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Management Science
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Business Analytics for Flexible Resource Allocation UnderRandom Emergencies
Mallik Angalakudati, Siddharth Balwani, Jorge Calzada, Bikram Chatterjee, Georgia Perakis,Nicolas Raad, Joline Uichanco
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Mallik Angalakudati, Siddharth Balwani, Jorge Calzada, Bikram Chatterjee, Georgia Perakis, Nicolas Raad, Joline Uichanco(2014) Business Analytics for Flexible Resource Allocation Under Random Emergencies. Management Science 60(6):1552-1573.http://dx.doi.org/10.1287/mnsc.2014.1919
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MANAGEMENT SCIENCEVol. 60, No. 6, June 2014, pp. 1552–1573ISSN 0025-1909 (print) ó ISSN 1526-5501 (online) http://dx.doi.org/10.1287/mnsc.2014.1919
©2014 INFORMS
Business Analytics for Flexible Resource AllocationUnder Random Emergencies
Mallik AngalakudatiPacific Gas and Electric Company, San Ramon, California 94583, [email protected]
Siddharth BalwaniBloomReach, Mountain View, California 94041; and Leaders for Global Operations, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, [email protected]
Jorge CalzadaNational Grid, Waltham, Massachusetts 02451, [email protected]
Bikram ChatterjeePacific Gas and Electric Company, San Ramon, California 94583, [email protected]
Georgia PerakisSloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, [email protected]
Nicolas RaadNational Grid, Waltham, Massachusetts 02451, [email protected]
Joline UichancoOperations Research Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, [email protected]
In this paper, we describe both applied and analytical work in collaboration with a large multistate gas utility.The project addressed a major operational resource allocation challenge that is typical to the industry. We study
the resource allocation problem in which some of the tasks are scheduled and known in advance, and someare unpredictable and have to be addressed as they appear. The utility has maintenance crews that performboth standard jobs (each must be done before a specified deadline) as well as respond to emergency gas leaks(that occur randomly throughout the day and could disrupt the schedule and lead to significant overtime).The goal is to perform all the standard jobs by their respective deadlines, to address all emergency jobs in a timelymanner, and to minimize maintenance crew overtime. We employ a novel decomposition approach that solves theproblem in two phases. The first is a job scheduling phase, where standard jobs are scheduled over a time horizon.The second is a crew assignment phase, which solves a stochastic mixed integer program to assign jobs tomaintenance crews under a stochastic number of future emergencies. For the first phase, we propose a heuristicbased on the rounding of a linear programming relaxation formulation and prove an analytical worst-caseperformance guarantee. For the second phase, we propose an algorithm for assigning crews that is motivated bythe structure of an optimal solution. We used our models and heuristics to develop a decision support tool that isbeing piloted in one of the utility’s sites. Using the utility’s data, we project that the tool will result in a 55%reduction in overtime hours.
Keywords : resource allocation; stochastic emergencies; scheduling; gas pipeline maintenance; utility; optimizationHistory : Received September 23, 2012; accepted January 13, 2014, by Noah Gans, special issue on business
analytics. Published online in Articles in Advance April 3, 2014.
1. IntroductionAllocating limited resources to a set of tasks is aproblem encountered in many industries. It has appli-cations in project management, bandwidth allocation,Internet packet routing, job shop scheduling, hospitalscheduling, aircraft maintenance, air traffic manage-ment, and shipping scheduling. In the past decades, thefocus has been primarily on developing methods foroptimal scheduling for deterministic problems. Theseapproaches assume that all relevant information is
available before the schedule is decided and the param-eters do not change after the schedule is made. In manyrealistic settings, however, scheduling decisions haveto be made in the face of uncertainty. After decidingon a schedule, a resource may unexpectedly becomeunavailable, a task may take longer or shorter time thanexpected, or there might be an unexpected release ofhigh-priority jobs (see Pinedo 2002 for an overview ofstochastic scheduling models). Not accounting for theseuncertainties may cause an undesirable impact, say,
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in a possible schedule interruption or in overutilizingsome resources. Birge (1997) demonstrated that inmany real-world applications, when using stochas-tic optimization to model uncertainties explicitly, theresults are superior compared to using a deterministiccounterpart.
In this paper, we study the problem of scheduling aknown set of jobs when there is an uncertain numberof emergency jobs that may arrive in the future. Thereare many interesting applications for this type ofproblem. For instance, Lamiri et al. (2008) describe theproblem of scheduling surgeries in hospital intensivecare units, where operating rooms are shared by twoclasses of patients: elective patients and emergencypatients. Emergency cases arrive randomly but must beserved immediately upon arrival. Elective cases can bedelayed and scheduled for future dates. In schedulingthe elective surgeries, the hospital needs to plan forflexibility (say, by having operating rooms on standby)to handle random arrivals of emergency cases.This paper is motivated by a project with a major
electric and gas utility. We worked on improvingscheduling of services for the utility’s gas businesssegment, which faces uncertainty in its daily opera-tions. The gas business segment of the utility generatesseveral billion dollars in revenue annually. The utilityoperates a distribution system (large networks of gaspipelines in several U.S. states) that delivers natural gasto its customers. A major part of daily operations of thegas utility is the maintenance of the pipeline networks.This entails executing two types of jobs: (i) standardjobs and (ii) emergency gas leak repair jobs. The firsttype of jobs includes new gas pipeline construction,maintenance and replacement of gas pipelines, andcustomer requests. The key characteristics of standardjobs are that they have deadlines by when they mustbe finished, they are known several weeks to a fewmonths in advance of their deadlines, and they areoften mandated by regulatory authorities or requiredby customers. The second type of job is to respondto reports of gas leaks. In the United States, morethan 60% of the gas transmission pipes are at least40 years old (Burke 2010). Most of them are composedof corrosive steel or cast iron. Gas leaks are likely tooccur on corroding bare steel or aging cast iron pipesand pose a safety hazard especially if they occur neara populated location. If undetected, a gas leak mightlead to a fire or an explosion. Such was the case in SanBruno, California, in September 2010, where a corrosivepipe ruptured, causing a massive blast and fire thatkilled eight people and destroyed 38 homes in the SanFrancisco suburb (Pipeline and Hazardous MaterialsSafety Administration 2011). To reduce the risk of suchaccidents occurring, company crews regularly monitorleak prone pipes to identify any leaks that need imme-diate attention. In addition, the company maintains an
emergency hotline for reports of suspected gas leaks.It is the company’s policy to respond to a report within24 hours of receiving it. The key characteristics ofemergency gas leak jobs are that they are unpredictable,they need to be responded to immediately, they requireseveral hours to complete, and they happen with fre-quency throughout a day. The leaks that do not posesignificant risk to the public are fixed later withinregulatory deadlines dictated by the risk involved.These jobs are part of the standard jobs.The company keeps a roster of maintenance crews
to execute standard jobs and to respond to emergen-cies. The company has experienced significant crewovertime driven by both controllable factors (such asworkforce management, scheduling processes, andinformation systems) and uncontrollable factors (suchas uncertainty related to emergency leaks, diverseand unknown site conditions, and uncertainty in jobcomplexity). Maintenance crews historically workeda significant proportion of their hours on overtime.From our analysis, one of the major causes of overtimeis suboptimal job scheduling and planning for theoccurrence of emergencies. Currently, the company hasno standard procedures or does not use quantitativemethods for job scheduling and crew assignment. Paststudies undertaken by the company suggested that abetter daily scheduling process that optimizes dailyresource allocation can provide a significant oppor-tunity for achieving lower costs and better deadlinecompliance.In this paper, we study the utility’s resource allo-
cation problem along with associated process andmanagerial factors. However, the models proposedand insights gained from this paper may have widerapplicability in settings where resources have to beallocated under stochastic emergencies.
