4.8.1 The Newsboy Problem

In the Newsboy Problem, a resupply decision has to be made at the beginning of

a period (e.g. a spring sales season) for a single commodity whose demand is not

known in advance. The demand d is modelled as a random variable with a continuous

cumulative distribution function Fd (δ). Let c be the purchasing cost or the variable

manufacturing cost, depending on whether the goods are bought from an external

supplier or produced by the company. Moreover, let r and u be the selling price and

the salvage value per unit of commodity, respectively. Of course,

r > c > u.

There is no ¬xed reorder cost nor an initial inventory. In addition, shortage costs are

assumed to be negligible. If the company orders q units of commodity, the expected

revenue ρ(q) is

∞ ∞

ρ(q) = r min(δ, q) dFd (δ) + u max(0, q ’ δ) dFd (δ) ’ cq

0 0

∞

q q

=r δ dFd (δ) + q dFd (δ) + u (q ’ δ) dFd (δ) ’ cq.

q

0 0

∞

By adding and subtracting r q δ dFd (δ) to the right-hand side, ρ(q) becomes

∞ q

ρ(q) = rE[d] + r (q ’ δ) dFd (δ) + u (q ’ δ) dFd (δ) ’ cq, (4.35)

q 0

where E[d] is the expected demand. It is easy to show that ρ(q) is concave for q 0,

and ρ(q) ’ ’∞ for q ’ ∞. As a result, the maximum expected revenue is achieved

when the derivative of ρ(q) with respect to q is zero. Hence, by applying the Leibnitz

rule, the optimality condition becomes

r(1 ’ Fd (q)) + uFd (q) ’ c = 0,

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142 SOLVING INVENTORY MANAGEMENT PROBLEMS

where, by de¬nition, Fd (q) is the probability Pr(d q) that the demand does not

exceed q. As a result, the optimal order quantity S satis¬es the following condition:

r ’c

S) = .

Pr(d (4.36)

r ’u

Emilio Tadini & Sons is a hand-made shirt retailer, located in Rome (Italy), close

to Piazza di Spagna. This year Mr Tadini faces the problem of ordering a new bright

colour shirt made by a Florentine ¬rm. He assumes that the demand is uniformly

distributed between 200 and 350 units. The purchasing cost is c = ‚¬18 while the

selling price is r = ‚¬52 and the salvage value is u = ‚¬7. According to Equation

(4.36), Pr(d S) = (S ’ 200)/(350 ’ 200) for 200 S 350. Hence, Mr Tadini

should order S = 313 units. According to Equation (4.35), the expected revenue is

350 1

ρ(q) = 52 — 275 + 52 (q ’ δ) dδ ’ 18q = 34q,

350 ’ 200

q

q

for 0 200,

350 1

ρ(q) = 52 — 275 + 52 (q ’ δ) dδ

350 ’ 200

q

q 1

+7 (q ’ δ) dδ ’ 18q

350 ’ 200

200

= ’0.15q + 94q ’ 6000,

2

for 200 < q 350, and

350 1

ρ(q) = 52 — 275 + 7 (q ’ δ) dδ ’ 18q = ’11q + 12 375,

350 ’ 200

200

for q > 350. Hence, the maximum expected revenue is equal to ρ(313) = ‚¬8726.65.

4.8.2 The (s, S) policy for single period problems

If there is an initial inventory q0 and a ¬xed reorder cost k, the optimal replenishment

policy can be obtained as follows. If q0 S, no reorder is needed. Otherwise, the

best policy is to order S ’ q0 , provided that the expected revenue associated with this

choice is greater than the expected revenue associated with not producing anything.

Hence, two cases can occur:

(i) if the expected revenue ρ(S) ’ k ’ cq0 associated with reordering is greater

than the expected revenue ρ(q0 ) ’ cq0 associated with not reordering, then

S ’ q0 units have to be reordered;

(ii) otherwise, no order has to be placed.

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SOLVING INVENTORY MANAGEMENT PROBLEMS 143

I(t)

q

l

IS

tl tl

tl

t

Figure 4.10 Reorder level inventory policy.

As a consequence, if q0 < S, the optimal policy consists of ordering S ’ q0 units

if ρ(q0 ) ρ(S) ’ k. In other words, if s is the number such that

ρ(s) = ρ(S) ’ k,

the optimal policy is to order S ’ q0 units if the initial inventory level q0 is less than

or equal to s, otherwise not to order. Policies like this are known as (s, S) policies.

The parameter s acts as a reorder point, while S is called the order-up-to-level.

If q0 = 50 and k = ‚¬400 in the Emilio Tadini & Sons problem, ρ(s) = ρ(S)’k =

‚¬8526.65 so that s = 277. As q0 < s, the optimal policy is to order S ’ q0 = 253

units.

4.8.3 The reorder point policy

In the reorder point policy (or ¬xed order quantity policy), the inventory level is kept

under observation in an almost continuous way. As soon as its net value I (t) (the

amount in stock minus the unsatis¬ed demand plus the orders placed but not received

yet) reaches a reorder point l, a constant quantity q is ordered (see Figure 4.10).

