located in London (Ontario, Canada) with highly automated systems operating with

a very low work-in-process. The raw materials originate from mines located close

to the factory. Consequently, the ¬rm does not need to stock a large amount of raw

materials (as a rule, no more than a two-week demand). As far as the ¬nished product

inventories are concerned, Wolferine makes use of EOQ models (see Section 4.4.2).

In the autumn of 1980, the ¬rm operated with a production level close to the plant™s

capacity. At that time the interest rate was around 10%. Using this value in the EOQ

model, the company set the ¬nished goods inventory level equal to 833 tons. During

the subsequent two years, an economic recession hit the industrialized countries. The

interest rate underwent continuous and quick variations (up to 20% in August 1981),

the demand of ¬nished products went down by 20%, and the price level increased

sharply. According to the EOQ model, the ¬nished goods inventory level should

have been lower under those conditions. In order to illustrate this result, let n be the

number of products; k an order ¬xed cost (assumed independent from the product);

di , i = 1, . . . , n, the annual demand of the product i; p the interest rate (increased to

¯

take into account warehousing costs); ci , i = 1, . . . , n, the price of product i; I the

average stock level at time period t0 (January 1981). On the basis of Equation (4.28),

n

1 2kdi

¯

I (t0 ) = .

pci

2

i=1

We can express parameter p as the sum of a bank interest rate p1 and of a rate p2

associated with warehousing costs:

p = p1 + p2 .

Moreover, let δ1 , δ2 and δ3 be the variation rates (assumed equal for all products) of

price, demand and interest rate at time period t, respectively. The average stock level

is equal to

n

2[k(1 + δ1 )][di (1 + δ2 )]

1

¯

I (t) =

[p1 (1 + δ3 ) + p2 ][ci (1 + δ1 )]

2

i=1

(1 + δ2 )p

¯

= I (t0 ) . (8.18)

p1 (1 + δ3 ) + p2

According to Equation (8.18), if demand decreases (δ2 < 0) and the interest rate

increases (δ3 > 0), the stock level should be lower. However, the managers of Wolfer-

ine continued to operate as in 1981. As a result, the ITR (see Section 4.1) suddenly

decreased. Moreover, to protect the manufacturing process against strikes at the mines,

the ¬rm also decided to hold an inventory of raw materials. Consequently, when the

recession ended in 1983, the ¬rm had an exceedingly large stock of both raw materials

and ¬nished products.

TLFeBOOK

322 LINKING THEORY TO PRACTICE

% container

loading success

100

90

80

70

60

50

40

30

20

% of used

10

capacity

73 78 83 87 92 94 100

Figure 8.15 Percentage of success in loading an aircraft as a function of

the percentage of load capacity (simulation made by FedEx).

8.11 Airplane Loading at FedEx

FedEx is one of the leading express carriers in the world, with a freight traf¬c estimated

at about 2 million parcels per day. Its sales of¬ces are located in 187 countries and

the company uses a ¬‚eet of 437 airplanes and about 30 000 trucks and vans.

In the USA, parcels whose origin and destination exceed a given distance are

consolidated in containers and sent by air. An airplane may ¬‚y between a pair of

destinations or may follow a multi-stop route where containers are loaded or unloaded

at intermediate stops.

In order to use airplane capacity ef¬ciently, a key issue is to devise good loading

plans, taking into account a number of aspects: the load must be balanced around the

centre of gravity of the aircraft, the total weight in the various areas of the aircraft

must not exceed given thresholds in order to limit the cutting forces on the plane,

etc. These aspects are critical, especially for some planes, such as the Airbus A300,

a low fuel consumption aircraft. In addition, when loading an airplane assigned to a

multi-stop route, containers to be unloaded at intermediate stops must be positioned

close to the exit.

In order to allow airplanes to take off on time, allocating containers on-board must

be done in real time, i.e. containers must be loaded on the aircraft as soon as they arrive

at the airport. As a matter of fact, 30% to 50% of the containers are already on board

when the ground staff has a complete knowledge of the features of the containers

to be loaded. Of course, once some containers have been loaded, it may become

impossible to load the subsequent containers. As a result, when new containers arrive

at the airport, it is sometimes necessary to de¬ne a new loading plan in which some

containers previously loaded are unloaded. This situation arises frequently when the

total load is close to the capacity of the aircraft, as shown in Figure 8.15, where the

percentage of success in loading an airplane is reported as a function of the percentage

of the load capacity used.

TLFeBOOK

LINKING THEORY TO PRACTICE 323

The objective pursued by FedEx consists of loading the largest number of con-

tainers as possible. If no container is yet loaded, the following solution procedure is

implemented.

Step 1. Let m be the number of containers to be loaded, n the number of positions

in the loading area, q the number of areas into which the plane is divided, pi , i =

1, . . . , m, the weight of container i; Pj , j = 1, . . . , n, the maximum weight that

can be loaded in position j ; dj , j = 1, . . . , n, the distance from position j to the

centre of gravity O, M min and M max the minimum and maximum moments of the

loads with respect to O, Lk , k = 1, . . . , q, the total maximum weight that can

be placed in area k; fj k , j = 1, . . . , n, k = 1, . . . , q, the fraction of position

j contained in area k. Also let xij , i = 1, . . . , m, j = 1, . . . , n, be a binary

decision variable equal to 1 if container i is placed in position j , and 0 otherwise,

uj , j = 1, . . . , n, a binary decision variable equal to 1 if position j is used, and 0

otherwise. A feasible solution is de¬ned by the following set of constraints:

n

xij = 1, i = 1, . . . , m, (8.19)

j =1

m

j = 1, . . . , n,

xij muj , (8.20)

i=1

m

j = 1, . . . , n,

pi xij P j uj , (8.21)

i=1

m n

M max ,

dj pi xij (8.22)

i=1 j =1

mn

M min ,

dj pi xij (8.23)

i=1 j =1

m n

k = 1, . . . , q,

pi fj k xij Lk , (8.24)

i=1 j =1

xij ∈ {0, 1}, i = 1, . . . , m, j = 1, . . . , n, (8.25)

uj ∈ {0, 1}, j = 1, . . . , n. (8.26)

Constraints (8.19) guarantee that each container is allocated to a position. Con-

straints (8.20) state that if a position j, j = 1, . . . , n, accommodates a container,

then its uj variable must be equal to 1. Constraints (8.21) ensure that the total

weight loaded in any position does not exceed a pre-established upper bound. Con-

straints (8.22) and (8.23) impose that the total moment, with respect to point O,

TLFeBOOK

324 LINKING THEORY TO PRACTICE

is within the pre-established interval. Constraints (8.24) ensure the respect of the

weight bounds in each section.

¯

Step 2. If problem (8.19)“(8.26) is infeasible, a container i is eliminated from the

loading list and the problem is solved again (where, of course, xij = 0, j =

¯

1, . . . , n). Step 2 is repeated until a feasible solution is found.

