Chapter 1



ISyE 6201: Manufacturing Systems

Instructor : Spyros Reveliotis

Spring 2006

Solutions for Homework #2

A.

Chapter 2

9. The key difference between EOQ and (Q,r) is that demand is stochastic in (Q,r) but deterministic in EOQ. The base stock model also has stochastic demand, but unlike the (Q,r) model, it assumes that replenishment lot sizes are always equal to one.

10. The statement “the reorder point, r, affects customer service, while the replenishment quantity, Q, affects replenishment frequency” is true in rough terms but is not precisely true because:

• Q certainly does govern order frequency (i.e., if annual demand is D, then the number of orders per year is F=D/Q).

• Once Q is fixed, then r determines the customer service (i.e., the probability of an order being filled out of stock-also known as Type II Service requirement).

• However, if we change Q for a given r, customer service changes. The reason is that if Q is made large, then the number of times per year that inventory falls to a level where stockouts occur is reduced. Thus, for a fixed r, service level increases with Q.

The dependence of the customer service on Q is also manifested in the derived formulae for B(Q,r) and S(Q,r). As we showed in class, the computation of the reorder point r is decomposed from the computation of the order quantity Q, only when we employ some (pertinent) approximations for B(Q,r) and S(Q,r).

Finally, it should be noticed that Q does not affect the customer service in case that the latter is defined through a Type I Service requirement; i.e., by the probability, p, that the system experiences no stockout during any single cycle. This requirement is satisfied by setting r such that G(r)=p, where G() is the cdf for the demand over the replenishment lead time.

11. Increasing the variability of the demand process tends to require a higher level of safety stock (or equivalently, a higher reorder point) because, in general, a more variable demand will have a cdf G() with a “heavier” right-hand tail, i.e., the probability that the demand will get its extreme higher values will increase. Hence, satisfying a requirement of the type G(r)=p, for some p value that is either provided explicitly or computed from the problem cost structure, will result in higher values of r.

12. Essentially you have to solve the model for both cases, and see which case resulted in the lowest overall cost.

Other factors you should consider in deciding to change vendors include issues related to vendor reliability, quality, cost of certifying the new vendor (if you have a vendor certification program), etc.

Chapter 17

1. If capacity is not an issue and the cost of a purchase order is independent of volume, then the assumptions of EOQ (modified to allow for a delay delivery) may be approximately satisfied for procurement. In production systems, the EOQ assumption of a fixed setup cost is not very valid because this cost will be determined by the current available capacity of the production system, which depends itself on the system load, which is in turn determined by the adopted replenishment policy.

5. Type I service is the probability the system does not stock out during a replenishment cycle, while Type II service is the fraction of demands filled from stock (not backordered). As discussed in Question 2-10, Type I service is more analytically tractable and allows us to get “simple” expressions for the controls in a (Q,r) type model. On the other hand, Type I service is much more conservative than Type II service, and will therefore usually result in higher inventory levels than its Type II counterpart.

6. Using approximations for fill rate and backorder level enables us to get simple expressions for Q and r. But, if we were to use these approximations to evaluate performance, we could wind up with solutions that do not meet performance targets. Therefore, even though we use expressions based on approximations of performance measures to compute Q and r, we use the actual performance measures to see how good the resulting solution is, and, if necessary, we adjust Q and r appropriately.

9. The four main causes of the bullwhip effect are batching, over-reacting forecasts, pricing that encourages ordering in higher quantities, and gaming behavior to guard against perceived shortages.

(a) In a consumer products network, all of the above factors can play a role, although forecasting, pricing, and gaming behavior are likely to be particularly significant because there are independent decision makers involved.

(b) In a spare parts network, batching is probably the dominant factor, although gaming behavior may also be important (e.g., if regional facilities can manipulate their portions of scarce parts by inflating orders.)

(c) In a military supply network, batching and gaming behavior are the predominant effects. Since the system is centrally controlled, forecasting and pricing should not play significant roles.

B.

Chapter 2

10. This is a single period stochastic inventory model – the “news vendor problem.” Since the shirts sell for $20 and cost $5 any unit we are short will “cost” us $15. Since shrits that are not sold at the event can be discounted and sold for $4, the excess cost is $1. Thus,

co = 1

cs = 15

At this point it is clear that printing too few shirts is worse than printing up too many so their policy is not a good one.