1.1. Literature Review and Our ContributionsOur work makes theoretical contributions in several keyareas, as well as contributions to the utility’s operations.We contrast our contributions with previous workfound in the literature.Modeling and Problem Decomposition. We develop a
multiperiod model for the utility’s operations understochastic emergencies. Before knowing the number ofemergencies, the utility has to decide a standard job’sschedule (which date it will be worked on) and itscrew assignment (the crew assigned to it). We modelthe problem as a stochastic mixed integer program(MIP).
Several practical limitations discussed later in §3.1(such as computational intractability, the utility’s restric-tive computing resources, and employees’ warinessof a different decision-making process) prevented theutility from implementing the multiperiod stochas-tic MIP model. Therefore, we propose a two-phase
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decomposition, which addresses the original model’slimitations. The first phase is a job scheduling phase,where standard jobs are scheduled so as to meet all thedeadlines but while evenly distributing work over alldays (§4). This scheduling phase solves a deterministicMIP. The second phase is a crew assignment phase, whichtakes the standard jobs scheduled for each day fromthe first phase and assigns them to the available crews(§5). Since the job schedules are fixed, the assignmentdecisions on different days can be made independently.The assignment decisions must be made before arrivalsof emergencies; hence, the assignment problem on eachday is solved as a two-stage stochastic MIP.
LP-Based Heuristic for the Scheduling Phase. We proposea heuristic for the NP-hard job scheduling problembased on solving its linear programming (LP) relax-ation and rounding the solution to a feasible schedule.The scheduling phase problem is equivalent to schedul-ing jobs on unrelated machines with the objective ofminimizing makespan (Pinedo 2002). In our problem,the dates are the “machines.” The makespan is themaximum number of hours scheduled on any day.Note that a job can only be “processed” on dates beforethe deadline (the job’s “processing set”). Schedulingproblems with processing set restrictions are known tobe NP-hard.Other popular heuristics in the literature are list
scheduling rules (Kafura and Shen 1977, Hwang et al.2004). However these are applicable for problems withparallel machines. For unrelated machines, a well-known algorithm by Lenstra et al. (1990) performsa binary search procedure, in each iteration solvingthe LP relaxation of an integer program and thenrounding the solution to a feasible schedule. Ouralgorithm is also based on solving an LP relaxation, butit applies for unrelated machines with processing setrestrictions. Furthermore, it does not require initializingthe algorithm with a binary search and therefore onlysolves an LP once. Since this heuristic is based onlinear programming, in practice it solves very quicklywith commercial off-the-shelf solvers.
Performance Guarantee for the LP-Based Heuristic. Weprove a data-dependent performance guarantee forthe proposed LP-based heuristic (Theorem 1). Lenstraet al. (1990) prove that the schedule resulting from theirLP-based algorithm is guaranteed to have a makespanof no more than twice the optimal makespan. Theirproof relies on graph theory. On the other hand, thebound we derive uses a novel technique based onstochastic analysis. Moreover, when the algorithm isinitialized with a binary search, we can prove, usinggraph theoretic and stochastic arguments, a perfor-mance guarantee that is the minimum of two anda data-driven factor (Theorem EC.1 in the electroniccompanion). Since with real utility data, the data-driven
factor is less than two, we improve upon the bound byLenstra et al. (1990) in realistic settings.Algorithm for Crew Assignment Under a Stochastic
Number of Emergencies. The assignment phase problemis a two-stage stochastic MIP. In the first stage theassignment of standard jobs to crews is determined,and in the second stage (after the number of emer-gencies is known) the assignment of emergencies tocrews is decided. Most literature on problems of thistype develop iterative methods to solve the problem.For instance, a common method is based on Benders’decomposition embedded in a branch-and-cut proce-dure (Laporte and Louveaux 1993). However, if thesecond stage has integer variables, the second-stagevalue function is discontinuous and nonconvex, andoptimality cuts for Benders’ decomposition cannot begenerated from the dual. Sherali and Fraticelli (2002)propose introducing optimality cuts through a sequen-tial convexification of the second-stage problem. Thereare other methods proposed to solve stochastic modelsof scheduling under uncertainty. For instance, Lamiriet al. (2008) introduce a local search method to planfor elective surgeries in the operating room schedul-ing problem. Godfrey and Powell (2002) introduce amethod for dynamic resource allocation based on non-linear functional approximations of the second-stagevalue function based on sample gradient information.However, since they are developed for general two-stage stochastic problems, these solution methods donot give insights on how resources should be allocatedin anticipation of an uncertain number of emergencies.We exploit the structure of the problem and of
the optimal assignment and propose a simple andintuitive algorithm for assigning the standard jobsunder a stochastic number of emergencies (AlgorithmStoch-LPT). This algorithm can be thought of as ageneralization of the Longest-Processing-Time First(LPT) algorithm in the scheduling literature (Pinedo2002). We prove that this algorithm terminates with anoptimal crew assignment for some special cases.Models and Heuristics for Resource Allocation with
Random Emergencies. Our paper is motivated by thespecific problem of a gas distribution company. How-ever, the models and algorithms we develop in thispaper may have wider applicability to other settingswhere resources need to be allocated in a flexible man-ner to be able to handle random future emergencies.As a specific example, in the operating room planningproblem described in the introduction, the resources tobe allocated are operating rooms. Elective surgeriesand emergency surgeries are equivalent to standardjobs and gas leak repair jobs, respectively, in ourproblem.
Business Analytics for a Large U.S. Utility. We collabo-rated with a large multistate utility on improving thescheduling of operations in its gas business segment.
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The job scheduling and crew assignment optimizationmodels described above are motivated by the com-pany’s resource allocation problem under randomlyoccurring emergencies. The job scheduling heuristicand the crew assignment heuristics described earlierare motivated from practical requirements, includ-ing the company’s need for fast solution methods.We developed a Web-based planning tool based onthese heuristics that is being piloted in one of thecompany’s sites.We also used our models to help the utility make
strategic decisions about its operations. In simulationsusing actual data and our models, we highlight howdifferent process changes impact crew utilization andovertime labor costs. In this paper, we analyzed threeprocess changes: (i) maintaining an optimal inventoryof jobs ready to be scheduled, (ii) having detailedcrew productivity information, and (iii) increasing crewsupervisor presence in the field. We demonstrate thefinancial impact of these new business processes on ahypothetical utility company.
1.2. OutlineIn §2, we give a background of the gas utility and itsoperations. In §3, we present the job scheduling andcrew assignment problem, and motivate the two-stagedecomposition. In §4, we discuss the job schedulingphase, introduce an LP-based heuristic, and develop adata-driven performance guarantee of the heuristic.In §5, we discuss the crew assignment phase. In thissection, we prove structural properties of the optimalsolution, propose a crew assignment heuristic, andshow that it terminates with an optimal solution forsome cases. In §6, we discuss the development of theplanning tool, the pilot project, and how we usedsimulation and the models we developed for businessanalytics at the gas business of a large multistateutility company. Proofs not shown in the paper canbe found in the electronic companion (available athttp://ssrn.com/abstract=2412239).
