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Wishing you a merry Christmas (with MP)

As you sit down to eat your Christmas dinner, have you considered how the fare came to be on your table? Not where you bought it or how it was cooked, but how it came to be in the shops at the right time?

Supplying Perishable Goods for Christmas

Just as there is the August bulge in car sales for the new registration letter, so there is an enormous bulge in the sales of fresh turkeys and brussels sprouts in the week before Christmas.

But whereas cars can be stored, as indeed can frozen turkeys and frozen brussels sprouts, this is not an option with fresh produce. Of course, there are problems with supplying stored commodities, but these are straightforward compared with those of supplying perishable commodities.

Turkeys . . .

Vegetarians may wish to skip this and the next two paragraphs, after which they can pick up the story with their brussels sprouts. From a farmer's point of view, turkeys are machines which convert their food into body weight. The more you feed them, the more body weight they put on. To a first approximation, there is some minimum feed rate which they need to remain at constant weight, but above this a fixed proportion of additional feed goes straight through to increased body weight, up to some maximum feeding rate which depends on their size.

The market demands turkeys of a variety of different sizes. Demand can be predicted for each of the sizes in the days leading up to Christmas. Given a flock of birds of known current weights, decisions must be taken as to how much to feed them to bring them to the required sizes at the specified dates. As bigger birds cost more to maintain than smaller ones, there is a tendency to fatten birds just before killing them (and, indeed, just afterwards, with the help of water and polyphosphates!).

Another very important consideration is the capacity of the processing facilities. These run flat out in the run-up to Christmas. However, this does not mean that the number of fresh turkeys available each day corresponds to the capacity of the processing facilities. Turkeys are sold as "fresh" either chilled or hanging, and in either case a number of days may have passed between killing and purchase. This provides some extra flexibility in the system, and therefore opportunities for optimization (to the suppliers' benefit, not your taste buds').

and Brussels Sprouts

Now for the sprouts. The problem here is in many ways more difficult. Whereas the farmer can control the growth of turkeys, it is the weather which controls the growth of crops; and this is uncertain. Farmers must sow their sprouts in the Spring and then wait. If there is insufficient rain, they can irrigate, but they cannot prevent the plants from growing too big too soon.

The main decisions to be taken are which variety to plant, the spacing and the date of sowing or planting out. Varieties differ in their yield, their response to adverse weather and the numbers and sizes of their buttons. (They also differ in their flavour, but that is not usually a consideration!) Spacing affects both total yield and the sizes of buttons produced: the closer the plants are together, the smaller each plant will be but the greater the crop from a given area. Date of planting affects yield because the longer the plants have to grow, the larger they become. However, the earlier they are planted, the greater the risk of damage or loss from late frosts.

Decisions about planting are also related to decisions about how the sprouts are to be sold. Farmers may contract with supermarkets to provide specified tonnages of sprouts of given sizes over particular periods. The tighter the specification agreed to, the higher the guaranteed price. In practice, a farmer will feel the need to plant a greater "safety margin" the tighter his contractual obligations and he will dispose of these surplus sprouts at market prices when the time comes.

All of these factors must therefore be taken into account. A Linear Programming model can be built which represents the options and finds the best solution. If the model incorporates only a single 'central' forecast of the weather, the result will often be that a single variety is selected and it is recommended to be planted as soon as the final frosts in the weather forecast are "known" to have finished.

A more robust solution can be obtained by having a number of scenarios for the weather, together with the probabilities of their happening. A single set of principal decision variables is used but with separate variables to represent the outcome in each scenario. The economic returns are calculated in each scenario and these are then aggregated to give the expected return. The model will prefer to use a mix of varieties and planting dates in order to be sure of meeting contractual demand under the range of possible weathers rather than generating the best return under one.

Sadly, the thread which is common to these applications is the unimportance of taste. This is not because mathematical programming cannot handle taste (it is used in blending problems in the food industry) but because it is irrelevant to the decisions here. C'est la vie. Bon appetit!

Related articles include Farm Management by MP and The Travelling Milk Collector's Problem. To find other articles, refer to the MP in Action page.