If some containers have already been loaded, the previous procedure is modi¬ed

¯ ¯

as follows. Let I be the set of the containers already loaded and let ji , i ∈ I , be the

position assigned to container i. Then, additional constraints,

¯

xiji = 1, i ∈ I, (8.27)

are added to (8.19)“(8.26). If this problem is feasible, the procedure stops since

the partly executed loading plan can be completed. Otherwise, the constraint (8.27)

¯

associated with container i ∈ I allocated to the position closest to the entrance/exit

is removed (this corresponds to unloading the container from the aircraft). This step

is repeated until a feasible solution is obtained.

8.12 Container Loading at Waterworld

Like other commodities derived from wood, paper is typically produced a long way

from the main points of consumption and then often transported by sea in containers.

As a rule, the transportation cost accounts for a signi¬cant part of the selling price.

Moreover, competition among manufacturers is ¬erce so that pro¬ts are very low. In

this context, it is crucial to reduce transportation costs as much as possible. To this

purpose, a key issue is to consolidate loads ef¬ciently.

Paper is usually transported in rolls having a diameter ranging between 0.5 m and

1.5 m, or in large size sheets. In the ¬rst case, rolls are loaded directly into containers,

while in the second case, the sheets are loaded onto pallets which are put in containers.

Generally, an order is composed of a set of different products of varying size, thickness

and density. A typical order can ¬ll several tens of containers.

Waterworld determines its own loading plans through an ad hoc heuristic procedure

outlined below.

8.12.1 Packing rolls into containers

Rolls vary in diameter and height. Roll packing amounts to loading rolls into con-

tainers with the circular base parallel to the bottom surface. The packing procedure

is divided into two steps.

Step 1. In the ¬rst step, each roll is characterized by its circular section and the aim

is to determine the least number of rectangles (representing bottom surfaces) that

can accommodate such sections (referred to as objects in the following). Let (X, Y )

be a Cartesian reference system with origin in the bottom left corner of a container.

TLFeBOOK

LINKING THEORY TO PRACTICE 325

L

W W

Figure 8.16 Two ways circular objects can be loaded in the Waterworld problem.

The objects are introduced in the container one at a time, the position of each

object being de¬ned in such a way that »1 x + »2 y is minimized ((x, y) are the

coordinates of the centre of the object, »1 and »2 are two positive constants). The

choice of (x, y) must result in a feasible solution, i.e. the object must be entirely

kept in the container and must not be overlapping the objects already inserted (see

Figure 8.16). The following combinations of parameters »1 and »2 were used:

(a) »1 = 1 and »2 = 0; (b) »1 = 0 and »2 = 1; (c) »1 = »2 = 0.5.

Step 2. With each cluster of rolls formed in the ¬rst step is associated a weight equal to

the height of the tallest roll of the cluster. Then, a classical 1-BP (see Section 5.4.3)

is solved heuristically in order to determine the least number of containers (each

characterized by a capacity equal to container height) that can accommodate all

clusters (see Figure 8.17).

8.12.2 Packing pallets into containers

Paper sheets are mounted on pallets in such a way that the total height of two stacked

pallets is equal to the container height. These pallet pairs are then packed through a

2-BP heuristic (see Section 5.4.3). The procedure followed by Waterworld is a slight

variant of the BL method (see Section 5.4.3).

8.13 Air Network Design at Intexpress

Intexpress is a ¬rm whose core business is express freight delivery all over North

America. The services provided to customers are (a) delivery within 24 hours (next

day service); (b) delivery within 48 hours (second day service); (c) delivery within 3“5

days (deferred service). In order to provide quick deliveries, long-haul transportation

is made by plane.

TLFeBOOK

326 LINKING THEORY TO PRACTICE

H

L

Figure 8.17 Roll packing in the Waterworld example.

The Intexpress logistics system comprises a set of shipment centres (SCs), of a

single hub, of a ¬‚eet of airplanes and of a ¬‚eet of trucks. Loads are consolidated

both in the SCs and in the hub. In particular, goods originating from the same SC are

transported as a single load to the hub while all the goods assigned to the same SC are

sent jointly from the hub. Freight is ¬rst transported to an originating SC, where it is

consolidated; it is then transported to a ¬nal SC by air, by truck (ground service), or

by using a combination of the two modes. Finally, freight is moved from the ¬nal SC

to the destination by truck. Of course, an SC to SC transfer by truck is feasible only

if distance does not exceed a given threshold. Air transportation is performed by a

company-owned ¬‚eet (dedicated air service) or by commercial airlines (commercial

air service). The outgoing freight is collected in the evening and delivered the morning

after. Every day a company-owned aircraft leaves the hub, makes a set of deliveries,

then travels empty from the last delivery point to the ¬rst pick-up point, where it

makes a set of pick-ups and ¬nally goes back to the hub. Each SC is characterized by

an ˜earliest pick-up time™ and by a ˜latest delivery time™. Moreover, all arrivals in the

hub must take place before a pre-established arrival ˜latest delivery time™ (cut-off time,

COT) in such a way that incoming airplanes can be unloaded, sorted by destination,

and quickly reloaded on to outgoing aircraft.

Since it is not economically desirable nor technically feasible that airplanes visit

all SCs, a subset of SCs must be selected as aircraft loading and unloading points (air-

stops, ASs). An SC that is not an AS is connected to an AS by truck (ground feeder

route). Figure 8.18 depicts a possible freight route between an origin“destination pair.

Commercial air services are less reliable than dedicated air services and their costs

are charged depending on freight weight. These are used when either the origin and

its closest SC are so far apart that no quick truck service is possible, or when the

overall transportation demand exceeds the capacity of company-owned aircraft.

Planning the Intexpress service network consists of determining

• the set of ASs served by each aircraft of the ¬rm;

TLFeBOOK

LINKING THEORY TO PRACTICE 327

Shift with a dedicated aircraft

Origin

Consolidate land shift

Local pick-up/delivery

SC

AS

AS

AS

Hub

AS

SC

Destination

AS

Figure 8.18 A possible freight route between an origin“destination pair at Intexpress.

• truck routes linking SCs (which are not ASs) to ASs;

• the transportation tasks performed by commercial airlines.

The objective pursued is the minimization of the operational cost subject to ˜earliest

pick-up time™ and ˜latest delivery time™ constraints at SCs, to the COT restriction, etc.

The solution methodology used by Intexpress is made up of two stages: in the

¬rst stage, the size of the problem is reduced (preprocessing phase) on the basis of a

qualitative analysis; in the second stage, the reduced problem is modelled and solved

as an IP program. In the ¬rst stage,

• origin“destination pairs that can be serviced by truck (in such a way that all

operational constraints are satis¬ed) are allocated to this mode and are not

considered afterwards;

• origin“destination pairs that cannot be served feasibly by dedicated aircraft or

by truck are assigned to the commercial ¬‚ights;

• low-priority services (deliveries within 48 hours or within 3“5 days) are made

by truck or by using the residual capacity of a company-owned aircraft;

• the demands of origin/destination sites are concentrated in the associated SC.

