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## Chapter 4 Models for Known Demand

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**1. Price discounts from suppliers**• Variable costs • Valid total cost curve • Finding the lowest valid cost • Rising delivery cost • Summary**Variable costs**• In the last chapter we assumed that all costs are fixed - so they have constant, known values that never change. • In this chapter we start by seeing what happens when the costs vary with the quantity ordered. You can often see this with discounted unit prices, where a supplier quotes lower prices for larger orders. • A particular computer, for example, might cost £2,500, but this falls to £2,250 for orders of ten or more, and to £2,000 for orders of 50 or more.**Valid total cost curve**• The most common variation in cost occurs when a supplier offers a reduced price on all units for orders above a certain size. There is often more than one discounted price, giving the pattern of unit cost shown in Figure 4.1. The basic unit cost is UC1, but this reduces to UC2 for orders bigger than Qa, to UC3, for orders bigger than Qb, to UC3 for orders bigger than Qc, and so on.**If we look at the most expensive unit cost, UC1, we can draw**a graph of the total cost per unit time against the order size, as we did to find the economic order quantity. In this case, though, the curve will only be valid for order quantities in the range zero to Qa.**Finding the lowest valid cost**• The optimal value of Q that corresponds to the lowest point on the valid cost curve.**We can express the holding cost as a proportion, I, of the**unit cost, and for each unit cost UCi, the minimum point of the cost curve comes with Qoi. We also know that:**For each curve with unit cost UQ this minimum is either**“Valid” or “invalid”: • A valid minimum is within the range of valid order quantities for this particular unit cost. • An invalid minimum falls outside the valid order range for this particular unit cost.**Every set of cost curves will have at least one valid**minimum, and a variable number of invalid minima, as shown in Figure 4.4.**Two other interesting features in the valid cost curve.**First, the valid total cost curve always rises to the left of a valid minimum. This means that when we search for an overall minimum cost it is either at the valid minimum or somewhere to the right of it. Second, there are only two possible positions for the overall minimum cost: it is either at a valid minimum, or else at a cost break point (as shown in Fig. 4.5).**Rising delivery cost**• We can extend this method for considering unit costs that fall in discrete steps to any problem where there is a discrete change in cost.**Finite replenishment rate**Stock from production • If the rate of production is greater than demand, goods will accumulate at a finite rate - so there is not instantaneous replenishment, but a finite replenishment rate. • Goods only accumulate when the production rate is greater than demand.**If we call the rate of production P, stocks will build up at**a rate P - D, as shown in Figure 4.11. This increase will continue as long as production continues. This means that we have to make a decision at some point to stop production of this item - and presumably transfer facilities to making other items. The purpose of this analysis is to find the best time for this transfer, which is equivalent to finding an optimal batch size.**So we have replenishment at a rate P and demand at a rate D,**with stock growing at a rate P - D. After some time, PT, we decide to stop production. Then, stock is used to meet demand and declines at a rate D. After some further time, DT, all stock has been used and we must start production again. Figure 4.12 shows the resulting variation in stock level, where we assume there is an optimal value for PT (corresponding to an optimal batch size) that we always use.**Optimal batch size**• The overall approach of this analysis is the same as the economic order quantity, so we are going to find the total cost for one stock cycle, divide this total cost by the cycle length to get a cost per unit time, and then minimize the cost per unit time**Consider one cycle of the modified saw-tooth pattern shown**in Figure 4.13. If we make a batch of size Q, the maximum stock level with instantaneous replenishment would also be Q. • The maximum stock level is lower than Q and occurs at the point where production stops.**Looking at the productive part of cycle we have:**A = (P – T) × PT The total production during the period is : Q = P × PT or PT = Q/P