on the value of optimal myopic solutions for dynamic

On the Value of Optimal Myopic Solutionsfor Dynamic Routing and Scheduling

Problems in the Presence ofUser Noncompliance

WARREN B. POWELL, MICHAEL T. TOWNS, AND ARUN MARAR

Department of Civil Engineering and Operations Research, Princeton University, Princeton, New Jersey 08544

The most common approach for modeling and solving routing and scheduling problems in adynamic setting is to solve, as close to optimal as possible, a series of deterministic, myopicmodels. The argument is most often made that, if the data changes, then we should simplyreoptimize. We use the setting of the load matching problem that arises in truckload truckingto compare the value of optimal myopic solutions versus varying degrees of greedy, suboptimalmyopic solutions in the presence of three forms of uncertainty: customer demands, travel times,and, of particular interest, user noncompliance. A simulation environment is used to testdifferent dispatching strategies under varying levels of system dynamism. An important issuewe consider is that of user noncompliance, which is the effect of optimizing when users do notadopt all of the recommendations of the model. Our results show that (myopic) optimalsolutions only slightly outperform greedy solutions under relatively high levels of uncertainty,and that a particular suboptimal solution actually outperforms optimal solutions under a widerange of conditions.

The challenge of optimizing routing and schedul-ing problems in a real-time setting has been receiv-ing increased attention (see, for example, PSARAFTIS,1988). The most common approach to solving theseproblems is to solve a model using the data as theyare known at a certain point in time, and then re-optimize as new data become available. Researchersin logistics typically work to find the best possiblesolution for a problem instance. In some cases, thismay involve solving little more than an assignmentproblem or network problem (e.g., POWELL, 1996) ora more complex integer programming problem (suchas those described in DESROSIERS, SOLOMON, andSOUMIS, 1995). In harder problems, such as classicalvehicle routing, optimal solutions are very hard tofind, raising the question of whether the additionaleffort is worth it. In related areas, such as machinescheduling, it is common to solve these problemsusing greedy heuristics (see GRAHAM, 1966; ALBERS,1997; and the general book PINEDO, 1995, and thereferences cited there). This field has received con-siderable recent attention from the computer science

community in the form of the study of on-line algo-rithms (SLEATOR and TARJAN, 1985).

Routing and scheduling in a dynamic environ-ment has been studied by a number of authors. Oneline of research has sought to analyze the propertiesof greedy heuristics for stochastic, dynamic prob-lems (BERTSIMAS and VAN RYZIN, 1991; BERTSIMASand RYZIN, 1991; BERTSIMAS and SIMCHI-LEVI, 1996;BERTSIMAS, CHERVI, and PETERSON, 1995). Othershave sought to develop explicit stochastic, dynamicmodels that incorporate both here-and-now informa-tion and some forecasts of future activities (see, forexample, DROR, LAPORTE, and TRUDEAU, 1989; LA-PORTE and LOUVEAUX, 1990; STEWART and GOLDEN,1983; TRUDEAU and DROR, 1992; POWELL, 1996).

One of the challenges of classical vehicle routingproblems is that it is looking to solve a stochasticversion of a problem where the deterministic versionremains computationally intractable and, for mostproblems, impossible to solve to optimality. Relatedresearch has focussed on problems where the deter-ministic versions are relatively easy to solve, mak-

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Transportation Science, © 2000 INFORMS 0041-1655 /00 /3401-0067 $05.00Vol. 34, No. 1, February 2000 pp. 67–85, 1526-5447 electronic ISSN

ing the stochastic version somewhat easier to ana-lyze. One problem in this class looks at managingfleets of vehicles over time, and the closely relatedproblem of assigning drivers to full-load movements.One of the first deterministic models of a fleet man-agement problem was posed by WHITE (1972), whichwas followed by a series of papers suggesting andsolving different approximations of a multistage,stochastic dynamic program (JORDAN and TURN-QUIST, 1983; POWELL, 1986, 1987, 1988; andFRANTZESKAKIS and POWELL, 1990). Recently,CHEUNG and POWELL (1996) showed that, for thisproblem class, explicit stochastic models can signif-icantly outperform deterministic models in rollinghorizon experiments. Despite this research, mostmodels in practice are basically deterministic andmyopic. All of these papers were oriented toward themanagement of large fleets of vehicles.

Recent research (SCHRIJVER, 1993; Powell, 1996)has focussed on the problem of matching individualdrivers to full-truckload movements, a problemcalled the load matching problem in truckload truck-ing. This problem class is much closer to vehiclerouting problems in its basic characteristics, withthe important exception that there is no in-vehicleconsolidation. In addition, the loads are typicallyvery long in duration, often requiring one or moredays to complete. The advantage of this problemstructure is that a single instance of the problem isa network assignment problem, considering thematching of drivers to loads. This problem is easy tosolve to optimality, and easy to update with real-time information. Schrijver (1993) used a greedyheuristic for matching drivers to loads in a study ofthe value of real-time communications. The researchin Powell (1996) compared a myopic model to anapproximation of the stochastic, dynamic problem,and showed that the stochastic, dynamic model out-performed the myopic model in rolling horizon ex-periments.

In this paper, we consider only myopic models ofthe load matching problem for truckload trucking.Following common engineering practice, our plan isto solve a given instance of the problem to optimal-ity, and then reoptimize as new information be-comes known. This is the approach that is widelyused in commercial applications. Our interest is incomparing optimal solutions of sequences of deter-ministic, myopic models, obtained by solving thenetwork assignment problem, to various approxima-tions including greedy solutions that simply assigndrivers to the best load. We study these algorithmsunder three forms of system stochasticity. First, weconsider the random process of shippers calling inloads. Second, we investigate the effects of network

performance by modeling randomness in traveltimes, which creates uncertainties around the ar-rival time of a driver at the destination. And third,we introduce and investigate the general issue ofuser noncompliance. Because no model is perfect,users routinely override the solutions of driver as-signment models. As a result, an additional level ofrandomness arises in the actual use of the recom-mendations of these models. In practice, severalcompanies report that average usage of these mod-els is typically below 60%, and good performance isconsidered around 70%.

We believe that the issue of user noncompliance isfundamental to all models, static and dynamic, andthat it raises the question of whether so-called opti-mal algorithms are indeed optimal. For example, ifreal-world schedules do not use 100% of the solutionproduced by a crew scheduling model (in a static,planning environment), was the original solutioneven optimal? In this paper, we show that, in thepresence of user noncompliance, optimal solutionswill be outperformed, over time, by algorithms thatare more local in nature. We feel this has broadimplications for the design of algorithms in othersettings (vehicle routing, crew scheduling), whichseek to find optimal solutions to sequences of deter-ministic, myopic problems.