2. Gas Utility Operationsand Background
In this section, we give a background of the utility’soperations and organization. The discussion serves tomotivate the model, assumptions, and our choice ofheuristics later in the paper.The gas utility operates and maintains a large net-
work of gas pipes. In the United States, the naturalgas industry is heavily regulated. Regulatory author-ities mandate gas utilities to replace their cast ironpipes into more durable steel or PVC pipes through amain replacement program. To meet the guidelines ofthis program, the utility has a dedicated departmentcalled the Resource Management Department that sets
yearly targets for standard jobs to be performed inthe field and monitors the progress relative to thesetargets throughout the year. All targets are yearly andcompany-wide.
Standard jobs that occur within a geographical region(usually a town or several neighboring towns) areassigned to a yard. A yard is the physical company sitethat houses maintenance crews who are dispatchedto complete the standard jobs. After the ResourceManagement Department decides on a company-widetarget for standard jobs, it is translated into monthlytargets for each yard based on yard size, numberof workers available, and other characteristics of theregion the yard serves. Several years ago, the utilityexpanded in the United States from a string of mergersof small independent local utilities operating in towns.As a result, even today, separate yards belongingto the company operate independently. Small yardscan have 10 crews, and large yards can have up to30 crews, with each crew composed of two or three crewmembers. Each standard job has a deadline set by theResource Management Department to ensure that thetargets are met and the company does not incur heavyregulatory fines for not meeting the requirements of themain replacement program. The company maintains acentralized database of standard jobs that lists eachjob’s deadline, status (e.g., completed, pending orin progress); location, job type, and other key jobcharacteristics; and also information on all past jobscompleted. A large yard can complete close to 500standard jobs in one month. The focus of the projectand hence of this paper is on yard-level operations,which we describe below.
2.1. Daily Yard OperationsEach yard has a resource planner who is charged withmaking decisions about the yard’s daily operations.At the start of each day, the resource planner reviewsthe pending standard jobs and their upcoming dead-lines and decides which jobs should be done by theyard that day. The resource planner also determineswhich crews should execute these jobs. Shortly after, themaintenance crews are dispatched to their first assign-ments. Throughout the day, the yard might receivereports of emergency gas leaks that also need to behandled by maintenance crews. These gas leaks arefound by company crews (operated by a departmentindependent from the yards) dedicated solely to moni-toring leak prone pipes to identify any leaks that needimmediate attention. Leaks found that do not posesignificant risk to the public are fixed later withinregulatory deadlines (usually within 12 months) dic-tated by the risk involved. These less severe leaks arecategorized as standard jobs. Emergencies are highlyunpredictable; a given yard can have between zeroto six emergencies per day (Figure 1). They also are
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Figure 1 Historical Distribution of the Number of Emergencies in aGiven Yard for April 2011
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long duration jobs since, for regulatory compliance, theutility requires its crews to dedicate eight hours forresponding to emergencies.Once an emergency is reported to a yard, any idle
crew is immediately dispatched to it. If there are no idlecrews, the first crew to become idle after finishing itscurrent job is immediately dispatched to the emergency.Once started, a standard job will not be paused evenwhen an emergency arrives because of the significantstartup effort for the job. Startup activities includetravel to the site, excavating the street to access the pipe,and arranging mandatory police presence at the site.This established procedure for handling emergencies
Figure 2 Crew Hours Worked Based on an Actual Schedule by a Yard’s Resource Planner
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complies with the regulatory guideline that any reportof an emergency has to be responded to within 24 hoursof receiving it.
Resource planners make crew assignment decisions(assignment of standard jobs to crews) at the beginningof the day. Then they monitor the arrival of emergenciesthroughout the day. However, once the crew assignmentis made, it is usually fixed for the rest of the day. Theydo not reassign a standard job to another crew onceit has been initially assigned. Yards rarely postponestandard jobs in the case of multiple emergencies.This is because the set of jobs to be done in a givenday needs to be known ahead of time to arrange forwork permits, police detail protection at the work site,crew equipment, and other logistical requirements forperforming the job.
2.2. Costs of Operations at YardsMaintenance crews have eight-hour shifts but can workbeyond their shifts if necessary. Any hours workedin excess of the crew’s shift is billed as overtime andcosts between 1.5 to 2 times as much as the regularhourly wage. Discussions with management revealthat it is preferable for maintenance crews to workovertime to complete standard job assignments ratherthan postpone standard jobs and risk incurring anyregulatory fines for not meeting deadlines. Based ondata from the company’s yards, maintenance crewshave been working a significant proportion of theirhours at overtime. Between 25% to 40% of an averagecrew member’s total work hours is on overtime.Figure 2 shows the actual crew hours worked in
April 2011 for one of the company’s average-sized
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yards (with 25 weekday crews and 4 weekend crews).From this figure, we observe that the hours spentworking on standard jobs are unevenly divided amongthe workdays. This variability of scheduled standardjob hours from day to day observed in the figureis due to several factors, including variation in thenumber of crews (e.g., reduced operations during week-ends, leaves, or unplanned absences) and standardjobs taking longer to finish than anticipated due toinclement weather or a complication during excava-tion (e.g., difficulty in locating the gas line). Resourceplanners are unable to anticipate variation in standardjob durations since they only rely on experience andintuition to plan yard operations rather than historicaldata. As part of our project, our team built a regres-sion model to predict standard job durations usingjob characteristics such as the job type, the size anddiameter of the pipe, the age of the pipe, and the joblocation.Another major cause of overtime is the stochastic
number of emergencies per day. Resource plannersdecide on crew assignments without accounting forunplanned emergencies. As seen in Figure 1, there isa huge variation in the number of emergencies evenwithin the same month. The crew assignment decisionsare made without the use of quantitative models, basedonly on the resource planner’s experience and subjec-tive input from supervisors. Therefore, often resourceplanners do not provide slack capacity (i.e., idle crewhours) to respond to any emergencies that might occurlater in the day. The variability of emergencies putresource planners in a reactive mode to meet dead-lines as well as to handle emergencies, resulting insuboptimal resource allocation.
3. Modeling and ProblemDecomposition
In this section, we discuss how we developed a stochas-tic optimization model for multiperiod planning ofyard operations under random emergencies. The modeldecides the job schedule (i.e., determining which dateeach standard job is done) and at the same timedecides the crew assignment (i.e., once a standardjob is scheduled on a date, determining which crewis assigned to complete the job). Later in this sec-tion, we discuss a novel decomposition motivatedby the practical limitations encountered during theproject.In what follows, we discuss all the assumptions in
our model, motivated from the yard operations.
Assumption 1. The number of crews available per day
is deterministic, although this number can vary daily.
Assumption 2. There is no preemption of standard
jobs.
Assumption 3. Standard jobs have deterministic dura-
tions. They do not necessarily have equal durations.
Assumption 4. The number of emergencies per day is
stochastic. Emergencies have equal durations.
Assumption 5. Crew assignment does not take distances
(geography) into consideration.
Assumption 6. The day can be divided into two parts
(pre-emergency and post-emergency). In pre-emergency, the
standard jobs are assigned to the available crews. Then the
number of emergencies are realized. In post-emergency, these
emergencies are assigned to the crews.
Some of these assumptions were imposed for sim-plicity of the model. One of the requirements of theutility was to have a simple model for reasons wediscuss later in §3.1.