If G(x) represents the distribution function for the demand, from the news vendor model we have

G(Q*) = cs/(cs + co) = 15/16 = 0.9375 = s

where s is the “service rate.” The question now is what distribution function to use. We estimate the mean to be 12,000 and know there is a “significant amount of uncertainty.” Since 12,000 is a large number, the normal distribution should be a reasonable approximation. If X is a random variable representing the demand during the event we can write an expression for Q* in terms of the mean and standard deviation of the demand.

P{X≤Q*} = cs/(cs + co) = s or P{(X-12000)/σ ≤ (Q*-12000)/σ} = s

so that

(Q*-12000)/σ = zs = 1.53 or Q* = 12000+1.53σ

If the actual demand distribution were Poisson, the standard deviation would be the square root of the mean or σ = (12000)1/2 = 109.54, and when this value is plugged to the above approximation based on the normal distribution, it gives: Q* = 12,167.6 ~ 12,168.

11. cs = 500, co = 200

(a) G(Q*) = cs/(cs + co) = 500/(500+200) = 0.714

Q* = μ + z0.714σ = 10,000 + (0.57)2500 = 11,425

(b) The news vendor model does consider lost sales vs. materials cost, and does address uncertain demand forecast. But it does not consider:

• Longer term impact of shortages, such as market share considerations, a negative corporate image (e.g., if they can’t deliver), etc.

• Risk attitude – is expected value appropriate as a decision criterion?

• The possibility of more complicated contracts, e.g., contracts which may allow the eventual purchasing of a quantity within a certain range.

12. The chairs are made in-house and so we are attempting to determine the appropriate parameters for a base-stock system. We assume that the wholesalers order once per month.

(a) The holding cost is h=$5 while the backorder cost is b=$20. The distribution of demand during a month is well approximated by a normal distribution with a mean of 1,000 chairs and a standard deviation of 200 chairs. Then, if X represents the demand during one month,

G(R*) = b/(h+b) = 20/(5+20) = 0.8

and so

G(R*) = P{X≤R*} = P{(X-1000)/200 ≤ (R*-1000)/200} = 0.8

The value of the standard normal with 0.8 probability is obtained from a standard normal table or using the Excel function NORMSINV(0.8) and yields 0.84. Then the order up to point is computed as

R* = 1000+0.84(200) = 1168.

(b) If the sale is lost (as opposed to backordered ) then the shortage cost must be the profit that would have been made which is $100. The computation is then similar,

G(R*) = 100/(5+100) = 0.9524

z0.9524 = 1.67

R* = 1000+1.67(200) = 1334

(c) Since the cost of being short is higher in the second case, we want to carry more inventory to avoid that possibility.

14.

(a)

[pic]

Formulae for some of the quantities:

θ ’ Dl σD ’ [pic](because demand is POISSON)

F = [pic] I = [R - θ + B(R)] c = [r+1 - θ + B(r+1)] c

S(R) = G(R-1) = G(r)

Holding cost per year = 12*hI Order cost per year = 12*FA

The colored cells are looked up from the table below, which uses the following formulae for the basestock model:

p(R) = θRe-θ/R! (cdf of Poisson random variable)

G(R) = [pic] (by definition on pg. 69 of the textbook)

B(R) = θp(R) + {[θ−R] [1-G(R)]} (eqn 2.63 on pg. 100)

[pic]

(b) EOQ = [pic] , so QA = 4, QB = 26.

Using the previous table, we compute the fill rates for the (Q,r) model using

S(Q, r) =[pic] (eqn 2.35 on pg. 78)

To achieve a fill rate of at least 98%, the minimum r for Type A and Type B modules are 12 and 18 units, respectively.

Now since Q>1, the inventory investment and backorder level are computed by

I(Q,r)*c = [(Q+1)/2 + r - θ + B(Q,r)]*c

and B(Q,r) = [pic]

(These two formulae apply to parts (c) and (d) as well since the order quantities Q are greater than 1.)

[pic]

The higher values of Q make it possible to achieve the same service with lower r values. The inventory is higher due to increased cycle stock caused by bulk ordering. However, the total cost is reduced by about 75% due to the reduction in order cost.

(c) In the backorder model, we find G(R*) using G(R*) = b/(b+h). This critical ratio defines the z-value in the same way as we did in Problem 2-10. Using r* = θ + zσ, the reorders points are found to be 9 and 22 for A and B, respectively. Then, look up for the Si values from the fill rates table, and we get

[pic]

This change lowers service for part A (expensive one) and raises it for part B, so the same average service is achieved with lower total inventory. Note that it is even below the base stock inventory level where Q = 1.