To address these issues in a systematic fashion,we have developed a simulation architecture to sim-ulate truckload dispatching operations. This archi-tecture consists of three modules: a network optimi-zation model for assigning drivers to loads, a fleetsimulator/dispatcher module, and a demand gener-ation module. The network optimization model is aproduction system that is in use at a major motorcarrier. The modules communicate with each otherthrough files, closely simulating the actual flow ofdata at a carrier (where the optimization model typ-ically runs on a separate workstation, communicat-ing with the carrier’s primary management informa-tion system through a database). Our experimentsused the actual data from a large motor carrier,scaled down to simulate a smaller fleet (of 400trucks), a step that accelerated the running of eachsimulation (which need to run much faster than realtime).

The primary contributions of this paper are asfollows.

● We show that different forms of system stochas-ticity have the effect of reducing the value ofoptimality when solving sequences of subprob-lems in dynamic settings, and we show that asolution that is suboptimal for a particular prob-

68 / W. B. POWELL, M. T. TOWNS, AND A. MARAR

lem instance actually provides higher solutionquality over an extended simulation.

● We introduce and study for the first time theissue of user noncompliance, and show that thisform of uncertainty can, in particular, reducethe value of optimal solutions. Noncompliance isan easy to measure barometer of the quality ofan optimization model as perceived by the user.Even relatively low levels of user noncompliance(say, 30% modification of model recommenda-tions) indicate that partially greedy solutionscan outperform optimal solutions. At the sametime, similar levels of user noncompliance sig-nificantly reduced the gap between an optimalsolution and a greedy solution.

● We experimentally quantify the effects of differ-ent forms of system uncertainty, providing ameasure of the value of reducing these forms ofuncertainty on actual system performance.

The paper is organized as follows. Section 1 de-scribes different perspectives of optimality in thecontext of dynamic routing and scheduling. Section2 presents an overview of dispatching operations inthe truckload motor carrier industry, which we useas our problem setting. Section 3 presents the opti-mization model that is used to solve the problem.Next, Section 4 discusses some of the issues thatarise in global versus local optimization. These is-sues are central to any optimization problem wherethe results will be implemented over time, duringwhich the status of the system may change. Then,Section 5 gives an overview of our simulation archi-tecture, which takes advantage of a production loadmatching system that is in use at a major motorcarrier. This is followed in Section 6 by a descriptionof the specific modules. Section 7 describes the ex-perimental design, and the results are reported inSection 8. Section 9 summarizes the major conclu-sions of the paper.

1. PERSPECTIVES ON OPTIMALITY

TO PLACE THIS PAPER in the proper context withinthe research community, it is important to brieflydiscuss the meaning of the word “optimal.” It iscommon engineering practice to formulate a mathe-matical model, typically with numerous simplifica-tions to ensure tractability, and then to seek anoptimal solution to this model. We all accept that theword optimal is used in the narrow sense of solvinga specific mathematical representation, withoutmaking any claims regarding the impact on the realproblem, because these impacts fall outside of whatcan be measured experimentally using scientifically

rigorous standards. In the context of dynamic prob-lems, it is necessary to solve a sequence of problemsover time as new information arises, thereby simu-lating the real process.

In this setting, we must explicitly acknowledgethree models: the sequences of subproblems that aresolved over time (each of which is a model with itsown objective function), the larger simulation (withits own cost stream), and the even larger problem inthe real world, which must reflect data errors andother biases.

Let � � {�0, �1, . . . , �t, . . . , �T} be a sequenceof random outcomes over a horizon � � {0, 1, . . .,T}, where � � �. We would then define a probabil-ity space (�, �, �) where � is the �-algebra on �,and � is a probability measure defined over �. Let�t represent the sequence of sub-�-algebras contain-ing the information known up to time t, where �t ��t�1. Let ft

�(xt��t) be the objective function at timet given an information set �t at time t. The super-script � captures both the structure of the function ftand any parameters used to create the function (in-cluding, for example, planning horizons, discountfactors, and any cost factors introduced so that solv-ing ft at time t will produce a solution with goodlong-run behaviors). We can think of ft

�, � � � asthe set of all possible functions (or policies, if youwish) that we could use at time t. Now let xt

�(�t) �arg minxt�X ft

�(xt��t) be a sequence of �t-adapteddecisions (meaning that the function x� can only useinformation in �t). (Note: The concept of �t-adapteddecisions is similar, but not identical, to the conceptof nonanticipativity used in the stochastic program-ming community, which is expressed as a constraintrequiring that one decision be chosen in time period0 for all possible outcomes in the future. Instead ofsaying that a decision is �t-adapted, some authorsprefer to say that the decision is �t-measurable.)

The decision function xt�(�t) returns a set of deci-

sions that are to be implemented at time t given theevents �t. We write x�(�) to represent the decisionmade for a particular realization �, with the implicitunderstanding that the function x� is �t-measur-able. In determining this vector, we need to considerboth here-and-now costs at time t, and possibly ap-proximate estimates of costs that may be incurred inthe future, but which are subject to solving xt�

� for t�� t. For example, we may decide that a vehicleshould go from customer i1 to customer i2 right now;this is because we have formed a tour that consistsof (i1, i2, i3, i4). The objective function ft

� would beformulated to capture these future decisions, andmay also include approximations of what would hap-pen even farther in the future (as was done in Pow-ell, 1996). However, only the vector xt

� is actually

69MYOPIC SOLUTIONS FOR DYNAMIC ROUTING /

implemented. Let ct represent the cost of these de-cisions. Then, the costs incurred at time t would bectxt

�, and the total costs over the entire simulationwould be F(x, �) � �t�� ctxt

�(�), where x �(x0

�, x1�, . . . , xT

�).We are now ready to talk about different types of

optimality. Our goal as modelers is to solve

maxx�X

F� x � E� �t��

ctxt�� ,

where X is the space of all feasible �-adapted con-trols xt

�, � is the planning horizon, and the expec-tation operator is over all outcomes in �. In ourhierarchy of optimization problems, this is ourlarger problem. In practice, we solve F(x) using aseries of subproblems ft

�(xt) in an effort to find asolution that maximizes F(x). F(x), in turn, is noth-ing more than a proxy for the even larger problem inthe real world, which must also capture issues thatwe are unable to put into our model.

