# Simultaneous determination of multiproduct batch and full truckload shipment schedules

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Available online 20 August 2008

Keywords:

Full truckload shipments

Supply chain coordination

Inventory model

A fundamental premise of the well-known economic lot-scheduling problem (ELSP) is

that the nished products are consumed at continuous rates, i.e. their respective cycle

utilized various mathematical programming techniques,

mixed integer programming (Delporte and Thomas, 1977).Efforts along these lines, however, have experienced

ms ise arelems.mial-

time algorithms if NP is identical to P (polynomial class),

decades have been dedicated towards developing heur-

Contents lists available at ScienceDirect

w.e

Int. J. Productio

ARTICLE IN PRESS

Int. J. Production Economics 118 (2009) 1111170925-5273/$ - see front matter & 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.ijpe.2008.08.015istics for obtaining near-optimal solutions, rather thanoptimization (e.g. Madigan, 1968; Stankard and Gupta,1969; Doll and Whybark, 1973; Goyal, 1973; Haessler andHogue, 1976; Saipe, 1977; Elmaghrabi, 1978; Haessler,

Tel.: +12158951449; fax: +12158952907.E-mail address: banerjea@drexel.edusuch as linear programming (Maxwell, 1964), dynamicprogramming (Bomberger, 1966; Elmaghrabi, 1978) and

or (b) proved to be permanently intractable. Therefore,most efforts in solving the ELSP during the last threefeasible production schedule at minimum total relevantcost are the fundamental issues of the ELSP. The ELSP hasreceived considerable research attention over the years.Past attempts to solve this problem optimally have

NP-hard (Luenberger, 1973). The NP class of problesolvable in nondeterministic polynomial time. Therno efcient algorithms known for this class of probThis class of problems can be (a) solved by polynoThe classical economic lot-scheduling problem (ELSP)involves the production of multiple products in a singlefacility or machine, which can process only one item at atime. The determination of each products lot size and a

gets larger and have concluded that the optimization ofthe ELSP becomes either inefcient or impossible for evenrelatively small problems.

Insuring the feasibility of a schedule in the ELSPappears to be a major factor of complexity, since it is1. Introductionhowever, employing complex distribution networks, nished goods inventories from

manufacturing plants are usually shipped in bulk to succeeding stages along the

distribution process. Moreover, existing transport economies often tend to favor full

truckload (TL), rather that partial or less than truckload (LTL) shipments, for economical

movement of such goods. The scenario examined here, however, involves a set of

products, for which individual TL shipments are uneconomical. As a remedy, we

construct a model for taking advantage of TL rates by combining LTL quantities of the

items into a full load. We adopt the common cycle approach for the ELSP, in conjunction

with a common replenishment cycle, as a coordination mechanism that is simple to

analyze and implement. This is integrated with a periodic full truckload shipping

schedule. Such effective coordination of production and shipment schedules is likely to

result in a more streamlined supply chain. The concepts developed are illustrated

through a simple numerical example.

& 2008 Elsevier B.V. All rights reserved.

increasing computational inefciency as the problem sizeEconomic lot scheduling inventories are depleted on the basis of unit transactions. In todays supply chains,Simultaneous determination of mutruckload shipment schedules

Avijit Banerjee

Department of Decision Sciences, Drexel University, Philadelphia, PA 19063, U

a r t i c l e i n f o a b s t r a c t

journal homepage: wwproduct batch and full

lsevier.com/locate/ijpe

n Economics

e.g.

weofillu

thissup

2.

2.1.

1.2.

3.

4.5.

ARTICLE IN PRESS

A. Banerjee / Int. J. Production Economics 118 (2009) 1111171121979; Park and Yun, 1984; Panayiotopoulos, 1983; Gengand Vickson, 1988; Davis, 1990) and by constructingfeasibility conditions (e.g. Vemuganti, 1978; Boctor,1982; Hsu, 1983; Davis, 1990).

Needless to say that within the framework of supplychain management, the ELSP has an important role to playin terms of coordinating the activities of the variousmembers of such systems. In recent years, a great deal ofresearch has focused on issues of coordinating supplyand demand within this context (see, e.g. Goyal andGupta (1989) and Thomas and Grifn (1996) for surveys).Much of the existing research on ELSP, however, assumesthat the inventory depletion rate for each of the multipleitems is uniform, which would be the case if customerdemands are satised directly from the manufacturingfacility. In most of todays global rms, supply chainsconsist of complex distribution networks, involvingproduction plants, vehicle terminals, warehouses, distri-bution centers, retail outlets, etc. In such cases, thenotion of uniform product demands at a productionfacility is an incorrect representation of the real world.In reality, inventory depletions at a manufacturing plantoccur in discrete, sizeable lots, as a result of bulkshipments, that take advantage of transportation econo-mies of scale. Thus, for achieving streamlined supply chainstructures, it is necessary to link or coordinate theproduction schedule with the outbound shipment sche-dule. Unfortunately, as mentioned above, there is a dearthof research that addresses the ELSP in terms of suchlinkages.

