A Linear Programming problem can be formulated to determine how much coal to purchase from each of the mining companies and how it should be allocated to the generating units so as to minimize the total cost.
Let Xij = Tons of coal purchased from supplier i and used by generating unit j.
As there are 7 suppliers and 5 generating units,
i=1(RAG), 2 (Peaboy), 3(American),4(Consol),5(Cyprus),6(Addington),7(Waterloo)
j=1(MF Unit 5), 2(MF Unit 7), 3(BJ unit 1), 4(EB Unit 2), 5(Zimmer Unit1)
Thus, we have total 35 decision variables as shown in table below:
Decision variables (Amount of coal shipped from I to j)
Miami Fort Unit 5 (1)
Miami Fort Unit 7 (2)
Beckjord Unit 1 (3)
East Bend Unit 2 (4)
Zimmer Unit 1 (5)
Objective function would be to minimize total cost of buying and burning coal.
Objective function coefficient Cij = Cost of buying coal from supplier i +Cost of shipping Xij units form supplier I to generating unit j + cost of burning the coal at generating unit j.
For example: fir X11, C11 =22+5+10 =37
The following table shows values of objective function coefficients for all the decision variables.
Miami Fort Unit 5
Miami Fort Unit 7
East Bend Unit 2
Zimmer Unit 1
Final objective function:Minimize Z= 37X11 + 39.75X21 + 38 X31 + 45.25X41 + 50X51 + 38.25X61 + 46X71
+ 37X12 + 39.75X22 + 38 X32 + 45.25X42 + 50X52 + 38.25X62 + 46X72
+ 36.75X13 + 40.50X23 + 39.75X33 + 45.85X43 + 49.75X53 + 39X63 + 45.6X73
+ 32X14 + 35.75X24 + 34X34 + 42.25X44 + 45X54 + 37.25X64 + 42X74
+ 32.75X15 + 36.5X25 + 33.75X35 + 41.85X45 + 45.75X55 + 37X65 + 43.6X75
There are two types of constraints: supply constraints and demand constraints.
The supply constraints limit the amount of coal that can be bought under the various contracts.
There are 7 suppliers so there are 7 supply constraints:
First three constraints are for the suppliers with fixed tonnage contract. Thus the constraint inequalities are =.
X11+X12+X13+X14+X15 =350,000 (RAG)
Last 4 constraints are for the suppliers with variable tonnage contract. This means that the maximum amount purchased is the amount specified in the contract. Thus, these constraints have