In the arena of routing and scheduling, it is verycommon to use myopic solutions that are then up-dated as new information arrives. Although this isthe simplest possible option for most classes of rout-ing and scheduling problems, it remains a combina-torically difficult problem. Thus, even finding theoptimal solution to this relatively simple problem isa nontrivial task. Because ft

�(xt) represents a singleinstance of a classical optimization problem, it isunderstandable that a considerable amount of effortwill be devoted toward optimizing this problem, re-alizing that it is simply a single snapshot of a largerproblem. Of course, even the larger problem, repre-sented by F(x), is little but an approximation of aneven larger problem that encompasses informationthat is not captured within the model. Because dy-namic routing models must run in production, thiseven larger problem, which must reflect the insightsand biases of real users, must begin to be addressedwithin the context of our research.

In this paper, we model the effect of the evenlarger problem through a process we call user com-pliance. In this process, we present a recommenda-tion to the user, at which point the recommendationis accepted or rejected. We make the assumptionthat the cause for rejecting a recommendation re-flects information outside the domain of the model,and, hence, we model it as an exogenous randomvariable. Specifically, for a particular decision, sayxij, we introduce a random upper bound uij(�) � 1with probability pd, and 0 with probability (1 pd).Only when we choose xij � 1 do we then learn thevalue of uij(�) where uij(�) � 1 represents the useraccepting the recommendation, whereas uij(�) � 0

represents a rejection. In the event of a rejection, itwould be possibly to exclude a solution and reopti-mize. In our work, we followed industry practice inthis setting and used the dual variables from theoptimal solution to suggest alternatives.

Because the physical process representing a useraccepting a decision is extremely complex, it is cus-tomary to model a process such as this as a randomvariable. This modeling approach is comparable toadding a noise term at the end of a function tocapture all the nonmeasurable effects. We feel thatthis representation of the user compliance issue rea-sonably captures the effects of the model adoptionprocess, and offers the benefit of being simple andexperimentally analyzable. Given that we cannotobserve the real process of why users reject modelrejections (and we have tried very hard to do justthis), the next best alternative, and far superiorthan ignoring the issue entirely, is to model theacceptance process as a random variable.

In the presence of user noncompliance, as well asother forms of uncertainty, it is reasonable to askthe question: what is the value of an optimal solu-tion of the function ft

�(xt)? Our work focuses on aproblem where finding the optimal solution, givenconventional modeling practice, is actually quiteeasy, allowing us to compare solutions that solveft

�(xt) optimally to those that solve it only approxi-mately. We consider only myopic models, in partbecause these are the models that are most widelyused in practice, and more importantly, because wedo not want to get into the tremendously rich andcomplex arena of developing approximations of fu-ture activities. Given the emerging research into theapplication of vehicle routing models in dynamicsettings, which focuses primarily on the heuristicsolution of myopic models, we feel that our choice ofresearch methodology is appropriate and producesinteresting results. Most significantly, we feel thatthe issue should be considered when testing differ-ent search algorithms in the context of harder prob-lems such as dynamic vehicle routing. For example,in the presence of user noncompliance, time-con-suming search heuristics may prove to add little orno value in a dynamic environment.

We do not claim that our simple model of usercompliance precisely captures the complex physicsof human thought. In support of our approach, wedraw on the long history of mathematical modelsthat use random variables to capture highly complexprocesses. We do claim, however, that the estimatesthat we obtain of the value of optimal solutions in adynamic environment are more accurate using ourmodel of user compliance than they would be if weignored the issue altogether. We further claim that


an algorithm that provides a better solution to ourmodel of user compliance is likely to be a morerobust algorithm that will provide better results inpractice.

2. THE DYNAMIC ASSIGNMENT PROBLEM INTRUCKLOAD OPERATIONS

WE FIRST PRESENT an introduction to the dynamicassignment problem and its relationship to truck-load motor fleet management (see Schrijver, 1993and Powell, 1996 for a much more complete discus-sion). The dynamic assignment problem involves theallocation of resources to perform tasks over a dis-crete-time dynamic network. Only some fraction ofthe relevant tasks in the network is known initially,with additional tasks becoming known in real timeas time progresses. This means that the optimalsolution for the problem is constantly changing asnew information becomes known. The result may bethat the viability of a previously optimal solution isreduced significantly in light of new information.

In truckload applications, drivers and loads arespread almost continuously over space. At a point intime, the challenge is to match drivers to loads, asillustrated in Figure 1. The problem is complicatedby the fact that there is much more to the problemthan simply finding the closest driver to a load.When finding a driver, a planner has to balancedeadhead (empty) miles, the availability of thedriver (some of the drivers may only be availablelate in the day), the pickup and delivery windows ofthe load, where the driver lives and whether the loadwill get him home on time, the skill and experienceof the driver (some shippers require top-rated driv-ers), whether the driver is a single driver (which canonly drive 10 hours at a stretch) or a team (whichcan drive almost continuously), and other character-istics. In addition, the planner needs to consider the

possibility of loads that will be called in later in theday.

The operations of a truckload carrier are rela-tively simple in concept. A shipper calls the dis-patching office to request the pickup of a load at aspecific location within a specific time window. Thistime window may begin on the day of the call-in or itmay begin some time in the future. The load must bedelivered to a specified location within a specifictime window. Other relevant information, such asspecial driver and trailer requirements, is also sup-plied by the shipper.

The dispatching situation for a major truckloadcarrier generally involves multiple human dispatch-ers each working within a specified geographic re-gion. In the absence of decision support systems,assignment decisions are made more or less inde-pendently of the actions of other dispatchers. Such apattern of multiple dispatchers acting this way inrelative isolation from one another tends to producea geographic patchwork of locally greedy optimiza-tions. These local optimizations mostly neglect theeffects downstream in space and time of assign-ments. The result is a solution that is suboptimalfrom a mathematical standpoint as compared to asolution derived from an optimization decision sup-port system.

3. THE LOAD MATCHING OPTIMIZATION MODEL

THE LOAD MATCHING problem assumes that at agiven point in time t, we have a set of loads �t anda set of drivers �t, and we need to match drivers toloads. Some of the loads may not be available forpickup until some time in the future (possibly a dayor more) and some of the drivers are currently pull-ing loads, and also will not be available until somepoint in the future (the driver estimated time ofarrival (ETA)).

For each potential assignment of a driver d � �tto a load l � �t we can assign a cost cdlt

a (this costwill change with time). In addition, we must con-sider the cost of refusing to service a load, and thecost of holding a driver idle. The problem here is todecide which drivers to assign to which loads tominimize the total costs (maximize total profits)throughout the network. An example of the driverassignment problem (using only assignment costs) isshown in Figure 2.

