I3 + D3 = 0
(7) F ≤ 14,000 February Production Capacity
(8) M ≤ 14,000 March Production Capacity
(9) A ≤ 18,000 April Production Capacity
(10) s1 ≤ 6,000 February Storage Capacity
(11) s2 ≤ 6,000 March Storage Capacity
(12) s3 ≤ 6,000 April Storage Capacity

Optimal Solution: Cost = $6,450

February March April

Production Level 12,000 14,000 16,500

Increase in Production 0 2,000 2,500

Decrease in Production 3,000 0 0

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