The reorder size q is computed through the procedures illustrated in the previous

¯

sections, by replacing d with d. In particular, under the EOQ hypotheses:

¯

2k d

q= .

h

The reorder point l is obtained by requiring that the inventory level be nonnegative

during tl , with probability ±. This is equivalent to assuming that demand should not

exceed l during the interval tl . In the following, it is assumed that

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144 SOLVING INVENTORY MANAGEMENT PROBLEMS

• the demand rate d is distributed according to a normal distribution with expected

¯

value d and standard deviation σd ;

¯

• d and σd are constant in time;

• the lead time tl is deterministic or is distributed according to a normal distribu-

tion with expected value t¯l and standard deviation σtl ;

• the demand rate and the lead time are statistically independent.

¯

The average demand rate d can be forecasted with one of the methods illustrated

in Chapter 2, while the standard deviation σd can be estimated as the square root of

¯

MSE. Analogous procedures can be adopted for the estimation of tl and σtl .

Let z± be the value under which a standard normal random variable falls with

probability ± (e.g. z± = 2 for ± = 0.9772 and z± = 3 for ± = 0.9987). If tl is

deterministic, then

√

¯

l = dtl + z± σd tl , (4.37)

√

¯

where dtl and σd tl are the expected value and the standard deviation of the demand

in an interval of duration tl , respectively. If tl is random, then,

¯¯ ¯

l = d tl + z± σd t¯l + σt2 d 2 ,

2

l

¯¯ ¯

2¯

where d tl and σd tl + σt2 d 2 are the expected value and the standard deviation of

l

the demand in a time interval of random duration tl , respectively.

The reorder point l minus the average demand in the reorder period constitutes a

safety stock IS . For example, in case tl is constant, the safety stock is

√

¯

IS = l ’ d tl = z± σd tl . (4.38)

Papier is a French retail chain. At the outlet located in downtown Lyon, the expected

demand for mouse pads is 45 units per month. The value of an item in stock is ‚¬4, and

the ¬xed reorder cost is equal to ‚¬30. The annual interest rate is 20%. The demand

forecasting MSE is 25. Lead time is 1 month and a service level equal to 97.7% is

required. On the basis of Equation (4.2), the holding cost is

h = 0.2 — 4 = 0.8 euros/year per item = 0.067 euros/month per item.

Therefore, from Equation (4.13),

2 — 30 — 45

q— = = 200.74 ≈ 201 items.

0.067

Moreover, σd can be estimated as follows:

√

σd = 25 = 5.

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SOLVING INVENTORY MANAGEMENT PROBLEMS 145

I(t)

S

T T

qi qi+1

qi qi+1

IS

tl tl

t

ti ti+1 ti+2

Figure 4.11 Reorder cycle inventory policy.

From Equation (4.37), the reorder point l is

l = 45 + 2 — 5 = 55 units.

Consequently, the safety stock IS is

IS = 55 ’ 45 = 10 units.

4.8.4 The periodic review policy

In the reorder cycle policy (or periodic review policy) the stock level is kept under

observation periodically at time instants ti (ti+1 = ti + T , T 0). At time ti ,

qi = S ’ I (ti ) units are ordered (see Figure 4.11). The parameter S (referred to as

the order-up-to-level) represents the maximum inventory level in case lead time tl is

negligible.

The periodicity T of the sampling (review period) can be chosen using procedures

analogous to those used for determining q — in the deterministic models. For instance,

under the EOQ hypotheses,

2k

T= . (4.39)

¯

hd

The parameter S is determined in such a way that the probability that the inventory

level becomes negative does not exceed a given value (1 ’ ±). Since the risk interval

¯

is equal to T plus tl , S is required to be greater than or equal to the demand in T + tl ,

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146 SOLVING INVENTORY MANAGEMENT PROBLEMS

with probability equal to ±. If the lead time tl is deterministic, then

¯

S = d(T + tl ) + z± σd T + tl , (4.40)

√

¯

where d(T + tl ) and σd T + tl are the expected value and the standard deviation of

the demand in T + tl , respectively. If the lead time is a random variable, then

¯ ¯

S = d(T + tl ) + z± σd (T + t¯l ) + σt2 d 2 ,

¯ 2

l

¯ ¯

where d(T + tl ) and σd (T + t¯l ) + σt2 d 2 are the expected value and the standard

¯ 2

l

¯

deviation of the demand in T + tl , respectively.

¯

The difference between S and the average demand in T + tl makes up a safety stock

IS . For example, if the lead time is constant,

IS = z± σd T + tl . (4.41)

Comparing Equation (4.41) with Equation (4.38), it can be seen that the reorder

cycle inventory policy involves a higher level of safety stock. However, such a policy

does not require a continuous monitoring of the inventory level.

In the Papier problem, the parameters of the reorder cycle inventory policy, com-

puted through Equations (4.39) and (4.40) are

2 — 30

T= = 4.47 months,

0.067 — 45

√

S = 45 — (4.47 + 1) + 2 — 5 — 4.47 + 1 = 269.54 units.

The associated safety stock, given by Equation (4.41), is

√

IS = 2 — 5 — 4.47 + 1 = 23.39 units.

4.8.5 The (s, S) policy

The (s, S) inventory policy is a natural extension of the (s, S) policy illustrated for the

one-shot case. At time ti , S ’ I (ti ) items are ordered if I (ti ) < s (see Figure 4.12). If

s is large enough (s ’ S), the (s, S) policy is similar to the reorder cycle inventory

method. On the other hand, if s is small (s ’ 0), the (s, S) policy is similar to a

reorder level policy with a reorder point equal to s and a reorder quantity q ∼ S. On

=

the basis of these observations, the (s, S) policy can be seen as a good compromise

between the reorder level and the reorder cycle policies. Unfortunately, parameters

T , S and s are dif¬cult to determine analytically. Therefore, simulation is often used

in practice.

