nsf grant number: dmi-0200439 nsf program name ... · oped more general algorithms for the...

NSF grant number: DMI-0200439

NSF Program Name: Manufacturing Enterprise Systems.

Models and Algorithms for Integrated Multi-Stage

Production/Distribution Problems

Philip Kaminsky

Hyun-soo Ahn

Osman Engin Alper

University of California, Berkeley

Abstract: We summarize a series of modelsand results related to the effective integrationof production and transportation decision mak-ing in a multi-stage environment. For a deter-ministic two stage supply chain, we develop andanalyze effective optimal algorithms. For con-tinuous time stochastic version of this model, wepartially characterize the optimal policy struc-ture, and develop heuristics. For a discrete timestochastic version of this model, we partiallycharacterize the optimal policy structure. Fi-nally, we introduce a model integrating produc-tion and transportation outsourcing that we arecurrently analyzing.

1. Introduction: For many firms, costs asso-ciated with transportation and distribution andinventory holding costs are a large percentageof total product costs. Indeed, US industryspends more than $350 Billion on transportationand more than $250 Billion on inventory holdingcosts annually (Lambert and Stock, 2001). Inmany industries, parts and subcomponents aremanufactured across a variety of sites. For ex-ample, in the personal computer industry, where

lowering product cost is of paramount impor-tance, the location of final assembly is often dif-ferent from the location of component assembly.Similarly, in the pharmaceutical industry, man-ufacturing facilities are expensive to build, andpharmaceutical products have a limited prof-itable life span (since after patent protectionexpires, generic manufacturers can manufacturethe same product). Thus, multi-purpose plants,which can perform several different manufactur-ing steps for many different products, are typ-ically built. Once a network of these plants isconstructed, new products are manufactured se-quentially at several different plants, dependingon the particular processes required for manu-facture.

These examples highlight the importance ofcoordinating both production and inventorypolicies in multi-stage supply chains. However,although there have been tremendous academicand practical efforts focused on minimizing theinventory level for components and parts withina facility while maintaining efficient production,there has been significantly less work on coor-dinating production and distribution simulta-

Proceedings of 2005 NSF DMII Grantees’ Conference, Scottsdale, Arizona Grant #DMI-0200439

neously, particularly when production faces ca-pacity constraints. Efforts in this direction arecomplicated by the non-linear nature of trans-portation costs. Indeed, over the last twentyyears, driven by the adoption of JIT, CONWIP,and flexible manufacturing systems, manufactur-ers have devoted significant effort to setup reduc-tion, and thus manufacturing setup costs play asmaller and smaller role in manufacturing deci-sion making. Most transportation, however, ex-hibits natural economies of scale; in many cases,an empty truck doesn’t cost much less to oper-ate than a full one so it is crucial for successfuldecision making approaches for multi-stage man-ufacturing supply chains to explicitly account forthese non-linear transportation costs.

The objective of our ongoing research projectis therefore to develop models of multi-stageproduction/distribution systems, analyze thesemodels, and use this analysis to develop use-

ful managerial insights, and effective algo-

rithms for planning, controlling, and operatingthese systems. In this paper, we survey a vari-ety of recent results, starting with a simple deter-ministic model, and continuing on to a variety ofstochastic models. We conclude by sketching anoutline of our ongoing exploration of third partylogistics contracts, and their impact on produc-tion planning.

2. Literature Review: There has been a va-riety of literature on multi-stage inventory sys-tems. Clark and Scarf (1960) show that a basestock policy is optimal in a finite horizon periodicproblem with no fixed cost and no capacity con-straint. Federgruen and Zipkin (1984a,1984b)show that the optimality of the order-up-topolicy is still valid in the case of discountedand average infinite horizon problems. Chen

(1994, 1998) demonstrates the effectiveness ofthe (R, nQ) policy in a similar setting, whileChen and Song (2001) characterize the structureof the optimal policy in a multi-stage inventorymodel with Markov modulated demand. A fewauthors have considered the control of capaci-tated production systems. Federgruen and Zip-kin (1986a, 1986b) consider an inventory policyin a capacitated production system under sta-tionary demand with no fixed cost. Glassermanand Tayur (1994, 1995, 1996) study a multistagestationary production-inventory model with ca-pacitated production and demonstrate the effec-tiveness of IPA (infinitesimal perturbation anal-ysis) as a tool to obtain an optimal policy. Ka-puscinski and Tayur (1998) extend the modelto periodic demand. Parker and Kapuscinsky(2002) consider a periodic review problem of atwo-stage capacitated echelon inventory systemwith zero or deterministic leadtime and showthat the optimal policy is a modified base stockpolicy. However, in their model, the replenish-ment decision is free of economies of scale inshipping quantity so that it is reasonable to con-jecture that a variant of an order-up-to policyshould be optimal. In most of these papers, theoptimal policy is either a base stock policy, or apolicy with monotone structure. An exceptionto this is Duenyas et al. (2003), who considera periodic review model with fixed cost and lostsales when the system has “must-meet” deter-ministic demand and random demand at eachperiod. They show that the optimal shippingquantity and rationing policy are not necessarilymonotone, because the inventory on hand maynot satisfy existing demands, and can be con-served for deterministic demand in subsequentperiods. Our analysis indicates that the optimalpolicy can be very complex, as well as counter-intuitive, when both production and distribution


are considered in a system with non-linear dis-tribution costs and finite production rate. Weare not aware of any research which considerssimilar systems.

There has been another stream of literatureon the control of production systems. Ha (1997)characterizes the structure of the optimal pol-icy in make-to-stock system as a switching curvetype policy. Carr and Duenyas (2000) considera make-to-stock/make-to-order system. Ahn etal. (1999), Iravani et al. (2003), Duenyas andPatane-Anake (1998) characterize the optimalpolicy in a serial production line with single ormultiple resources. Duenyas and Tsai (2002)consider a two-stage production system wherestage one can sell its finished goods and providethe components to the downstream stage at thesame time. However, in contrast to our proposedresearch, very little research considers multipleunits shipped at a fixed cost.