Assumption 1 is due to staffing decisions not beingpart of yard operations since they are being madeby the Resource Management Department basedon company-wide projections of work for the year.Assumption 2 reflects the actual situation in yard oper-ations due to significant startup effort for standardjobs (see §2.1 for further discussion). Assumption 3 isbecause standard job durations are accurately predictedby a regression model using factors such as the jobtype and the age of the pipe. We observe minimalvariation between the regression’s predicted values andactual values of job durations. Assumption 4 is due tothe utility requiring its crews to devote a fixed amountof time on emergencies. However, the total numberof emergencies in each day is stochastic based onthe variation seen in yards (Figure 1). Assumptions 5and 6 are reasonable from a practical point of viewsince the factors they ignore are second order in themodel. Assumption 5 is assumed since travel timebetween jobs is usually much less than the duration ofjobs. Assumption 6 means that we ignore the specifictime that an emergency arrives. This is a reasonableassumption since regulation only requires a crew tobe dispatched to an emergency within 24 hours (andnot immediately as the emergency arrives). Therefore,if crews are already working on standard jobs whenan emergency arrives, the emergency does not haveto be responded to until a crew finishes its currentjob. However, it is possible that a crew might be idleif an emergency arrives after the crew finishes all itsstandard job assignments. Therefore, the problem set-ting can include dynamics in a given day and accountfor specific arrival times of emergencies. In §5.2, weprovide a rolling horizon implementation for a dynamicassignment of standard jobs and emergencies thatdepends on specific arrival times of emergencies.Next, we present our model for yard operations.
Consider a set of standard jobs that need to be com-pleted within a time horizon (e.g., one month). Each
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standard job has a known duration and a deadline.Without loss of generality, the deadline is assumed tobe before the end of the planning horizon. Within agiven day, a random number of emergencies may bereported. Reflecting actual yard operations, the numberof emergencies is only realized once the standard jobschedule and crew assignments for that day have beenmade. The following is a summary of the notationused in our model.
T : length of planning horizon;Kt : number of crews available for work on day t,
where t = 11 0 0 0 1T ;n: total number of known jobs;di: duration of job i, where i= 11 0 0 0 1n;íi: deadline of job i, with íi T , where i= 11 0 0 0 1n;dL: duration of each emergency;
L4ó5: number of emergencies under outcome ó;Ït : (finite) set of all outcomes in day t, where
t = 11 0 0 0 1T ;Pt4 · 5: probability distribution of events on day t,
Pt2 Ït 7! 60117.
We can estimate the probability distribution of thenumber of emergencies, which is different for each yardand each month, from historical yard data. For example,Figure 1 can be used as the probability distribution fora yard in the month of April.
At the start of the planning horizon, the job schedulehas to be decided. At the start of each day, the crewassignments need to be decided before the numberof emergencies is known. This is because the calls foremergencies occur later in the day, but the crews mustbe dispatched early in the morning to their assignedstandard jobs before these reports are received. After thenumber of emergencies is realized, the model decideson an assignment of the emergencies to the crews.Let the binary decision variable Xit take a value
of 1 if and only if the job i is scheduled to be doneon day t. Let the binary decision variable Yitk takea value of 1 if and only if job i is done on day t bycrew k. If outcome ó is realized on day t, let Ztk4ó5be the second-stage decision variable denoting thenumber of emergencies assigned to crew k. It dependson the number of standard jobs that have alreadybeen assigned to all the crews on day t. The variables4Xit5it , 4Yitk5itk are the first-stage decision variables.The variables 4Ztk4ó55tkó are the second-stage decisionvariables.For each day t, a recourse problem is solved. In
particular, given the day t crew assignments, Yt4=
4Yitk5ik, and the outcome of the number of emergencies,L4ó5, the objective of the day t recourse problemis to choose an assignment of emergencies, Zt4ó5
4=4Ztk4ó55k, so as to minimize the maximum number of
hours worked over all crews. Thus, the day t recourseproblem is
Ft4Yt1L4ó55
4=minimizeZt 4ó5
maxk=110001Kt
⇢dLZtk4ó5+
nX
i=1
diYitk
�
subject toKtX
k=1
Ztk4ó5= L4ó5
Ztk4ó5 2⇢+1 k= 11 0 0 0 1Kt1
(1)
where the term in the brackets of the objective functionis the total hours (both standard jobs and emergencies)assigned to crew k. We refer to Ft as the day t recoursefunction. The constraint of the recourse problem is thatall emergencies must be assigned to a crew.
The objective of the first-stage problem is to minimizethe maximum expected recourse function over all daysin the planning horizon:
minimizeX1Y
maxt=110001T
Et6Ft4Yt1L4ó557
subject toíiX
t=1
Xit=11 i=110001n1
KtX
k=1
Yitk=Xit1 i=110001n1 t=110001T 1
Xit 2801191 i=110001n1 t=110001T 1
Yitk2801191 i=110001n1 t=110001T 1
k=110001Kt1
(2)
where Ft4Yt1L4ó55 is described in (1). The constraintsare (i) job i must be scheduled before its deadline íi,and (ii) if a job is scheduled for a certain day, a crewmust be assigned to work on it. The optimization prob-lem (2) can be rewritten as a mixed integer program.Section EC.1 of the electronic companion provides theMIP formulation.The min–max objective function in (1) was chosen
after many discussions with the utility management, theyard employees, and other key stakeholders. An earlierversion of the model had the objective of minimiz-ing the total expected labor cost of completing thejobs (which is equivalent to minimizing total expectedovertime since straight hours are a sunk cost). How-ever, the min–max objective led to better acceptanceamong all the key company stakeholders compared toa cost-related or overtime-related objective. Moreover,management views high overtime labor cost as a symp-tom of a problem in their operations and not the rootcause that it wanted to solve. Management believesthat the underlying problem that the company is facedwith is an uneven distribution of both planned andunplanned work to the yard’s crews. Another reason
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for the choice of objective is that, in the earlier versionof the model that minimizes the expected overtimehours, the resulting solution was not acceptable to thecompany because of “fairness” issues and due to therisk of noncompliance with regulation. Even thoughthe solution with the expected overtime objective hasless overtime cost, in events with multiple leaks, itsometimes assigns one crew significantly more workhours compared to others (this crew incurs all theovertime hours while the rest do not).1 In practice,yards need to comply with union rules on the amountof work a single crew can do; therefore, yards preferto distribute work as evenly as possible among theavailable crews.2 Motivated by these considerations,we decided to choose the objective of minimizing theexpected maximum work hours.Our model assumes that standard jobs cannot be
postponed if multiple emergencies appear in one day.However, it is possible to explicitly incorporate jobpostponement in a dynamic programming model. Notethat such models are difficult to solve computationally(see a discussion by Godfrey and Powell 2002 ondifficulties of solving multistage problems). In practicalapplications, the most natural solution strategy is touse a rolling-horizon, solving the static problem at eachtime period using what is known at that period and aforecast of future events over some horizon. In §4.3, wecompare the rolling-horizon implementation to otherdynamic models for job scheduling.
3.1. Practical Limitations of the Joint JobScheduling and Crew Assignment Problem
Crew utilization is impacted by the combination ofthe job schedule and of the daily crew assignments.Hence, in model (2) the job scheduling decisions andcrew assignment decisions are made jointly. However,there were several practical issues that prevented theimplementation of the joint job scheduling and crewassignment problem in yard operations, which wediscuss below.
First, the full optimization problem is intractable tosolve with actual yard data within a reasonable amountof time. In actual yard settings, crew assignmentsneed to be determined within at most a few minutes.If there are no emergencies, the problem is known tobe NP-hard (Pinedo 2002). The presence of a stochastic
1 To illustrate this, consider a yard with a one-day horizon; threemaintenance crews; eight standard jobs (with three hour durationseach); and three emergency outcomes (zero emergencies with proba-bility 0.2, one emergency with probability 0.6, or two with probability0.2). If the objective is to minimize expected total overtime, then forthe event of two emergencies, one crew will be working for 22 hours,whereas the other two crews work for 8 hours. If the objective isto minimize the expected maximum work hours, then no crew isworking more than 16 hours in all outcomes.2 The number of crews is not a decision of the yard. See Assumption 1.