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SOLVING INVENTORY MANAGEMENT PROBLEMS 147

I(t)

S

qi qi qi+2 qi+2

s

qi+1 = 0

tl tl

t

ti+2

ti ti+1

Figure 4.12 (s, S) policy.

Pansko, a Bulgarian chemical ¬rm located in Plovdiv, supplies chemical agents to

state clinical laboratories. Its product Merofosphine has a demand of 400 packages

per week, a variable cost of 100 levs per unit, and a pro¬t of 20 levs per unit. Every

time the manufacturing process is set up, a ¬xed cost of 900 levs is incurred. The

annual interest rate p is 20%. If the commodity is not available in stock, a sale is lost.

In this case, a cost equal to the pro¬t of the lost sale is incurred. The MSE forecast

equals 2500. The lead time can be assumed to be constant and equal to a week. The

inventory is managed by means of an (s, S) policy with a period T of two weeks. The

values s and S are selected by simulating the system for all combinations of s (equal

to 800, 900, 1000, 1100 and 1200, respectively) and S (equal to 1500, 2000 and 2500,

respectively). According to the results reported in Table 4.1, s = 1100 and S = 2000

are the best choice. This would result in an average cost per week equal to 612.7 levs.

4.8.6 The two-bin policy

The two-bin policy can be seen as a variant of the reorder point inventory method

where no demand forecast is needed, and the inventory level does not have to be

monitored continuously. The items in stock are assumed to be stored in two identical

bins. As soon as one of the two becomes empty, an order is issued for an amount equal

to the bin capacity.

Browns supermarkets make use of the two-bin policy for tomato juice bottles. The

capacity of each bin is 400 boxes, containing 12 bottles each. In a supermarket close

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148 SOLVING INVENTORY MANAGEMENT PROBLEMS

Table 4.1 Average cost per week (in levs) in the Pansko problem. The average ¬xed cost,

the average variable cost and the average shortage costs are reported in brackets.

S

s 1500 2000 2500

800 1120.8 625.0 994.9

(337.8 + 168.5 + 614.5) (224.8 + 236.8 + 163.3) (152.3 + 330.2 + 512.3)

900 644.7 622.9 908.9

(447.6 + 184.6 + 12.4) (225.0 + 236.8 + 161.0) (162.9 + 339.3 + 406.6)

1000 625.0 623.0 724.3

(450.0 + 184.9 + 0.0) (225.0 + 236.8 + 161.0) (197.9 + 375.7 + 150.5)

1100 635.0 612.7 634.6

(450.0 + 184.0 + 0.0) (229.7 + 239.1 + 143.9) (222.2 + 403.3 + 9.0)

1200 635.0 622.7 631.8

(450.0 + 185.0 + 0.0) (291.2 + 276.3 + 55.1) (224.9 + 406.8 + 0.0)

Table 4.2 Daily sales of tomato juice (in bottles) during the ¬rst week of

December last in a Browns supermarket.

Day Sales Inventory level

1 Dec 850 8510

2 Dec 576 7934

3 Dec 932 7002

4 Dec 967 6035

5 Dec 945 5090

6 Dec 989 4101

7 Dec 848 3253

to Los Alamos (New Mexico, USA) the inventory level on 1 December last was 780

boxes of 12 bottles each. Last 6 December, the inventory level was less than 400

boxes and an order of 400 boxes was issued (see Table 4.2). The order was ful¬lled

the subsequent day.

4.9 Selecting an Inventory Policy

It is quite common for a warehouse to contain several hundreds (or even thousands)

of items. In such a context, goods having a strong impact on the total cost have to be

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SOLVING INVENTORY MANAGEMENT PROBLEMS 149

managed carefully while for less important goods it is wise to resort to simple and

low-cost techniques.

The problem is generally tackled by clustering the goods into three categories (indi-

cated with the symbols A, B and C) on the basis of the average value of the goods

in stock. This method is often called the ABC technique. Category A is made up of

products corresponding to a high percentage (e.g. 80%) of the total warehouse value.

Category B is constituted by a set of items associated with an additional 15% of

the warehouse value, while category C is formed by the remaining items. Goods are

subdivided into these categories as follows: ¬rst, commodities are sorted by nonin-

creasing values with respect to the average value of the goods in stock; the items are

then selected from the sorted list, to reach the pre-established cumulated value levels.

On the basis of the 80“20 principle (or Pareto principle), category A usually con-

tains a small fraction (generally, 20“30%) of the goods whereas category C includes

many products. This observation suggests that the goods of categories A and B should

be managed with policies based on forecasts and a frequent monitoring (e.g. category

A by means of the reorder level inventory method and category B through the reorder

cycle inventory policy). Products in category C can be managed using the two-bin

policy that does not require any forecast.

The Walloon Transportation Consortium (WTC) operates a Belgian public trans-

portation service in the Walloon region. Buses are maintained in a facility located in

Ans, close to a vehicle depot. The average inventory levels, the unit values and the

total average value of the spare parts kept in stock are reported in Table 4.3. It was

decided to allocate to category A the products corresponding approximately to the

¬rst 80% of the total value of the stock, to category B the items associated with the

following 15%, and to category C the remaining commodities (see Table 4.4). It is

worth noting that category A contains about 30% of the goods, while each of the cat-

egories B and C accounts for about 35% of the inventory. The cumulated percentage

of the total value as a function of the cumulated percentage of the number of items

(Pareto curve) is reported in Figure 4.13.

