Case Study: The Coke Supply Chain Problem

Submitted: 20 Feb 2008

Operations Research Topics: LinearProgramming, IntegerProgramming, MasterSlaveConstraints, FacilityLocationProblem

Application Areas: Logistics, Industrial Planning, Fuel Manufacturing

Contents

Problem Description

Adapted from a real-world problem

Over 29% of world's coke is made in China. Recently market liberalization has lead to town and village enterprises, with uncertainty in future markets resulting in short-sighted resource use. There are currently many small cokemaking plants with relatively primitive technology. This has resulted in an industry with low transportation costs (as coke is supplied by many small local producers) but high pollution and energy use.

Figure 1 Coke from en.wikipedia.org/wiki/Image:Koks_Brennstoff.jpg

800px-Koks_Brennstoff.jpg

The government wants to move cokemaking to Shanxi, where they can establish bigger facilities. This plan will be more efficient, with less pollution, but will also result in higher transport costs (calculated from a GIS database). The plants in Shanxi will be built with one or more oven batteries, each making 75,000 tonnes of coke a year. The process these plants will use to convert coal to coke is "thermal decomposition". This process results in 1 tonne of coke produced for every 1.3 tonnes of coal processed.

The (simplified) problem we will model has the following details:

  1. There are six coal mines, each able to supply up to a certain amount of coal each year to the plants for processing
  2. There are six customers, each with a certain demand for coke each year
  3. There are six possible plant locations, each able to process a certain amount of coke each year

coke_map.jpg

Each plant has 6 possible sizes, with coke processing levels and associated construction costs shown below:

Size Cost
(kT/yr) (MRMB)
75 4.4
150 7.4
225 10.5
300 13.5
375 16.5
450 19.6

Each Mine has the following coal supply per year:

Mine 1 2 3 4 5 6
Coal Supply (kT/yr) 25.8 728 1456 40 36.9 1100

Each Customer has the following coke demand per year:

Customer 1 2 3 4 5 6
Coke Demand (kT/yr) 83 5.5 6.975 5.5 720.75 5.5

The transportation costs (RMB/kT) between the mines and the plants are as follows:

RMB/kT Plant 1 Plant 2 Plant 3 Plant 4 Plant 5 Plant 6
Mine 1 231737 46813 79337 195845 103445 45186
Mine 2 179622 267996 117602 200298 128184 49046
Mine 3 45170 93159 156241 218655 103802 119616
Mine 4 149925 254305 76423 123534 151784 104081
Mine 5 152301 205126 24321 66187 195559 88979
Mine 6 223934 132391 51004 122329 222927 54357

The transportation costs (RMB/kT) between the plants and the customers are as follows:

RMB/kT Plant 1 Plant 2 Plant 3 Plant 4 Plant 5 Plant 6
Customer 1 6736 42658 70414 45170 184679 111569
Customer 2 217266 227190 249640 203029 153531 117487
Customer 3 35936 28768 126316 2498 130317 74034
Customer 4 73446 52077 108368 75011 49827 62850
Customer 5 174664 177461 151589 153300 59916 135162
Customer 6 186302 189099 147026 164938 149836 286307

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Extra for Experts

This problem is similar to the Cosmic Computers Problem, but the facility location is a little different. Rather than face a decision about whether to build a plant or not, we need to determine what type of plant to build. This gives rise to the following variable:

\[ y_{ps} = 1 \text{ if plant $p$ is built with size $s$, 0 otherwise} \]

While solving the Coke Supply Chain Problem, the expression

\[ \sum_{s \in \text{sizes}} y_{ps} \]
will often be fractional. This means that the capacity of the plant
\[ \sum_{s \in \text{sizes}} C_s y_{ps} \]
where $C_s$ is the capacity of a plant of size $s$ will be between the capacities of two different size plants, e.g., 280 kT/yr is between 225 kT/yr plant and a 300 kT/yr plant. We could use constraints to remove the fractionality, e.g.,
 \[ \begin{array}{r@{\,}l@{\quad\quad}l} \displaystyle \sum_{s \in \text{sizes}} C_s y_{ps} &\leq 225 & \text{(branching down)} \\ \displaystyle \sum_{s \in \text{sizes}} C_s y_{ps} &\geq 300 & \text{(branching up)} \end{array} \]

These branches will ensure plants have the right capacities, but will still not resolve fractionalities in the $y_{ps}$. However, if we want the capacity of plant $p$ to be $\leq$ 225, then we can remove all the $y_{ps}$ that allow for the capacity to be larger than 225:

\[ \sum_{s \in \text{sizes} | C_s > 225} y_{ps} = 0 \]

Conversely, if we want to branch up, i.e., the capacity is $\geq$ 300, then we can remove all the $y_{ps}$ that allow the capacity to be smaller than 300:

\[ \sum_{s \in \text{sizes} | C_s < 300} y_{ps} = 0 \]

Note For these branches to work properly, the possibility of building no plant must be modelled as building a plant with no capacity (i.e., $C_s$ = 0) for no cost.

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Student Tasks

  1. Write AMPL model, data and script files (coke.mod, coke.dat and coke.run respectively) to solve the Coke Production Problem. Write a management summary for your solution.

    What to hand in Your 3 AMPL files. Your management summary.

  2. Experts Only Solve the LP relaxation of the Coke Supply Chain Problem integer programme. Using the branches described in Extra for Experts, use AMPL to perform branching on at least 4 nodes and draw the resulting branch-and-bound tree.

    Hint See Extra for Experts in the Cosmic ComputersProblem for an example of constraint branching in AMPL.

    What to hand in Your AMPL code for implementing the branches. Your drawing of your branch-and-bound tree.

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