There has also, of course, been a long his-tory of research into deterministic single stage

single item lot sizing models, starting with theseminal works of Wagner and Whiten(1958) forthe uncapacitated model, and Florian and Klein(1971) for the capacitated model. Aggarwal andPark (1990), Federgruen and Tzur (1991), andWagelmans et al. (1992) developed faster ex-act algorithms for the uncapacitated case, whileLove (1973) and Baker et al. (1978) devel-oped more general algorithms for the capaci-tated case. Various authors have consideredmulti-stage production models under determin-istic constant demand. Muckstadt and Roundy(1993) survey and summarize many of these re-sults. The work presented in this paper allowsfor time-varying multi-period demand. A vari-ety of research has been devoted to heuristic andmathematical programming-based methods fordeterministic multilevel, multi-product lot siz-

ing problems. Baker (1993) summarizes variousheuristic approaches which appeared in the lit-erature up to that point, and subsequently, Har-rison and Lewis (1996), Katok et al. (1998), andArmentano et al.(2001) have proposed additionalheuristics for these complex problems. Althoughthese papers don’t explicitly mention transporta-tion costs, some of the formulations are generalenough to encompass our model. In addition,Chan et al. (1997) and the references thereinconsider a concave cost network flow version ofa multistage inventory/transportation problem(without capacitated production), and developheuristics for the problem. However, in contrastto the heuristics these authors have focused on,we focus on polynomial approaches to obtain theoptimal solution.

3. The Deterministic Model: The initialdeterministic model we consider consists of a twostage supply chain which faces a deterministicstream of external demands for a single product.We assume an infinite supply of raw materialsat stage one, and capacitated production at bothstages. Items are manufactured at stage one, andthen held in inventory at stage one prior to ship-ping. Items are transported to stage two, wherethey are again held in inventory. Additional ca-pacitated production is completed at stage two(that is, value is added to each item, but nonew items are created), items are held in finishedgoods inventory after this stage, and this inven-tory is used to meet final demand. Each period,production levels in stage one and stage two, aswell as transportation levels between stage oneand stage two, must be determined.

Formally, we consider a two-stage, n periodmodel of a supply chain, illustrated in Figure 1.As described above, an infinite supply of raw


stage 1 -

±°²¯i1 -

®

©ªtransportation -

±°²¯i2 - stage 2 -

±°²¯if -

externaldemand

Figure 1: Model 2SPDP

material is available at stage one. Each period,x1

t , t = 1, 2, ..., n units are produced at stage 1,at a cost of ct

i, where x1t ≤ C1

t . Units can be heldin inventory at the stage 1 buffer, where a hold-ing cost h1

t , t = 1, 2, ..., n is charged per unit atperiod t. In addition, st units are also shipped toa buffer located before stage two (with no ship-ping lead time). If a shipment occurs, a fixedcost of ft is charged independent of the numberof units shipped, and a variable cost of vt perunit is charged. Units can be held in inventoryin the buffer before stage two, where a holdingcost of h2

t is charged per unit at period t. Al-ternatively, x2

t ≤ C2t units can enter production,

at a cost of c2t per unit. After production, items

can be held in finished goods inventory at thepost stage two buffer where a holding cost hf

t

is charged per unit at period t or, or they canbe shipped to meet demand dt. We note thatin this model, all demand must be met, and callthis the two stage production distribution prob-lem (2SPDP).

We summarize each period’s order of eventsbelow:

1. Stage one production x1t is determined, and

production cost ct1 is charged per unit man-

ufactured.

2. The shipping quantity st is determined andunits are shipped. Fixed and variable ship-ping cost (ft + stvt if st > 0) is charged.

3. Holding cost h1t is charged on the i1t units

remaining in the post stage one buffer.

4. Stage two production x2t is determined, and

production cost ct2 is charged per unit man-

ufactured.

5. Holding cost h2t is charged on the i2t units

remaining in the pre stage two buffer.

6. x2t units are added to the finished goods

buffer.

7. Demand is filled from the finished goodsbuffer.

8. Holding cost hft is charged on the units re-

maining in the finished goods buffer.

We define the following parameters: let dt bethe external demand at period t, dtn be the cu-mulative demand from t to period n. In addi-tion, let h1

t be the holding cost per unit in thepost stage one buffer at period t, h2

t be the hold-ing cost per unit in the pre stage two buffer atperiod t, hf

t be the holding cost per unit in thefinished goods buffer at period t, C1

t be the pro-duction capacity at stage 1 at period t, C2

t be theproduction capacity at stage 2 at period t, ft bethe fixed cost for shipping at period t, vt be thevariable cost for shipping at period t, c1

t be theper unit production cost at stage 1 at period t,and c2

t be the per unit production cost at stage2 at period t.

Also, let x1t be the production quantity at

stage one at period t, x2t be the production quan-

tity at stage two at period t, st be the shippingquantity at period t, yt be the shipment indica-tor variable at period t, i1t be the inventory levelat the stage one buffer at the end of period t, i2t


be the inventory level at the pre stage two bufferat the end of period t and ift be the inventorylevel at the post stage two buffer at the end ofperiod t. We model our problem as follows:

minn

∑

t=1

(I1t h1

t +I2t h2

t +Ift hf

t +ytft+stvt+c1t x

1t +c2

t x2t )

s.t.

x1t ≤ C1

t t = 1 . . . n (1)

i1t = i1t−1 − st + x1t t = 1 . . . n (2)

st ≤ ytdtn t = 1 . . . n (3)

x2t ≤ C2

t t = 1 . . . n (4)

i2t = i2t−1 − x2t + st t = 1 . . . n (5)

ift = ift−1 + x2t − dt t = 1 . . . n (6)

ift , i1t , i2t , x

1t , x

2t , st ≥ 0 t = 1 . . . n (7)

yt ∈ {0, 1} t = 1 . . . n (8)

Constraints (1) and (4) are capacity con-straints for both stages. Constraints (2), (5),and (6) are the inventory balance equations forthe three buffers. Constraint (3) ensures that thefixed cost is incurred when items are shipped.

In addition, we have also considered a generalversion of this model, in which rather than be-ing characterized by a fixed and a linear cost,the shipping costs can be a general concave costfunction: ft(st).

Kaminsky and Simchi-Levi(2001) present effi-cient algorithms for the model described above,both with the fixed and linear transportationcost, and the concave transportation cost underthe following general assumptions:

• increasing holding costs:

h1t < h2

t < hft ∀t ∈ 1...T.