Loads are characterized by a) an origin and des-tination, b) a pickup time window and a deliverytime window, and c) a vector of attributes, whichmight include such items as type of driver requiredand load priority. Drivers are characterized by a) alocation at time of availability, b) a time of availabil-

Fig. 1. A simplified spatial assignment network.


ity, which may be in the future (if the driver iscurrently assigned to a load) or the past (if the driveris waiting for assignment), and c) a vector of at-tributes including special information like timespent on duty, experience level, and home domicile.

We can formulate this assignment problem as asimple static assignment model as follows.

�t �the set of all loads known to the system attime t

�t �the set of all drivers in the network at time tcdlt

a �the cost of assigning driver d to load l attime t

cltr �the cost of not assigning any drivers to load l at

time t (the refusal cost)cdt

h �the cost of not assigning driver d to any load attime t (the holding cost).

Our decision variables are

xdlta � � 1 if driver d is assigned to cover

load l at time t0 otherwise,

ylt � � 1 if load l is refused at time t0 otherwise,

xdth � � 1 if driver d is not assigned to any load

0 otherwise.

The objective function is then given by, for a givenpoint in time t,

minxa, xh, y

ft�xt � � �d��t

�l��t

cdlta xdlt

a � �d��t

cdth xdt

h � �l��

cltr ylt�

(1)

subject to

�d��t

xdlta � ylt � 1 l � � t (2)

�l��t

xdlta � xdt

h � 1 d � � t (3)

xdlta , ylt , xdt

h � 0 l � �, d � � t .

We let �l be the dual variable for Eq. 2 and �d be thedual variable for Eq. 3. Note that ft(xt) representsthe costs that would be incurred at time t if xt werechosen. This function is purely greedy and makes noeffort to incorporate factors that might improve theselection of xt to find solutions that would producehigher values for the multistage problem. In Section1, the notation ft

�(xt) refers to a function that incor-porates not only the immediate effect of a solution(represented by ft(xt) above) but any other adjust-ments that might be included to produce better long-run solutions. In the remainder of this paper, we useonly ft(xt) because we are considering only solutionsof myopic models.

The optimal solution of Eq. 1 can be found bysolving a simple assignment problem as depicted inFigure 2. In more complex problems, such as thosethat involve routing a driver through a sequence ofpoints (for example, if the loads are short, or if thereis in-vehicle consolidation), we may have to resort toheuristic solutions. In a dynamic setting, it is rea-sonable to consider the use of suboptimal or evengreedy solutions. For this reason, we let � be theclass of algorithmic strategies (or policies), and letxt

�, � � � be the solution obtained using policy �.The cost function cdlt

a for a truckload motor carrierhas to capture a number of performance measures.A successful carrier must minimize empty miles,maximize on-time service for loads, maintain a highlevel of driver satisfaction, and maximize equipmentproductivity. In the truckload arena, where driversoften spend several weeks away from home, driversatisfaction is measured by our ability to get driversback to their home by a certain time. Other issuesinclude whether a driver has the right training tohandle the needs of a particular load; whether histruck is too heavy to handle a particular load; andwhether he is considered sufficiently reliable tomeet the needs of an account.

We denote all these factors using a multidimen-sional cost function that is then collapsed into asingle utility function. Suppressing the time param-

Fig. 2. A simple driver assignment problem. The cdlt on se-lected arcs represent the arc cost for the assignment of driver d toload l at time t.


eter t, we may define

� �set of cost factors to be considered inthe assignment of a driver (such asempty miles, on-time service, and soon)

ad �vector of attributes of driver dbl �vector of attributes of load lrl �revenue, or reward from picking up a

load� �vector of parameters that are used to

translate a performance measure(such as arriving late to pick up aload) into a cost.

cmta (ad, bl, �)�cost of factor m � � of assigning a

driver with attribute vector. ad to aload with attributes bl at time t,given utility parameter vector �

The net total contribution of assigning driver d toload l at time t can now be expressed as

cdlta �� rl �

m��

cmta �ad , bl , �. (4)

In this paper, we are using the cost function that isin production at a major motor carrier. The param-eters have been carefully chosen to reflect what thecarrier believes is an accurate measure of the factorsthat impact the performance of the behavior. Thus,when we translate a late pickup into a $50 cost, wetreat this as a real cost, comparable to incurring $50in transportation costs.

4. GLOBAL VERSUS LOCAL OPTIMIZATION OF THESUBPROBLEM

THE EASE WITH WHICH the driver assignment modelcan be solved has spawned a cottage industry ofsoftware vendors who supply models of this sort tothe truckload industry. A major selling point wasthe ability to perform global optimization in an in-dustry where the driver assignment problem wassolved manually, using methods that tended to pro-duce locally good solutions. Researchers are activelydeveloping similar globally optimal solutions forother problem classes in vehicle routing and sched-uling (see Desrosiers et al., 1995 and FISHER, 1995for recent surveys of the field). Of course, the valueof a globally optimal solution depends, as always, onthe quality of the data. These references all refer toglobal optimization in the context of providing amathematically optimal solution to an individualsubproblem ft(xt). For the remainder of this paper,all references to global optimization refer to ourability to find the optimal solution to ft(xt) as op-

posed to a heuristic or suboptimal solution (as mightbe used in harder problems such as vehicle routing).

Real-time problems offer a special challenge to theoptimization community because it is difficult to getquality data in a timely way. For example, humanspossess a certain amount of “head knowledge,”which is information that they have acquired bytelephone, conversations, visual inspection, and ex-perience that is not in the computer. In addition,there is information that no one knows. The mostsignificant form of uncertain information is the de-mands of the customers, but we would also includeweather delays, breakdowns, and failure of driversto perform a given task.

We believe these factors are fundamental to anyapplication. The result is that the recommendationsmade by a model are not always implemented. Werefer to this effect as user noncompliance, whichmeasures the degree to which users actually imple-ment the recommendations of a model. User non-compliance often reflects information the user hasthat the system does not. The presence of user non-compliance implies that globally optimal solutionsare, in fact, not optimal at all. Humans reflect thisproperty intuitively by making decisions that aregood in a local sense (both spatial and temporal) butwhich may not, in theory, properly take into accounttheir impact on other parts of the system (in space ortime). For this reason, software developers have ar-gued that global optimization models can outper-form humans, overcoming the limitations humanshave in dealing with large problems.

