Q1) Below are the number of cars, which are still under warranty, that show up for repairs with engine problems at authorized car repair shops for XYZ brand model 3. The XYZ company executives are worried what is going to happen with the number of cars with engine problem, not only it is possibly going to ruin their brand’s reputation but also they have to pay the repair cost since the cars are under manufacturer’s warranty. They provided us with real data for the months 1 through 7 of the year. Also they provided the forecasts they made using an exponential smoothing with α = 0.2. (they used D1 for D0 and F0 values in calculating for ES forecast.)

Month (t) / Actual # of cars with Engine problems (Dt) / ES Forecast (Ft)
120 / 120,00
1 / 120 / 120,00
2 / 163 / 120,00
3 / 170 / 128,60
4 / 181 / 136,88
5 / 195 / 145,70
6 / 212 / 155,56
7 / 220 / 166,85

a.  How good is this forecast? Why do you think so? (5 points)

There are a number of ways for quantitavely measuring how good is a forecast. Main three of them are MAD(Mean Absolute Deviation), MAPE (Mean Absolute Percentage Deviation) and MSE (Mean Square Error). You can use any of them but it is always better to check at least two and see if they agree. For our problem,

Month (t) / Actual # of cars with Engine problems (Dt) / ES Forecast (Ft) / Error / |Error| / |Error|/Dt / (Error)^2
0 / 120 / 120,00
1 / 120 / 120,00 / 0,00 / 0,00 / 0,00 / 0,00 / 134,8214
2 / 163 / 120,00 / -43,00 / 43,00 / 0,26 / 1849,00 / 149,9286
3 / 170 / 128,60 / -41,40 / 41,40 / 0,24 / 1713,96 / 165,0357
4 / 181 / 136,88 / -44,12 / 44,12 / 0,24 / 1946,57 / 180,1429
5 / 195 / 145,70 / -49,30 / 49,30 / 0,25 / 2430,10 / 195,25
6 / 212 / 155,56 / -56,44 / 56,44 / 0,27 / 3185,11 / 210,3571
7 / 220 / 166,85 / -53,15 / 53,15 / 0,24 / 2824,86 / 225,4643
MAD / 41,06
MSE / 1992,80
MAPE / 0,22

It seems like, based on errors, we are constantly underestimating and deviations seem to grow in one direction. This cannot be a good forecast. Also MAPE tells us that on the average we are 22% (or 25% if you do not include period 1) away from the true value.

b.  The management seems to think that a double exponential smoothing method applied to this data will give much better results? Do you agree, why? (5 points)Using S0 = 120 and G0 = 60 calculate double exponential smoothing forecast for periods 1-3 using alpha=0.4 and beta =0.5 (10 points)

Double Exponential Smoothing should give much better results. If you plot the data, you will see that there is a positive linear trend, and Double ES is a good method for this kind of data, whereas ES is not because ES is used with stationary data. When the double ES method with given parameters are applied. Alpha=0.4, beta=0.5, G0=60, S0=120.

Month(t) / Actual # of cars with Engine problems(Dt) / St / Gt / Ft / Error
120 / 60 / 120
1 / 120 / 156 / 33 / 180 / 60
2 / 163 / 178,6 / 27,8 / 189 / 26
3 / 170 / 191,84 / 20,52 / 206,4 / 36,4
4 / 181 / 199,816 / 14,248 / 212,36 / 31,36
5 / 195 / 206,4384 / 10,4352 / 214,064 / 19,064
6 / 212 / 214,92416 / 9,46048 / 216,8736 / 4,8736
7 / 220 / 222,630784 / 8,583552 / 224,3846 / 4,38464

Q2) Izmir Brass Ornaments (IBO) Company produces special handmade ornaments made out of brass. They are in the process of planning labor force requirements and production levels for the next 4 quarters. The marketing department of IBO provided following forecasts

Quarter Demand

1  380

2  630

3  220

4  160

Assume 280 initial employees. Employees are hired for at least 1 full quarter. Hiring cost is 1200TL/employee, Firing cost is 2500 TL/employee, Inventory Cost is 1000TL/unit/quarter. One worker produces 1 ornament/quarter. IBO has currently 80 in stock and would like to end with 20 in inventory at the end of the year.

a.  Determine a constant workforce plan and cost of this plan (no stockouts are allowed) (10 points)

Unit/Worker Net Cum Net Min.