A route is a partial solution characterized by

• a sequence of stops of a company-owned aircraft ending or beginning in the

hub (depending on whether it is a collection or a delivery route, respectively).

• a set of SCs (not ASs) allocated to each aircraft AS.

Therefore, two routes visiting the same AS in the same order can differ because

of the set of SCs (which are not ASs), or because of the allocation of these SCs

to the ASs. If the demand of a route exceeds the capacity of the allocated aircraft,

TLFeBOOK

328 LINKING THEORY TO PRACTICE

the exceeding demand is transported by commercial ¬‚ights. The cost of a route is

therefore the sum of costs associated with air transportation, land transportation, and

possibly a commercial ¬‚ight if demand exceeds dedicated aircraft capacity.

The IP model solved in the second stage is de¬ned as follows. Let K be the set of

available airplane types; U k , k ∈ K, the set of the pick-up routes that can be assigned

to an airplane of type k; V k , k ∈ K, the set of the delivery routes that can be assigned

to an airplane of type k; R the set of all routes (R = k∈K (U k ∪ V k )); nk , k ∈ K,

the number of company-owned airplanes of type k; S the set of SCs, oi , i ∈ S, the

demand originating at the ith SC; di , i ∈ S, the demand whose destination is the ith

SC; cr , r ∈ R, the cost of route r; qi , i ∈ S, the cost paid if the whole demand oi

is transported by a commercial ¬‚ight; si , i ∈ S, the cost paid if the whole demand

di is transported by a commercial ¬‚ight; ±ir , i ∈ S, r ∈ R, a binary constant equal

to 1 if route r includes picking up traf¬c at the ith SC, and 0 otherwise; δir , i ∈ S,

r ∈ R, a binary constant equal to 1 if route r includes delivering traf¬c to the ith SC,

and 0 otherwise; γir , i ∈ S, r ∈ R, a binary constant equal to 1 if the ¬rst (last) AS

of pick-up (delivery) route r is the ith SC, and 0 otherwise. The decision variables of

binary type are vi , i ∈ S, equal to 1 if demand oi is transported by commercial ¬‚ight,

and 0 otherwise; wi , i ∈ S, equal to 1 if demand di is transported by commercial

¬‚ight, and 0 otherwise; xr , r ∈ R, equal to 1 if (pick-up or delivery) route r is selected,

and 0 otherwise.

The integer program is as follows.

Minimize

cr xr + (qi vi + si wi ) (8.28)

k∈K r∈U k ∪V k i∈S

subject to

xr ±ir + vi = 1, i ∈ S, (8.29)

k∈K r∈U k

xr δir + wi = 1, i ∈ S, (8.30)

k∈K r∈V k

xr γir ’ xr γir = 0, i ∈ S, k ∈ K, (8.31)

r∈U k r∈V k

nk , k ∈ K,

xr (8.32)

r∈U k

∈ F, r ∈ R,

xr (8.33)

∈ {0, 1}, r ∈ R,

xr (8.34)

∈ {0, 1}, i ∈ S,

vi (8.35)

∈ {0, 1}, i ∈ S.

wi (8.36)

The objective function (8.28) is the total transportation and handling cost. Con-

straints (8.29) and (8.30) state that each SC is served by a dedicated route or by a

TLFeBOOK

LINKING THEORY TO PRACTICE 329

commercial route; constraints (8.31) guarantee that if a delivery route of type k ∈ K

ends in SC i ∈ S, then there is a pick-up route of the same kind beginning in i. Con-

straints (8.32) set upper bounds on the number of routes which can be selected for

each dedicated aircraft type. Finally, constraints (8.33) express the following further

restrictions. The arrivals of the airplanes at the hub must be staggered in the period

before the COT because of the available personnel and of the runway capacity. Simi-

larly, departures from the hub must be scheduled in order to avoid congestion on the

runways. Let na be the number of time intervals in which the arrivals should be allo-

cated; np the number of time intervals in which the departures should be allocated;

fr , r ∈ R, the demand along route r; at the maximum demand which can arrive to

the hub in interval t, . . . , na ; At the set of routes with arrival time from t on; Pt the

set of routes with departure time before t; pt the maximum number of airplanes able

to leave before t. Hence, constraints (8.33) are

t = 1, . . . , na ,

fr xr at , (8.37)

k∈K r∈U k ©At

pt , t = 1, . . . , np .

xr (8.38)

r∈V k ©Pt

k∈K

Constraints (8.37) ensure that the total demand arriving at the hub is less than or

equal to the capacity of the hub in each time interval t = 1, . . . , na , while constraints

(8.38) impose that the total number of airplanes leaving the hub is less than or equal

to the maximum number allowed by runaway capacity in each time interval t =

1, . . . , np .

Other constraints may be imposed. For instance, if goods are stored in containers

one must ensure that once a container becomes empty it is brought back to the orig-

inating SC. To this end, it is necessary that the aircraft arriving at and leaving from

each SC be compatible. In the Intexpress problem, there are four types of airplanes,

indicated by 1, 2, 3 and 4. Aircraft of type 1 are compatible with type 1 or 2 planes,

while aircraft of type 2 are compatible with those of type 1, 2 and 3. Therefore, the

following constraints hold:

xr ±ir + xr δir i ∈ S,

’ 0, (8.39)

r∈V 1 ∪V 2

r∈U 1

xr ±ir ’ xr δir i ∈ S,

0, (8.40)

r∈U 1 ∪U 2 r∈V 1

xr ±ir + xr δir

’ i ∈ S,

0, (8.41)

r∈V 1 ∪V 2 ∪V 3

r∈U 2

xr ±ir ’ xr δir i ∈ S.

0, (8.42)

r∈U 1 ∪U 2 ∪U 3 r∈V 2

Moreover, some airplanes cannot land in certain SCs because of noise restrictions

or insuf¬cient runaway length. In such cases, the previous model can easily be adapted

by removing the routes r ∈ R including a stop at an incompatible SC.

TLFeBOOK

330 LINKING THEORY TO PRACTICE

The variables in the model (8.28)“(8.32), (8.34)“(8.42) are numerous even if the

problem is of small size. For example, in the case of four ASs (a, b, c and d), there

are 24 pick-up routes (abcd, acbd, adbc, etc.) each of which has a different cost and

arrival time at the hub. If, in addition, two SCs (e and f ) are connected by truck to one

of the ASs a, b, c and d, then the number of possible routes becomes 16—24 = 384 (as

a matter of fact, for each AS sequence, each of the two SCs e and f can be connected

independently by land to a, b, c or d). Finally, for each delivery route making its last

stop at an AS d, one must consider the route making its last stop in a different SC

g ∈ S \ {d}. Of course, some of the routes can be infeasible and are not considered

in the model (in the case under consideration the number of feasible routes is about

800 000).