(d) Standard deviation of lead time demand σ ’ [pic], where σL is the standard deviation of lead time. Note that D and σL must use the same base time unit (e.g. D = 7 units/month, σL= 0.25 month). With the new σ, the reorder points are recomputed using the same formula r* = θ + zσ as in part (c).

[pic]

The variability in the lead times inflates the reorder points – in this case for part B (rounding can result in no change). It should also be pointed out that the predictions of service, backorder level, and inventory level are no longer exact, since the employed formulae were derived under the assumption of fixed lead times.

15. Formulae for some of the quantities:

θ ’ Dl σ ’ [pic](because demand is POISSON)

F = [pic] I(Q,r)*c = [(Q+1)/2 + r - θ + B(Q,r)]*c

Holding cost per year = 12*hI Order cost per year = 12*FA

The fill rates table is at the end of this problem’s solution.

(a,b)

Note that Type 1 service underestimates true service by a lot, leading to a much larger r and higher inventory.

(c)

Type 2 service is very accurate because when Q is this large, the term B(r+Q) is negligible.

Note that when Q is reduced, we get slightly higher service at a much smaller inventory investment. But of course, we order twice as often. If we neglect the cost or capacity considerations of placing orders, we can always minimize inventory costs y choosing Q=1. But if we consider either order frequency (capacity) or fixed order cost, then EOQ may give a perfectly reasonable Q.

Formulae used in the fill rates table:

p(r) = θre-θ/r! (cdf of Poisson random variable)

G(r) = [pic] (by def on pg. 69 of the textbook)

B(r) = θp(r) + {[θ−r] [1-G(r)]} (eqn 2.63 on pg. 100. This is the backorder level formula

for the base stock model. The values of B(r) are

computed because they are used in the following B(Q,r)

formula, which is a (Q,r) model formula.)

B(Q, r) =[pic] (eqn 2.38 on pg. 78)

Type 1 service = G (r) (eqn 2.36 on pg. 78)

Type 2 service = [pic] (eqn 2.37 on pg. 79)

Exact S(Q,r) = [pic] (eqn 2.35 on pg. 78)

Fill rates table for Problem 2.15:

[pic]

Chapter 17

5.

(a) By looking at Figure 17.4, we can see that the point F=12, S=0.85, I=$2500 lies well above the corresponding efficient frontier, which doesn’t make it look very efficient. Presumably, the current policy has been set up in a very ad-hoc fashion, and we could expect an improvement by implementing an optimized (Q,r) policy. However, we must be careful in our expectations. Actual service is affected by factors not considered by the model, such as non-stationary demand (i.e., a demand distribution that evolves with time), obsolescence, data entry errors, etc. So actual performance may not match exactly that predicted by the model. In this case, it looks like inventory could be reduced by ~50% by going to an optimized policy; the actual amount of improvement will depend on how bad the current policy is and how significant the un-modeled factors are.

(b) In this multi-product situation, we want to find an ordering plan that minimizes the inventory cost while maintaining an average fill rate of 95%. Note that the inventory cost decreases by increasing order frequency. Therefore, we set the order quantity for each product to Qi=1, to maximize the order frequency. Using the algorithm for the multiproduct (Q,r) stockout model (on pg. 606 of the textbook), we can find that when k=$1.402, the corresponding order points, rj, will give an average fill rate of 95%. The following table gives the calculations for the inventory investment.

However, even with Q=1, S=95% requires more than $1000 to achieve, so the president’s targets are not feasible unless we change some of the other parameters determining the system dynamics, like leadtimes or demand variability.

6.

(a) Mean lead time demand considering supplier delays is E[L] = l+W = 7 + 0.465 = 7.465 days. Since the demand rate experienced by Windsong is one per day, the expected demand over the lead time will be 7.465 units.

(b) Variance of lead time σL2 = [S/(1-S)] * W2 = [0.897/(1-0.897)] * 0.4652 = 1.883

Standard deviation of lead time demand is σm ’ [lσD2 + Dd 2σL2]1/2. But lσD2 = l E[Dd] = θ, becaue daily demand is POISSON. Therefore, σm = (7.465 + 1.883)1/2 = 3.057, which is greater than (7.4657)1/2 = 2.7323 (i.e., what it would be if demand were Poisson)

(c) The fill rate, assuming Poisson demand for r =10, is [pic]. Actual service will be lower than this because variability of demand is greater than that of the Poisson.

C. Extra Credit

[pic]

[pic]

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