Quarter (000) Demand Dem. Work Force

1 1 300 300 300

2 1 630 930 465

3 1 220 1150 384

4 1 180 1330 333

Hence the min. constant workforce is 465 workers.

The cost of the resulting plan is:

Cum Cum Ending

Quarter Production Net Demand Inventory

1 465 300 165

2 930 930 0

3 1395 1150 245

4 1860 1330 530

Total 940

We must also add back in the 20 required to be on hand in the fourth quarter. Hence the total cost of this plan is:

(1,200)(465 - 280) + (1000)(940 + 20) = $1,182,000.

b.  If IBO company can backorder excess demand at 2000TL/unit/quarter, determine the minimum number of workers needed so that only a stockout occurs in period 2. What is the cost of the new plan? (10 points)

If we use the minimum number of workers required through period 3 of 1150/3=384, it will satisfy the conditions stated.

Quarter Cum Prod Cum Net Dem. Ending Inv.

1 384 300 84

2 768 930 -162

3 1152 1150 2

4 1536 1330 206

Total cost = (1,200)(384-280)+(1,000)(312)+(2,000)(162) = $760,800.

Q3) Karçelik A.Ş. for a particular type of compressor used in the production of refrigerator must decide among three suppliers. Source 1 will sell the compressors for $3.0 per compressor, independently of the number of compressors ordered. Source B will sell the compressors for $2.80 each but will not consider an order for fewer than 2500 compressors, and Source C will sell the compressors for $2.40 each but will not accept an order for fewer than 3000 compressors. Assume an order setup cost of $100 independent of the source selected and an annual requirement of 20000 compressors which is uniformly distributed throughout the year. Assume a 20 percent annual interest rate for holding cost calculations.

a)  Which source should be used, and what is the size of the standing order? (8 points)

b)  What is the optimal value of the holding and setup costs for compressors when the optimal source is used? (7 points)

c)  If the replenishment lead time for compressors is two months, determine the reorder point based on the on-hand level of inventory of compressors. (5 points)

Q5) The inventory of a purchased item is under continuous review. The fixed cost of an order is $100, the cost per unit is $5, the annual inventory carrying cost rate, is 20 percent, and the expected annual demand is 10000 units. There is a cost of not satisfying demand, which is $10/unit. Lead time demand is uniformly distributed between 300 and 500. The managers are interested in finding an inventory system, that will minimize the expected total cost of inventory, shortage and setup.

a) What kind of an inventory system would you recommend, and why? (6 points)

b) Find the reorder point if the probability of a shortage during any lead time is to be 0,05. (6 points)

c) For optimal total expected cost, as mentioned above, how many would you order (4 points), when would you reorder (4 points)and how often (time between two consecutive reorders) (3 points) would you order?

Q4) Billy’s Bakery bakes fresh bagels each morning. The daily demand for bagels is a random variable with a distribution estimated from prior experience given by;

Number of Bagels Sold in One Day Probability

0 0.05

5 0.10

10 0.15

15 0.20

20 0.20

25 0.15

30 0.10

35 0.05

The bagels cost Billy’s 8 cents to make, and they are sold for 35 cents each. Bagels unsold at the end of the day are purchased by a nearby charity soup kitchen for 3 cents each.

a)  Based on the discrete distribution above, how many bagels should Billy’s bake at the start of each day? (Answer should be a multiple of 5.) (8 points)

c0 = .08 - .03 = .05

cu = .35 - .08 = .27

Critical ratio = = .84375

From the given distribution, we have:

Q f(Q) F(Q)

0 .05 .05

5 .10 .15

10 .10 .25

15 .20 .45

20 .25 .70

- - - - .84375

25 .15 .85

30 .10 .95

35 .05 1.00

Since the critical ratio falls between 20 and 25 the optimal is Q = 25 bagels.

b) Determine the optimal number of bagels to bake each day using normal approximation. (Hint: You must compute the mean and the variance of the demand from the discrete distribution above.) (8 points)

m = åxf(x) = (0)(.05) + (5)(.10) +...+(35)(.05) = 18

s2 = åx2f(x) - m2 = 402.5 - (18)2 = 78.5

s = = 8.86

The z value corresponding to a critical ratio of .84375 is 1.01. Hence,

Q* = sz + m = (8.86)(1.01) + 18 = 26.95 ~ 27.

c) What is the type 2 service level (fill rate) for the solution in part (a)? (6 points)

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