The solution methodology is a classical branch-and-bound algorithm in which at

each branching node a continuous relaxation of (8.28)“(8.32), (8.34)“(8.42) is solved.

The main disadvantage of this approach is the large number of variables. Since the

number of constraints is much less than the number of variables, only a few variables

take a nonzero value in the optimal basic solution of the continuous relaxation. For

this reason, the following modi¬cation of the method is introduced. At each iteration,

in place of the continuous relaxation of (8.28)“(8.32), (8.34)“(8.42), a reduced LP

problem is solved (in which there are just 45 000 ˜good™ variables, chosen by means of

a heuristic criterion); then, using the dual solution of the problem built in this manner,

the procedure determines some or all of the variables with negative reduced costs,

introducing the corresponding columns in the reduced problem (pricing out columns).

Various additional devices are also used to quicken the execution of the algorithm.

For example, in the preliminary stages only routes with an utilization factor between

30% and 185% are considered. This criterion rests on two observations: (a) because

of the reduced number of company aircraft, it is unlikely that an optimal solution

will contain a route with a used capacity less than 30%; (b) the cost structure of the

air transportation makes it unlikely that along a route more than 85% of the traf¬c is

transported by commercial airlines.

The above method was ¬rst used to generate the optimal service network using

the current dedicated air ¬‚eet. The cost reduction obtained was more than 7%, cor-

responding to a yearly saving of several million dollars. Afterwards, the procedure

was used to de¬ne the optimal composition of the company™s ¬‚eet (¬‚eet planning).

For this purpose, in formulation (8.28)“(8.32), (8.34)“(8.42), it was assumed that nk

was in¬nite for each k ∈ K. The associated solution shows that ¬ve aircraft of type 1,

three of type 2 and ¬ve of type 3 should be used. This solution yielded a 35% saving

(about 10 million dollars) with respect to the current solution.

8.14 Bulk-Cargo Ship Scheduling Problem at the

US Navy

The Tanker Division of the US Navy is in charge of planning the fuel replenishment

of the US naval bases in the world. This task is accomplished by a ¬‚eet of 17 tankers

TLFeBOOK

LINKING THEORY TO PRACTICE 331

owned by the US Navy and, if needed, of specialized companies (spot carriers). Every

three months a team of the Tanker Division works out a replenishment plan including

both a scheduling of the military tanks and a possible list of tasks for the spot carriers.

The objective is to minimize the operational expenses of the military tanks plus the

hiring costs of additional tankers provided by spot carriers. The main operational

constraints are related to delivery times, which must be included in pre-established

time windows, and the impossibility for some ships to moor in certain ports. The

replenishment plans must also consider the current position and initial status of the

¬‚eet. Plans are revised daily in order to consider new demands, or delays due to

unfavourable weather conditions.

Each fuel request consists of a type of product, a demand, a delivery point and a

delivery time. On the basis of fuel availability at the depots, the team establishes how

each request must be satis¬ed.After this preliminary analysis, the demand is expressed

by a set of elementary fuel transfers. An elementary transfer is composed of two or

three pick-ups in a same port or in adjacent ports, and of two or three deliveries to

adjacent demand points. Typically, a tank is able to make two elementary transfers

in a month. The formulation of the model takes into account the fact that because of

the operational constraints, a tank can only complete a limited number of elementary

transfers (generally, no more than 10) in the planning period.

Let n be the number of elementary transfers; m the number of ports, K (= 17) the

number of military tanks; Sk , k = 1, . . . , K, the set of the feasible working plans

for tank k; aij k , i = 1, . . . , n, j ∈ Sk , k = 1, . . . , K, a binary constant equal

to 1 if working plan j for tank k includes elementary transfer i, and 0 otherwise;

fi , i = 1, . . . , n, the cost paid if transfer i is committed to a spot carrier; gj k ,

j ∈ Sk , k = 1, . . . , K, the variation of cost incurred if tank k executes working plan

j , as opposed to the situation in which the tank lies idle for all the planning period.

Further, let xj k , j ∈ Sk , k = 1, . . . , K, be a binary decision variable equal to 1 if

working plan j is selected for tank k, and 0 otherwise; yi , i = 1, . . . , n, a binary

decision variable equal to 1 if transfer i is made by a spot carrier, and 0 otherwise.

The problem was formulated as follows.

Minimize

K n

gj k xj k + fi yi (8.43)

k=1 j ∈Sk i=1

subject to

K

aij k xj k + yi = 1, i = 1, . . . , n, (8.44)

k=1 j ∈Sk

k = 1, . . . , K,

xj k 1, (8.45)

j ∈Sk

xj k ∈ {0, 1}, j ∈ Sk , k = 1, . . . , K, (8.46)

yi ∈ {0, 1}, i = 1, . . . , n. (8.47)

TLFeBOOK

332 LINKING THEORY TO PRACTICE

The objective function (8.43) is equal to the total transportation cost. Constraints

(8.44) impose that each transfer is realized by a military tank or by a spot carrier.

Constraints (8.45) establish that, for each tank, at most one working plan must be

selected. Model (8.43)“(8.47) is therefore a set partitioning problem with some addi-

tional constraints (relations (8.45)). The US Navy has almost 30 elementary transfers

in each planning period, to which correspond various thousands of feasible work-

ing plans. Using a general-purpose IP solver it is generally possible to solve these

instances exactly.

The optimal solution yielded an average monthly cost of $341 900, inferior by

$467 000 to the manual scheduling. The yearly saving was about 1.5 million dollars.

Further cost reductions were obtained by simulating various compositions of the

military ¬‚eet. This analysis suggested a reduction of the current ¬‚eet, no longer using

¬ve tanks and using mainly the spot carriers. This solution generated a cost reduction

of 3.2 million dollars per year.

8.15 Meter Reader Routing and Scheduling at Socal

Socal (Southern California Gas Company) distributes, with its own pipe network,

domestic and industrial gas in an area including all of southern California (USA).

The task of surveying user consumption is accomplished by motorized operators.

Every day a meter reader makes an initial car trip from the of¬ces of the company up

to a parking point; afterwards, he walks along several streets (or segments of streets)

where he does some surveying; ¬nally, he reaches the parking point from which he

returns to the company by car. According to the working contract, if the duration of

a shift exceeds a pre-established maximum value T max , a meter reader receives an

additional remuneration, proportional to the overtime. For reasons of equity, Socal

prefers solutions in which all service routes have a similar duration. The planning

of the meter-reading activities then consists of partitioning the set of street segments

into subsets yielding service routes with a duration close to T max .

In 1988 Socal mandated a consulting ¬rm, Distinct Management Consultants, Inc.

(DMC) to evaluate the costs and the potential bene¬ts resulting from the use of an

optimization software for the planning of meter-reading activities. A pilot study was

conducted in a sample region including the townships of Culver City, Century City,

Westwood, West Hollywood and Beverly Hills. This area corresponds to about 2.5%

of the entire service territory and to 242.5 working days per month. The interest of

Socal in this project was prompted by the sizeable expense (about 15 million dollars

per year) resulting from meter-reading activities.