• non-speculative assumptions:

c1t +h1

t > c1t+1 and c2

t +h2t > c2

t+1 ∀t ∈ 1...n−1.

We also assume that the following cost functionholds for the case with the fixed and linear trans-portation cost,

x(vt+h2t−h1

t )+ft > xvt+1+ft+1 ∀t ∈ 1...n−1, x > 0.

and for the case with concave cost,:

fi(x)+(h2−h1)x > fj(x) ∀i < j, x > 0.

In all cases, we prove the following property,which substantially reduces the complexity ofthe problem:

Property 1. The optimal values for x2t , t =

1, 2, ..., T are determined by the following recur-sive equations:

x2t = min{C2

t , dt +

T∑

i=t+1

(di − x2i )} (9)

Indeed, this result enables us to develop a sim-pler alternate model with the same optimal pro-duction and shipping quantities. As in the modeldescribed above, an infinite supply of raw mate-rial is available at stage one. Each period, x1

t

units are produced at stage one, where x1t ≤ C1

t ,at a cost of c1

t . Units can be held in inventoryat the stage one buffer, where a holding cost h1

t

is charged per unit at time t. st units can alsobe shipped to another buffer (we refer to thisas buffer two), and a fixed cost of ft is chargedif st > 0. Units can be held in inventory inbuffer two, where a holding cost of h2

t is chargedper unit at time t or shipped to meet demandd′t = x2

t , t = 1, 2, ..., n, where x2t is determined

using Equation (9). Thus, we have eliminated


a stage from the original model. We call thisequivalent model 2SPDP′.

For the case with fixed and linear trans-portation cost, we further identify the followingstructural properties:

Property 2. In any optimal solution to 2SPDP′,shipping only occurs in periods t where theinventory in the stage two buffer, i2t−1 = 0.

Property 3. In any optimal solution to 2SPDP′,the shipping quantity at time t, st is equalto some partial sum of future demands∑a

i=t di, a ≥ t.

If we are given a production schedule, i.e, ifthe variables x1

t , t = 1, 2, ..., n are already de-termined, we can efficiently solve the problem ofdetermining what quantity to ship in each periodusing a dynamic programming approach .

Given an efficient approach to this shippingproblem, we define a block [s,t] to be a set ofconsecutive periods such that i1s−1 = 0, i1t = 0,and i1a > 0, s ≤ a < t.

This definition leads to the following twoproperties:

Property 4. In any optimal schedule, for anyblock, production will be at capacity in allperiods except possibly for the first period.

Property 5. In any block, the total productionin that block is equal to some partial sum ofconsecutive demands.

These properties, combined with the definitionof a block, implies that any block [s, t] serves alldemands in some interval a, a + 1, ..., b.

We have used all of the properties definedabove to develop a dynamic programming ap-

proach to solving this problem. By carefullyavoiding duplication of effort, we developed analgorithm which allows us to solve the originalproblem in O(n4) time, even when productioncapacity varies over time.

This problem is much more difficult in the gen-eral concave cost case. In particular, Properties2 and 3 do not hold, so the approach we de-veloped for the fixed+linear case will not work.However, we have developed a new (more com-plex) set of structural properties, and a newO(n8) algorithm for the constant capacity case(see Kaminsky and Simchi-Levi (2001)).

These results have recently been generalizedby van Hoesel et al (2004).

4. Continous Time Stochastic Model: Forour first stochastic model, in Ahn and Kaminsky(2004), we consider a two stage push-pull supplychain. The orders arrive at stage 2, the finalstage, according to a Poisson process with rateλ. Two separate operations, which take placeat different locations, are required to convertthe raw materials to finished goods. We assumethat an infinite supply of raw material is avail-able at stage 1. A single server whose processingtime follows i.i.d exponential distribution withrate µi, i = 1, 2 is available at each of the twostages. Items are produced at stage 1, the pushstage, and can then be either held as inventoryat stage 1, or shipped to stage two, in whichcase shipping cost are incurred. Items will beprocessed at stage two, the pull stage, only to

meet outstanding orders. Non-negative holdingcosts are incurred on any intermediate inventory,and penalty costs are incurred for orders waitingto be filled. The holding cost is incurred at arate hi (i = 1, 2) per unit time per item, whilepenalty is incurred at rate b per unit time per


outstanding order. We assume that holding costsare non-decreasing (i.e., h1 ≤ h2 ≤ b). Shippingis assumed to be immediate, and a fixed cost, K,is incurred for each shipment to reflect shippingeconomies of scale. This model is illustrated inFigure 2.

We model this problem as a Markov DecisionProcess, and attempt to minimize the long runaverage cost per unit time. Without loss of gen-erality, we uniformize λ, µ1 and µ2 such thatλ + µ1 + µ2 = 1. The states of the MDP areS(t) = (n1(t), n2(t), n3(t)) where n1(t) denotesthe number of units held in inventory at stage 1at time t, n2 denotes the number of units heldin inventory at stage 2 (before the completion ofstage 2 operation) at time t and n3 denotes thenumber of outstanding orders at time t. We as-sume that only Markovian stationary determin-istic policies are under consideration (and callthe class of these policies Π). In other words,the policies in Π specify the decision as a func-tion of the current state only. Since processingtimes and interarrival times are exponential, itis easy to see that {S(t)} is a continuous timeMarkov chain where the transition rate at anystate is bounded by 1, and therefore {S(t)} isuniformizable. By using the results of Lippman(1975), we can translate the original continu-ous time optimization problem into an equiva-lent (discrete time) Markov decision process withstate s = (n1, n2, n3). Decision can be madeupon the arrival of a new order or at any servicecompletion. It is easy to see that stage 2 will al-ways be busy as long as it is has product to pro-cess and outstanding orders to fill (i.e., n2 > 0and n3 > 0). At any decision epoch, the produc-tion decision at stage 1 must be made (whetheror not to initiate production of a unit at stage 1),and if n1 > 0, the distribution decision must bemade (i.e., the quantity greater than or equal to

zero which should be shipped must be decided).Using the uniformization technique, we write theequivalent discrete time dynamic programming.

