Below are the details of your coursework assignment. Note that this covers optimisation as well as Monte-Carlo simulation.

You need to submit a short report (less than 5 pages including a cover sheet), together with the spreadsheet you have used to carry out your calculations. Please ensure that the report can be read on its own, without reference to the spreadsheet.

WinStone Tiles supplies customers in the building products industry with a variety of imported stone tiles from three warehouses.

Context

All tiles are sold in packs of eight. New stock is delivered to the warehouses every two weeks. It is possible midway through this period – so with one more week of demand to be met – to move stock between warehouses to rebalance. Transport costs are expensive, but the costs of running out are also high. Generally, WinStone Tiles will move stock around if one of the warehouses has lower stock than the expected demand over the week for some product. Your task is to find an optimum arrangement of stock movement between warehouses and to estimate the expected costs given your transport plan.

It is not economic to arrange delivery of items from one warehouse to customers usually served by another, so the rebalancing has to be done in advance of knowing what the actual demand is going to be.

If demand cannot be met from stock, then there is a cost: the customer is contacted individually and a special delivery organised, but there is also some reputational damage. If there is a shortfall of x packs of tiles then the cost is estimated to be £10x.

Transport costs involve a fixed cost plus an amount that depends on the route and the number of packs transferred. The cost for the transfer of x packs from warehouse 1 to warehouse 2 (or vice versa) is £(20+5x). The cost for the transfer of x packs from warehouse 1 to warehouse 3, or vice versa is £(20+7x). The cost for the transfer of x packs from warehouse 2 to warehouse 3, or vice versa is £(20+6x). Costs are zero if there is no transfer.

The average weekly level of demand for each type of tile at the three warehouses has been calculated over the last six months and is given by the following table (in packs).

Consider a decision on rebalancing, taken one week before the delivery is made, with inventory levels (in packs) at the three warehouses as shown in the table below.

Part 1 (40%)

Assuming there is no uncertainty in demand – so that the demand over a week is always at its average level – calculate the optimum choice of transfer amounts between each of the three pairs of warehouses in order to minimise the sum of transport and stockout costs. You should assume that transfer amounts are in whole numbers of packs.

There are several different ways to set up a spreadsheet for this problem, but some approaches will cause difficulties for the solver. I recommend:

1. Consider moves from 1 to 2 and from 2 to 1 separately (and similarly for the other pairs of warehouses) rather than allowing negative transfers.
2. Use the standard approach to decision variables for which there is fixed cost for a non-zero (this will involve a set of binary decision variables and a constraint that the amount transferred is less than M times this indicator variable for some choice of large M.)
3. Your solution will probably involve a nonlinear function in order to have a penalty for a stockout without giving a benefit for positive inventory. This will make it appropriate to use the evolutionary solver. Depending on how this is set up, the solver may not find it easy to find the optimal solution; I suggest running some checks (for example try the nonlinear solver from multiple starting points)

In your report you need to be explicit about exactly how the optimisation problem has been formulated, and the method you have used. We want to be able to give credit for the approach you take even if the answer you get is wrong

Part 2 (30%)

Suppose that actual demand for tiles at each warehouse varies around its average level according to the following table, which shows the probabilities of weekly demand variation around the average number.

Thus, there is a probability of 0.07 that the demand for Product A at warehouse 1 is 120, a probability of 0.24 that the demand at warehouse 1 is 110, a probability of 0.38 that it is 100, etc. Demand variation at one warehouse does not have an impact on the variation at another.

Calculate the optimum choice of transfer amounts between each of the three pairs of warehouses to minimise the sum of transport costs and expected stockout costs. Again, you should assume that transfer amounts are in whole numbers of packs.

HINT: You need to calculate the expected stockout costs by looking at the expected stockout costs for each warehouse separately and then adding these together. Dealing with each warehouse separately avoids you having to consider combinations of, for example, demand at warehouse 1 is 10% down, demand at warehouse 2 is 20% up, and demand at warehouse 3 is as expected. You do not need Monte Carlo for this part of the coursework.

In your report you need to explain how the optimisation problem has been set up.

Part 3 (30%)

Given your optimal choice of transfer amounts from Part 2, assume that the actual demand distribution for each warehouse has a normal distribution with standard deviation of 10, but with numbers rounded to integer numbers of packs. The demand at different warehouses is independent. Use a Monte Carlo simulation with 5000 experiments to estimate the distribution of stockout costs. What is the average stockout cost from your simulation? What is the 95th percentile of stockout costs from your simulation?

In your report you need to describe how the simulation is set up as well as giving your answers. We want to be able to give credit for the approach you take even if the answer you get is wrong.

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