DMC built a model of the problem integrating the data contained in the Socal

archives with those derived from a GIS. In the case of streets having users on both sides,

it was assumed that the meter reader would cover both sides separately. Therefore, the

problem was represented by a multigraph G(V , E) in which the road intersections

are described by vertices in V and street segments are associated with single edges or

to pair of parallel edges. The street sides containing some consumers formed a subset

TLFeBOOK

LINKING THEORY TO PRACTICE 333

1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

Figure 8.19 An example of a multigraph used for modelling the Socal problem

(bold edges are required, while those with a dotted line are not required).

R ⊆ E (see Figure 8.19). With each edge (i, j ) ∈ E was associated a traversing time

tij , and with each edge (i, j ) ∈ R, was associated a service time sij (> tij ).

The duration t (R ) of a work shift corresponding to a subset of the service edges

R ⊆ R has a cost equal to the sum of three terms: (a) the time spent to deal with

administrative issues at the company headquarter (generally 15 min); (b) the travel

time by car from the headquarter to a parking site; (c) the duration T (R ) of the

solution of the RPP (see Section 7.6.2) associated with the service on foot of all edges

of R . Socal prefers work shifts associated with subsets R ⊆ R whose duration is

included between T min = 7.9 hours and T max = 8 hours (balanced work shifts).

A partition {R1 , . . . , Rn } of R is feasible if each pair of parallel service arcs belongs

to the same subset Ri , i = 1, . . . , n (feasible partition). The search for a feasible bal-

anced partition (or, at least, ˜almost™balanced) of R can be viewed as the minimization

of a penalty associated with a violation of the desired length [T min , T max ] of each

work shift (edge partitioning problem).

Denoting by U and L the penalties associated with the two types of violation, the

objective function to minimize is

n n

max{t (Ri ) ’ T , 0} + L max{T min ’ t (Ri ), 0}.

max

U

i=1 i=1

DMC solved the problem by means of the following procedure.

Step 0. The workloads associated with T (R ), R ⊆ R, are approximated as W (R ) =

(i,j )∈R tij . This approach is justi¬ed by the presence of consumers on both sides

¯

of most of the street segments. An evaluation of the average useful time T of a route

(the time a meter reader will take on average to travel on foot) was calculated as

the difference between T max and the average time required for the administrative

tasks and for driving.

TLFeBOOK

334 LINKING THEORY TO PRACTICE

Table 8.10 Comparison between the results provided by DMC procedure and

the manual solution used by Socal.

Socal DMC Savings

routes routes (%)

Number of routes 242,5 236 2.7

Average shift duration (hours) 7.82 7.95

Shift interval duration [7.03; 8.56] [7.77; 8.11]

Overtime (minutes) 293.4 110.4 62.3

Total deadheading time 693.4 63.1 90.9

(without service)

Total travel time by car 545.5 133.2 75.6

Step 1. The number n of shifts was evaluated by rounding down the ratio between

¯

the total workload W (R ) of the service area and T .

Step 2. The load of remaining workload (W (R )’nT max ) was assigned to an ˜incom-

plete™ auxiliary route, built with a heuristic criterion along the border of the service

area.

Step 3. The n ˜complete™ shifts were de¬ned by means of a local search procedure

(see Section 7.3.2). The algorithm ¬rst selected n seed vertices appropriately spaced

out and then progressively generated the routes by adding to each of them couples

of parallel edges in order to constantly have balanced work shifts. The solution

obtained in this manner was improved, where possible, with some exchanges of

couples of edges (or of groups of couples of edges) among the work shifts.

The solution generated by this procedure for the sample district was characterized,

compared to the one obtained manually by Socal, by a reduced number of meter

readers, by a remarkable reduction of overtime and by an increase in the average

useful time of an operator (see Table 8.10). Extrapolating the savings associated with

these improvements, DMC estimated at about $870 000 the saving expected after the

use of the optimization method for the entire area served by Socal.

The previous analysis convinced the Socal management to order from DMC a

decision support system with a user-friendly interface, based on the optimization

method just described. The system development required one year and was installed

at the beginning of 1991.

8.16 Annotated Bibliography

The ExxonMobil case (see Section 8.2) is derived from:

1. Ferrarese G 2003 Un sistema di supporto alle decisioni per la piani¬cazione

delle spedizioni in un™azienda manifatturiera. Bachelor thesis, University of

Lecce, Lecce (Italy).

TLFeBOOK

LINKING THEORY TO PRACTICE 335

The P¬zer case (see Section 8.3) is derived from documents courtesly provided by

the company.

The Railion case (see Section 8.4) is derived from the company™s website: www.

railion.nl

The Gioia Tauro case (see Section 8.5) is derived from:

2. Chiodo A 2000 Modelli di ottimizzazione nella gestione dei terminali marittimi.

Master thesis, University of Calabria, Rende (CS), Italy.

The Hamilton-Wentworth case (see Section 8.6) is derived from:

3. Huang G H, Baetz B W, Patry G C and Terluk V 1996 Capacity planning for

an integrated waste management system under uncertainty: a North American

case study. Waste Management 15, 523“546.

The Adriatica Accumulatori case (see Section 8.7) is inspired by:

4. Stampacchia P 1988 Le tecniche di estrapolazione e di correlazione per la

previsione delle vendite. In Il sistema d™impresa (ed. S Sciarelli), pp. 359“367.

Cedam, Milan.

The case of DowBrands (see Section 8.8) is taken from:

5. Powell RE, Gao L and Muggenborg SD 1993 Designing an integrated distri-

bution system at DowBrands. Interfaces 23(3), 107“117.

The case of Hardcastle (see Section 8.9) is taken from:

6. Crainic TG, Dejax P and Delorme L 1989 Models for multimode multicom-

modity location problems with interdepot balancing requirements. Annals of

Operations Research 18, 279“302.

The case of Wolferine (see Section 8.10) is taken from:

7. Bell PC and Noori H 1985 Managing inventories through dif¬cult economic

times: a simple model. Interfaces 15(5), 39“45.

The case of FedEx (see Section 8.11) is taken from:

8. Thomas C, Campbell K, Hines G and Racer M 1998 Airbus packing at Federal

Express. Interfaces 28(1), 21“30.

The case of Waterworld (see Section 8.12) is taken from:

9. Fraser HJ and George JA 1994 Integrated container loading software for pulp

and paper industry. European Journal of Operational Research 77, 466“474.

The case of Intexpress (see Section 8.13) is taken from:

10. Barnhart C and Schneur RR 1996 Air network design for express shipment

service. Operations Research 44, 852“863.

TLFeBOOK

336 LINKING THEORY TO PRACTICE

The case of the US Navy (see Section 8.14) is taken from:

11. Fisher ML and Rosenwein MB 1989 An interactive optimization system for

bulk-cargo ship scheduling. Naval Research Logistics 36, 27“42.