To develop insight into the structure of opti-mal solutions for this model, we performed com-putational testing using the value iteration algo-rithm with a variety of parameters. For the valueiteration, we truncated the state space such thatni ≤ 100 by redirecting any transition to a stateoutside the truncated state space to the near-est state. Through an extensive computationalstudy, we observed the following. In all cases, itis optimal to not ship if n2 > 0. (We prove arelated result below) For fixed values of n1, theshipping decision increases monotonically in n3.That is, there is some level of n3, which is a func-tion of n2, below which it is optimal to not ship,and above which it is optimal to ship. We notethat in all observed cases, this level was greaterthan 0. However, for fixed values of n1, the ship-ping quantity fluctuates in increasing n3. Forfixed values of n3, shipping quantity increases inn1, and then cycles. The maximum and mini-mum values of the optimal shipping quantitiesincrease with increasing n3. This result is sur-prising, since it implies that in some cases it isoptimal to ship a smaller quantity as the numberof outstanding orders increases. Note that we ex-pended considerable effort in eliminating trunca-tion of countable state space and round-off erroras potential reasons for this striking behavior.For example, we tried conducting a value itera-tion on a considerably larger state space, and ex-trapolating the relative value function for statesoutside the boundary, but the optimal policy wasstill non-monotonic.

Figure 3 illustrates that optimal shippingquantity is not always monotone in the amountof on-hand inventory at stage one or the numberof outstanding orders. In this example, it is op-


Figure 2: 2-stage Push-Pull System

timal to ship 17 units at state (19,0,3), but 14units at state (27,0,3), even though this secondstate has more units to ship from stage one.

We characterize a sufficient condition for not

shipping in this model:

Lemma 1. For any state (n1, n2, n3), it is opti-

mal not to ship as long as n2 > 0.

This result can be easily shown using an inter-change argument.

Unfortunately, the lack of monotonicity in theshipping quantities makes additional structuralproperties of this model difficult to determine.Thus, we are motivated to consider a variety ofrestrictions of this model. In the following sub-section, we consider a version of this model forwhich the possible set of shipping quantities islimited. In Section 4.3, we characterize the im-pact of this restriction on the optimal objectivevalue.

4.1 Restricting the Possible Shipping

Quantities: As we observed above, the opti-mal policy of original formulation does not pos-sess the monotonicity that we have hoped for. Inaddition, our original formulation, which consid-ers shipping every quantity between zero and thecurrent inventory level at each decision epoch, is

computationally inefficient. We therefore con-sider a simplified version of the problem, inwhich the firm ships the minimum of the max-imum transportation capacity and the currentinventory level whenever the decision to ship ismade. We note that this may be a reasonablerestriction in practice, since firms may experi-ence a physical limit on the quantity that can beshipped at a time (one truckload, for example).In addition, we assume that shipments only oc-cur when inventory is zero at stage two, whichwe proved above is optimal in our original model,although not necessarily in this model. As we re-port in Section 4.3, computational testing showsthat this restricted model performs about as wellas the original model if the shipping quantity isselected appropriately.

At state s = (n1, n2, n3), the set of feasiblepolicies, As is given by

As =

{

(i, q)|i =

{

1, if produce;

0, otherwise.,

q ∈{

0,1{n2=0} min[n1, Q]}

}

.

Using the uniformization technique, the opti-mal average cost, g and the relative value func-


Figure 3: An example of optimal shipping quantity when n3 = 0, 1, 2 and 3.


tion, v(·), must satisfy the following equations:

v(n1, n2, n3) + g =2

∑

i=1

hini + bn3

+min

K + (h2 − h1)Q+ λv(n1 − Q, Q, n3 + 1)+ µ2v(n1 − Q, Q − 1, n3 − 1)+ µ1 min[v(n1 − Q + 1, Q, n3),v(n1 − Q, Q, n3)],λv(n1, 0, n3 + 1) + µ2v(n1, 0, n3)+ µ1 min[v(n1 + 1, 0, n3), v(n1, 0, n3)]

for n1 ≥ Q, n2 = 0, n3 > 0.

v(n1, n2, n3) + g =2

∑

i=1

hini + bn3

+min

K + (h2 − h1)n1 + λv(0, n1, n3 + 1)+ µ2v(0, n1 − 1, n3 − 1)+ µ1 min[v(1, n1, n3), v(0, n1, n3)],λv(n1, 0, n3 + 1) + µ2v(n1, 0, n3)+ µ1 min[v(n1 + 1, 0, n3), v(n1, 0, n3)]

for n1 < Q, n2 = 0, n3 > 0.

v(n1, n2, n3) + g =2

∑

i=1

hini + bn3

+ λv(n1, n2, n3 + 1) + µ2v(n1, n2 − 1, n3 − 1)

+ µ1 min[v(n1 + 1, n2, n3), v(n1, n2, n3)]

for n2 > 0, n3 > 0.

v(n1, n2, n3) + g =2

∑

i=1

hini + bn3

+ λv(n1, n2, 1) + µ2v(n1, n2, 0)

+ µ1 min[v(n1 + 1, n2, 0), v(n1, n2, 0)]

for n2 ≥ 0, n3 = 0.

In the next subsection, we characterize the op-timal policy of this model.

4.2 The Optimal Policy for the Capac-

itated Shipping Problem – Counterintu-

itive Observations: As we mentioned above,the restriction on shipping quantity simplifiescomputational efforts. Furthermore, as the ship-ping decision itself is monotone in the originalproblem and shipping quantity initially increasesthen fluctuates, we had anticipated that themonotonicity would continue to hold when weimposed a reasonable monotone shipping quan-tity which resembled the spirit of the optimalshipping quantity, such as min(n1, Q). However,even this simplification does not result in a sim-ple optimal policy. In fact, in many examples,the optimal policy is not only complex, but alsocounterintuitive, and very different from resultsfound in the literature for more simple make-to-order models. This counterintuitive behaviorseems to be due to the non-linear shipping costs.Later in the paper, we computationally comparethis model to an analogous model with linearshipping costs. In this subsection, we discussthe structure (or lack of structure) of the opti-mal policy when shipping quantity is boundedby Q. Clearly, the overall performance of thesystem will be greatly affected by the choice ofQ. In the next section, we discuss an heuristicapproach to selecting a good Q value.