Figure 3 Relative MIP Gap in Gurobi’s Branch and Bound
39.5040.0040.5041.0041.5042.0042.5043.00
0 20 40 60 80 100 120 140 160
Gap
(%
)
Solving time (hours)
Notes. The gap represents the difference between the current upper and lowerbounds on the optimal cost in the branch-and-bound procedure. When the gapis zero, the current solution is optimal.
number of emergencies makes the problem even morecomputationally intractable when solving the determin-istic equivalent problem3 using commercial off-the-shelfsolvers. This is due to the structure of a stochasticMIP (Ahmed 2010). We demonstrate this by solvingthe deterministic equivalent problem with actual yarddata4 using Gurobi. Gurobi uses a branch-and-boundalgorithm that systematically enumerates all candidatesolutions, where large subsets of suboptimal candidatesare discarded as a group, by using upper and lowerestimated bounds of the optimal cost. The best integersolution found at any point in the search is called theincumbent solution. The cost of the incumbent is avalid upper bound to the optimal cost. The differencebetween the current upper and lower bounds is knownas the gap. A gap of zero means that the incumbentsolution is optimal. A large gap means that the solvercannot determine if the incumbent solution is optimal,and hence the branch-and-bound search continues.Figure 3 shows the gap in Gurobi’s branch-and-boundsearch plotted against the solving time. Note that evenafter 140 hours, Gurobi still is only able to reduce thisgap to about 40%. Therefore, for a yard that requiresgood solutions within a few minutes, implementingthe joint problem (2) is impractical.
A second practical issue our project faced is that atthe onset of the project, the resource planners werehesitant to use quantitative decision models in replac-ing decisions that they have previously been makingwithout guidance from any data or models, especially ifthe model follows a different decision-making processfrom their own.
3 With a finite number of scenarios, two-stage stochastic linearprograms such as (2) can be modeled as large linear programmingproblems with many variables and constraints by enumerating allscenarios and introducing a second-stage decision variable for eachscenario. This formulation is called the deterministic equivalentproblem.4 Actual yard data had 481 standard jobs, 20 crews per weekday,5 crews per weekend, 0 to 6 emergencies per weekday, 0 to 3 emer-gencies per weekend.
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A third issue is that because of concerns aboutintegration with the company’s current databases andother strategic matters, the company chose not to investin a commercial integer programming solver for aimplementation of the project throughout the wholecompany. Therefore, any models and heuristics wedevelop needed to be solved using Excel’s Solver orPremium Solver.These practical issues motivated us to consider a
decomposition of the joint problem into one in whichthe two decisions (job scheduling and crew assignment)are made sequentially. First is the job scheduling phase,which approximately schedules the jobs on the planninghorizon assuming only an average number of emergencieson each day. The goal is to meet all the standard jobdeadlines while evenly distributing work (i.e., the ratioof scheduled work hours to the number of crews) overthe planning horizon. Once the schedule of jobs isfixed, then the crew assignment problem can be solvedindependently for each day. In the crew assignmentphase, the standard jobs are assigned to crews assuminga stochastic number of emergencies. The goal is to minimizethe expected maximum hours worked by any crew.Note that the two-phase decomposition results in twolayers of resource allocation problems. The first layeris a longer-term problem where the “resources” arethe days that needed to be allocated to the standardjobs. The second layer is a one-day planning problemwhere the “resources” are the crews that needed tobe allocated to the standard jobs and the randomemergencies.
We discuss each phase of the decomposition in §§4and 5. Note that the optimization problems resultingfrom the decomposition are more tractable becauseof the smaller problem dimensions. The decomposedproblem also has better acceptance among the yardemployees. This is because the decomposition mimicsthe sequential decision-making process done by theresource planners. Resource planners usually makescheduling decisions based on average emergenciesbecause of the longer time horizon, whereas theyconsider variability as a problem that needs to beaddressed in crew assignments within the day. Finally,we developed heuristics for solving each stage in thedecomposition that can be implemented in Excel.
4. Phase I: Job SchedulingIn this section, we discuss the job scheduling phase,where standard jobs of varying durations and dead-lines have to be scheduled on a planning horizon.We present a deterministic mixed integer program(MIP) whose solution is a feasible job schedule thatevenly distributes work over the horizon. We alsopresent a tractable algorithm for producing a job sched-ule. The algorithm is based on solving the LP relaxation,
which, based on actual problem sizes, can be solvedusing Excel Premium Solver. The schedule resultingfrom the heuristic is near optimal in computationalexperiments and in actual yard problems.In yard operations, there is a random number of
emergencies per day, and the number of crews canchange for different days. For instance, yards usuallyhave fewer crews working during weekends comparedto weekdays. Moreover, there are fewer company crewsmonitoring gas leaks during weekends, so there areusually fewer emergencies discovered during week-ends. We chose to model the job scheduling phase toschedule standard jobs assuming a deterministic num-ber of emergencies (equal to the average). That is, thestandard jobs are scheduled to meet all the deadlines,while balancing (over all the days) the average hoursscheduled scaled by the number of crews. The jobscheduling phase solves the following optimizationproblem:
minimizeX
maxt=110001T
⇢1Kt
✓dLEt6L4ó57+
nX
i=1
diXit
◆�
subject toíiX
t=1
Xit = 11 i= 11 0 0 0 1n1
Xit 2 801191 i= 11 0 0 0 1n1 t = 11 0 0 0 1T 0
(3)
The motivation behind scaling the average scheduledhours per day by the number of crews is so that theoptimal solution will schedule fewer hours on dayswhen there are only a few crews.
Note that the scheduling decisions are made withouta detailed description of the uncertainties. Rather, thisphase simply takes the expected value of the number ofemergencies per day. The stochasticity in emergencieswill be handled in the crew assignment phase describedin §5. Because of these modeling assumptions, theproblem can be cast as an MIP with only a smallnumber of variables and constraints.
Proposition 1. Scheduling phase problem (3) can be
cast as the mixed integer program:
minimizeC1X
C
subject to dLEt6L4ó57+nX
i=1
diXit KtC1
t = 11 0 0 0 1T 1íiX
t=1
Xit = 11 i= 11 0 0 0 1n1
Xit 2 801191 i= 11 0 0 0 1n1 t = 11 0 0 0 1T 0
(4)
This problem is related to scheduling jobs to unrelatedmachines with the objective of minimizing makespanwhen there are processing set restrictions (Pinedo2002). The makespan is the total length of the schedulewhen all machines have finished processing the jobs.
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In our setting, “machines” are equivalent to the dates81121 0 0 0 1T 9. Each job i is restricted to be scheduledonly on dates (or “machines”) before the deadline,i.e., on “machines” 81121 0 0 0 1 íi9. In our setting, themakespan of machine t is the ratio of scheduled hoursto number of crews for day t.Note that even the simpler problem of scheduling
jobs on parallel machines is well known to be NP-hard (Pinedo 2002). List scheduling heuristics (wherestandard jobs are sorted using some criterion andscheduled on machines one at a time) are commonlyused to approximately solve scheduling problems withparallel machines (Kafura and Shen 1977, Hwanget al. 2004, Glass and Kellerer 2007, Ou et al. 2008).For the case of unrelated machines, a well-knownalgorithm by Lenstra et al. (1990) performs a binarysearch procedure, in each iteration solving the linearprogramming relaxation of an integer program, andthen rounds the solution to a feasible schedule. Usinga proof based on graph theory, they show that theschedule resulting from their algorithm is guaranteedto have a makespan of no more than twice the optimalmakespan.