The case of Socal (see Section 8.15) is taken from:

12. Wunderlich J, Collette M, Levy L and Bodin L 1992 Scheduling meter readers

for Southern California gas company. Interfaces 22(3), 22“30.

8.17 Further Case Studies

A large number of logistics case studies can be found in operations research and

management science journals, such as Operations Research, the European Journal of

Operational Research, the Journal of the Operational Research Society, Omega and

Interfaces. In what follows, a few remarkable articles are introduced and summarized.

Various options are examined for shipping tomato paste from Heinz™s processing

facilities on the west coast of the USA to factories in the Midwest in:

Kekre S, Purushottaman N, Powell TA and Rajagopalan S 1990 A logistics

analysis at Heinz. Interfaces 20(5), 1“13.

An MILP model is solved to determine a worldwide manufacturing and distribution

strategy at Digital Equipment Corporation in:

Arntzen BC, Brown GG, Harrison TP and Trafton LL 1995 Global supply chain

management at Digital Equipment Corporation. Interfaces 25(1), 69“93.

A continuous approximation method is used for designing a service network for the

US Postal Service in:

Rosen¬eld DB, Engelstein I and Feigenbaum D 1992 An application of sizing

service territories. European Journal of Operational Research 63, 164“172.

A decision support system is developed to reduce the cost of transporting materials,

parts and components from 20 000 supplier plants to over 160 General Motors plants

in:

Blumenfeld DE, Burns LD, Daganzo CF and Hall RH 1987 Reducing logistics

costs at General Motors. Interfaces 17(1), 26“47.

A system for dispatching and processing customer orders for gasoline and distillates

at ExxonMobil Corporation is illustrated in:

Brown GG, Ellis CJ, Graves GW and Ronen D 1987 Real time, wide area

dispatch of Mobil Tank Trucks. Interfaces 17(1), 107“120.

A combined location-routing problem is illustrated in:

TLFeBOOK

LINKING THEORY TO PRACTICE 337

Rosen¬eld DB, Engelstein I and Feigenbaum D 1992 Plant location and vehicle

routing in the Malaysian smallholder sector: a case study. European Journal of

Operational Research 38, 14“26.

A routing problem associated with the collection and delivery of skips is described

in:

De Meulemeester L, Laporte G, Louveaux FV and Semet F 1997 Optimal

sequencing of skip collection and deliveries. Journal of the Operational Re-

search Society 48, 57“64.

A complex vehicle routing problem occurring in a major Swiss company is analysed

in:

Rochat Y and Semet F 1994 A tabu search approach for delivering pet food and

¬‚our in Switzerland. Journal of the Operational Research Society 45, 1233“

1246.

TLFeBOOK

TLFeBOOK

Index

3-BP-L algorithm, 194, 195 automated storage and retrieval

system (AS/RS), 4, 161“165

ABC auxiliary graph, 155, 252, 281, 285,

classi¬cation, 151 290

technique, 149

back order, 122

accuracy, 64

backhaul customer, 250, 251

add“drop, 231

balance-and-connect heuristic, 287

Adriatica, 312, 313

balanced graph, 282

after-sales service, 1

bar code, 6, 166, 167

agent, 10

basic solution, 210, 220, 223, 330

aggregate demand, 28

batch, 164, 171, 172, 181“183

air

formation, 158, 181

¬‚eet, 330

picking, 164, 181, 183

network design, 325

sizing, 182

transportation, 11, 12, 13, 203,

belt conveyor, 4, 162, 164, 182, 183

326, 328, 330

benchmarking, 19

air“truck (birdyback) transportation,

Benders decomposition, 99, 102, 105

13

berth, 310, 311

air-stops, 326

best ¬t (BF), 188, 189, 196

aisle, 160, 161, 163, 165, 169“174,

best ¬t decreasing (BFD), 189, 190,

183“185, 196

195, 196

alternating heuristic, 237, 238

big M method, 212

arc routing problem (ARP), 249, 281,

bin, 147, 185“192, 194, 196, 201, 267

297

bipartite complete directed graph, 77

assembly, 1, 4, 5, 18, 157, 180, 216 bipartite directed graph, 283, 284

assignment problem (AP), 253“255 bottom left (BL), 192, 195, 324, 325

asymmetric travelling salesman bottom“up, 27

problem (ATSP), 252, 256 Box“Jenkins method, 29, 62

branch-and-bound, 76, 79, 99, 188,

Introduction to Logistics Systems Planning

226, 231, 330

and Control

breakbulk terminal, 200

G. Ghiani, G. Laporte and R. Musmanno

broker, 10, 199

© 2004 John Wiley & Sons, Ltd

ISBN: 0-470-84916-9 (HB) Brown method, 48, 49

0-470-84917-7 (PB)