First, by studying a variety of computationalexamples, we observe that neither production orshipping decisions are monotone in most of thestate variables. In particular,

1. Even if it is optimal to ship at state(n1, 0, n3), it is not necessarily optimal toship at state (n1, 0, n3 + 1).

2. Even if it is optimal to ship at state(n1, 0, n3), it is not necessarily optimal toship at state (n1 +1, 0, n3), even in the case


when n1 > Q so the shipping quantity re-mains the same.

3. The production policy at stage 1 is not nec-essarily monotone in the inventory level atstage 1, that is, even if it is optimal not toproduce at state (n1, n2, n3), it may be op-timal to produce at state (n1 + 1, n2, n3).

4. The production policy at stage 1 is not nec-essarily monotone in the number of out-standing orders. Even if it is optimal to pro-duce at state (n1, n2, n3), it is not necessar-ily optimal to produce at state (n1, n2, n3 +1).

While in some examples the optimal policy ismonotone (mostly when fixed shipping cost isclose to zero or small), there are many otherexamples showing that such monotonicity doesnot hold in general. As can be seen in Fig-ure 4, the optimal policy can not be character-ized by a simple switching curve. First, considerthe lack of monotonicity in n3. In the example,when n1 = 10, it is optimal not to ship whenn3 ∈ {0, 1, 5, 6, 7}, but optimal when n3 takes onany other value. Although this seems counter-intuitive, consider the following possible expla-nation. In the optimal policy, units shipped fromstage 1 cover current and (possibly) future out-standing orders. Any shipment decision balancesincreased holding cost in stage two with backo-rder cost. In addition, any shipment less than Qalso accounts for increased shipping costs. Also,once a shipment is made, the next shipment can-not be made until inventory at stage 2 is onceagain zero, and the time until inventory at stagetwo is once again zero depends on the backorderlevel during the first shipment, and the amountshipped. For very low backorder levels (n3 = 1

in this example), increased holding and trans-portation costs offset increased backorder costs,so it is optimal not to ship. For certain low back-order levels (n3 = 2 in this example), the benefitfrom shipping, decreased backorder costs, mayoutweigh the benefits of waiting and producingmore before shipping. This effect is enhanced bythe fact that backorder level is likely to be rela-tively low at the time of the next shipment, sincethe amount being shipped is relatively small, andmuch of it is already allocated to existing backo-rders. However, for other small backorder levels(n3 = 5, 6, 7 in this example), the benefit of de-laying shipment (consequently shipping more ina later time) dominates the benefit of shippingimmediately. Finally, as the backorder level getslarger, it again makes sense to ship immediately,as the backorder level at the next shipment willbe large regardless of whether or not some quan-tity is immediately shipped.

The non-monotonicity of shipping in n1 can beexplained similarly. When there are few unitsin stage 1, shipping immediately is suboptimalsince doing so fails to take advantage of theeconomies of scale in shipping. However, as thenumber of units in stage 1 increases, the ship-ping cost per unit decreases. However, at somepoint, increasing stage 1 holding cost may beginto dominate, so that it therefore becomes opti-mal to wait until additional orders arrive beforeshipping.

Figure 5 shows an example where the optimalproduction policy is not monotone in backorderlevels. For instance, it is optimal to produce forn3 ≤ 3 or n3 ≥ 8, but optimal not to producefor 4 ≤ n3 ≤ 7 when n1 = 7. For low back-order levels, it is optimal to produce, but notto ship since backorder cost is not high enoughto justify immediate shipping (with resultant in-creased per unit shipping charges). However, as


Figure 4: An optimal policy at n2 = 0 in a capacitated shipping problem.


the backorder level gets larger, it makes senseto ship immediately for n3 ≥ 4 as in Figure 5.While it is optimal not to produce after a ship-ment for medium backorder levels (n3 = 4, 5, 6, 7in this example), it is optimal to produce after ashipment for high backorder levels. This is truebecause with high backorder, the time until thenext time inventory is again zero at stage twodecreases. Thus, production must continue toensure that there is enough material to includewith the next shipment. On the other hand, withlower backorder, it will be longer until the nextshipment is required. Indeed, the time at whichinventory at stage two is zero after a shipmentstochastically decreases in the number of out-standing orders.

This phenomenon is affected by the relativeproduction and arrival rates. When the produc-tion rate at stage 1 is higher than the rate atstage 2, it is more likely to be optimal to waitbefore producing at stage one, as the time un-til another shipment is required is greater. Asbackorder costs increase, the value of waiting un-til the next shipment decreases, so production ismore likely, although only if relative processingrates suggest that this production is likely to beshipped relatively soon.

This example contradicts the intuition estab-lished by the optimal solution to many make-to-order models. The conventional intuition sug-gests that the optimal production rate shouldincrease as the number of outstanding orders in-creases so that the system can clear orders assoon as possible. Therefore, as the backordercost becomes high, it becomes urgent to reduceoutstanding orders. However, at the push-pullboundary, the finite capacity at stage 2 damp-ens such urgency. When there is enough time toproduce units at stage 1 while stage 2 is busy,the production can be delayed. On the other

hand, the benefit of increasing shipping quanti-ties increases when backorder cost is small andshipping cost is high. Therefore, it becomes op-timal not to ship, but to produce when there arefew backorders.

The non-monotonicity of optimal productionpolicy in n1 is even more intriguing. Observe inFigure 6 that it is optimal to produce at state(9, 0, 2) although it is optimal not to produce atstate (8, 0, 2). Most previous analysis of relatedmodels implies that it is optimal to produce un-til the inventory level reaches a certain point (al-though this point may be state dependent). Thisexample contradicts that intuition, since it is op-timal to produce at a particular state, (9, 0, 2),but not at a state with lower inventory, (8, 0, 2).In general, this counterintuitive behavior seemsto occur when n2 is low. In fact, when n2 = 0or 2, the production policy is characterized by amonotone switching curve in the opposite direc-tion to what intuition may suggest

Although this seems perplexing, consider thefollowing possible explanation. When the fixedshipping cost is extremely high as in this exam-ple, it is optimal to ship Q units when a shipmentoccurs. Therefore, stage 1 will always produce Qunits before a shipment occurs. Now, if holdingcosts are similar at stage 1 and stage 2, it makeslittle difference if inventory is held at stage 1 orat stage 2. Thus, when the inventory level atstage 1 is close to Q, it becomes optimal to pro-duce more units (up to Q), then ship to stage 2,rather than storing inventory at stage 1. How-ever, when the inventory level at stage 1 is low,it is optimal to wait for more outstanding or-ders before producing additional units to ship.Thus, in this case, the optimal policy is mono-tone in the counter-intuitive direction (i.e., themore inventory there is, the more valuable it isto produce more.) However, as the level of inven-


Figure 5: The non-monotonicity of production policy in n3 when n2 = 0.


