Exam Inventory Management (35V5A8) Part 1: Theory (1 hour) Closed book: Inspection of book/lecture notes/personal notes is not allowed; computer (screen) is off. Maximum number of points for this part: 30. Answers can be given in Dutch or in English; formulate concisely. Make clear assumptions for results that you need to solve a problem if you were not able to obtain these results from an earlier part of the problem. (12) 1. (a) Under some conditions the (R, S)-control system and the (s, Q)-control system are equivalent. List these conditions. Solution: This question was skipped from the exam, because it is not clear enough. The following relationships between the parameters of the two models should hold: L (s,q) = L (R,S) + R, s = S, Q = E[D R ]. Furthermore, we have to neglect undershoots in the (s, Q)-control model (unit-sized or continuous demand). (b) Under which circumstances lead the (s, Q) and (s, S) inventory policies to the same ordering decisions? Solution: If Q = S s and demand is continuous or unit-sized. (c) Under what conditions do the net stock and inventory position never coincide in an (R, S) control policy. Ignore any initialization issues. Solution: If there is a positive demand during each review period and L > R. (6) 2. Describe the Silver-Meal heuristic for determining lot sizes in the case of time-varying deterministic demand. Give at least two reasons why the Silver-Meal heuristic is often preferred above the exact Wagner-Whitin method. Solution: See SPP and slides. 1. No fixed time horizon N required. 2. Solution does not change when used in a rolling-horizon environment. 3. Less computational effort. 4. More intuition. Exam Inventory Management January 15, 2010 Page 1 of 6
(6) 3. Explain what is meant by the undershoot in a (s, Q) inventory control system. Why is it convenient that the undershoot and the demand during lead time are independent? Under what circumstances is it reasonable to assume that they are independent? Solution: The undershoot is the amount with which the inventory position drops below the reorder point s as a result of a (customer) demand. The net stock just before delivery equals s U D L, where the undershoot U and demand during lead time D L are random variables. The analysis becomes much more complicated if U and D L are dependent. If the customer interarrival times exponentially distributed, then U and D L are independent. This is a reasonable assumption if there are many (potential) customers. (6) 4. A very generic version of the P 2 -service equation for the (s, Q)-control system with complete backordering is given by (1 P 2 )Q = ESPRC, with P 2 the service target, Q the order quantity and ESPRC the expected shortage per replenishment cycle. Argue why 1 P 2 should be replaced by (1 P 2 )/P 2 if we are dealing with complete lost sales. Solution: In the case of complete backordering the expected demand per RC, E[D RC ], is equal to Q. However, in the case of complete lost sales E[D RC ] = Q + ESPRC and the result follows if we replace Q by Q + ESPRC. Exam Inventory Management January 15, 2010 Page 2 of 6
Exam Inventory Management (35V5A8) Part 2: Practice (2 hours) Open book: Inspection of book and lecture/personal notes is allowed. Maximum number of points for this part: 50. Answers can be given in Dutch or in English; formulate concisely. Excel and MATLAB may be used. E-mail and Internet are not allowed with the exception of Blackboard. Motivate all your answers. If necessary indicate how you have obtained your results using the software. Make clear assumptions for results that you need to solve a problem if you were not able to obtain these results from an earlier part of the problem. (20) 1. A warehouse adopts a (R, S)-control policy where all items satisfy a P 2 -criterion with a target of 0.95. A group of items have the same supplier with a common lead time of 3 weeks. In order to obtain economy of scale effects, an equal length of the review period for each item is taken, namely 4 weeks, and all items are ordered on the same (review) moment. You may assume that a year contains exactly 52 weeks. A gamma distribution appropriately describes the demand process, however, the company s software uses a normal demand distribution. After consulting management it appears that the total number of stock-out occasions per year is an important performance indicator for the company. Information about these items is given in the following table, where D denotes the average yearly demand and σ the standard deviation of the yearly demand. Item D σ Value (e) 1 1600 150 70 2 1100 400 90 3 900 500 100 (a) Without computing anything, give your opinion on the warehouse s inventory policy with respect to the criterion used to determine order-up-to levels. Solution: Given the fact that ETSOPY is an important performance indicator it makes more sense to use a B 1 cost criterion, or to minimize the ETSOPY subject to a constraint on the total (safety) stock. Exam Inventory Management January 15, 2010 Page 3 of 6
