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 malthouse and/or purchased at the market.
Forecasted malt need for respective quarters of the next year is as
follows:

[tons] 








Maximum production capacity of the own malthouse is 160 t per quarter.
Malt production costs depend on the malt quantity produced. In previous
years the following relationship was recorded:
[tons] 
[Sk/ton] 


50

10000

70

8500

100

7500

120

8200

150

9500

Market price of the malt depends on the time of purchase. Estimated
price in quarters is as follows:

[Sk/ton] 

8300


8300


8200


8100

Limmited 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:
x_{1 } quantity of malt to be produced in the 1st quarter
x_{2 } quantity of malt to be produced in the 2nd quarter
x_{3 } quantity of malt to be produced in the 3rd quarter
x_{4 } quantity of malt to be produced in the 4th quarter
y_{1}  quantity of malt to be purchased in the 1st quarter
y_{2}  quantity of malt to be purchased in the 2nd quarter
y_{3}  quantity of malt to be purchased in the 3rd quarter
y_{4}  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 + 0.830489x^{2}
Quadratic programming problem:
Objective function
min z = (16361,84  169.805x_{1} + 0.830489x_{1}^{2})x_{1 }+ (16361,84  169.805x_{2} + 0.830489x_{2}^{2})x_{2 }+ (16361,84  169.805x_{3} + 0.830489x_{3}^{2})x_{3 }+ (16361,84  169.805x_{4} + 0.830489x_{4}^{2})x_{4 }+ 8300y_{1} + 8300y_{2} + 8200y_{3} + 8100y_{4}
Malt need (production & purchase)  
1st Q. 
x_{1}

+ y_{1}


200


2nd Q. 
x_{2}

+ y_{2}


220


3rd Q. 
x_{3}

+ y_{3}


250


4th Q. 
x_{4}

+ y_{4}


160


Malthouse production limits  
1st Q. 
x_{1}


160


2nd Q. 
x_{2}


160


3rd Q. 
x_{3}


160


4th Q. 
x_{4}


160


Malt purchase 
+8300 y_{1}

+ 8300y_{2}

+ 8200y_{3}

+ 8100y_{4}


2500000


x_{1}, x_{2},
x_{3}, x_{4}, y_{1}, y_{2}, y_{3},
y_{4}


0

Problem has been solved by Excel's Sover  see file BreweryQPP.xls.
Optimal solution is as follows:
x_{1} = 132.08  quantity of malt to be production in the 1st
quarter
x_{2 }= 132.08  quantity of malt to be production in the 2nd
quarter
x_{3 }= 131.53  quantity of malt to be production in the 3rd
quarter
x_{4 }= 130.97  quantity of malt to be production in the 4th
quarter
y_{1} = 67.91  quantity of malt to be purchased in the 1st
quarter
y_{2} = 87.91  quantity of malt to be purchased in the 2nd
quarter
y_{3} = 118.47  quantity of malt to be purchased in the 3rd
quarter
y_{4} = 29.03  quantity of malt to be purchased in the 4th
quarter
Total production and purchase costs are: 6924975,96 Sk