Figure 6: The non-monotonicity of production policy in n1 (λ = .05,µ1 = .50, µ2 = .45, h1 = 1.0,h2 = 1.05,b = 1.10, K = 1000.0 ,and Q = 12)


tory at stage 2 increases, this effect diminishes,as the time until the next shipment increases. InFigure 6, for example, this “opposite directionmonotonicity” disappears when n2 ≥ 6.

Although the optimal policy to this problem isclearly complex, we are able to partially analyti-cally characterize its structure. We show that itis always optimal to ship if there are at least Qunits of inventory and at least Q units of backo-rder.

Lemma 2. For any state (n1, 0, n3) such that

n1 ≥ Q and n3 ≥ Q, it is optimal to ship inde-

pendent of production policy.

Proof. We prove the claim by a sample path ar-gument that any policy with delayed shippingcan be improved by a policy with immediateshipping. Without loss of generality, we assumethat t = 0. Let w be a sample path in a proba-bility space P large enough to contain all futurearrivals and service times. Suppose that in theoptimal policy Π∗, shipping is delayed by ts > 0.Let T denote the first time that n2(t) becomeszero after shipping Q units at time ts on a givensample path, w. Also, let pΠ∗

(t) be the cumula-tive units produced in stage 1 by time t,DΠ∗

(t)be the cumulative units produced in stage 2 attime t under policy Π∗, and A(t) be the cumu-lative number of arrivals to be system by timet. It can be easily shown that the stage 2 re-mains idle until time ts (that is when the firstshipment occurs) and works to produce Q con-secutive units without idling. The process willreaches state (n1 +PΠ(T )−Q, 0, n3 +A(T )−Q)under policy Π∗ at time T and the total costaccumulated over this interval [0, T ], denoted asCΠ∗

((0, T ];w), can be expressed as the follow-

ing:

CΠ∗

((0, T ];w) =

∫ ts

0

[

h1(n1 + pΠ∗

(t))

+ b(n3 + A(t))]

dt +

∫ T

ts

[

h1(n1 + pΠ∗

(t) − Q)

+ h2(Q − DΠ∗

(t)) + b(n3 + A(t) − DΠ∗

(t))]

dt

Consider another policy Π on the same sam-ple path, that mimics the production of pol-icy Π, but ships Q units at time zero insteadof ts. Under this policy, Q units are producedwithout idling and idling occurs in the interval(T − ts, T ] at stage 2 (i.e., DΠ(t) = DΠ∗

(t + ts)for t ∈ [0, T − ts] and DΠ(t) = DΠ∗

(T ) = Q fort ∈ [T − ts, T ].) After T , policy Π mimics theproduction and shipping decisions of policy Π∗.For every sample path, it can be easily shownthat the sample paths under both policies coin-cide after T . The total cost accrued over thisinterval [0, T ], denoted as CΠ ((0, T ];w), is

CΠ ((0, T ];w) =

∫ T

0

[

h1(n1 + pΠ(t) − Q) + b(n3

+ A(t) − DΠ(t))]

dt +

∫ T−ts

0

[

h2(Q − DΠ(t))]

dt

=

∫ T

0

[

h1(n1 + pΠ∗

(t) − Q)]

dt +

∫ T−ts

0

[

b(n3

+ A(t) − DΠ∗

(t + ts)) + h2(Q − DΠ∗

(t + ts))]

dt

+

∫ T

T−ts

[b(n3 + A(t) − Q)] dt.

Comparing the cost associated with policy Π∗

and the cost associated with policy Π, we have

CΠ∗

((0, T ];w)−CΠ ((0, T ];w) =

∫ ts

0Q(h1+b)dt ≥ 0.

The result holds for every sample path. There-fore, for any policy under which the shipping is


delayed, it is possible to construct an immediateshipping policy which is better for every samplepath. This contradicts the optimality of policyΠ∗.

Of course, the performance of this restrictedmodel relative to the original model depends onthe selection of a good Q value. In Section 4.3,we demonstrate that if we select the best possibleQ value, the gap between the average cost of thispolicy and the average optimal cost is very small.

4.3 Computational Analysis: Our compu-tational experiments are designed to characterizethe loss associated with restricting the shippingquantity to the minimum of some Q value andthe available inventory and characterize the im-pact of non-linear shipping costs. We tested avariety of combinations of system parameters:

• h1, h2, b ∈ {(1, 2, 5), (1, 4, 15)}

• K ∈ {250, 1000, 5000}

• λ, µ1, µ2 ∈ {(.2, .4, .4), (.15, .6, .25),(.15, .25, .6)}

We vary costs so that they increase a relativelysmall amount and a relatively large amount be-tween stages, we consider a variety of fixed costs,and we consider processing rates which are muchfaster at stage one, and much faster at stage two.All of the examples are solved through a valueiteration on a sufficiently large truncated statespace to alleviate any boundary effect.

For each of the combinations of system pa-rameters listed above, we tested a variety of Qvalues Q = 1, 2, ..., 70. Table 1 displays these re-sults. Note that we essentially lose nothing bymaking this assumption. Indeed, in all cases theobjective value for the best possible Q is almost

identical to the optimal objective value for theoriginal model. We have to consider four deci-mal digits to see a difference in most cases. Inaddition, this result is relatively insensitive to Q.Figure 7 graphs objective value versus Q valuefor one sample problem (The problem instanceis: h1 = 1, h2 = 2,b = 5, K = 250, λ = .2,µ1 = .4,µ2 = .4). Observe that the objectivevalue is very close to optimal for a large rangeof Q values. The performance of all the sampleproblems we considered was similar.

To determine the impact of non-linear ship-ping costs on total system cost, in the last col-umn of Table 1, we present system cost if ship-ping cost is linear, and equal to K/Q where Q isthe optimal Q value for the restricted shippingproblem. In all cases, this system is significantlyless expensive than the system with fixed ship-ping costs.