4.1. LP-Based Job Scheduling HeuristicIn what follows, we describe an algorithm for approxi-mating the solution for the job scheduling problem (3),which we refer to an Algorithm LP-schedule. Similar toLenstra et al. (1990), this algorithm is also based onsolving the LP relaxation and rounding to a feasibleschedule. However, we do not require initializing thealgorithm with a binary search procedure, thereforesolving the LP relaxation only once. We are able toprovide a data-dependent performance guarantee forLP-schedule (Theorem 1) that we derive using a noveltechnique based on stochastic analysis.The following is a high-level description of Algo-
rithm LP-schedule. (For details, refer to the electroniccompanion.) Consider the LP relaxation of the schedul-ing phase MIP (4) where all constraints of the formXit 2 80119 are replaced by Xit � 0. Let 4CLP 1XLP 5 be thesolution to the relaxed problem. The algorithm takesXLP and converts it into a feasible job schedule using arounding procedure. The idea is to fix the jobs thathave integer solutions while re-solving the schedul-ing problem to find schedules for the jobs that havefractional solutions. However, a job i with a fractionalsolution can now only be scheduled on a date t whenthe corresponding LP solution is strictly positive; i.e.,XLP
it 2 40115. The rounding step solves an MIP; however,it only has O4n+ T 5 binary variables instead of theoriginal scheduling phase integer problem, which hadO4nT5 binary variables (the proof of this is similar tothat in Lenstra et al. 1990).
The following theorem states that the schedule result-ing from Algorithm LP-schedule is feasible (in that
it meets all the deadlines), and its maximum ratio ofhours scheduled to number of crews can be bounded.
Theorem 1. Let COPTbe the optimal objective cost of
the scheduling phase problem (4), and let CLPbe the optimal
cost of its LP relaxation. If XHis the schedule produced by
Algorithm LP-schedule, then XHis feasible for the scheduling
phase problem (4) and has an objective cost CH, where
CH COPT ⇥✓1+ 1
CLP
✓min
t=110001TKt
◆É1
·s12
✓ nX
i=1
d2i
◆41+ lnÑ5
◆1 (5)
where Ñ=maxt=110001T Ñt and Ñt4= ó8r = 11 0 0 0 1T 2 XLP
ir > 0and XLP
it > 09ó.The proof is based on stochastic analysis and can
be found in the electronic companion. We provide anoutline of the proof: Introduce X as the randomizedschedule derived by interpreting the LP solution XLP
as probabilities. For example, if XLPi1 =XLP
i2 = 005, thenjob i is equally likely to be scheduled on day 1 andday 2 in the random schedule. All outcomes of Xare all the possible roundings of XLP . Note that thealgorithm produces the rounding XH with the smallestcost (the maximum ratio of scheduled hours to numberof crews). Therefore, if we can prove that there exists apositive probability that X has a cost bounded by theright-hand side of (5), then this proves Theorem 1.We do this by defining Bt as the “bad” event that Xhas a day t ratio of scheduled hours to number ofcrews greater than the right-hand side of (5). We needto show that with positive probability none of thebad events B11B21 0 0 0 1BT occur. Note that each badevent is mutually dependent on at most Ñ other badevents. Then if there exists an upper bound on Pr4Bt5for all t, we can use Lovász’s Local Lemma (Erdos andLovász 1975) to prove that there is a strictly positiveprobability that none of the “bad” events occur. We useMcDiarmid’s Inequality (McDiarmid 1989) to derive anupper bound on Pr4Bt5.We now demonstrate using (5) that the schedule
resulting from the algorithm has a cost closer to optimalif the job deadlines are more restrictive (i.e., morestandard jobs are due earlier in the horizon) or ifthe job durations are less variant (i.e., standard jobshave very similar durations). Suppose that there areK crews per day. Note that CLP takes its smallestvalue when all jobs are due on the last day, withCLP = 4
Pni=1 di5/4KT 5. On the other extreme, if all the
deadlines are on the first day, CLP takes its largest valuewith CLP = 4
Pni=1 di5/K. Hence, the bound is smaller
under more restrictive deadlines. Furthermore, notethat Ñt represents (based on the LP solution) the numberof days that share a fractional job with day t. With more
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restrictive deadlines, we would expect Ñt to be smaller,implying that Ñ=maxt Ñt is smaller. Finally, considerthe case where CLP = 4
Pni=1 di5/4ÅK5, for some constant
Å> 0 (note that this is the case when deadlines areeither all on the first day or all on the last day). Thenthe bound simplifies to 1+Å4òdò2/òdò15
q12 41+ lnÑ5,
where d is the vector of job durations. Interpretingòdò2/òdò1 as the coefficient of variation in job durations,we can infer that the bound is smaller if there is lessvariance in the job duration data.
In both randomly generated job scheduling probleminstances as well as actual yard problems, we observethat the data-dependent multiplicative factor in (5)is less than two. But in some cases, the factor mightbecome large, for instance as T increases. However, wecan modify the algorithm, ensuring that the resultingschedule has a cost of no more than ÅCOPT , where Å isthe minimum of two and a data-dependent expression(Theorem EC.1 in the electronic companion). Hence,the bound will not explode in asymptotic regimes. Themodification is to initialize the algorithm with binarysearch procedure (described in §EC.4 of the electroniccompanion).We next compare Algorithm LP-schedule to the
optimal schedule on actual yard data for one month.In that month, there were 481 standard jobs withdurations ranging from three hours to nine hours.On weekdays, there were 20 crews available per day,and the number of emergencies ranged from zero to sixper day. On weekends, there were five crews availableper day, and emergencies ranged from zero to three perday. The job scheduling problem (4) implemented inGurobi did not terminate with an optimal solution afterseveral days. However, since COPT is bounded below byCLP , we used the LP relaxation solution to determinethat our algorithm results in a schedule that is at most506% different from the optimal job schedule. In theremainder of this section, we compare LP-schedule toother scheduling heuristics on simulated data and onactual yard data.
4.2. Comparing Algorithm LP-Schedule to aSensible Resource Planner
We randomly generate 100 problem instances and com-pare the schedule resulting from Algorithm LP-scheduleto a schedule that a sensible resource planner might oth-erwise produce following some rules-of-thumb. An algo-rithm that follows the sensible resource planner’s ruleswill be referred to as Algorithm SRP. SRP’s rules havebeen determined after consulting with several of theutility’s resource planners. In SRP, the standard jobsare sorted with increasing deadlines so that the jobwith the earliest deadline comes first in the list. ThenSRP will determine a cutoff value for work hours perday. Starting from the first day in the horizon, SRP willgo through the sorted list of jobs. If the current job
has a deadline of today or if the current work hoursscheduled for today is less than the cutoff, SRP willschedule the current job for today and remove it fromthe list. Otherwise, it does not schedule it today andmoves on to the next day. The cutoff used by SRP is thetotal work hours divided by the number of days.In each problem instance, there are seven days
in the planning horizon, and three crews availableeach day. There are 70 standard jobs to be scheduled(with durations randomly generated between zeroto eight hours, and deadlines randomly chosen fromthe seven days). The size of the problem instance ischosen so that the job scheduling problem solves tooptimality within a reasonable amount of time. For eachproblem instance, we apply both LP-schedule and SRP,noting the cost of both schedules, i.e., the maximumratio of average scheduled hours to number of crews.A schedule is near optimal if its cost is close to theoptimal cost from solving the scheduling problem (4).For each problem instance, we compute the percentagedifference of the heuristics’ cost to the optimal cost.Algorithm LP-schedule has a sample mean (taken over100 instances) for the percentage difference equal to3.6%. The 95% confidence interval for this samplemean is 6209%1402%7. On the other hand, SRP hasa sample mean for the percentage difference equalto 9.7%. The 95% confidence interval for this samplemean is 6901%11002%7. We can infer that schedulingwith Algorithm LP-schedule results in a cost closerto the optimal cost than scheduling by SRP since therange of values of LP-schedule’s confidence intervalfalls below the range of values for SRP’s confidenceinterval. Additionally, Algorithm LP-schedule managesto improve computational efficiency. Solving for theoptimal schedule in (3) often requires several hours,which is not viable in actual yard operations. On theother hand, Algorithm LP-schedule only takes a fewseconds to solve.