We can quickly illustrate the impact of uncertaindata on solution quality using a simple illustrationshown in Figure 3. Here, we show driver A coveringload 1, driver B covering load 2, and driver C cover-ing load 3. Driver C would prefer to cover load 1, butif this were done, driver A would not have veryattractive options. A few minutes later, a new load iscalled in that is better for driver A, allowing driver Cto switch to load 1 after all. At this point, we learnthat driver B would rather cover load 3 instead ofload 2.

Given the same example, it would not be uncom-mon for a human to assign driver C to load 1 anddriver B to load 3, leaving both driver A and load 2unassigned. Because we assumed that driver A didnot really have any attractive assignments, theplanner might keep driver A unassigned while hop-ing for something better. This would be an instanceof using a greedy solution now to achieve an overallbetter solution later.

In practice, humans making decisions on an oper-ational basis spend relatively little time trying tofind the best solution for a given situation. Instead,


they spend most of their time simply verifying thatthe data they are looking at is, in fact, correct. Giventhe inherent uncertainty in the data (which mayhave nothing to do with forecasts of future customeractivities), it is perhaps not surprising that theylimit their searches for good solutions to small, ver-ifiable subsets of data. Below, we show that such astrategy can actually outperform a globally optimalsolution.

5. SIMULATOR ARCHITECTURE AND DEVELOPMENT

WE HAVE DEVELOPED a specialized simulator archi-tecture to test hypotheses regarding the behavior ofdifferent dispatching strategies in a dynamic envi-ronment. A primary goal of our architecture was totake advantage of a production load-matching sys-tem that has been developed and implemented for alarge truckload motor carrier. For this reason, weadopted a modular architecture consisting of threecomponents: a network optimization module, a fleetsimulator/dispatcher module, and a demand-gener-ator module. The interaction between these threemodules is represented graphically in Figure 4.Each of the three modules runs independently, co-ordinated through the use of a common clock object(just as a real system would operate). Of course, thesimulation runs at a rate faster than real time.

The simulator architecture allows a wide varietyof dispatching situations to be simulated. When the

dispatcher determines that a driver or load needs tobe matched, it sends a query to the optimizer askingfor information about possible assignments. The op-timizer, in simulated real time, sends over a smallfile containing a number of possible loads that adriver may be assigned to, along with the full set ofcosts and duals. Depending on parameter settings,the dispatcher may implement exactly what the op-timizer recommends, or we may allow it to makeother choices.

As the simulation progresses, two types of eventscan occur: drivers become available for assignmentto available loads, and new loads come into the net-work for assignment. A schematic of the driver as-signment-availability process is shown in Figure 5.The letter labels in the diagram refer to the variousmodules used at each step in this process. As adriver becomes available (in simulation time) forassignment, the simulator module (A) requests fromthe optimization module (B) driver–load arc-costinginformation for all loads with which the driver canbe feasibly paired. This costing information is of thetype required for the costing functions described inSection 3 and includes travel distances, availabilitytimes, and load characteristics. In addition, the mostrecent set of dual variables from the network algo-rithm is supplied from which the arc reduced costsmay be calculated. The optimizer also supplies theflow on each arc, because more than one arc mayhave zero reduced cost (in such an event, the simu-lator will always choose the arc to which the opti-mizer assigned flow). From this information, a dis-patching function inside the simulator module (C)makes a driver–load assignment decision (whichmay include no assignment at all), and updates all

Fig. 3. Revised optimal assignment in light of new servicedemand information.

Fig. 4. The multimodular fleet simulation system. The diagramshows the main modules and communication pathways betweenthem.


relevant driver, equipment, and load information,such as future driver time of availability, and loadavailability status. The assignment selection, infor-mation regarding any currently uncovered (rejected)loads, and any other desired information is thenwritten to a log file for later analysis by the user (D).Driver and load status update information are writ-ten to flat files for input by the optimizer module (E),which then supplies a reoptimization of the network.

This driver assignment process is completely gen-eral in that it can be used with any optimizationalgorithm, or even with no algorithm at all, so long

as the appropriate cost and assignment informationis given to the simulator module. The assignmentdecision-making logic is also quite flexible in thatthe user can specify exactly how and what costinginformation is to be used. This, for example, allowsthe user to specify that the simulated dispatcherexactly implement the solution of the optimizationalgorithm being used and then compare the result-ing solution to that obtained by using some arbi-trarily defined costing function. Such a functionmight be used to represent a human dispatcher’sinternal utility function.

Fig. 5. Sequence of multimodular simulator actions occurring when a driver becomes available for assignment.


6. SIMULATOR MODULES AND PROCEDURES

IN THIS SECTION, we describe some relevant detailsspecific to the three modules that comprise the fleetsimulation architecture: the optimizer, the simula-tor module (or interchangeably, the fleet simulator/dispatcher module), and the demand generator. Wealso discuss the use of the dispatcher selection prob-ability parameter, pd, and the implementation ofrandom travel times in the simulation architecture.

6.1 Optimizer

The driver assignment optimizer consists of a net-work generator and a network simplex code for solv-ing the matching problem. The code is a productiondispatch system that is in use at several carriers.The cost functions that determine the penalties forviolated delivery windows or for not returning adriver to his home on time are the same costs thatare used in production. For this reason, we havetreated these costs as real costs, because they havebeen chosen to accurately capture the tradeoff be-tween meeting these goals and running additionalempty miles (which is a hard, well-defined cost).

6.2 Demand Generator

The demand generator uses an actual historicaldemand file, from which it draws a random sample.As the simulation progresses, each load is read in,and a random number is drawn to determinewhether it is to be included in this run. If it is, thenthe generator waits until it is the right time, com-paring the actual call-in time of the load to thesimulated wall-clock time. When it is the right time,the generator sends the load over to the dispatchsimulator.

The use of a real dataset means that we have anaccurate picture of actual call-in processes, and realdata for the attributes of the loads, including pickupand delivery windows. By using only a sample ofthese loads, we were able to run repeated simula-tions using different random number seeds, allow-ing us to obtain repeated observations of a particu-lar optimization process. In our work, we used a 30%sample of the load, reflecting the fact that our fleetsize was approximately 30% of the size of the actualfleet used by the carrier. When we run repeatedsimulations with the same set of parameters, thenwe are simply choosing different samples of loads.

6.3 Fleet Simulator/Dispatcher

The simulator module performs two main func-tions. The first is the actual simulation of a truck-load fleet, and the second is the simulation of ahuman dispatching operation. The main steps in

these functions are presented in Figure 6. The fol-lowing notation is used in this figure.

tst �the simulated start time (in seconds) of thesimulation run

tend�the simulated end time (in seconds) of the sim-ulation run

tcur �the simulated current time (in seconds) of thesimulation run

d �a given driver�t �the set of all drivers in the network at time t�t

a �the set of all drivers available for assignmentat time t, (�t

a � �t).