budget constraint, 1, 136, 138, 226

TLFeBOOK

340 INDEX

bullwhip effect, 22 complementary slackness conditions,

220

bundle constraints, 209, 225

computational complexity, 184, 189,

business

192, 271

cycle, 34

concave, 77, 79, 90, 92, 96, 132, 141,

logistics, 73

155

business-to-business (B2B), 17

conjugate gradient method, 137

business-to-consumers (B2C), 17

connectivity constraints, 253, 258

consolidation, 10

capacitated plant location (CPL), 80,

consolidation-type transportation,

84, 85, 88, 94, 98, 116, 118

200

capacity constraint, 83, 91, 97, 98,

consumer, 3, 5, 7, 10, 28, 87, 332, 333

136, 137, 210, 265, 267, 280, 316

consumer goods, 10

capital cost, 122, 124

container, 2, 3, 5, 11, 13, 14, 17, 67,

car, 2, 196, 243, 308, 312“314, 332,

185, 192, 193, 200, 201, 204, 212,

333

224, 239, 307, 308, 310, 311,

carrier, 3, 9, 11, 14, 18, 67, 81, 194,

317“320, 322“325, 329

196, 199“201, 204, 224, 229“231,

on ¬‚atcar, 14

234, 235, 239, 241, 242, 248, 307,

transshipment hub, 3, 308

315, 322

continuous approximation method,

causal

20, 336

method, 29, 30, 61, 313

contract transportation, 9

variable, 30

control chart, 65“67

central distribution centre (CDC), 4,

conveyor, 4, 162

14, 18, 22, 73, 74, 118, 157, 304,

convoy, 3, 12, 310, 311, 318

306, 314“316 corner, 237, 324

centrality of a terminal, 236 cost versus level of service, 15

centralized warehousing, 9 crane, 163, 311

Chebychev metric, 163 crew, 2, 9, 88, 200, 201, 203, 204, 302

Chinese postman problem (CPP), crossdock, 4, 8

249, 281, 282, 284 crossdocking, 8

circuit, 249, 252“254, 274, 291, 296 cumulative error, 65

class-based storage policy, 167 customer, 4“6, 8, 14, 15, 17, 19,

clothing industry, 10 25“28, 62, 74“78, 85, 111, 113,

cluster ¬rst, route second, 269, 270, 114, 116, 117, 122, 123, 149, 157,

281 199, 200, 203, 224, 229, 234, 235,

coal, 3, 12, 77, 300, 307 242, 247“250, 254, 265, 267“269,

coef¬cient of variation, 11, 12 271“273, 280, 293, 294, 296, 302,

commodity, 26, 61, 73, 97, 98, 104, 308, 315, 317, 318, 325

119, 123, 137, 138, 141, 149, 199, degree constraints, 267

200, 204, 207, 225, 226, 228, 324 demand, 7, 25, 78, 97, 121

common transportation, 9 order, 5, 25, 122, 157, 159, 160,

company-owned warehouse, 157 164, 165, 181, 235, 247, 248,

competitive location model, 76 336

TLFeBOOK

INDEX 341

service, 7, 9, 17, 73, 123, 241 history, 29, 48, 67

pattern, 25“29, 33“35, 37, 41, 43,

zone, 9, 18, 116

47, 51, 54“56, 60, 61, 73, 306,

customized transportation, 200, 201

314

cycle, 211, 215, 252, 291

point, 73, 77, 97, 98, 103, 104,

cyclical variation, 34

115, 117, 315, 318, 331

data aggregation, 115, 119 rate, 123, 127, 130, 136, 144, 292,

decentralized warehousing, 9 293, 296

decision, 4, 5, 10, 18, 25“27, 73“75, volume, 4, 27, 74

111, 141, 171, 197, 200, 201, 203, destination, 11, 12, 81, 200, 201, 206,

204, 206, 224, 229, 236, 237, 247, 207, 212, 221, 229, 234, 236, 240,

292, 294, 295, 318 291, 292, 307, 308, 317, 322, 326,

maker, 29, 115, 165 328

-making, 18“20, 29 deterministic model, 123

support, 18, 165, 186, 194, 334 dial-a-ride, 247

variable, 77, 78, 82, 88, 94, 98, diameter of a terminal, 236

104, 105, 112, 114, 131, 165, direct shipment, 8

169, 172, 176, 182, 205, 225, directed

234, 240, 250, 252, 257, 267, Chinese postman problem, 283,

274, 283, 316, 319, 323, 328, 287

331 graph, 108, 131, 196, 207, 208,

decomposition, 29, 33“35, 37, 40, 71, 210, 212, 221, 230, 232, 233,

72, 270 252, 254, 265, 282“284, 288,

principle, 294 290, 291, 296

dedicated rural postman problem, 287, 299

air service, 326 disposal, 3, 5, 22, 312

storage policy, 167 distribution

defective, 17 centre, 1, 2, 7, 8, 22, 76, 97, 98,

deferred service, 292, 325 103“105, 157, 160, 302

degree, 5, 258, 262, 263, 281“283, channel, 9, 10, 306, 307

285 company, 7, 22, 77, 247, 248

constraints, 253, 258, 267 cost, 88, 117

of dynamism, 292, 293 decision, 4

Delphi method, 28, 29 function, 141

demand logistics, 314

aggregation, 116 management, 17, 184, 302

allocation, 83, 85, 89, 95, 98, network, 21, 104, 118, 151, 314,

101“103, 106, 206 315

divisibility, 75 pattern, 116

entry, 30 plan, 26, 216, 248

forecast, 26, 27, 35, 40, 42, 48, 49, problem, 255, 256, 261, 262

64, 67, 147, 313 strategy, 8, 336

forecasting, 28, 46, 59, 60, 144, system, 3, 74, 303, 305, 314, 315

312 distributor, 5, 9, 17, 21, 43

TLFeBOOK

342 INDEX

elementary technique, 29, 30, 33,

dock, 160, 166, 196, 236, 304, 310

42“44, 50, 55

door, 204, 236“238

emergency service, 247, 292

door-to-door, 10“12, 206, 290

end-of-line

double moving average method, 52,

port, 3, 310

53

terminal, 200

DowBrands, 314, 315

end-pairing procedure, 282, 285, 287,

dual, 99, 100, 106, 211

291

cycle, 163

end-user, 2, 5“7, 10, 17, 22, 302

Lagrangian multiplier, 219

µ-optimal solution, 86, 102, 107

Lagrangian relaxation, 218

equipment, 10

problem, 101, 211

error, 30, 49, 63“67, 117, 118, 153,

simplex, 220

212, 272, 313

solution, 100“103, 106, 330

pattern, 66, 67

variable, 99, 219

Eulerian

dummy

graph, 281, 282

arc, 212

multigraph, 262, 264, 284“286,

customer, 299

288

directed graph, 213

subgraph, 262

driver, 241

tour, 281, 282, 287

facility, 94, 95

European Union (EU), 1

load, 241, 243

even graph, 282

problem, 212

event, 14

vertex, 213

expected

dump site, 1

demand, 141, 144

dynamic

revenue, 76, 141, 142

driver assignment problem

value, 28, 34, 37, 40, 47, 66, 67,

(DDAP), 241, 243, 244

144, 146, 153, 316

network, 208

exponential smoothing method, 48

programming, 131, 184, 196

ExxonMobil Chemical, 2, 300, 301

e-commerce, 16, 17 facility, 1“4, 7“9, 17, 22, 25, 73“80,

e-logistics, 17 83“85, 90“97, 108, 109, 111“119,

earliest 123, 149, 151, 157, 170, 199, 207,

pick-up time, 326, 327 236, 294, 308, 312, 318, 336

time, 273 consolidation, 11

econometric model, 29, 61 location, 17, 18, 25, 73, 74, 76,

economic order quantity (EOQ), 129, 107, 108, 119, 294

132, 136, 137, 143, 145, 149, 151, operating cost, 77“79, 90

153“155, 321 typology, 74