4.10 Multiple Stocking Point Models

Good inventory policies for multiple interdependent stocking points can be very dif-

¬cult to devise. In this section a very simple model is described and analysed. In a

decentralized logistics system, a market is divided into n identical sales districts, each

of which is allocated to a warehouse, while in a centralized system every customer

is serviced by a unique facility. Under the EOQ hypotheses, the average inventory

levels of the two systems are linked by the following square-root law.

Property. If the EOQ hypotheses hold, and each warehouse in the decentralized

¯

system services the same demand, then the total average inventory level I (n) in the

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150 SOLVING INVENTORY MANAGEMENT PROBLEMS

Table 4.3 Spare parts stocked by WTC.

Product Average Average unit Total average

code stock value (in euros) value (in euros)

AX24 137 50 6 850

BR24 70 2 000 140 000

BW02 195 250 48 750

CQ23 6 6 000 36 000

CR01 16 500 8 000

FE94 31 100 3 100

LQ01 70 2 500 175 000

MQ12 18 200 3 600

MW20 75 500 37 500

NL01 15 1 000 15 000

PE39 16 3 000 48 000

RP10 20 2 200 44 000

SP00 13 250 3 250

TA12 100 2 500 250 000

TQ23 10 5 000 50 000

WQ12 30 12 000 360 000

WZ34 30 15 450

ZA98 70 250 17 500

Cumulated % of the total value

100.00

90.00

80.00

70.00

60.00

50.00

40.00

Class A Class B Class C

30.00

20.00

10.00

0.00

Cumulated % of the

0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 100.00

number of products

Figure 4.13 Pareto curve in the WTC problem.

decentralized system is

√ (1)

¯ ¯

I (n) = nI ,

¯

where I (1) is the average inventory level in the centralized system.

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SOLVING INVENTORY MANAGEMENT PROBLEMS 151

Table 4.4 ABC classi¬cation of the spare parts in the WTC problem.

Total Total

Fraction Cumulated % average cumulated Cumulated

Product of total of the number value value % of the

code inventory of products (in euros) (in euros) total value Class

WQ12 3.3 3.30 360 000 360 000 28.87

TA12 10.8 14.10 250 000 610 000 48.92

LQ01 7.6 21.70 175 000 785 000 62.95 A

BR24 7.6 29.30 140 000 925 000 74.18

TQ23 1.1 30.40 50 000 975 000 78.19

BW02 21.1 51.50 48 750 1 023 750 82.10

PE39 1.8 53.30 48 000 1 071 750 85.95

RP10 2.1 55.40 44 000 1 115 750 89.47 B

MW20 8.2 63.60 37 500 1 153 250 92.48

CQ23 0.6 64.20 36 000 1 189 250 95.37

ZA98 7.6 71.80 17 500 1 206 750 96.77

NL01 1.6 73.40 15 000 1 221 750 97.98

CR01 1.8 75.20 8 000 1 229 750 98.62

AX24 14.8 90.00 6 850 1 236 600 99.17 C

MQ12 2.0 92.00 3 600 1 240 200 99.45

SP00 1.4 93.40 3 250 1 243 450 99.72

FE94 3.3 96.70 3 100 1 246 550 99.96

WZ34 3.3 100.00 450 1 2470,00 100.00

Proof. In the EOQ model the average inventory level is half the order size. Therefore,

1√

¯

I (1) is equal to 2 2kd/ h, where d is the demand of the whole market. In a decen-

¯

tralized system, I (n) is the sum of the average inventory levels of the facilities, each

1√

¯

of which services 1/n of demand d. Hence, I (n) = 2 n 2k(d/n)/ h.

Kurgantora distributes tyres in Russia and Kazakhstan. The distribution network

currently includes 12 warehouses, each of which serves approximately the same

demand. In an attempt to reduce the total inventory level by 30%, the company

has decided to close some warehouses and allocate their demand to the remaining

facilities. Applying the square-root law, we see that the number of stocking points

should be reduced to 6, since

√

¯ ¯

I (12) = 12 I (1) ,

√

¯ ¯

I (n ) = n I (1) ,

√√

¯ ¯

I (n ) /I (12) = n / 12 = 0.7,

n = 5.88.

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152 SOLVING INVENTORY MANAGEMENT PROBLEMS

4.11 Slow-Moving Item Models

As shown in the previous sections, a major issue for fast-moving product inventory

management is determining how often reorders should take place. On the other hand,

if demand is very low (e.g. a few units in 10“20 years), as in the case of spare parts

of a complex machinery (slow-moving products), the main issue is determining the

number of items to be purchased at the beginning of the machinery™s life cycle.

In this section we examine a model in which item purchase cost and shortage

penalties are taken into account while holding cost and salvage value (i.e. the value

of unused spare parts at the end of the machinery lifetime) are negligible. Let c and

u be the purchase cost of an item at the beginning of the planning horizon and during

the planning horizon, respectively (c < u). If n units of product are purchased at the

beginning of the planning period, and m units are demanded in the planning period,

the total cost is

C(n, m) = cn, if n m;

C(n, m) = cn + u(m ’ n), if n < m.

