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Page 1 of 6

Exam Inventory & Production Management

Course35V3A4 June 3, 2016

Part 1: Closed book

Total 30 points

You have ± 45 minutes to answer the following questions. Motivate all your answers short, clear and to the point. Please write readable.

Question 1 (5 points)

a) Why is Simple Exponential Smoothing (SES) not a suitable forecasting technique for intermittent demand patterns?

b) Explain what is meant by the statement “a forecasting system should balance the costs of forecasting against the costs of forecasting errors”.

c) The variance of the forecast error of demand during lead time is an important parameter in many inventory models. This variance is often estimated by extrapolation: the estimate for the one-period-ahead forecast error variance is multiplied by the length of the lead time. This is often an underestimate of the true variance, even when the lead time is deterministic. Explain why.

Question 2 (4 points)

For each of the following inventory control models give the expression of the expected shortage at the start of the replenishment cycle. Give the expected value in terms of the random variables 𝐷𝐿 (lead-time demand), 𝐷𝑅 (demand during review period), 𝐷𝑅+𝐿 (demand during review period and lead time), 𝑈 (undershoot) and the control

parameters 𝑅, 𝑠, 𝑄 and 𝑆. Do not make assumptions about probability distributions and clearly distinguish between 𝑠 and 𝑆!

a) (𝑅, 𝑆) b) (𝑠, 𝑄) c) (𝑠, 𝑆) d) (𝑅, 𝑠, 𝑆)

Question 3 (6 points)

a) A company wants to minimize the sum of the holding costs and stockout costs for an item. The costs per stockout occasion are estimated to be in the range of 10 to 20 euro. A manager considers analyzing the financial records of the company to come up with a more accurate estimate of the stockout costs. Explain how the percentage cost penalty function can be used to assess the added value of a more accurate estimate of the stockout costs.

b) Suppose that a company sets the reorder point of a certain item according to a cycle service level target of 95%. Explain what the implied stockout cost per unit short means in this context.

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Page 2 of 6 Question 4 (5 points)

Discuss the major shortcomings of Manufacturing Resources Planning (MRP).

Question 5 (5 points)

In a well-managed JIT system, all inventories will be eliminated; there will be no raw materials inventory, no work-in-process inventories for any production period and no finished good inventory. Do you agree with this statement? Justify briefly your answer, explain.

Question 6 (5 points)

Define Drum-Buffer-Rope mechanism in manufacturing run under TOC concept; explain the benefits and implementation difficulties.

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Page 3 of 6

Part 2: Open book

Total 40 points

Hand in your answers to Part 1 before starting Part 2!

You have ± 135 minutes to answer the following questions. Motivate all your answers short, clear and to the point. Please write readable.

Question 7 (10 points)

A touch screen LCD manufacturer produces three types of screens for smartphone companies. The information concerning the products and production process is given in the following tables. Using the Theory of Constraint, find out the bottleneck and

determine how many units of each screen must be produced to maximize the earning.

Products 6 inch Screen 5.5 inch Screen 5 inch Screen

Weekly Demand in Units 1,750 1,250 500

Selling Price/Unit € 190.00 € 200.00 € 100.00

Material Cost/Unit 100.00 80.00 60.00

Resource minutes required per unit

6 inch Screen 5.5 inch Screen 5 inch Screen

Machine A 2 0 1.5

Machine B 1.5 1.8 0

Machine C 0.5 1.3 3

Total resource minutes per machine is 4800.

Question 8 (10 points)

An item is processed in two machines. The production rate in the first machine is twice as large as in the second machine. The demand per unit of time is constant and half the value of the production rate in the second machine.

Given a large setup time, the production process is as follows. In front of machine 1, a batch of size Q is placed. Items from this batch are processed in machine 1, and then successively processed in machine 2. When all Q items have been processed in both machines, the same batch is transported to an end-customer-inventory where demand takes place. No backorders are allowed.

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Page 4 of 6 Use the following notation:

i = Product type, i=1…n,

p1 = production rate in the first machine, p2 = production rate in the second machine,

Q = batch size,

A1 = setup cost in the first machine, A2 = setup cost in the second machine,

h1 = the holding cost of an item before machine 1,

h2 = the holding cost of an item just after being processed in machine 1, h3 = the holding cost of an item just after being processed in machine 2,

Dt = demand per unit of time, t.

a) Develop a mathematical model to optimize the Q with respect to inventory holding costs and setup costs. (7 points)

b) Can the average inventory level between the two machines, expressed in terms of Q, be estimated without a model? (3 points)

Question 9 (14 points)

Consider an item controlled by the inventory control system (𝑅, 𝑠, 𝑆) with review period 𝑅 = 1 week and full backordering. The item is supplied by a very reliable supplier with a fixed lead time of two weeks. The following estimates regarding the moments of the weekly demand are available:

𝐸(𝐷𝑅) = 100, 𝐸(𝐷𝑅2) = 10,225, 𝐸(𝐷𝑅3) = 1,067,500 A business analyst wants to determine the reorder point that should be used to obtain a fill rate of 98%. The difference between order-up-to level and reorder point 𝑆 − 𝑠 has been fixed at 400 units. Ideally, the business analyst would like to solve the service equation using Tijms and Groenevelt’s heuristic procedure. Unfortunately, this requires a complex numerical procedure involving second-order loss functions. The business analyst only has a pocket calculator and a normal distribution table (see the table at the end of this exam) available to calculate the reorder point. So instead of Tijms and

Groenevelt’s procedure, he decides to use a (𝑠, 𝑆) model using normally distributed demand where shortages at the start of the replenishment cycle are ignored.

a) Calculate the expected value and standard deviation of the undershoot at the moment a replenishment order is placed under the (𝑅, 𝑠, 𝑆) control system.

b) The business analyst uses the results about the undershoot from part (a) in his alternative (𝑠, 𝑆) model. Is that a reasonable approach?

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Page 5 of 6

c) Write down the service equation that the business analyst needs to solve in terms of parameters, random variables and expectations (don’t use the normality assumption yet).

d) Compute the reorder point 𝑠 by solving this service equation.

e) Give an estimate for the average on-hand stock if this reorder point is applied.

Question 10 (6 points)

Consider an (𝑠, 𝑄) system with complete backordering and 𝑄 = 1. Assume Poisson demand during lead time (𝐷𝐿) with mean 𝜇:

𝑝𝑗 = 𝑃(𝐷𝐿 = 𝑗) = 𝑒−𝜇𝜇𝑗 𝑗!.

The indifference curve for the reorder point 𝑠 for in case of the 𝐵2-stockout cost type is represented by the equation

𝑝𝑠+1

𝑠𝑗=0𝑝𝑗 = 𝑟 𝐷𝐵2,

where 𝐷 is the expected yearly demand, 𝑟 the carrying charge and 𝐵2 the fractional stockout cost per unit short.

a) Explain what it means when a reorder point 𝑠 satisfies the equation above, and why it can be used to obtain the optimal reorder point for this stockout type.

b) Using the Poisson distribution of 𝐷𝐿, develop an expression for the expected number of units on backorder at any random point in time.

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Page 6 of 6 Normal distribution table

The following table lists the probability density function, the cumulative distribution function and the loss function for the standard normal distribution, respectively.

k 𝜑(𝑘) Φ(𝑘) 𝑁𝐿(𝑘)

0.00 0.3989 0.5000 0.3989 0.02 0.3989 0.5080 0.3890 0.04 0.3986 0.5160 0.3793 0.06 0.3982 0.5239 0.3697 0.08 0.3977 0.5319 0.3602 0.10 0.3970 0.5398 0.3509 0.12 0.3961 0.5478 0.3418 0.14 0.3951 0.5557 0.3328 0.16 0.3939 0.5636 0.3240 0.18 0.3925 0.5714 0.3154 0.20 0.3910 0.5793 0.3069 0.22 0.3894 0.5871 0.2986 0.24 0.3876 0.5948 0.2904 0.26 0.3857 0.6026 0.2824 0.28 0.3836 0.6103 0.2745 0.30 0.3814 0.6179 0.2668 0.32 0.3790 0.6255 0.2592 0.34 0.3765 0.6331 0.2518 0.36 0.3739 0.6406 0.2445 0.38 0.3712 0.6480 0.2374 0.40 0.3683 0.6554 0.2304 0.42 0.3653 0.6628 0.2236 0.44 0.3621 0.6700 0.2169 0.46 0.3589 0.6772 0.2104 0.48 0.3555 0.6844 0.2040 0.50 0.3521 0.6915 0.1978 0.52 0.3485 0.6985 0.1917 0.54 0.3448 0.7054 0.1857 0.56 0.3410 0.7123 0.1799 0.58 0.3372 0.7190 0.1742 0.60 0.3332 0.7257 0.1687 0.62 0.3292 0.7324 0.1633 0.64 0.3251 0.7389 0.1580 0.66 0.3209 0.7454 0.1528 0.68 0.3166 0.7517 0.1478 0.70 0.3123 0.7580 0.1429 0.72 0.3079 0.7642 0.1381 0.74 0.3034 0.7704 0.1334 0.76 0.2989 0.7764 0.1289 0.78 0.2943 0.7823 0.1245 0.80 0.2897 0.7881 0.1202 0.82 0.2850 0.7939 0.1160 0.84 0.2803 0.7995 0.1120 0.86 0.2756 0.8051 0.1080 0.88 0.2709 0.8106 0.1042 0.90 0.2661 0.8159 0.1004 0.92 0.2613 0.8212 0.0968 0.94 0.2565 0.8264 0.0933 0.96 0.2516 0.8315 0.0899 0.98 0.2468 0.8365 0.0865 1.00 0.2420 0.8413 0.0833

k 𝜑(𝑘) Φ(𝑘) 𝑁𝐿(𝑘)