The costing of potential assignments described inthis figure is done using a modified reduced costformula. This formula uses a dual variable discount-ing factor to vary the degree of dual variable utili-zation in assignment decisions. The formula used isas follows:

c� dlt� � cdlt � �� lt �dt, (5)

where

c�dlt()�the adjusted reduced cost for assigningdriver d to load l at time t

cdlt �the arc cost for assigning driver d to load lat time t

�dt �the dual variable for the node for driver d attime t

�lt �the dual variable for the node for load l attime t

�the dual variable discount factor.

Potential assignments for a given driver are rankedfor selection purposes from lowest to highest costusing c�dlt(). For full global optimization, we set �1.0, which gives us the equivalent network simplexreduced cost. For full greedy optimization, we set � 0.0, which gives a ranking according to arc costs.Values of in the interval (0, 1) are used to repre-sent intermediate strategies.

6.4 The Dispatch Function

As discussed above, the presence of user noncom-pliance with the optimization system’s recommen-dations is a major factor in the implementation ofoptimization decision-support systems. To simulatethis process, we define

udlt�w

� �1 if the user accepts the recommendationxdlt � 1 at time t

0 otherwise


pd � the probability that a dispatcher will

accept a potential driver to load assignment

� P�udl�w � 1�.

For a given realization w, we can determine whichload a driver will be assigned to, allowing us todefine the dispatch function for a driver d � �t by

xdlt� �, w � �1 if l � argmin

l��t

c� dl�� udl��w � 1�

0 otherwise.(6)

Equation 6, then, defines a class of dispatch func-tions parameterized by . If � 1, then we obtainthe optimal solution of ft; if � 0, then we obtain a

Fig. 6. Main fleet simulator/dispatcher module procedure.


greedy solution. 0 � � 1 produces a range ofintermediate solutions. Of course, many other typesof decision functions are also possible. For example,we could use a stable marriage algorithm (see, forexample, GUSFIELD and IRVING, 1989) or a localsearch heuristic. Our function provides an elegantknob that takes us smoothly from greedy local solu-tions to optimal solutions.

For example, the first assignment will be selectedby the simulated dispatcher function with probabil-ity pd and rejected with probability (1 pd). If it isrejected, the second-best assignment will then beselected with probability pd and rejected with prob-ability (1 pd). This process continues until eitheran assignment selection is made, or the number ofpossible assignments rejected equals a specifieduser parameter n. Should n potential loads be re-jected (with probability (1 pd)n), the given driveris assigned to remain idle for a specified time period,tidle. For our experiments, we set n � 5 and tidle � 60minutes of simulation time.

A problem that exists when using this logic is thepotential for the selection of excessively expensiveassignment arcs that would never actually be se-lected by a real human dispatcher. For example,when using a value of pd � 0.4, the chance ofselecting the fourth-best cost-wise assignment isabout 9%, and the fifth-best is about 5%. Because, inreality, there are often not five realistic assignmentarcs for a given driver, these low-ranked arcs mayhave very large arc costs and should be consideredinfeasible from a common-sense standpoint. The in-clusion of even a small number of such arcs in thesimulated dispatcher’s solution can skew objectivefunction values in an unrealistic fashion. To preventselection of such arcs, a cost cap of $500 was placedon arcs considered feasible for assignment.

6.5 Objective Function

We may now define the total net contribution (orprofit) incurred from running a simulation as

F�, pd, � � Epd� �t�tst

tend �d��t

a

�l��t

cdlt�� xdlt� �, w� ,

(7)

where we choose now to maximize profits over asimulation (instead of minimizing costs at a givenpoint in time). The expectation is over all randomevents, including the likelihood of user acceptance,given by the parameter pd (hence the explicit repre-sentation of pd in the expectation). We approximatethe expectation by running several simulations us-ing different random number seeds. Each run of the

simulation is a function of and pd. In addition, oursimulations considered variations of other problemcharacteristics, notably the presence of randomnessin travel times and the dynamics of the process bywhich customer demands become known to the sys-tem.

6.6 Random Travel Times

In transportation applications, one of the mostimportant sources of noise arises in travel times. Inthe load matching problem, travel times are impor-tant because they impact our ability to estimatewhen a driver will become available in the future,and whether this driver could be assigned to a loadwith a pickup window. To analyze the impact oferrors in travel times, we first define

dod�the distance in miles from load origin to desti-nation

teta�the expected travel time in hours�60dod/r

r �the average rate of speed (typically 40 mph).

We calculate a realization of an actual travel using

tactual � U�tmin , tmax

� the real travel time in hours,

where U(a, b) is a random variable uniformly dis-tributed over the interval [a, b]. The bounds of ourdistribution are determined using

tmin � the minimum possible travel time,

� teta �ws�teta

tmax � the maximum possible travel time,

� teta � �we�teta ,

where

ws �the number of standard errors (�teta) prior tothe expected time to include in the randomvariable window,

we �the number of standard errors (�teta) after theexpected time to include in the random variablewindow,

� �noise intensity parameter ranging from 0.0 to1.0. If � � 0, then there is no noise and we havedeterministic travel times.

During the simulated dispatcher’s costing of driver–load pairs, real travel times for transit between loadorigin and destination are obtained by the aboveformulation and stored in the simulator module’sdata structures. When the simulator module selectsa pairing for assignment, it sends an updated ETA


for the driver to the optimizer using teta. Thus, theoptimizer is planning future assignments using teta,whereas the simulator maintains the actual arrivaltime tactual.

In the event that tactual � teta, the driver willarrive at the load destination and become availablefor a new assignment earlier than expected. At thistime, the simulated dispatcher queries the optimizermodule to obtain costing information for feasibleloads for the given driver. This costing informationwill be based on teta. The simulated dispatcher thenrecosts potential assignment arcs from the optimizerbased on the early arrival, and makes an assign-ment selection using the procedure described in Sec-tion 6.3.

Alternatively, if tactual � teta, the driver will arriveat the load destination and be available for a newassignment later than expected. When the simula-tion clock reaches and then passes time teta, theoptimizer continually resets teta to be the currenttime.