¯

Let P (m) be the probability that m items are demanded. Then, the expected cost C(n)

in case n items are purchased is

∞ ∞

¯

C(n) = C(n, m)P (m) = cn + u (m ’ n)P (m).

m=0 m=n+1

Hence,

¯ ¯

C(n ’ 1) = C(n) ’ c + u[1 ’ F (n ’ 1)], (4.42)

¯ ¯

C(n + 1) = C(n) + c ’ u[1 ’ F (n)], (4.43)

where F (n) is the probability that n units (or less) are demanded. The minimum

expected cost is achieved if n— items are purchased at the beginning of the planning

period:

¯ ¯

C(n— ’ 1) C(n— ), (4.44)

¯ ¯

C(n— + 1) C(n— ). (4.45)

Finally, combining Equation (4.44) and Equation (4.42), the following relation is

obtained:

u’c

F (n— ’ 1) .

u

Similarly, combining Equation (4.45) and Equation (4.43) for n = n— gives

u’c

F (n— ) .

u

Consequently, the optimal number of items n— to be purchased is such that

u’c

F (n— ’ 1) F (n— ). (4.46)

u

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SOLVING INVENTORY MANAGEMENT PROBLEMS 153

Hydro Ltd uses ¬ve LIF03 generators in its hydroelectric power plants located in

Nigeria. Each piece of machinery has an average life of 20 years, during which the

expected number of engine failures is equal to 1.4. The cost of a spare part, purchased

when a generator is manufactured, is $60 000 while producing an additional unit costs

around $300 000. The failure process is modelled as a Poisson probability distribution

with expected value »,

» = 5 — 1.4 = 7 faults per life cycle.

Therefore, the probability P (n) that the number of demanded spare parts equals n

is given by

e’» »n

P (n) = , n = 0, 1, . . . ,

n!

while the cumulative probability F (n) is

F (0) = P (0),

n

e’» »k /k! = F (n ’ 1) + P (n),

F (n) = n = 1, 2, . . . .

k=0

The values of P (n) and F (n) for n = 0, . . . , 10 are reported in Table 4.5.

Since

u’c 300 000 ’ 60 000

= = 0.8,

c 300 000

on the basis of Equation (4.46), n— = 9 spare parts should be purchased and stocked.

4.12 Policy Robustness

The inventory policies illustrated in the previous sections often have to be slightly

modi¬ed in order to be used in practice. Fractional order sizes and shipment frequen-

cies have to be suitably rounded up or down (e.g. q — = 14.43 pallets should become

14 or 15 pallets). Fortunately, the total cost is not very sensitive to variations of the

order size around the optimal value. For the EOQ model the following property holds.

Property. In the EOQ model, errors in excess of 100% on the optimal order size

cause a maximum increase of the total cost equal to 25%.

Proof. Recall that the average cost in the EOQ model is given by

µ(q) ’ cd = kd/q + hq/2.

√

If q = q — = 2kd/ h (see Equation (4.13)), then

√

µ(q — ) ’ cd = 2kdh.

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154 SOLVING INVENTORY MANAGEMENT PROBLEMS

Table 4.5 Probability distribution of spare part demand in the Hydro Ltd problem.

n P (n) F (n)

0 0.0009 0.0009

1 0.0064 0.0073

2 0.0223 0.0296

3 0.0521 0.0818

4 0.0912 0.1730

5 0.1277 0.3007

6 0.1490 0.4497

7 0.1490 0.5987

8 0.1304 0.7291

9 0.1014 0.8305

10 0.0710 0.9015

If q = 2q — , then √

µ(2q — ) ’ cd = 5

2kdh.

4

Therefore,

µ(2q — ) ’ cd

= 1.25

µ(q—) ’ cd

and

1.25(µ(q — ) ’ cd), q ∈ [q — , 2q — ].

µ(q) ’ cd

Similarly, it can be shown that rounding the reorder size to the closest power of 2

(power-of-two policy) induces a maximum cost increase of about 6%.

In the Al-Bufeira Motors problem (see Section 4.4.2), the total cost associated with

¬ve shipments per year (i.e. q = 44 units),

800 — 220 216 — 44

µ(q — ) = + = 8752 dollars per year,

44 2

is higher than the optimal solution (µ(q — ) = 8719.63 dollars per year) only by 0.37%.

4.13 Questions and Problems

4.1 In most industrialized countries the average ITR is around 20 for dairy products

and around 5 for household electrical appliances. Discuss these ¬gures.

4.2 Modify the EOQ formula for the case where the stocking point has a ¬nite

capacity Q.

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SOLVING INVENTORY MANAGEMENT PROBLEMS 155

4.3 Modify the EOQ formula for the case where the holding cost is a concave

function of the number of items kept in inventory.

4.4 Devise an optimal inventory policy for the EOQ model with a ¬nite time horizon

TH .

4.5 Modify the EOQ formula for the case where the order size q is delivered by a

number of vehicles of capacity qv each having a ¬xed cost kv .

4.6 Draw the auxiliary graph used for solving the Sao Vincente Chemical problem

as a shortest-path problem.

4.7 Modify the Wagner“Within model for the case where the stocking point is

capacitated. Does the ZIO property still hold?

4.8 What is the optimal order quantity in the Newsboy Problem if the stocking

point is capacitated?

4.9 If type A products are overstocked, the total cost increases dramatically, while if

type C products are overstocked, the total cost does increase too much. Calculate

the cost increase whenever the inventory level of A products is increased by

20%. Repeat the calculation for C products.

4.10 Show that the power-of-two policy induces a maximum cost increase of about

6%.

4.14 Annotated Bibliography

An in-depth treatment of inventory management can be found in:

1. Zipkin PH 2000 Foundations of Inventory Management. McGraw-Hill, New

York.

