Practice exam comparable to the 2011 exam

Inventory control

Company Shear & Fool sells coffee machines. The demand X per replenishment lead time of 2 weeks is distributed as P (X = i) = 1=10 for i = 0; : : : 9. Shear & Fool use a base stock level with r = 5 and single-item replenishments as an inventory replenishment strategy. Assume that the inventory carrying cost is 10 Euro per unit per year, the backorder cost 40 Euro per unit per year.

A

Why does this strategy make sense for coffee machines?

B

Why is the demand not normally distributed?

C

Compute the service level.

D

Compute the expected backorder level.

E

Compute the average inventory level.

F

Compute the yearly expected cost related to the inventory strategy used by S&F.

G

Given the above results, what would you recommend to S&F?

MRP

A paint production firm uses MRP to control the production and the inventories at two distribution centers. It takes one week to replenish the distribution centers from FGI at the factory, and it takes 3 weeks to replenish the FGI at the factory from production. Production and stock levels are counted in cans. Order lot sizes are 10 cans, production lot sizes are determined by the FOP rule with P = 2. The MRP tableaux provide the MPS and the inventory levels at each of the distribution centers.

A

Why does the paint production firm use an ERP system to control production and inventories?

B

Assuming that it is allowed to shift scheduled receipts to other periods if necessary (just as in the book), use the MRP tables on Pages 4–5 to compute the planned order releases for cans at the distribution centers, compute the gross requirements for cans at the factory, compute the sizes of the production orders at the factory.

Table 1: MRP table for distribution center 1. Gross requirements are in cans per week.

Table 2: MRP table for distribution center 2.

Table 3: MRP tableau for the factory.

The MRP Table on this page are graded.

Table 4: MRP table for distribution center 1. Gross requirements are in cans per week.

Table 5: MRP tableau for distribution center 2.

Table 6: MRP tableau for the factory.

Discussion Questions

A

Give a logistic advantage of forecasting

B

Let B be the mean number of backorders in a make-to-stock system, D the average yearly demand, and T the average time until demand is satisfied from inventory. How are B, D and T related?

C

The previous question implies that once two quantities are known, the third follows. How to obtain or compute these two independent quantities? Give a formula, or a reference to an equation in the book for each of these two quantities.

D

Consider the base-stock model, and suppose a fraction p

E

Give a reason why CONWIP may not be necessary to control the WIP downstream of a bottleneck loop.

Bottlenecks and Batching

Consider two production stations in series (such as the stations in Penny Fab 1). Jobs arrive as a Poisson process with rate 1 per hour. The first station works at rate 1:2 job per hour, the second at 1:1 job per hour. Processing times are deterministic at both stations.

A

What is the load at the two stations?

B

What is the queuing time and cycle time at station 1?

C

What is the queuing time and cycle time at station 2?

D

Which station has the longest queue? Which station is the bottleneck? Comment on your observations.

E

Suppose station 1 suffered from an outage for 10 hours. How long does it take, in expectation, to clear the queue that built up during the outage time?

Assume now that station 1 (not station 2) is subject to random failures. The repair time is 1 hour precisely, and the number of failures per day is Poisson distributed with an average of 3 per day.

F

What is now the average time in queue at station 1? (Hint: think carefully about the MTTF and MTTR. How many hours per day, on average, is station 1 down, and how many hours up?)

G

Compute again the average time in queue at station 2.

H

If you did your work correctly, you should notice that the queueing time at station 2 is actually shorter as compared to the situation without the outages, i.e., Question C. This is strange since there is more variation now due to the outages. Can you explain why the queueing time at station 2 becomes shorter?

I

Suppose that all items after station 2 have to pass a quality check. A fraction p of all items has to be considered lost, hence becomes rework for the entire system. What is the largest allowable value for p?

J

Compute the time effectiveness of the total system, including outages at station 1.

A paint factory has 10 dissolvers to prepare paint in batches. Orders for paint arrive at a rate ra and have to processed in batches on one of the dissolvers. The production time at a dissolver takes an average time te independent of the number of orders in the batch. Before a batch can start a clean up time of s hours on a dissolver is required.

K

Give a formula to compute the minimal batch size (on average) on a dissolver.

Answers

Answer Question 1 Inventory control

Company Shear & Fool sells coffee machines. The demand X per replenishment lead time of 2 weeks is distributed as P (X = i) = 1/10 for i = 0, . . . 9. Shear & Fool use a base stock level with r = 5 and single-item replenishments as an inventory replenishment strategy. Assume that the inventory carrying cost is 10 Euro per unit per year, the backorder cost 40 Euro per unit per year.

Answer A

It is essential to show that you understand that the ordering cost must be small relative to the other cost involved, since this justifies to use the base-sto ck model.

Answer B

Because its distribution is given above, i.e., discrete and uniform.

Answer C

S then I considered this as a grave error, especially If you used the normal distribution here to compute in view of the previous question.

Answer D

• 
  

  Answer: Use B(r) =	9
  	I=R+1(i − r − 1)pI

Answer E

I(r) = r + 1 − θ + B(r). θ = E(X) = 0.1 · 0 + · · · 0.1 · 9.

Answer F

Y (r) = hI(r) + bB(r).

Answer G

The service level is rather low. Also the cost of backordering is low relative to the inventory costs. Hence, increase the reorder level r.