Randomness in travel times can have the effect ofencouraging greedy solutions. Consider the situa-tion where drivers 1 and 2 are available now, anddriver 3 will become available later in the afternoon.Driver 1 is 40 miles away from a load, driver 2 is 90miles away, and driver 3, when he arrives, wouldonly be 20 miles away. Assume that driver 3’s esti-mated time of arrival is 3 P.M., and the load must bepicked up before 5 P.M. A network model wouldchoose driver 3 to cover the load based on cost andservice considerations. In this case, we might assigndriver 1 to another load. Later, we might discoverthat the ETA for driver 3 is inaccurate, and that hewill be delayed several hours. In this case, we haveno choice but to assign driver 2 to the load, at a muchhigher cost.

7. EXPERIMENTAL DESIGN

OUR PRIMARY INTEREST is in determining the valueof optimal (myopic) solutions over local solutionsunder different sources and degrees of uncertainty.We measure the degree to which a decision is globalversus local through the dual discount parameter .We consider three sources of randomness: a) ran-domness in the dynamic arrival of loads to the sys-tem; b) randomness in the travel times, captured aserrors in our estimates of driver ETA’s; and c) ran-domness in the implementation of recommendationsby the user, which we refer to as the user noncom-pliance problem. With each source of randomness,we focus on finding the value of that minimizes thesimulation cost function. In the process, we also

obtain estimates of the cost of each source of uncer-tainty.

We measure the value of global over local solu-tions through the dual utilization factor . Thus, ourexperiments will focus on determining the optimalvalue of under different degrees and sources ofrandomness.

The experimental data used for our runs was ob-tained from a major truckload motor carrier. A sim-ulated fleet consisting of 400 drivers was con-structed by randomly picking from a data setconsisting of the carrier’s full staff of about 1500drivers. An initial set of 100 loads was constructedfor input into the system. Subsequent to these initial100 loads, the demand generator was set to provideapproximately 225 loads per day of simulation time.Both the initial and subsequently generated loadsets were constructed by sampling from a set ofactual historical service demands called into thecarrier over a one-week period. Both the driver andload sets were distributed throughout the continen-tal U.S. Each run simulated seven days of actualdispatching, run at a speed of 60 times real time,resulting in a run time of 2.8 hours per run. For eachcombination of parameter values simulated, withthe exception of the advance demand booking exper-iment discussed in Section 8.1, we conducted fivesimulation runs. For the advance demand bookingexperiment, 10 runs were conducted for each param-eter value studied. This was done to clearly delin-eate solution behavior for this special case.

For each run, we statistically estimated a second-order polynomial relating total simulation profits(given by Eq. 7) to the dual discount factor . Weretained this second-order specification throughoutthe analysis.

Our experimental design is based on running sim-ulations with different values of and pd, alongwith other sources of randomness, and drawing con-clusions regarding the appropriate decision function(as specified by the parameter in Eq. 6). Thus, weneed to think about what it means to compare val-ues of F(, pd, �) in Eq. 7 for different values of pd

and . Three issues need to be discussed. First is therole of the utility parameter vector �, and what itmeans to optimize over a mixture of soft and hardcosts. Second is the issue of comparing model runsfor different values of the user acceptance parame-ter pd. Finally, we discuss the important issue ofcomparing model runs for different values of .

The utility vector � controls the relative impor-tance of different dimensions of the cost function. Aswe discussed above, our costs include both quantifi-able transportation costs, and other soft costs re-lated to customer service and driver management.


Our research addresses the challenge of finding adecision function x� that maximizes F(, pd, �)within the class of decision functions defined by Eq.6, parameterized by . At this point, we take asgiven the utility function parameterized by the vec-tor �. Thus, a dollar of lateness counts the same as adollar from traveling empty. In the context of ourresearch, this is the correct way to run the simula-tion. The choice of the vector � is set by management(in our case, this is the vector running in productionat a major motor carrier). It is not our job to questionwhether we have the correct value of �, which hasbeen chosen to balance the importance of issues suchas on-time service and routing drivers through theirhome with operating costs. Our simulations cover aweek of dispatching; tradeoffs of empty miles andservice, in contrast, are learned over years of watch-ing customers switch to competitors because of poorservice. Our challenge, then, is to maximizeF(, pd, �) for a given choice of �. For this reason,we have made no effort to run simulations withdifferent values of �.

The second issue concerns comparing model runs

using different values of the user acceptance proba-bility, pd. This question raises a very subtle issuewith regard to interpreting the objective functionF(, pd, �). Although we have a well-defined objec-tive function, the issue of user compliance bringsinto focus the difference between the objective func-tion we are maximizing, and the one the users aremaximizing. Assume that we wanted to compareF(, p1

d, �) to F(, p2d, �) where p1

d � p2d. We might

wish to estimate the increase in profits if the useracceptance level increased. In fact, this comparisonis not meaningful, and we cannot use our results toestimate the value of higher user compliance. To seethis, consider a run with p1

d � 1 versus p2d � 0.7. At

p1d � 1, we are accepting all the recommendations of

the model. In general, these recommendations arethe highest profit options and will produce the high-est profits over a simulation. If p2

d � 0.7, then weare, in effect, saying that 30% of the time, a top-ranked recommendation is not acceptable. Thus, thesupposedly high profit loads that appear at the topof the list for a driver are, in fact, low profit (mea-sured in terms of our utility function) for reasons

Fig. 7. Objective function versus dispatcher dual discount factor with advance demand booking and perfect user compliance(pd � 1.0).


that are not captured by the model. It would betempting to use our simulations to estimate thevalue of always using the model versus only some-times using the model. We feel that our simulationsdo not allow us to draw any conclusions along theselines.

Finally, we address the third and most importantissue of comparing values of F(, pd, �) for differentvalues of . Here, we are using the same user accep-tance probability pd. For a fixed value of pd we will,for any given value of , be searching, on average,the same distance down any list of loads offered to adriver. As long as pd is held fixed, we are capturingthe same degree of dissatisfaction with the higherranked options for a driver. When we change whileholding pd fixed, we are, in effect, solving the sameproblem with different algorithms.

The parameters � and pd, then, represent charac-teristics of the problem. , in contrast, is a charac-teristic of the algorithm. We may compareF(, pd, �) directly for different values of , and wemay also compare the optimal value of for differentvalues of pd and �, but we may not compareF(, pd, �) directly for different values of pd or �.Because a major focus of our research is the impactof user noncompliance, it is interesting to estimatethe best value of for different values of pd. We didnot feel that it would be very interesting to ask thesame question for different parameter vectors �.

8. EXPERIMENTAL RESULTS

WE NOW PRESENT the results of our simulations.First, Section 8.1 describes the results of our exper-iments that focus on the effects of randomness in thebooking process. It is in these runs that we produceresults that show that, if the demands are known inadvance, and travel times are deterministic, anduser compliance is 100%, then the optimal myopicsolution ( � 1) produces the best results. This is animportant piece of validation.