This paper is an attempt to ll this research gap and re-examine the ELSP for developing a procedure for integrat-ing the production schedule of multiple items into ashipment plan. In other words, the primary focus here liesin coordinating the manufacturing process with thetransportation function. In shipping goods from manu-facturing facilities to subsequent stages of the supplychain, truck shipments are perhaps the most commonlyused means. Such transportation can be either fulltruckload (TL) or less than truckload (LTL) shipments.Although truck shipping rate structures are often complex,generally speaking, TL shipments are substantially lessexpensive, on a cost per unit basis, than LTL shipments(see, e.g. Chopra and Meindl, 2004). Under certaincircumstances, though, the demand rate of a productmay not be sufciently large to warrant periodic indivi-dual TL shipments, since such large delivery lots mayresult in excessive inventory holding costs, negating theadvantage of low transportation costs. Nevertheless, ifthere is a group of such relatively low-demand productsthat are shipped from a source to various demandlocations, it may be possible to combine several partialtruckloads of individual products to constitute a fulltruckload, thus obtaining the relative advantage of TLrates, in conjunction with lower inventory levels at thedestination locations. In this study, we examine a scenariosuch as this.

In our proposed procedure for coordinating theproduction schedule of multiple products (in an ELSPenvironment) with their shipments to several demandlocations, the common production cycle approach (see,details).6. Each of the products is transported via truck and sold

at one or more given demand location(s). For any ofthese products, however, its market demand rate isnot sufciently high to warrant full TL shipments. Onthe other hand, LTL shipments are relatively moreexpensive and undesirable.

7. Thus, TL shipments are made, where each truckload(capacitated by total weight and/or volume) containsa mix of all the products, which are delivered at theappropriate location(s).

8. For each TL shipment, a xed cost is incurred,regardless of the mix of products carried. In contrast,LTL shipment rate structures often entail a unitvariable shipping cost, based on the weight and/orthe volume of shipment.

9. During each production cycle, an integer number, K, ofthese TL shipments are made at equal intervals oftime.tion can indeed be optimal or near-optimal undercertain conditions (see, Jones and Inman (1989) forsufcient capacity to produce all the products exists.Secondly, from a practitioners viewpoint, the com-mon cycle method is relatively easy to understandand implement, especially when stable productionand transportation schedules need to be coordinatedon a routine basis. Finally, the common cycle solu-assumption, the rationale for its adoption is threefold.First, this approach ensures a feasible solution ifproduction cycle. Although this is a restrictiveng-distribution scenario, are made in this paper:

The operating environment is deterministic.Multiple products are produced in a capacitated batchproduction environment, with different productionrates for the various products. We assume, withoutloss of generality, zero setup times for all the products.If, in reality, the setup times are nonzero, they can beeasily incorporated into our model, outlined later.Only a single product may be produced at any giventime.Stockouts or shortages are not permitted.We adopt the common cycle solution to the ELSP,where each product is produced exactly once in everyTturiAssumptions and notation

Assumptions

he following assumptions, describing the manufac-shipment decisions, such as the one suggested inpaper, may result in a more coherent and efcient

ply chain.subsandindividual items. For simplicity of implementation,restrict the production cycle to an integer multiplethe delivery cycle. Under many circumstances, asstrated later through a numerical example andequent sensitivity analysis, coordinated productiondelitheMaxwell, 1964) is combined with the notion of avery (or replenishment) cycle that is common to all

ifor product i ($/batch)

to minimize the total inventory holding cost.T

I1

2 P3

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A. Banerjee / Int. J. Production Economics 118 (2009) 111117 1131

2KQ1

KQ1P1

KQ1

KQ2P2

Q3P3

K 1 Q1

D1

Q1

K 2 Q1D1

Q1

Q1D1

Q1

=KQ1=D1com

he average inventory values for the three items areputed as follows:hi inventory holding (carrying) cost for product i($/unit/time unit)

Qi amount of product i contained in each TLshipment (units)

K a positive integer, representing the number ofshipments per production cycle;

T the shipment interval in time units (common toall products and locations)

KQi the production lot size (in units) for product iKT production cycle length in time unitsC the TL capacity, i.e. maximum total load (or

volume) allowable per truckloadwi weight (or volume) of each unit of product iIi average inventory level (units) of product iTRC total relevant cost ($) per time unit

3. Model development

For illustrative and model development purposes, theinventory-time plots pertaining to a scenario involvingthree products (n 3) and three full TL shipments perproduction cycle (K 3), of length KT time units, aredepicted in Fig. 1.

From this gure it is clear that for a common deliverycycle time of T (i.e. T QiDi, 8i), each truckload contains Qiunits of product i, i 1, 2, 3. The total weight (or volume)in a load, w1Q1+w2Q2+w3Q3, is obviously limited to thetruck capacity, C. Also, the products should be sequenced10. At each of the various demand locations, stocks arereplenished via a periodic review, order-up-to levelinventory control system. For coordination purposes,all the items at all the demand locations share acommon xed review period.

