PAS365/PAS6022 Industrial Statistical Modelling Stock Control Problems

Transcription

1 PAS365/PAS60 Industrial Statistical Modelling Stock Control Problems 1. In the deterministic stock control model which leads to the Wilson- Harris formula (constant demand D per year, holding cost H per unit per year and re-order cost I), suppose that, following an order for a lot of size q, stock is replenished steadily, rather than instantaneously, at a constant rate p per year, where p > D, until delivery is completed. If no shortages are permitted, show that the total yearly cost when lot size q is used is Hq ( 1 D ) + DI p q and deduce that the optimal lot size is DI ( ) H 1 D p Discuss the sensitivity of the total cost to using a sub-optimal lot size q.. In the deterministic model with constant demand D per year, holding cost H per unit per year and re-order cost I, suppose that shortages are permitted and that shortages cost K per unit per year. If S is the maximum stock level and q the lot size (with instantaneous delivery), show that the cost per unit time may be written ( H + K C(q, S) = q ) ( S 1 Kq ) +( H + K. HKq (H + K) ) DI DIHK + q H + K. Deduce the optimal S (for given q) and the optimal q from this. How sensitive is the cost to sub-optimal S (for given q)? What is the effect if stocks are replenished steadily at constant rate p per year? 3. Suppose that, in the standard framework, there is some economy in having larger orders, resulting in the re-order cost being I 1 if q < q 0 and I if q q 0, where I < I 1 and q 0 is given. What lot size should be used? In particular if D = 500, H =, I 1 = 30 and I = 0, investigate how the optimal lot size varies with q 0. 1

2 4. Four different items are kept in store for continuous use in a manufacturing process. Shortage is not permitted and stock may be replenished instantaneously following an order. The demand for each item is steady. The various costs and demands are tabulated below. Item i I i H i D i Find the economic lot sizes for the four items when the total number of orders per year is constrained not to exceed 10 (averaged over many years). 5. A rural motor dealer stocks two kinds of vehicle: cars and tractors. Each car in his showroom ties up 5,000 in capital and each tractor 1,500. Demand for each kind of vehicle is steady at 15 cars per month and 8 tractors per month. In pounds, storage costs, H, per vehicle per month and re-order costs, I, for new stock (regardless of quantity) are H I Cars 0 75 Tractors New deliveries are essentially instantaneous. If the dealer wishes to have no more than 10,000 tied up in capital at any one time, what, from the point of view of his stock system, are the best order quantities for cars and tractors? 6. The daily demand for bags of cement at a builder s merchant are independent Poisson random variables with mean 6. It costs 5p per day per bag to store the cement, and each new delivery from the cement works incurs a handling charge of 15. If the lead time is 3 days, what buffer stock will be needed to ensure that the chance of a stockout is

3 less than 0.01 and, if the Wilson-Harris order quantity is used, what is the corresponding average daily cost of the system? Suppose instead that the merchant decides to avoid backlogging by purchasing bags which may be delivered immediately, at a handling charge of per bag, as necessary whenever a stockout occurs. What buffer stocks should he now hold? What is the average daily cost in this case? 7. Consider a stochastic (s, q) lot-size/re-order-point model with holding cost H per year, re-order cost I, shortage cost K per item (regardless of the duration of the shortage), random lead-time demand (L) and expected demand per year D. If H = 1, I = 10, K = 5, D = 100 and (L) follows a normal distribution N(0, 16), find the optimal values of s and q, assuming (a) backorders are permitted, (b) backorders are lost. Compare the costs with those incurred if instead the system were operated with a lot size given by the Wilson-Harris formula and re-order point s given by the formula s = E( (L)) + B, where B is a buffer stock chosen to reduce the chance of lead-time demand exceeding s to A confectioner buys Easter eggs from a wholesaler for 1.05 each and sells them for 1.40 each. He is fairly certain that after Easter he will be able to sell off at half price any eggs that have not been sold earlier. If he has the opportunity to place only one order with the wholesaler, find the number he should order to maximise expected profit, and the expected profit, if D, the pre-easter demand, has Poisson distribution with mean 30. The confectioner realises that it might be more profitable to try to sell off any surplus after Easter initially at a price of 1 before lowering his price to clear the stock. If demand after Easter when the price is 1 is D, which is independent of D with Poisson distribution, mean 10, and he orders q eggs, find an expression for his net profit in terms of q, D and D. Hence find the optimal q, and the maximised expected profit. 3

4 9. A supermarket must decide how much French bread to order each day. For shopping days Monday to Friday experience has shown that daily demand follows approximately a N(300, 500) distribution. A loaf of bread sells for 35p and costs the supermarket 9p. A loaf not sold by the end of the day is placed on a separate shelf the following day for sale at 0p. All bread on this shelf is invariably sold. What is the optimal number of loaves for the supermarket to order? If the selling price of fresh bread could be set at any level s pence per loaf and if the relationship between expected demand and s were known (all other conditions remaining as before), indicate, without detailed calculations, how the best selling price and the optimal number to order could be found. 10. An amateur toy maker has in stock 100 metres of 1 metre wide plywood suitable for making jigsaw puzzles. He has the choice of making puzzles of sizes 1 metre square, 100cm 50cm or 50cm 50cm. The selling prices of these three sizes are 3.50, 1.50 and 50p respectively. The pre- Christmas demands for the three sizes of jigsaw can be approximately described as independent exponential random variables with means 40, 50 and 100 respectively. Unsold puzzles will be donated to charity. How should the toy maker divide his wood so as to maximise his expected profit? If he can buy more wood at k per square metre, how much should he buy and what is the maximum price per square metre that he should be prepared to pay? 11. A dealer in musical instruments has the opportunity to buy a consignment of harpsichords at 750 each. He plans to sell them at 1,000 each and any that remain after six months he will return to the manufacturer at a loss of 0% of the purchase price. Never having dealt in harpsichords before, he guesses that the demand for them over the next six months could equally likely be anything from zero to two dozen. How many harpsichords should the dealer buy? Before placing his order he seeks advice from experienced harpsichord specialists who assure him that in his area the demand is likely to be adequately described by a Poisson random variable with mean 1. How valuable is this information to the dealer? 4

5 1. Consider a stochastic (τ, S) periodic review model with penalty cost per item K, review and re-order cost I, holding cost H, random lead time and demand, with expected demand per unit time D. In the model above, suppose that H = 1, I = 100, K = 50, D = 10, that the demand is deterministic and the random lead time L is distributed as N(, 0.16). Find the optimal values for the review period and order level, τ and S, assuming that backorders are permitted, but that shortage costs depend only on the number of items short, and not on the duration of the shortage. 5