FORMULATION:
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.
DECISION VARIABLES:
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)
RAG(1)
X11
x12
X13
X14
X15
Peabody(2)
X21
X22
X23
X24
X25
American(3)
X31
X32
X33
X34
X35
Consol(4)
X41
X42
X43
X44
X45
Cyprus(5)
X51
X52
X53
X54
X55
Addington (6)
X61
X62
X63
X64
X65
Waterloo(7)
X71
X72
X73
X74
X75
OBJECTIVE FUNCTION:
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
BeckjordUnit 1
East Bend Unit 2
Zimmer Unit 1
RAG
37.00
37.00
36.75
32.00
32.75
Peabody
39.75
39.75
40.50
35.75
36.50
American
38.00
38.00
39.75
34.00
33.75
Consol
45.25
45.25
45.85
42.25
41.85
Cyprus
50.00
50.00
49.75
45.00
45.75
Addington
38.25
38.25
39.00
37.25
37.00
Waterloo
46.00
46.00
45.60
42.00
43.60
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
CONSTRAINTS:
There are two types of constraints: supply constraints and demand constraints.
SUPPLY 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)
X21+X22+X23+X24+X25=300,000 (Peabody
X31+X32+X33+X34+X35=275,000 (American)
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