Next, Section 8.2 considers the impact of uncer-tain travel times. Finally, Section 8.3 presents theexperiments that summarize most carefully thevalue of greedy, optimal, and intermediate solutionsin the presence of varying levels of user compliance.

8.1 Effect of Dynamic Service DemandBooking

Our first set of experiments included a set of runswith the lowest level of uncertainty among all ourexperiments. In this run, we assumed all loads to becalled in during the day were known at 7 A.M. onthat day (but we do not know about tomorrow’sloads). We assumed deterministic travel times andperfect user compliance. We then ran simulationswith different values of , expecting to find that thebest value of was very close to 1.0. For this set of

Fig. 8. Objective function versus dispatcher dual discount factor, deterministic and stochastic travel times (pd � 1.0).


experiments, we performed 10 samples for eachvalue of that was tested.

The results are shown in Figure 7, which com-pares total profits versus for two sets of data: thefirst assumes all the daily bookings are known at thebeginning of the day, and the second retains theoriginal booking profile of loads. It is not surprisingthat we get higher profits when all the demands areknown at the beginning of the day. Most significantis that, when demands are known in advance, thecurve steadily increases with , leveling off for val-ues of over approximately 0.60. For this run, wecould not reject the hypothesis that the optimumdual discount factor was equal to 1.0, as we wouldexpect. When we simulate dynamic booking (thelower curve), performance noticeably dropped forvalues of over 0.80, with an apparent optimumclose to 0.75. This is our first evidence that, in thepresence of random customer booking, the globaloptimum solution does not give the best overall re-sults.

8.2 Effect of Network Stochasticity

The second set of experiments we performed ex-amined the effect of stochasticity in fleet operations

in the form of random travel times between loadorigin and destination locations. The random traveltimes were implemented as described in Section 6.6.The objective function values for these runs werecompared to those obtained using only expectedtravel times.

For these experimental runs, we set pd � 1.0. Thenoise intensity parameter, �, was set to 1.0. Thetime window start and end parameters, ws and we,respectively, were both set to 1.5. This means that a600-mile move that takes an average of 16 hoursmight require anything from 10 to as much as 22hours. While drivers in a real-world situation tendto run late more often than early, we wanted theexpectation of the random travel time to equal theexpected travel time. This was done to avoid influ-encing the objective function value by changing therelative number of driver–load assignments madeduring a simulation. Had the window been set toalways make travel times at least as long as theexpected time, for example, the fact that driverswould always be at best on-time and usually latewould mean that fewer assignments could be madein the course of a simulation.

The results are shown in Figure 8. These show

Fig. 9. Objective function versus dispatcher dual discount factor, various probabilities of selection.


that randomness in travel times has a relativelymodest (but significant to a motor carrier) impact onoverall profits. We note that the impact of is di-minished in the presence of random travel times.More importantly for our research, the shape of thecurve as a function of appears not to have animpact on the optimum dual discount factor. Ofcourse, both runs were done in the process of dy-namic booking, so we are, in effect, observing thatthe randomness in customer bookings is more im-portant than randomness in travel times. The opti-mum value of still appears to be around 0.75.

8.3 Effect of User Noncompliance

Our last set of experiments looked at the effect ofuser noncompliance on system performance. We re-port these results in two ways. First, Figure 9 sum-marizes system profits as a function of for differentvalues of pd. Our first conclusion from these runs isthat the optimum value of is close to 0.75 for allthree values of pd. Our second and more significantconclusion is that, for the lowest levels of user com-pliance, pd � 0.4, the dual discount has relativelylittle effect on overall profits. Said differently, thevalue in listing loads sorted in the order from best toworst (measured with respect to cdlt()) is signifi-cantly reduced. As user compliance rises, the valueof a proper listing of loads increases, although the

globally optimal solutions appear to slightly under-perform solutions using � 0.75.

A different perspective on the results was ob-tained by plotting total profits as a function of dif-ferent values of pd, using � 0, 0.75, and 1.0, shownin Figure 10. The results show that, as the level ofuser compliance drops, the value of a globally opti-mal solution over a greedy solution drops dramati-cally. In the truckload motor carrier industry, it isnot unusual to see values of pd � 0.6 or even lower.A value of pd � 0.7 is considered very high. Theseresults show that, for these levels of user compli-ance, greedy solutions work almost as well as glo-bally optimal solutions. Furthermore, using � 0.75gives the best results over all levels of user compli-ance, although, for the lowest levels, it really doesnot matter what you do. Reassuringly, if user com-pliance is high (over 90%) then the value of globalnetwork information (either with � 1.0 or 0.75) isquite high, and demonstrates the usefulness of op-timization models.

9. CONCLUSIONS

THE EXPERIMENTS in this paper investigate the im-pact of different forms of uncertainty on the value ofoptimal (myopic) solutions in a particular class ofrouting and scheduling problems. The results show

Fig. 10. Objective function versus dispatcher selection probability, various dual discounts.


that uncertainty does reduce the value of globallyoptimal solutions, bringing into question whetherrigorously optimal solutions are useful in a dynamicsetting.

Our research did not compare myopic and dy-namic models, but rather looked only at models thatconsider only information that is known (or reason-ably well known, in the case of random travel times)at a given point in time. This said, even a myopicmodel plans for decisions in the future, as long as itis based on known information. In our case, wewould consider the global optimization of all driversto all loads, including drivers that would not arriveuntil later in the day or the next day. Our dualdiscount, then, implicitly was putting a weight of 1.0on the decision we were about to make, and a re-duced weight on decisions that we might make at alater point in time (although it might only be a fewseconds later).

We are unaware of any other research that hasraised the issue of user compliance, and measuredits impact on the choice of algorithm. Even staticmodels are subject to user editing and modificationprior to implementation. Given such postoptimalitytampering, it is clear that the original solution is nolonger optimal, because it assumes that all the de-cisions in the solution will be implemented. Wewould argue that such considerations should bebuilt into the design and analysis of algorithms forall classes of routing and scheduling problems. Theresult may be problems that are actually easy tosolve, producing both faster run times and morerobust solutions.

ACKNOWLEDGMENT

THIS RESEARCH WAS SUPPORTED in part by grantAFOSR-F49620-93-1-0098 from the Air Force Officeof Scientific Research. The authors would also like togratefully acknowledge the careful reading by ananonymous referee who helped to highlight impor-tant issues in this study.

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(Received: August 1997; revisions received: July 1998, September1998; accepted: October 1998)


on the value of optimal myopic solutions for dynamic

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