5. A Discrete Time Stochastic Model: Inmany cases, a discrete time model is more appli-cable. In addition, we were motivated to explorethe discrete time model to help us understandwhich of the observations made in the continu-ous time case are general, and which seem to bespecifically related to the continous time natureof that model. In this case, we consider a man-ufacturing system consisting of two stages fac-ing stochastic demand through n periods. Aninfinite supply of raw material is available atstage one. Production at stage one is uncapaci-tated while the one at stage two is capacitated.Items are manufactured at stage one, and thenheld in inventory at stage one prior to shipping.Transported items are held in inventory at stagetwo until additional production is completed andthen the external demand is met.

Our problem is to determine the production


h1 h2 b K λ µ1 µ2 Optimal Restrict. Ship Linear

1 2 5 250 0.2 0.4 0.4 22.2961 22.2966 10.5471

1 4 15 250 0.2 0.4 0.4 38.6485 38.6502 29.4737

1 2 5 1000 0.2 0.4 0.4 37.7050 37.7056 14.5638

1 4 15 1000 0.2 0.4 0.4 59.7015 59.7025 34.9888

1 2 5 4000 0.2 0.4 0.4 68.0791 68.0804 26.5500

1 4 15 4000 0.2 0.4 0.4 101.1060 101.1076 50.8105

1 2 5 250 0.15 0.6 0.25 21.7331 21.7344 14.4781

1 4 15 250 0.15 0.6 0.25 42.1668 42.1690 33.8936

1 2 5 1000 0.15 0.6 0.25 34.5979 34.599685 18.6450

1 4 15 1000 0.15 0.6 0.25 60.1309 60.133752 40.1438

1 2 5 4000 0.15 0.6 0.25 59.9318 59.934515 26.9787

1 4 15 4000 0.15 0.6 0.25 95.6139 95.6182 52.6442

1 2 5 250 0.15 0.25 0.6 16.9865 16.9876 9.2840

1 4 15 250 0.15 0.25 0.6 26.3363 26.3505 21.8587

1 2 5 1000 0.15 0.25 0.6 29.8138 29.8154 11.1591

1 4 15 1000 0.15 0.25 0.6 43.6287 43.6309 24.3597

1 2 5 4000 0.15 0.25 0.6 55.4480 55.4504 26.1596

1 4 15 4000 0.15 0.25 0.6 78.3368 78.3406 41.6330

Table 1: Performance When Shipping Quantity is Restricted, and with Linear Shipping Costs

Figure 7: Q vs. Objective for restricted shipping.


Figure 8: Two-Stage System

level at first stage, as well as the transportationlevel between stage one and stage two at eachperiod. Notice that the first stage is operatedin a push-based manner, utilizing the economiesof scale in transportation. On the other handthe second stage is clearly a pull-based stage inwhich the production level is directly determinedby the external demand. Although production isunrestricted at stage one, shipment from stageone to stage two is restricted by the amount ofinventory at stage one. Similarly the productionat stage two is limited by the amount of inven-tory plus shipment at stage two as well as theproduction capacity at stage two.

5.1 The Sequence of Events: In each (dis-crete) time period, the following events takeplace in the following order:

1. Make shipment and production decisions.

2. Receive shipment at stage two inventory.

3. Realize demand, update inventory positionat stage two.

4. Receive production at stage one inventory.

5. Calculate the costs.

5.2 Notation and Formulation: We employthe following notation in our analysis.

I1t : Inventory level at stage one at the beginning

of period t.

I2t : Inventory level at stage two at the beginning

of period t.

Bt: Backlog level at stage two at the beginningof period t.

x1t : Production level at stage one at period t.

x2t : Production level at stage two at period t.

st: Shipment level from stage one to stage twoat period t.

h1: Unit holding cost of inventory at stage one.

h2: Unit holding cost of inventory at stage two.

p: Unit penalty cost of backlogged demand.

c1: Unit production cost at stage one.

c2: Unit production cost at stage two.

C: Production capacity of stage two.

K: Fixed cost for shipping.

N : Number of periods.

dt: External demand (random disturbance) re-alized at period t.

α: Discount factor.


~It: State vector.~ut: Control vector.Dt: Random disturbance space.St: State space.Ct: Control space.Ut: State constrained control space.π: Policy (control law).µt: Control function.

We employ this notation to define the follow-ing dynamic program:

dt ∈ Dt, where Dt ⊆ Z+.

~It = (I1t , I2

t , Bt) ∈ St, where St ⊆ Z3+.

~ut = (x1t , st) ∈ Ct, where Ct ⊆ Z2+.

~ut ∈ Ut(~It), where

Ut(~It) ={

(x1t , st) ∈ Ct|st ≤ I1

t

}

.

π = {µ0, ..., µN−1}, ~ut = µt(~It), where

µt(~It) ∈ Ut(~It), for all ~It ∈ St.

~It+1 = ft

(

~It, µt(~It), dt

)

, where

ft

(

(I1t , I2

t , Bt), (x1t , st), dt

)

=[

I1t + x1

t − st,

I2t + st − min(Bt + dt, I

2t + st, C),

Bt + dt − min(Bt + dt, I2t + st, C)

]

.

minJπ(~I0) = E

{

N−1∑

t=0

αtgt

(

~It, µt(~It), dt

)

}

, where

gt

(

(I1t , I2

t , Bt), (x1t , st), dt

)

= (I1t + x1

t − st)h1

+[


2t + st, C)

]

h2

+[


]

p

+ K1(st>0) + x1t c1 + min(Bt + dt, I

2t + st, C)c2.

JN ( ~IN ) = 0, and

Jt(~It) = min ~ut

{

x1t c1 + K1(st>0)

+ (I1t + x1

t − st)h1

+ h2E[


2t + st, C)

]

+ pE[


]

+ c2E[

min(Bt + dt, I2t + st, C)

]

+ αE

[

Jt+1

(

ft

(

~It, µt(~It), dt

)

)

]

}

.

5.3 Results: To gain insight into the structureof optimal policy for this model, we performedcomputational testing using the value iterationalgorithm with a variety of parameter values.For the value iteration, we truncated the statespace with a conservative upper bound, redirect-ing any transition to a state outside the trun-cated space to the nearest state in. Throughan extensive computational study, we observedthat neither optimal production nor shipping de-cisions are monotone in most of the state vari-ables. The following examples are taken fromthe optimal solution to a given set of particularparameters.