4.3. Dynamic Job SchedulingThe Phase I model results in a static job schedule. In thecase of the utility, yards rarely postpone standard jobsin the case of multiple emergencies (see discussionin §2). However, one can potentially solve the jobscheduling problem with a rolling horizon so thatstandard jobs can be rescheduled as more informationis revealed. In practice, static models are often solvedwith a rolling horizon rather than solving a dynamicprogram. This is because dynamic resource allocationmodels are computationally intractable, with solutionmethods often only approximating the value function(Godfrey and Powell 2002, Huh et al. 2013).
Using actual yard data for one month, we comparethree job scheduling heuristics: the static AlgorithmLP-schedule, the rolling horizon implementation ofLP-schedule, and Algorithm SRP. To evaluate the cost
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of these schedules, we compare their costs relative tothe cost of a perfect hindsight job schedule, which isthe optimal job schedule after knowing the sequenceof emergencies occurring each day. The cost of theperfect hindsight job schedule is clearly smaller thanany dynamic job schedule since it has the advantage ofcomplete information. The cost of the perfect hindsightmodel is unachievable in reality. However, we use it tobenchmark against the costs of the three heuristics.
The experiments use actual yard data for one month(481 standard jobs, 20 weekday crews, 5 weekendcrews). The static job schedule is the output of Algo-rithm LP-schedule applied to the one-month horizon.To implement the rolling horizon schedule, AlgorithmLP-schedule is reapplied every day, with a horizonstarting from the current day until the end of themonth. On the current day, the number of emergenciesis known (for our experiments, it is randomly drawnfrom the empirical probability distribution shown inFigure 1). The algorithm is applied using the knownnumber of emergencies for the current day and anexpected number of emergencies for the remainingdays. The SRP job schedule uses the rules describedin §4.2. We compare the three schedules to the perfecthindsight schedule where the sequence of emergenciesis known.
Figure 4 shows the histograms of the three heuristics’costs relative to the perfect hindsight cost generatedfrom 100 sample paths for emergencies. In the experi-ments, the total number of emergencies for the monthvaried between 48 to 88. Note that the cost of theschedule produced by the sensible resource planner isthe highest of the three, with an average percentagedifference of 123%. The average percentage differenceof the rolling horizon schedule is only 11.4%, whereasfor the static schedule it is 27.3%. For 80% of the samplepaths, the percentage difference of the rolling horizonschedule is no more than 16%. On the other hand,the 80th percentile of the static schedule percentagedifference is 39%. Based on these experiments, weconclude that if the yard could postpone standardjobs, then it can reap significant cost savings fromimplementing a dynamic schedule rather than a staticschedule. Moreover, its small percentage cost differencecompared to the perfect hindsight cost implies thata simple rolling horizon implementation of the jobscheduling problem already achieves most of thesegains compared to the best dynamic schedule or aschedule obtained from solving a dynamic program.
5. Phase II: Crew AssignmentIn this section, we focus on the second phase of thedecomposition, i.e., the crew assignment problemwithin one day. The basic problem is determining whichcrews should execute which standard jobs and which
crews to reserve for emergencies, given the stochasticnumber of emergencies. We develop an algorithm forcrew assignment, which we motivate from a propertyof the optimal crew assignment. We can show that, insome special cases, the algorithm terminates with anoptimal crew assignment. We also modify the algorithmfor a multiperiod setting that allows reassignmentsand evolution of forecasts for the emergencies withinthe day.Denote by I the set of standard job indices to
be assigned in one day (as determined in the jobscheduling phase). Let L be the stochastic number ofemergencies on that day. Suppose there are K crewsavailable on that day. The crew assignment problemassigns all standard jobs in set I to the available crews.However, these assignments must be made before thenumber of emergencies for the day is realized. Afterthe number of emergencies is known, all emergenciesmust be assigned to the crews. The objective is tominimize the expected maximum hours worked onthat day.
The crew assignment problem for one day solves atwo-stage stochastic mixed integer program. The first-stageproblem is
minimizeY
E6F 4Y 1L4ó557
subject toKX
k=1
Yik = 11 i 2 I1
Yik 2 801191 i 2 I1 k= 11 0 0 0 1K1
(6)
where F 4Y 1L4ó55 is defined as
F 4Y 1L4ó55 4=minimizeZ
maxk=110001K
⇢dLZk +
X
i2IdiYik
�
subject toKX
k=1
Zk = L4ó51
Zk 2⇢+1 k= 11 0 0 0 1K0
(7)
Note that the term in the brackets of the objective func-tion is the number of hours assigned to crew k duringoutcome ó and under the standard job assignments Y .The assignment phase problem can also be solved asa mixed integer program (see §EC.6 of the electroniccompanion).Even if the number of emergencies is known, the
deterministic crew assignment problem is NP-hard.With a stochastic number of emergencies, the problemis even more computationally intractable (Ahmed 2010).The difficulty with stochastic integer programming isthat some constraints are repeated with small changesfor each scenario. One of the important features ofcommercial off-the-shelf solvers is the generation ofcutting planes by analyzing the polyhedral relaxationof the problem and expediting branch-and-bound
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Figure 4 Histogram of Static, Dynamic, and SRP Cost Percentage Difference Relative to the Perfect Hindsight Cost
(%)
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60
80
100
120
0
5
10
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20
25
30
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40
3 7 11 14 18 22 25 29 33 36 More
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quen
cy
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(%)
0
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120
0
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search. Typically, such cuts are generated by analyzinga single row of the constraint system. A stochastic IPis computationally difficult since information frommultiple rows in its constraint set is lost (Ahmed 2010).With actual yard data, the crew assignment problem(6) implemented in Gurobi terminates with an optimalsolution only after several hours.
Motivated by the intractability of solving the prob-lem to optimality, we developed a crew assignmentalgorithm that exploits the specific structure of thecrew assignment problem, which we will describe laterin §5.1. This algorithm is simple and intuitive sinceit can be viewed as a stochastic variant of the LPTrule. The algorithm we developed also resulted in
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natural guidelines for resource planners to follow inmaking yard operations under a stochastic number ofemergencies.
5.1. Crew Assignment HeuristicWe conducted computational experiments on severalexamples to gain insight into the structure of theoptimal crew assignment solution to (6). Section EC.11of the electronic companion explains in detail theexperiments we conducted. An observation we makefrom the experiments is that in the optimal solution,if a crew is assigned to work on an emergency in agiven emergency outcome, that crew should also beassigned to work on an emergency under outcomeswith more emergencies. This is formalized in thefollowing proposition.