1.00 0.2420 0.8413 0.0833 1.02 0.2371 0.8461 0.0802 1.04 0.2323 0.8508 0.0772 1.06 0.2275 0.8554 0.0742 1.08 0.2227 0.8599 0.0714 1.10 0.2179 0.8643 0.0686 1.12 0.2131 0.8686 0.0659 1.14 0.2083 0.8729 0.0634 1.16 0.2036 0.8770 0.0609 1.18 0.1989 0.8810 0.0584 1.20 0.1942 0.8849 0.0561 1.22 0.1895 0.8888 0.0538 1.24 0.1849 0.8925 0.0517 1.26 0.1804 0.8962 0.0495 1.28 0.1758 0.8997 0.0475 1.30 0.1714 0.9032 0.0455 1.32 0.1669 0.9066 0.0436 1.34 0.1626 0.9099 0.0418 1.36 0.1582 0.9131 0.0400 1.38 0.1539 0.9162 0.0383 1.40 0.1497 0.9192 0.0367 1.42 0.1456 0.9222 0.0351 1.44 0.1415 0.9251 0.0336 1.46 0.1374 0.9279 0.0321 1.48 0.1334 0.9306 0.0307 1.50 0.1295 0.9332 0.0293 1.52 0.1257 0.9357 0.0280 1.54 0.1219 0.9382 0.0267 1.56 0.1182 0.9406 0.0255 1.58 0.1145 0.9429 0.0244 1.60 0.1109 0.9452 0.0232 1.62 0.1074 0.9474 0.0222 1.64 0.1040 0.9495 0.0211 1.66 0.1006 0.9515 0.0201 1.68 0.0973 0.9535 0.0192 1.70 0.0940 0.9554 0.0183 1.72 0.0909 0.9573 0.0174 1.74 0.0878 0.9591 0.0166 1.76 0.0848 0.9608 0.0158 1.78 0.0818 0.9625 0.0150 1.80 0.0790 0.9641 0.0143 1.82 0.0761 0.9656 0.0136 1.84 0.0734 0.9671 0.0129 1.86 0.0707 0.9686 0.0123 1.88 0.0681 0.9699 0.0116 1.90 0.0656 0.9713 0.0111 1.92 0.0632 0.9726 0.0105 1.94 0.0608 0.9738 0.0100 1.96 0.0584 0.9750 0.0094 1.98 0.0562 0.9761 0.0090 2.00 0.0540 0.9772 0.0085

k 𝜑(𝑘) Φ(𝑘) 𝑁𝐿(𝑘)

2.00 0.0540 0.9772 0.0085 2.02 0.0519 0.9783 0.0080 2.04 0.0498 0.9793 0.0076 2.06 0.0478 0.9803 0.0072 2.08 0.0459 0.9812 0.0068 2.10 0.0440 0.9821 0.0065 2.12 0.0422 0.9830 0.0061 2.14 0.0404 0.9838 0.0058 2.16 0.0387 0.9846 0.0055 2.18 0.0371 0.9854 0.0052 2.20 0.0355 0.9861 0.0049 2.22 0.0339 0.9868 0.0046 2.24 0.0325 0.9875 0.0044 2.26 0.0310 0.9881 0.0041 2.28 0.0297 0.9887 0.0039 2.30 0.0283 0.9893 0.0037 2.32 0.0270 0.9898 0.0035 2.34 0.0258 0.9904 0.0033 2.36 0.0246 0.9909 0.0031 2.38 0.0235 0.9913 0.0029 2.40 0.0224 0.9918 0.0027 2.42 0.0213 0.9922 0.0026 2.44 0.0203 0.9927 0.0024 2.46 0.0194 0.9931 0.0023 2.48 0.0184 0.9934 0.0021 2.50 0.0175 0.9938 0.0020 2.52 0.0167 0.9941 0.0019 2.54 0.0158 0.9945 0.0018 2.56 0.0151 0.9948 0.0017 2.58 0.0143 0.9951 0.0016 2.60 0.0136 0.9953 0.0015 2.62 0.0129 0.9956 0.0014 2.64 0.0122 0.9959 0.0013 2.66 0.0116 0.9961 0.0012 2.68 0.0110 0.9963 0.0011 2.70 0.0104 0.9965 0.0011 2.72 0.0099 0.9967 0.0010 2.74 0.0093 0.9969 0.0009 2.76 0.0088 0.9971 0.0009 2.78 0.0084 0.9973 0.0008 2.80 0.0079 0.9974 0.0008 2.82 0.0075 0.9976 0.0007 2.84 0.0071 0.9977 0.0007 2.86 0.0067 0.9979 0.0006 2.88 0.0063 0.9980 0.0006 2.90 0.0060 0.9981 0.0005 2.92 0.0056 0.9982 0.0005 2.94 0.0053 0.9984 0.0005 2.96 0.0050 0.9985 0.0004 2.98 0.0047 0.9986 0.0004 3.00 0.0044 0.9987 0.0004

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