1. The production level at stage one is notmonotone in outstanding orders. For exam-ple, while x = 9 at (9,8,8); x = 0 at (9,8,9).

2. The production level at stage one is notmonotone in inventory level at stage one.For example, while x = 0 at (10,8,5); x = 5at (11,8,5).

3. The production level at stage one is notmonotone in inventory level at stage two.For example, while x = 0 at (9,8,9); x = 9at (9,9,9).

4. The shipment level is not monotone in out-standing orders. For example, while s = 9at (9,8,12); s = 1 at (9,8,13).

5. The shipment level is not monotone in in-ventory level at stage two. For example,while s = 7 at (13,2,8); s = 13 at (13,3,8).


As can be observed from the examples, theoptimal policy is not only complex but alsocounterintuitive. This counterintuitive behaviorseems to stem from the nonlinear shipping costs.The production capacity at stage two furthercomplicates the structure of the optimal policy.

We were able to prove the following two propo-sitions, which are fairly intuitive.

Proposition 1. I2t ≥ C implies st = 0.

Proof. For sake of contradiction, suppose that inthe optimal policy, π, in period t the shipmentlevel is positive although I2

t ≥ C. Consider thepolicy, π, which is the same as π except st = 0,and st+1 = st+1 + st. But Jπ(~I0) − Jπ(~I0) =st(h2−h1) > 0, which contradicts the optimalityof π.

Proposition 2. I2t ≥ 2C implies xt = 0.

Proof. For sake of contradiction, suppose that inthe optimal policy, π, in period t the productionlevel is positive although I2

t ≥ 2C. Consider thepolicy, π, which is the same as π except xt = 0,and xt+1 = xt+1 + xt. Notice that the optimalshipment level at the next period is zero, st+1 =0, by the previous result; thus such a policy, π,exists. But Jπ(~I0) − Jπ(~I0) = xth1 > 0, whichcontradicts the optimality of π.

5.4 Modifications of the Initial Model:

The lack of monotonicity in both the shippingand production levels makes the structural prop-erties of this model very difficult to determine.Thus, we are motivated to consider various mod-ifications of this model in the hope of better char-acterizing optimal policy structure. We consid-ered the following:

1. In this model, we restrict shipment to ei-ther zero or the minimum of the inventory

at hand and Q, where Q is an additionalparameter specifying the shipment upperbound.

2. In this model, we restrict production to an(s,S) type policy. Hence the production de-cision is eliminated from the problem, re-ducing the dimension of the control spaceto one.

3. In this model, we restrict shipment to an(s,S) type policy; i.e. whenever the inven-tory at stage two drops below s min(S −I2t , I1

t ) is shipped, otherwise there is no ship-ment. Hence the shipment decision is elim-inated from the problem, reducing the di-mension of the control space to one.

4. In this model, we restrict shipment to eitherQ (given I1

t ≥ Q) or zero, where Q is an ad-ditional parameter specifying the shipmentlevel. Hence the shipment decision becomesa binary one.

For all the modified models, we performed thesame computational testing using the value iter-ation algorithm with a variety of parameter val-ues. Although the computational effort is sub-stantially reduced by all of the modified models,the structure of the optimal policy is hardly sim-plified by the first three of them. The optimalpolicy structure of the fourth modified model isremarkably well behaved compared to the others,and we were able to partially characterize theoptimal policy structure more extensively thanin the other cases. Unfortunately, the completecharacterization of the optimal policy structureeluded us in all of the models we considered. Itappears that both discrete and continuous timestochastic versions of this model behave quitecounter-intuitively, and optimal polices are inboth cases very complex.


6. Third Party Logistics Contracting: Fi-nally, building on prior research, over the nextyear we will investigate third party logistics con-tracting in a multi-stage production/distributionenvironment. As we have argued above, in asupply chain, production and distribution op-erations are generally the most important op-erational functions. It is crucial to integratethese two functions by planning and schedul-ing them jointly in a coordinated way to achieveoptimal operational performance of the supplychain. On the other hand, outsourcing non-corecompetencies has now become a widely acceptedpractice across many industries. Running an in-house trucking fleet is a difficult task and not acore competency. Hence, acquiring transporta-tion contracts and coordinating production anddistribution operations with them have becomecritical operational and tactical points of interestin a supply chain.

Transportation contracts in the modern eraoften specify in advance the frequency and vol-ume to be reserved by the carrier for a particularcustomer’s future deliveries. The most commontypes of transportation contracts in research lit-erature and practice are the following:

1. Long-term (or forward buy or fixed com-mitment) contracts, which specify a fixedamount of service to be delivered at somepoint in the future.

2. Option contracts, in which the buyer pre-pays the reservation price (or premium) forthe service capacity and then pays the ex-ecution (or exercise) price for each unit ofcapacity used.

3. Flexibility contracts, in which a fixedamount of supply is determined when the

contract is signed, but the amount to be de-livered and paid for can differ by no morethan a given percentage established uponsigning the contract. These contracts areused in practice to share risk between theparties, and are equivalent to a relevantcombination of a long-term contract and anoption contract.

We are interested in developing and analyzingmathematical models of supply chain networksintegrating production and distribution opera-tions through the use of transportation con-tracts. In this general setting we plan to considerthe following research problems:

1. We will consider a manufacturer in posses-sion of a transportation contract of one ofthe types described above over a given finitehorizon. This manufacturer faces stochas-tic demand, and has access to a spot mar-ket for expedited shipping. At each pe-riod, given the demand at that period themanufacturer have to decide between im-mediate shipment and consolidating ordersto utilize the transportation contract. Weaim to determine the optimal operation ofthe production and distribution functions ofthe manufacturer in this setting, while gain-ing important practical insights for supplychain operation.

2. Given the solution of the first problem, weare interested in jointly optimizing the ac-quisition and the operation of the trans-portation contracts. For this purpose, weassume a large pool of available transporta-tion contracts with various schedules andcontract parameters.

3. Given the solution of the second problem,now intend to consider the problem from the


transportation provider’s point of view andtackle the questions of designing and pricingtransportation contracts in a market with apool of buyers under competition.

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nsf grant number: dmi-0200439 nsf program name ... · oped more general algorithms for the...

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