Quantitative Methods

(See attached file for full problem description)

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Case Problem 2

Cinergy Corporation manufactures and distributes electricity for customers located in Indiana, Kentucky, and Ohio. The company spends $725 to $750 million each year for the fuel needed to operate its coal-fired and gas-fired power plants; 92% to 95% of the fuel used is coal. Cinergy uses 10 coal-burning generating plants: five located inland and five located on the Ohio River. Some plants have more than one generating unit. As the seventh-largest coal-burning utility in the United States, Cinergy uses 28-29 million tons of coal per year at a cost of approximately $2 million every day.
The company purchases coal using fixed-tonnage or variable-tonnage contracts from mines in Indiana (49%), West Virginia (20%), Ohio (12%), Kentucky (11%), Illinois (5%), and Pennsylvania (3%). The company must purchase all of the coal contracted for on fixed-tonnage contracts, but on variable-tonnage contracts it can purchase varying amounts up to the limit specified in the contract. The coal is shipped from the mines to Cinergy's generating facilities in Ohio, Kentucky, and Indiana. The cost of coal varies from $19 to $35 dollars per ton and transportation/delivery charges range from $1.50 to $5.00 per ton.
A model is used to determine the megawatt hours (mWh) of electricity that each generating unit is expected to produce and to provide a measure of each generating unit's efficiency, referred to as the heat rate. The heat rate is the total BTU's required to produce 1-kilowatt hour (kWh) of electrical power.

Coal Allocation Model

Cinergy uses a linear programming model, called the coal allocation model, to allocate coal to its generating facilities. The objective of the coal allocation model is to determine the lowest-cost method for purchasing and distributing coal to the generating units. The supply/availability of the coal is determined by the contracts with the various mines, and the demand for coal at the generating units is determined indirectly by the megawatt hours of electricity each unit must produce.
The cost to process coal, called the add-on cost, depends upon the characteristics of the coal (moisture content, ash content, BTU content, sulfur content, and grindability) and the efficiency of the generating unit. The add-on cost plus the transportation cost are added to the purchase cost of the coal to determine the total cost to purchase and use the coal.

Current Problem

Cinergy signed three fixed-tonnage contracts and four variable-tonnage contracts. The company would like to determine the least cost way to allocate the coal available through these contracts to five generating units. The relevant data for the three fixed-tonnage contracts are as follows:

Supplier Number of Tons
Contracted For Cost $/ton BTUs/lb
RAG 350,000 22 13,000
Peabody Coal Sales 300,000 26 13,300
American Coal Sales 275,000 22 12,600

For example, the contract signed with RAG requires Cinergy to purchase 350,000 tons of coal at a price of $22 per ton; each pound of this particular coal provides 13,000 BTUs.
The data for the four variable-tonnage contracts follow:

Supplier Number of Tons
Available Cost $/ton BTUs/lb
Consol, Inc. 200,000 32 12,250
Cyprus Amax 175,000 35 12,000
Addington Mining 200,000 31 12,000
Waterloo 180,000 33 11,300

For example, the contract with Consol, Inc., enables Cinergy to purchase up to 200,000 tons of coal at a cost of $32 per ton; each pound of this coal provides 12,250 BTUs.
The number of megawatt hours of electricity that each generating unit must produce and the heat rate provided are as follows:

Generating Unit Electricity Produced (mWh) Heat Rate (BTUs per kWh)
Miami Fort Unit 5 550,000 10,500
Miami Fort Unit 7 500,000 10,200
Beckjord Unit 1 650,000 10,100
East Bend Unit 2 750,000 10,000
Zimmer Unit 1 1,100,000 10,000

For example, Miami Fort Unit 5 must produce 550,000 megawatt hours of electricity, and 10,500 BTUs are needed to produce each kilowatt hour.

The transportation cost and the add-on cost in dollars per ton are shown here:

Transportation Cost ($/ton)
Miami Fort
Unit 5 Miami Fort
Unit 7 Beckjord
Unit 1 East Bend
Unit 2 Zimmer
Unit 1
RAG 5.00 5.00 4.75 5.00 4.75
Peabody 3.75 3.75 3.5 3.75 3.5
American 3.00 3.00 2.75 3.00 2.75
Consol 3.25 3.25 2.85 3.25 2.85
Cyprus 5.00 5.00 4.75 5.00 4.75
Addington 2.25 2.25 2.00 2.25 2.00
Waterloo 2.00 2.00 1.60 2.00 1.60

Add-On Cost ($/ton)
Miami Fort
Unit 5 Miami Fort
Unit 7 Beckjord
Unit 1 East Bend
Unit 2 Zimmer
Unit 1
RAG 10.00 10.00 10.00 5.00 6.00
Peabody 10.00 10.00 11.00 6.00 7.00
American 13.00 13.00 15.00 9.00 9.00
Consol 10.00 10.00 11.00 7.00 7.00
Cyprus 10.00 10.00 10.00 5.00 6.00
Addington 5.00 5.00 6.00 4.00 4.00
Waterloo 11.00 11.00 11.00 7.00 9.00

Managerial Report

Prepare a report that summarizes your recommendations regarding Cinergy's coal allocation problem. Be sure to include information and analysis for the following issues.

1. Determine how much coal to purchase from each of the mining companies and how it should be allocated to the generating units. What is the cost to purchase, deliver, and process the coal?
2. Compute the average cost of coal in cents per million BTUs for each generating unit (a measure of the cost of fuel for the generating units).
3. Compute the average number of BTUs per pound of coal received at each generating unit ( a measure of the energy efficiency of the coal received at each unit).
4. Suppose that Cinergy can purchase an additional 80,000 tons of coal from American Coal Sales as an "all or nothing deal", for $30 per ton. Should Cinergy purchase the additional 80,000 tons of coal?
5. Suppose that Cinergy learns that the energy content of the coal from Cyprus Amax is actually 13,000 BTUs per pound. Should Cinergy revise its procurement plan?
6. Cinergy has learned from its trading group that Cinergy can sell 50,000 megawatt hours of electricity over the grid (to other electricity suppliers) at a price of $30 per megawatt hour. Should Cinergy sell the electricity? If so, which generating units should produce the additional electricity?
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