# FORMULATION: Bend Unit 2 (4) Zimmer Unit 1

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.

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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,

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

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

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