A simpli¬ed approach is available in:

2. Lewis CD 1998 Demand Forecasting and Inventory Control: A Computer Aided

Learning Approach. Wiley, New York.

An introduction to the use of simulation methods with some applications on the

inventory management is:

3. Law L and Kelton WD 2000 Simulation Modelling and Analysis, 3rd edn.

McGraw-Hill, New York.

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5

Designing and Operating a

Warehouse

5.1 Introduction

Warehouses are facilities where inventories are sheltered. They can be broadly classi-

¬ed into production warehouses and DCs. This chapter deals with warehouse design

and operation, with an emphasis on DCs. In the following, a product is de¬ned as

a type of good, e.g. wine bottles of a speci¬c brand. The individual units are called

items (or stock keeping units (SKUs)). A customer order is made up of one or more

items of one or more products.

Flow of items through the warehouse. Warehouses are often used not only to

provide inventories a shelter, but also to sort or consolidate goods. In a typical DC,

the products arriving by truck, rail, or internal transport are unloaded, checked and

stocked. After a certain time, items are retrieved from their storage locations and

transported to an order assembly area. In the simplest case (which occurs frequently

in CDCs (see Chapter 1)), the main activity is the storage of the goods. Here, the

merchandise is often received, stored and shipped in full pallets (all one product)

and, as a result, material handling is relatively simple (see Figure 5.1). In the most

complex case (which occurs frequently in RDCs (see Chapter 1)) large lots of prod-

ucts are received and shipments, containing small quantities of several items, have to

be formed and dispatched to customers. Consequently, order picking is quite com-

plex, and product sorting and consolidation play a mayor role in order assembly (see

Figure 5.2).

Ownership of the warehouses. With respect to ownership, there are three main

typologies of warehouses. Company-owned warehouses require a capital investment

in the storage space and in the material handling equipment. They usually represent

the least-expensive solution in the long run in the case of a substantial and constant

demand. Moreover, they are preferable when a higher degree of control is required

to ensure a high level of service, or when specialized personnel and equipment are

Introduction to Logistics Systems Planning and Control G. Ghiani, G. Laporte and R. Musmanno

© 2004 John Wiley & Sons, Ltd ISBN: 0-470-84916-9 (HB) 0-470-84917-7 (PB)

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158 DESIGNING AND OPERATING A WAREHOUSE

Article A

Receiving

Holding

Picking

Shipping

Figure 5.1 The ¬‚ow of items through the warehouse; the goods are received and

shipped in full pallets or in full cartons (all one product).

Article A Article B Article C

Receiving Receiving Receiving

Holding Holding Holding

Picking Picking Picking

Batch forming

Packaging

Shipping

Figure 5.2 The ¬‚ow of items through the warehouse; the goods are received in full pallets

or in full cartons (all one product) and shipped in less than full pallets or cartons.

needed. Finally, they can be employed as a depot for the company™s vehicles or as a

base for a sales of¬ce. Public warehouses are operated by ¬rms providing services

to other companies on a short-term basis. As a rule, public warehouses have stan-

dardized equipment capable of handling and storing speci¬c types of merchandise

(e.g. bulk materials, temperature-controlled goods, etc.). Here, all warehousing costs

are variable, in direct proportion to the storage space and the services required. As

a result, it is easy and inexpensive to change warehouse locations as demand varies.

For these reasons, public warehouses can suitably accommodate seasonal inventories.

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DESIGNING AND OPERATING A WAREHOUSE 159

Receiving

17%

Holding Handling

15% 50%

Shipping

18%

Figure 5.3 Common warehouse costs.

Finally, leased warehouse space is an intermediate choice between short-term space

rental and the long-term commitment of a company-owned warehouse.

Warehouse costs. The total annual cost associated with the operation of a ware-

house is the result of four main activities: receiving the products, holding inventories

in storage locations, retrieving items from the storage locations, assembling customer

orders and shipping. These costs depend mainly on the storage medium, the stor-

age/retrieval transport technology and its policies. As a rule, receiving the incoming

goods and, even more so, forming the outgoing lots, are operations that are dif¬-

cult to automate and often turn out to be labour-intensive tasks. Holding inventories

depends mostly on the storage medium, as explained in the following. Finally, picking

costs depend on the storage/retrieval transport system which can range from a fully

manual system (where goods are moved by human pickers travelling on foot or by

motorized trolleys) to fully automated systems (where goods are moved by devices

under the control of a centralized computer). Common warehouse costs are reported

in Figure 5.3.

5.1.1 Internal warehouse structure and operations

The structure of a warehouse and its operations are related to a number of issues:

• the physical characteristics of the products (on which depends whether the

products have to be stored at room temperature, in a refrigerated or ventilated

place, in a tank, etc.);

• the number of products (which can vary between few units to tens of thousands);

• the volumes handled in and out of the warehouse (which can range between a

few items per month to hundreds of pallets per day).

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160 DESIGNING AND OPERATING A WAREHOUSE

Receiving zone

Storage zone

Shipping zone

Figure 5.4 Warehouse with a single receiving zone and a single shipping zone.

Typically, in each DC there are (see Figure 5.4)

• one or more receiving zones (each having one or more rail or truck docks),

where incoming goods are unloaded and checked;

• a storage zone, where SKUs are stored;

• one or more shipping zones (each having one or more rail or truck docks), where

customer orders are assembled and outgoing vehicles are loaded.