11. The product lot sizes, Qi, are treated as continuousvariables for all i.

2.2. Notation

The following notational scheme applies throughoutthis paper:

i an index used to denote a specic product, i 1,2,y,n

Di the demand rate for product i (units/time unit)Pi the production rate for product i (units/time

unit)A manufacturing setup cost per production batch(for derivation see, e.g. Joglekar, 1988).In general, for the n-products case:

Ii KXn1i1

DiQi2Pi

Xn1ji1

QjPj

24

35 Qn

Pn

Xn1i1

Di

K 1Xn1i1

Qi=2; for i 1;2; . . . ;n 1

and In Qn2

DnPn

2 K K 1

3.1. Objective function

The objective of our optimization model is to minimizethe total relevant cost per time unit, which is a function ofQ1, Q2,y,Qn (denoted by Q) and K. Thus, using the resultsobtained above, the objective function is

Minimize TRCQ ;K Xni1

DiAiKQi

KXn1i1

DihiQi2Pi

Xn1ji1

QjPj

24

35

QnPn

Xn1i1

Dihi K 1Xn1i1

Qihi2

Qnhn2

DnPn

2 K K 1

(1)

The rst term on the RHS of (1) represents the totalsetup cost per time unit for the n products. The remainingterms, capturing the inventory holding cost per time unitfor items 1, 2,y,n1 and n, respectively, are obtained byadding the average inventory levels, Ii, for i 1, 2,y,n1,and In, derived above, weighted by the respective holdingcost parameters, hi and hn.

3.2. Constraints

The relevant model constraints are outlined below:

1. The delivery cycle time is common to all products:

Q1D1

Q2D2

QnDn

T

or Qi TDi; i 1;2; . . . ;n (2)

2. The production capacity of the system is limited, i.e.

Xni1

DiPip1 (3)I3 2 P2 P3 D2

K 2 Q2D2

Q2

Q2D2

Q2=KQ2=D2

KD2Q22P2

D2

Q3P3

K 1Q2

2

Q3 D3 2 K K 1 I2 KD1Q12P1

Q2P2

D1

Q3P3

K 1Q1

2

1KQ2

KQ2

KQ2Q3

K 1 Q2

Q2

3.

4.

3.3

rep

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K=

A. Banerjee / Int. J. Production Economics 118 (2009) 111117114KQProduct 2

Inventory

P

DKQTDQ

=

KQProduct 1

KQ

InventorySchedule feasibility, i.e. total production time neededcannot exceed the manufacturing cycle time:

KXn1i1

QiPi

QnPnpKT

orXn1i1

QiPi

QnKPnpT (4)

Note that this constraint can be easily modied fornonzero setup times by adding these to its left-hand side.Truck capacity is limited by weight or volume, i.e.Xni1

wiQi C (5)

. Model simplication

The coordination mechanism of an equal inventorylenishment (or delivery) cycle time for all products

PQ

KQProduct 3

PKQ

KD

KQ=

Inventory

PKQ

KD

KQ=

Fig. 1. Inventory-time plTime

Tenables us to considerably simplify the non-linear mixedinteger programming problem expressed by (1)(5).From (2)

Qi TDi; 8i or Qj Q1D1

Dj; j 2;3; . . . ;n

Substituting the Qi values for all i in (5), we obtain

w1Q1 w2Q1D1

D2 wn

Q1D1

Dn C

or Q1 CD1=Pni1

wiDi

and Qj TDj; j 2;3; . . . ;n:

9>>=>>; (6)

Recalling that Di for i 1,2,y,n are known parameters,T and all Qi values can be determined a priori andrecursively using (6). Now, dene the parameters a, b

TDQ

=Time

T

TDQ

=

Time

T

ots (n 3, K 3).

i1n

an

s

g

$13,600 per year are incurred in shipping cost, bringing

ARTICLE IN PRESS

A. Banerjee / Int. J. Production Economics 118 (2009) 111117 115Substituting the values Qi, i 1, 2,y,n, obtained from(6), into (7) and eliminating constraints (2) and (5), theoriginal model expressed by (1)(5) can be considerablysimplied to

Minimize TRCK a=K bK g (8)

Subject to

Xni1

DiPip1; i 1;2; . . . ;n (9)

Xn1i1

QiPi

QnKPnpT (10)

K 1; 2; . . . ::; etc. (11)

Note that (8) is strictly convex in the integer K.Therefore, if TRC(K) is minimized at K K*, then

TRCKnpTRCKn 1 and TRCKnpTRCKn 1 (12)

Substituting (12) into (8) we obtain the followingoptimality conditions:

KnKn 1pabpKnKn 1 (13)

3.4. Solution procedure

Utilizing the optimality conditions (13), the simpliedmodel, (8)(11), can now be easily solved by a simpleprocedure, which is outlined below:

1. For in...

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