Anwswer MRP question 2

A paint production firm uses MRP to control the production and the inventories at two distribution centers. It takes one week to replenish the distribution centers from FGI at the factory, and it takes 3 weeks to replenish the FGI at the factory from production. Production and stock levels are counted in cans. Order lot sizes are 10 cans, production lot sizes are determined by the FOP rule with P = 2.

A

Answer: Since the inventories at the distribution centers are located at another place than the factory, it easy to use an electronic system to keep overview of the inventories.

B

The Tableaux have the answers filled in

Table 1: MRP tableau for distribution center 1

Table 2: MRP tableau for distribution center 2.

Table 3: MRP tableau for the factory.

Table 4: MRP tableau for distribution center 1. Gross requirements are in cans per week.

Table 5: MRP tableau for distribution center 2.

Table 6: MRP tableau for the factory.

Answer Question 3Discussion Questions

Answer A

Answer: Smooting of demand, so that production can be smoothed. Required capacity can be better organized, etc.

Answer B

Answer: See Section 9.6, time effectiveness, T = B/D, or B = DT .

Answer C

Answer: B see eq. 2.41. D can be computed from measured data.

Answer D

Answer: With probability p a customer stays if it sees an empty inventory, and the rest leaves. Given n customers that see an empty inventory, the number of customers that stay is binomially distributed with parameters n and p. The expected number that stay is then np. Thus, for the base stock model, when i−x−1 is the amount of customers that is potentially backlogged, only p(i − x − 1) are actually backlogged on average. Therefore

˜

where B(r) is the expected backorder level without loss.

Answer D

Since the stations downstream have higher rate than the bottleneck, queues will typically not become very long. Especially, if the rates are substantially higher than the bottleneck rate, there is no need to control the WIP downstream.

Answer Question 4 Bottlenecks and Batching

Answer A

u₁ = 1/1.2, u₁ = 1/1.1. Thus, the second station has the highest load.

Answer B

CT_Q = 2.1, CT = 2.92 hours. CT1 = ( c²_A + c²_E)/2u₁/(1 − u₁)t _E. Now c²_A = 1 and c²_E = 0.

Answer C

CT_Q = 1.4, CT = 2.3 hours. Realize that you need to compute the squared coefficie nt of variation c²_A of the arrival process at the second station. Use formula 8.10 for this.

If you did not use that the arrival process of station 1 is the departure process of station 2 (with appro-priate coefficient of variation) you do not earn your points.

Answer D

The first station has the longest queueing time, while the sec ond station is the bottleneck. Thus, in this case it is not true that the bottleneck is the station with the longest queue.

It is essential that you understand that variability also affects cycle times. If you just state that the CT at station 2 is larger since it is the bottleneck, you failed to show your understanding of this fact, hence, no points. In fact, if you missed this, you might as well have left the book unread. This is a true, big blunder.

Answer E

Suppose station 1 suffered from an outage for 10 hours. How long does it take, in expectation, to clear the queue that built up during the outage time?

Answer: The queue depletes at an average of r_E − r_A = 1.2 − 1 = 0.2 per hour. Thus, the time to clear the queue is the number in queue at the end of the outage divided by the average depletion rate. Thus, 10 ∗ r_A /(r_E − r_A ) = 10/0.2 = 100/2 = 50 hours.

Answer F

Assuming 24 hours per day, and 3 hours down on average, the amount of time available on a day is 21 hours. Since 3 outages occur per day, the MTTF must be 21/3 = 7 h, and the MTTR = 1 h as given in the problem.

In case you made other assumptions about the number of working hours on a day, and you showed to deal with this reasonably, I also accepted this.

Use 8.4 to compute the effective service time and use formula 8.6 to determine the SCV for the service times. Use that c²_R = 0, as repair times are by assumption deterministic. I get that CT_Q1 = 10.8 hours, and CT₁ = 11.7.

It is essential that you show that you are aware that the coefficient of variation depends on the (distribution of the) outage.

Answer G

Answer: Now CT_Q,2 = 0.96 and CT₂ = 1.87280972995 hours.

Answer H

Answer: Since the utilization at station 1 increased, due to the outages, the queues increase too. Since the service process at station 1 is completely deterministic, save the occasional outages, the departure process becomes more regular (less variation of the arrival process is passed on to the departure process). Therefore the arrival process at station 2 becomes more regular.

Answer I

If r_A is the arrival rate at the system, and assume that station 1 is bottleneck with effective processing time t_E. Then r_A · t_E/(1 − p) A t_E.

Some students argued like this: 1 − u is the fraction of time the station is idle. Hence, this must be the fraction of time we can waste on rework. Therfore, p ≤ 1 − u = 1 − r_At_E. Nice.

Answer J

Use the equations on page 337. D = r_A = 1. CT = 11.7 + 1.9 = 13.8 hours. Hence D · CT = 13.8. However, we need B/D = CT . Also, T₀^∗ = 1/1.2 + 1/1.1, so that the time effectiveness becomes CT /T₀^∗. Note that we should leave out the outages from T₀^∗ as it is the raw processing time not including detractors.
 

A paint factory has 10 dissolvers to prepare paint in batches. Orders for paint arrive at a rate rA and have to processed in batches on one of the dissolvers. The production time at a dissolver takes an average time tE independent of the number of orders in the batch. Before a batch can start a clean up time of s hours on a dissolver is required.

Answer K

The arrival rate for a single dissolver is r_A /10. Suppose a batch contains n orders. Then the arrival rate of batches is 0.1r_A/n. The load should be less than 1, hence 0.1r_A/n(s + t_E)

n > 0.1r_A(s + t_E).