The storage zone is sometimes divided into a large reserve zone where products

are stored in the most economical way (e.g. as a stack of pallets), and into a small

forward zone, where goods are stored in smaller amounts for easy retrieval by order

pickers (see Figure 5.5). The transfer of SKUs from the reserve zone to the forward

zone is referred to as a replenishment. If the reserve/forward storage is well-designed,

the reduction in picking time is greater than replenishment time.

5.1.2 Storage media

The choice of a storage medium is strongly affected by the physical characteristics of

the goods in stock and by the average number of items of each product in a customer

order. Brie¬‚y, when storing solid goods three main alternatives are available: stacks,

racks and drawers. In the ¬rst case, goods are stored as cartons or as pallets, and

aisles are typically 3.5“4 m wide (see Figure 5.6). Stacks do not require any capital

investment and are suitable for storing low-demand goods, especially in reserve zones.

In the second case, goods are stored as boxes or pallets on metallic shelves separated

by aisles. Here quick picking of single load units is possible. When SKUs are moved

by forklifts, the racks (see Figure 5.7) are usually 5“6 m tall and aisles are around

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DESIGNING AND OPERATING A WAREHOUSE 161

Receiving Reserve Forward Shipping

zone zone zone zone

Figure 5.5 Warehouse with reserve and forward storage zones.

Figure 5.6 Block stacking system.

3.5 m wide. Instead, as explained in the following, in automated storage and retrieval

systems (AS/RSs), racks are typically 10“12 m tall and aisles are usually 1.5 m wide

(see Figure 5.8). Finally, in the third case, items are generally of small size (e.g.

metallic small parts), and are kept in ¬xed or rotating drawers.

5.1.3 Storage/retrieval transport mechanisms and policies

A common way of classifying warehouses is the method by which items are retrieved

from storage.

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162 DESIGNING AND OPERATING A WAREHOUSE

Figure 5.7 Rack storage.

Figure 5.8 An AS/RS.

Picker-to-product versus product-to-picker systems. The picking operations can

be made

• by a team of human order pickers, travelling to storage locations (picker-to-

product system);

• by an automated device, delivering items to stationery order pickers (order-to-

picker system).

Clearly, mixed solutions are possible. For instance, in picker-to-belt systems, the

items are retrieved by a team of human order pickers and then transported to the order

assemblers by a belt conveyor. Picker-to-product systems can be further classi¬ed

according to the mode of travel inside the warehouse. In person-aboard AS/RS, pickers

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DESIGNING AND OPERATING A WAREHOUSE 163

y

x

Figure 5.9 Item retrieval by trolley.

are delivered to storage locations by automated devices which are usually restricted

to a single aisle. In walk/ride and pick systems (W/RPSs), pickers travel on foot or by

motorized trolleys and may visit multiple aisles (see Figure 5.9).

The most popular order-to-picker systems are the AS/RSs. An AS/RS consists of

a series of storage aisles, each of which is served by a single storage and retrieval

(S/R) machine or crane. Each aisle is supported by a pick-up and delivery station

customarily located at the end of the aisle and accessed by both the S/R machine and

the external handling system. Therefore, assuming that the speeds vx and vy along

the axes x and y (see Figure 5.10) are constant, travel times t satisfy the Chebychev

metric,

x y

t = max , ,

vx vy

where x and y are the distances travelled along the x- and y-axes.

AS/RSs were introduced in the 1950s to eliminate the walking that accounted for

nearly 70% of manual retrieval time. They are often used along with high racks and

narrow aisles (see Figure 5.11). Hence, their advantages include savings in labour

costs, improved throughput and high ¬‚oor utilization.

Unit load retrieval systems. In some warehouses it is possible to move a single

load at a time (unit load retrieval system), because of the size of the loads, or of

the technological restrictions of the machinery (as in AS/RSs). In AS/RSs, an S/R

machine usually operates in two modes:

• single cycle: storage and retrieval operations are performed one at a time;

• dual cycle: pairs of storage and retrieval operations are made in sequence in an

attempt to reduce the overall travel time.

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164 DESIGNING AND OPERATING A WAREHOUSE

Side aisle

Central aisle

Figure 5.10 An S/R machine.

Side aisle

Figure 5.11 Item storage and retrieval by an AS/RS and a belt conveyor.

There exist systems in which it is possible to store or pick up several loads at the

same time (multi-command cycle).

Strict order picking versus batch picking. In multiple load retrieval systems,

customer orders can be assigned to pickers in two ways:

• each order is retrieved individually (strict order picking);

• orders are combined into batches.

In the latter case, each batch may be retrieved by a single picker (batch picking).

Otherwise, the warehouse is divided into a given number of zones and each picker is

in charge of retrieving items from a speci¬c zone (zone picking).

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DESIGNING AND OPERATING A WAREHOUSE 165

5.1.4 Decisions support methodologies

Warehouses are highly dynamic environments where resources have to be allocated in

real-time to satisfy customer orders. Because orders are not fully known in advance,

design and operational decisions are affected by uncertainty. To overcome the inher-

ent dif¬culty of dealing with complex queueing models or stochastic programs, the

following approach is often used.

Step 1. A limited number of alternative solutions are selected on the basis of expe-

rience or by means of simple relations linking the decision variables and simple

statistics of customer orders (the average number of orders per day, the average

number of items per order, etc.).

Step 2. Each alternative solution generated in Step 1 is evaluated through a detailed

simulation model and the best solution (e.g. with respect to throughput) is selected.

