A case study - Brewery
A brewery management prepares a production plan for the next year. Malt - the basic raw material
for beer production will be produced in company's own malt-house and/or purchased at the market.
Forecasted malt need for respective quarters of the next year is as follows:
Period
1st quarter:
2nd quarter:
3rd quarter:
4th quarter:
Quantity
[tons]
200
220
250
160
Maximum production capacity of the own malt-house is 160 t per quarter.
Malt production costs depend on the malt quantity produced. In previous years the following
relationship was recorded:
Production
[tons]
x(i)
50
70
100
120
150
Costs
[Sk/ton]
c(i)
10000
8500
7500
8200
9500
Market price of the malt depends on the time of purchase. Estimated price in quarters is as follows:
Period
1st quarter:
2nd quarter:
3rd quarter:
4th quarter:
Price
[Sk/ton]
8300
8300
8200
8100
Limited budget of 2 500 000 Sk has been planned for malt purchase.
Formulate mathematical programming problem for the optimal plan of malt production/purchase
with the objective of minimal costs.
Problem formulation:
Variables:
x1
x2
x3
x4
- quantity of malt to be produced in the 1st quarter
- quantity of malt to be produced in the 2nd quarter
- quantity of malt to be produced in the 3rd quarter
- quantity of malt to be produced in the 4th quarter
y1
y2
y3
y4
- quantity of malt to be purchased in the 1st quarter
- quantity of malt to be purchased in the 2nd quarter
- quantity of malt to be purchased in the 3rd quarter
- quantity of malt to be purchased in the 4th quarter
Preparatory calculations:
Relationship between malt quantity production and production costs has been investigated. A
nonlinear relationship (parabola) has been used (see regression calculations in the file BreweryQPP.xls):
c = 16361,84 - 169.805x + 0.830489x2
Production costs [Sk/ton]
Production costs - production quantity relationship
10000
c = 0,8305x2 - 169,81x + 16362
9000
8000
7000
0
50
100
Production quantity [tons]
150
200
Nonlinear programming problem:
Objective function
min z
=
(16361.84 - 169.805x1 + 0.830489x12)x1 + (16361.84 - 169.805x2 + 0.830489x22)x2 +
(16361.84 - 169.805x3 + 0.830489x32)x3 + (16361.84 - 169.805x4 + 0.830489x42)x4
+ 8300y1 + 8300y2 + 8200y3 + 8100y4
Subject to
Malt need (production & purchase)
1st Q.
x1
2nd Q.
x2
3rd Q.
4th Q.
Malt-house production limits
1st Q.
x1
2nd Q.
x2
3rd Q.
4th Q.
Malt
purchase
+ y1
+ y2
x3
+ y3
x4
+ y4
x3
x4
+8300 y1
+ 8300y2 + 8200y3 + 8100y4
x1, x2, x3, x4, y1, y2, y3, y4
Problem has been solved by Excel's Sover - see file Brewery-QPP.xls.
Optimal solution is as follows:
x1 = 132.08
x2 = 132.08
x3 = 131.53
x4 = 130.97
- quantity of malt to be production in the 1st quarter
- quantity of malt to be production in the 2nd quarter
- quantity of malt to be production in the 3rd quarter
- quantity of malt to be production in the 4th quarter
y1 = 67.91
y2 = 87.91
y3 = 118.47
y4 = 29.03
- quantity of malt to be purchased in the 1st quarter
- quantity of malt to be purchased in the 2nd quarter
- quantity of malt to be purchased in the 3rd quarter
- quantity of malt to be purchased in the 4th quarter
Total production and purchase costs are: 6924975.96 Sk
≥
≥
≥
≥
200
220
250
160
≤
≤
≤
≤
160
160
160
160
≤
≥
2500000
0