(b) Determine the order-up-to levels (rounded to integer values) and total safety stock value (TSSV) according to the current policy. Use the formula S i = µ (i) R+L + SS i for the determination of the safety stocks. Hint: you have to use the MATLAB function RSnorP2. Solution: For all items compute µ (i) R = R/52 D i, σ (i) R = R/52σ i, ν (i) R = σ (i) R /µ (i) R. and evaluate ceil(rsnorp2(beta, nur(i), mur(i), 3/4)). Next, we compute SS i = S i µ (i) R+L T SSV = SS i v i = 54130 i (c) Determine the Expected Total Stockout Occasions Per Year (ETSOPY) of the current policy. Hint: first determine the implied P 1 service level. Which model for the demand distribution do you have to use? Solution: We compute P (i) = P (D (i) R+L > S i ) using a gamma distribution since this distribution appropriately describes the demand. The parameters for the gamma distribution are ρ = µ 2 R+L/σ 2 R+L and λ = µ R+L /σ 2 R+L. The result follows from gamcdf(s, rho, 1/lambda). Finally, ET SOP Y = i (1 P (i) )52/R = 4.38 Suppose that the warehouse changes its method for determining the order-up-to levels to the equal safety factors rule in such a way that the total safety stock value (TSSV) does not change. (d) Compute the Expected Total Stockout Occasions Per Year (ETSOPY) according to this equal safety factors rule. Solution: The equal safety factor that does not change TSSV follows from k = T SSV i σ (i) R+Lv i = 1.53 Next, we compute S i = µ (i) R+L + kσ (i) R+L and the implied P 1 service levels using the gamma distribution. Finally, ET SOP Y = 2.96 follows by using the same method as before. (30) 2. A company hires a consultant to reconsider the company s (s, Q)-control policy of one of its spare parts. The spare part is a critical component in one of the company s main Exam Inventory Management January 15, 2010 Page 4 of 6
production lines. The consultant inquires about the current policy and is provided with the following information. The order quantity is determined using the EOQ-formula with fixed ordering costs e 40 and carrying charge 0.2 (e/e/year). The value of the spare part is e 30 and the average annual demand equals 100 units. The item s lead time is 2 months, and the coefficient of variation of the demand during lead time equals ν = 1.2 (= σ L /µ L ). The company uses a P 2 service target of 99%. The reorder point s is determined using the standard text book P 2 -service equation given by G u (k) = Q σ L (1 P 2 ), where G u (k) is the (standard) normal loss function, k = (s µ L )/σ L is the safety factor and P 2 is the service target. (a) Compute the company s current order quantity and reorder point (both rounded). Solution: EOQ = 2AD/(vr) = 36.5 Q = 37. The reorder point can be computed by solving the service equation in MATLAB: s = loss_normal_inv((1-p2)*q, mul, sigmal) or s = mul + sigmal * loss_normal_inv((1-p2)*q/sigmal) where µ L = 2/12 100 and σ L = µ L ν. This rounded upwards this yields s = 51. After some analysis the consultant comes up with some criticism about the current approach. First of all, he questions the use of the normal distribution in this case, and argues that a gamma distribution would be a better choice. (b) Discuss why the gamma distribution is a better choice than the normal distribution in this case. What would the effect of this new approach be on the reorder point? Just indicate the direction of this effect (no computations required). Solution: The coefficient of variation is considerably larger than 0.5, which makes the normal distribution less appropriate because the probability of negative values. The gamma distribution does not have this drawback. The gamma distribution has a fatter tail distribution, which will result in a higher reorder point when the gamma distribution is used. Next, the consultant argues that current method does not correct for the expected shortage immediately after a replenishment and that this quantity is not insignificant even though a high service target is being used. (c) Explain why the mentioned correction could indeed be significant in this case, and discuss the expected effect of including this correction on the reorder point. Just indicate the direction of this effect (no computations required). Exam Inventory Management January 15, 2010 Page 5 of 6
Solution: The expected shortage after a replenishment is E[(D L s Q) + ]. Hence, if σ L (= 20) is relatively large compared to Q (= 37), then this quantity could be significant even though the service level is high. If the correction is neglected, then the expected shortage per RC is overestimated. Hence, including this correction results in a lower reorder point. (d) Compute another reorder point by taking into account the issues raised by the consultant. Solution: We have the use the gamma distribution and apply it in the service equation E[(D L s) + ] E[(D L s Q) + ] = Q/σ L (1 P 2 ). In Matlab, the reorder point s = 77 follows from s = ceil(sqgamp2(q, P2, mul, sigmal^2)) (e) Give a simple approximating formula for the average on-hand stock and compute corresponding value. Under what circumstances is this a reasonably accurate approximation? Solution: E[OH] = SS + Q/2 = s µ L + Q/2 = 78.8 is a reasonable approximation if the number of backorders is small compared to the average on-hand stock, i.e., if the service level is high. The consultant discovers that four identical components are required for the correct operation of a single production line. It appears that the mechanics who replace the spare parts usually replace all four components in a single production line as soon as one of them fails. The consultant computes the following relative frequency table for the individual demand observations over the last year. Components replaced 1 4 Frequency 0.1 0.9 (f) Use this information to compute the expectation and standard deviation of the undershoot. Solution: Let X denote the individual demands with the given discrete distribution. We can use the following approximations with corrections since X is a discrete distribution. E[U] = E[X2 ] 2E[X] 1 2 = 1.4595, [ E[X 3 ( ] E[X 2 ) 2 σ[u] = 3E[X] ] 1 ] 1/2 = 1.1293 2E[X] 12 Exam Inventory Management January 15, 2010 Page 6 of 6