5.2 Warehouse Design

Designing a warehouse amounts to choosing its building shell, as well as its layout

and equipment. In particular, the main design decisions are

• determining the length, width and height of the building shell;

• locating and sizing the receiving, shipping and storage zones (e.g. evaluating

the number of I/O ports, determining the number, the length and the width of

the aisles of the storage zone and the orientation of stacks/racks/drawers);

• selecting the storage medium;

• selecting the storage/retrieval transport mechanism.

The objective pursued is the minimization of the expected annual operating cost

for a given throughput, usually subject to an upper bound on capital investment.

In principle, the decision maker may choose from a large number of alternatives.

However, in practice, several solutions can be discarded on the basis of a qualitative

analysis of the physical characteristics of the products, the number of items in stock

and the rate of storage and retrieval requests. In addition, some design decisions are

intertwined. For instance, when choosing an AS/RS as a storage/retrieval transport

mechanism, rack height can be as high as 12 m, but when traditional forklifts are used

racks must be much lower. As a result, each design problem must be analysed as a

unique situation.

Following the general framework introduced at the end of Section 5.1, we will

make in the remainder of this section a number of qualitative remarks and we will

illustrate two simple analytical procedures.

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166 DESIGNING AND OPERATING A WAREHOUSE

5.2.1 Selecting the storage medium and the storage/retrieval

transport mechanism

The choice of storage and retrieval systems is in¬‚uenced by the physical characteristics

of the goods, their packaging at the arrival and the composition of the outgoing

lots. For example, in a single storage zone warehouse, palletized goods are usually

stocked on racks if their demand is high enough, otherwise, stacks are used.Automated

systems are feasible if the goods can be automatically identi¬ed through bar codes

or other techniques. They have low space and labour costs, but require a large capital

investment. Hence, they are economically convenient provided that the volume of

goods is large enough.

5.2.2 Sizing the receiving and shipment subsystems

The receiving zone is usually wider than the shipping area. This is because the incom-

ing vehicles are not under the control of the warehouse manager, while the formation of

the outgoing shipments can be planned in order to avoid congesting the output stations.

Determining the number of truck docks

Goods are usually received and shipped by rail or by truck. In the latter case, the

number of docks nD can be estimated through the following formula,

dt

nD = ,

qT

where d is the daily demand from all orders, t is the average time required to

load/unload a truck, q is the truck capacity, and T is the daily time available to

load/unload trucks.

Sintang is a third-party Malaysian ¬rm specialized in manufacturing electronic

devices. A new warehouse has been recently opened in Kuching. It is used for storing

digital satellite receivers, whose average daily demand is d = 27 000 units. Outgoing

shipments are performed by trucks, with a capacity equal to 850 boxes. Since the

average time to load a truck is t = 280 min and 15 working hours are available every

day, the warehouse has been designed with the following number nD of docks:

27 000 — 280

nD = = 10.

850 — 900

5.2.3 Sizing the storage subsystems

The area of the storage zone must be large enough to accommodate goods in peak

periods. On the other hand, if the storage zone exceeds the real needs of the ¬rm,

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DESIGNING AND OPERATING A WAREHOUSE 167

storage and retrieval times become uselessly high. This could decrease throughput or

increase material handling costs.

Determining the capacity of a storage area

The size of a storage area depends on the storage policy. In a dedicated storage

policy, each product is assigned a pre-established set of positions. This approach is

easy to implement but causes an underutilization of the storing space. In fact, the

space required is equal to the sum of the maximum inventory of each product in time.

Let n be the number of products and let Ij (t), j = 1, . . . , n, be the inventory level of

item j at time t. The number of required storage locations md in a dedicated storage

policy is

n

md = max Ij (t). (5.1)

t

j =1

In a random storage policy, item allocation is decided dynamically on the basis of

the current warehouse occupation and on future arrival and request forecast. Therefore,

the positions assigned to a product are variable in time. In this case the number of

storage locations mr is

n

mr = max Ij (t) md . (5.2)

t

j =1

The random storage policy allows a higher utilization of the storage space, but

requires that each item be automatically identi¬ed through a bar code (or a similar

technique) and a database of the current position of all items kept at stock is updated

at every storage and every retrieval.

In a class-based storage policy, the goods are divided into a number of categories

according to their demand, and each category is associated with a set of zones where

the goods are stored according to a random storage policy. The class-based storage

policy reduces to the dedicated storage policy if the number of categories is equal to

the number of items, and to the random storage policy if there is a single category.

Potan Up bottles two types of mineral water. In the warehouse located in Hangzhou

(China), inventories are managed according to a reorder level policy (see Chapter 4).

The sizes of the lots and of the safety stocks are reported in Table 5.1. Inventory levels

as a function of time are illustrated in Figures 5.12 and 5.13. The company is currently

using a dedicated storage policy. Therefore, the number of storage locations is given

by Equation (5.1):

md = 600 + 360 = 960.

The ¬rm is now considering the opportunity of using a random storage policy. The

number of storage locations required by this policy would be (see Equation (5.2))

mr = 600 + 210 = 810.

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168 DESIGNING AND OPERATING A WAREHOUSE

Table 5.1 Lots and safety stocks (both in pallets) in the Potan Up problem.

Product Lot Safety stock

Natural water 500 100

Sparkling water 300 60

I(t)

600

350

100

t

Figure 5.12 Inventory level of natural mineral water in the Potan Up problem.

I(t)

360

210