economies of scale, 5 fast-moving

edge partitioning problem, 333 goods, 69

electronic data interchange (EDI), 6, item, 123

product, 152

17

TLFeBOOK

INDEX 343

FedEx, 236, 322, 323 Frederickson, 290“292, 297

feeder, 310 freight

few-to-many transportation system, consolidation, 9

200 rail transportation, 3, 307

¬nished route, 315, 326, 327

goods, 1, 3, 7, 18, 27, 62, 121, terminal design, 236

305, 321 transportation, 3, 5“7, 9, 67, 199,

item, 27 200, 203, 304, 315, 317, 318

product, 3“5, 7, 12, 77, 88, 321

gap, 102, 184

¬nite

garbage

best ¬t (FBF), 191“193

collection, 1, 3, 247

¬rst ¬t (FFF), 191

incinerator, 1

¬rst

gate, 236, 237

come ¬rst served (FCFS) policy,

geographic information system

182, 292

(GIS), 17, 248, 332

¬t (FF), 188, 189, 196

Gioia Tauro, 3, 308, 310, 311

¬t decreasing (FFD), 189

global positioning system (GPS), 17,

¬xed

22, 248

cost, 7, 75, 79, 80, 93, 94, 97, 104,

globalization, 16, 17

112“114, 118, 121, 123, 138,

government, 1, 199, 303, 308

147, 148, 155, 172, 205, 227,

232, 234, 235, 251, 296, 315, graph, 71, 111, 184, 252, 281, 286

316, 319“321 Gross Domestic Product (GDP), 1, 30

order quantity policy, 143 ground

¬xed-charge network design (FCND) feeder route, 326

problem, 225, 226 service, 326

¬‚eet

Hamilton-Wentworth, 3, 312

composition, 204

Hamiltonian

planning, 330

cycle, 257, 261, 263“265, 267,

sizing, 18

270, 271

food industry, 10

path, 261

forecast, 4, 5, 18, 25, 28, 33, 37,

tour, 252

41“44, 46“50, 55, 56, 61, 62, 64,

handling cost, 167, 200, 201, 328

147, 149, 167, 251, 313

Hardcastle, 317, 318

control, 65

hazardous waste depot, 3, 312

error, 30, 48, 64

heuristic, 19, 20, 76, 84, 85, 89, 98,

forecasting

137, 138, 184, 187“189, 191, 192,

accuracy, 64, 65, 67

194, 226, 231, 237, 252, 261,

method, 25, 28“30, 49, 50, 54, 61,

268“270, 274, 278, 287, 290, 294,

63“66

296, 299, 324, 325, 330, 334

forklift, 4, 160, 165, 236, 237, 305

historical data, 21, 29, 48, 49, 56, 58,

forward

93

area, 171

holding cost, 8, 122

zone, 160

TLFeBOOK

344 INDEX

inventory-routing problem (IRP),

Holt method, 53

295, 296

hub, 81, 310

item aggregation, 116

and spoke system, 310

I/O port, 165, 174“178, 180, 185 joint

in-transit inventory, 7 cost, 123, 138

inbound order, 140

¬‚ow, 8, 77 just-in-time distribution, 8

transportation cost, 9

k-exchange, 264

incremental quantity discount, 132,

kerbside garbage collection, 3, 312

134, 135

key performance indicator (KPI), 19

industrial

cargo shipping, 200

labour cost, 8, 25, 73, 163, 166

goods, 6, 9, 10

Lagrangian

user, 7

heuristic, 84, 89, 98, 118

information

multiplier, 84“87, 98, 217, 218,

¬‚ow, 5, 6

259, 260

system, 8, 305

procedure, 80, 84, 217

technology, 6, 16, 17

relaxation, 84“86, 217, 259, 260

input“output model, 29, 61

land¬ll, 3, 312

insertion heuristic, 274, 275

laptop computer, 6

instantaneous resupply, 123, 128

largest gap heuristic, 184“186

integer programming (IP), 19, 315,

latest time, 273

327, 328, 332

lead time, 4, 7“9, 25, 121, 123, 124,

integrated location and routing, 294

130, 141, 144“147, 306

interest rate, 122, 124, 127, 129, 132,

least squares method, 31

134, 136, 137, 144, 147, 321

least-cost perfect matching, 262

intermodal

Leibnitz rule, 141

carrier, 11, 212

less-than-truckload (LTL), 11, 12,

train, 308

200, 201, 203, 234, 236, 243, 315,

transportation, 13, 317

316

Intexpress, 325“327, 329

life-cycle, 29, 62

inventory

linear

level, 6, 22, 25, 27, 124“126,

¬xed-charge network design

128“131, 138, 139, 143,

(LFCND) problem, 226, 228,

145“147, 149“151, 167, 168,

230, 232, 233

248, 296, 321

multicommodity minimum-cost

management, 6, 7, 25, 26, 64, 121,

¬‚ow (LMMCF) problem, 217,

123, 136, 152, 321

218, 220, 221, 232, 233, 244

policy, 62, 121, 123, 128, 132,

network ¬‚ow problem, 19

136, 141, 145, 146, 148, 149,

programming (LP), 19, 83, 98, 99,

153

174, 210, 217, 218, 228, 330

turnover ratio, 121, 321

regression method, 51

inventory-balance constraint, 131

TLFeBOOK

INDEX 345

make-to-

single-commodity minimum-cost

assembly (MTA) system, 5, 68

¬‚ow (LMCF) problem, 209,

order (MTO) system, 4, 5, 26

210, 216, 218

stock (MTS) system, 4, 5, 25, 68

linehaul customer, 250, 251

manufacturer, 4, 5, 7“10, 17, 71, 233,

liner cargo shipping, 200

324

local search, 19, 263

manufacturing

location

and assembly centre, 1, 4, 18

decision, 19, 74, 75, 78, 111, 294

centre, 3

model, 75“77, 95

cost, 16, 122, 141

problem, 73, 74, 76, 77, 84, 90,

plant, 2, 4, 9, 206, 300

95, 110, 115

process, 4, 77, 123, 132, 147, 302,

location“allocation, 74

321

location-covering, 108, 111, 112,

resource planning (MRP), 27

114, 115, 119

site, 2, 303, 304

location-routing, 75, 294

many-to-many transportation system,

logistician, 7, 9, 21, 26, 27, 29, 31

200

logistics

marginal cost, 79, 93, 94, 96, 97, 104

cost, 1, 2, 5, 7, 9, 73, 89, 108, 118,

market research, 28

121, 199, 315

master problem, 100

decision, 18, 19

material

management, 20, 299

¬‚ow, 1, 4, 74, 75, 77, 96, 237

network, 20, 26, 73

handling, 4, 157, 167, 236, 310,

service, 14, 15

311

system, 1“3, 6, 7, 9, 14, 15,

mean

17“20, 22, 25, 68, 74, 88, 93,

absolute deviation (MAD), 64, 65

149, 299, 303, 318, 326

absolute percentage deviation

long-haul

(MAPD), 64, 72

carrier, 247, 248

error, 67

freight transportation, 199

squared error (MSE), 64, 66, 69,

transportation, 17, 81, 200, 325

144, 147

long-term forecast, 26, 28

medium-term forecast, 26, 28

lost sales cost, 122

merchandise, 7, 11

lower bound (LB), 77, 80, 84, 85, 87,

meter reader routing and scheduling,

98, 99, 101, 102, 173, 188, 190,

332

194, 196, 197, 218“220, 228, 253,

minimum-cost

254, 258, 267, 296

¬‚ow, 207, 228, 239“241, 283

lumpy demand, 25

spanning r-tree problem

machine scheduling, 4, 181 (MSrTP), 258, 259, 296

mail spanning tree, 262“264

box, 294 mixed

delivery, 1, 247 Eulerian graph, 282

distribution, 76 graph, 108, 249, 254, 299

sorting centre, 1 rural postman problem, 299

TLFeBOOK

346 INDEX