Proposition 2. There exists an optimal solution
4Y ⇤1Z⇤4ó51ó 2Ï5 to the stochastic assignment problem (6)where each crew’s emergency assignment is monotonic in the
number of emergencies. That is, if L4ó15<L4ó25 for some
ó11ó2 2Ï, then Z⇤k4ó15Z⇤
k4ó25 for all k= 11 0 0 0 1K.
This proposition motivates our heuristic for crewassignment under stochastic emergencies. The heuris-tic results in crew emergency assignments that aremonotonic in the number of emergencies. We referto the heuristic as Algorithm Stoch-LPT since it is isa variant of the LPT algorithm under a stochasticnumber of emergencies. LPT applies when there areno emergencies, and the objective is to minimize themaximum work hours of the crews. In each iteration ofLPT, it keeps track of the current number of assignedwork hours (current load) for each crew. LPT initializesthe current load for each crew to be zero. Then itsorts the standard jobs in decreasing duration. Startingwith the longest duration job, each iteration of LPTassigns the current standard job to the crew with thesmallest current load, updating the current load afteran assignment is made.In what follows, we describe Algorithm Stoch-LPT
for stochastic emergencies, with the objective of mini-mizing the expected maximum work hours of the crews.The algorithm begins by first making assignmentsof emergencies in each outcome of L (the stochasticnumber of emergencies). For example, in an outcomewith two emergencies, the algorithm needs to assigntwo emergencies to the crews. For each outcome, thealgorithm assigns emergencies, starting with the out-come with the least emergencies, then the one with thesecond least, continuing until it assigns all emergenciesunder all outcomes. For the current outcome, Algo-rithm Stoch-LPT uses a procedure for assigning theemergencies that preserves the property of monotoniccrew emergency assignments described in Proposition 2.It assigns the emergencies in the current outcome tothe crews by the LPT rule. But in case of ties (where
more than one crew has the smallest current load), itchooses a crew whose current load is strictly smallerthan its load in the previous outcome’s assignment.After emergencies have been assigned for all out-
comes, the next step in Stoch-LPT is to assign thestandard jobs. The algorithm keeps track of the currentload of each crew in each outcome, which is initializedafter the crews’ emergency assignments. Then, like LPT,the algorithm sorts the standard jobs in decreasingorder of duration. Starting with the longest durationjob, each iteration of Stoch-LPT assigns the currentstandard job to a crew according to the following rule.Under each crew, determine the increase in expectedmaximum load that results from assigning the cur-rent job to that crew. Note that different assignmentsresult in different loads for crews in a given outcomeof L. The expected maximum load is computed bysumming over all outcomes the maximum load in theoutcome multiplied by the probability. The standardjob is assigned to the crew that has the smallest amountof increase in the expected maximum load. If there areany ties, the standard job is assigned to the crew withthe smallest expected current load. After a standardjob is assigned, the current load of each crew in eachoutcome is updated.
We next discuss an implication of having crew emer-gency assignments that are monotonic in the number ofemergencies. In reality, the leak outcome reveals itselfover time since leaks are discovered throughout the day.However, as the proposition states, if a crew is assignedto an emergency for an outcome with one leak, thenthis same crew is assigned at least one emergency foroutcomes with two, three, and more leaks. Therefore,the first leak that appears in any outcome is alwaysassigned to that crew. This way, one can “rank” thecrews that handle the emergencies. Thus, the crewassignment solution of the static model can be easilyimplemented in a real-time setting where leaks arrivethroughout the day. This ranking of crews based on theoptimal leak assignment is formalized in the followingproposition.
Proposition 3. Suppose 4Y 1Z4ó51 ó 2Ï5 is a feasible
solution to the stochastic assignment problem (6) with crew
emergency assignments that are monotonic in the number
of emergencies; i.e., if L4ó15<L4ó25 for some ó11ó2 2Ï,
then Zk4ó15 Zk4ó25 for all k= 11 0 0 0 1K. Then crews can
be relabeled as k11k21 0 0 0 1kK so that ZkjÉ14ó5�Zkj
4ó5 forall ó 2Ï.
Corollary 1. There exists an optimal solution
4Y ⇤1Z⇤4ó51 ó 2 Ï5 to the stochastic assignment prob-
lem (6) where the crews can be relabeled as k11k21 0 0 0 1kKso that Z⇤
kjÉ14ó5� Z⇤
kj4ó5 for all ó 2Ï and
Pi2I diY
⇤i1kjÉ1
Pi2I diY
⇤i1kj
.
In what follows, we investigate in several exampleshow the crew assignment produced by Algorithm
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Stoch-LPT compares to the optimal crew assignmentsolution to (6) and to crew assignments from otherheuristics. All examples use the same 15 standard jobsand seven crews but a different probability distributionof the number of emergencies. (See Tables EC.1 andEC.2 in the electronic companion for the data.) Alldistributions have an expected number emergenciesequal to one. We refer to the optimal crew assign-ment as OPT. We consider two other crew assignmentheuristics aside from Stoch-LPT. First is the heuristicthat solves a deterministic crew assignment modelassuming that the number of emergencies is equal to theexpectation E6L7 (in the examples, the E6L7= 1). Thisheuristic is motivated from practice since often practi-tioners facing uncertainty in their operations choose toignore it by planning their operations based on theaverage uncertainty. We refer to this heuristic as AVG.The second heuristic we consider approximates thedeterministic crew assignment optimization problemAVG (whose set of jobs includes the standard jobs andone emergency) by applying the LPT rule. We referto this heuristic as 1-LPT. Note that AVG and OPTboth solve NP-hard optimization problems. 1-LPTand Stoch-LPT are heuristics that approximate theproblems, respectively. Moreover, 1-LPT is a methodthat is a closer approximation of how resource plannerscurrently make their assignment decisions since theymake decisions based on averages and they do nothave the computational resources to solve optimizationproblems.Table 1 summarizes the expected maximum work
hours. It illustrates under which emergency leak distri-butions each heuristic performs best. Note that AVGhas near-optimal expected maximum hours in mostexamples, except under leak distribution 4. This dis-tribution has the highest probability of having morethan one leak. The assignment that AVG producesis not robust to an outcome with a large number ofleaks. Now, let us compare Stoch-LPT to 1-LPT. Notethat in most of the examples, Stoch-LPT has a smallerexpected maximum hour than 1-LPT. Moreover, 1-LPT
Table 1 Comparing Algorithm Stoch-LPT to the Optimal CrewAssignment and Other Heuristics
Expected % differencemaximum hours to OPT
Stoch- Stoch-OPT AVG 1-LPT LPT AVG 1-LPT LPT
Leak distribution 1 10066 10066 11050 11050 0000 7096 7096Leak distribution 2 11041 11042 12013 12006 0006 6028 5072Leak distribution 3 11078 12018 12075 12075 3039 8024 8023Leak distribution 4 11079 13070 14000 12067 16023 18073 7050Leak distribution 5 12018 12018 12077 12019 0003 4080 0011Leak distribution 6 12050 12095 13039 13044 3059 7013 7052Leak distribution 7 12085 12095 13040 13034 0075 4027 3079
also does poorly when there is a high probability of anoutcome with many leaks.
Section EC.10 of the electronic companion providesanalytical results on the performance of AlgorithmStoch-LPT relative to the optimal crew assignmentsolution when there are only two crews and twoemergency outcomes. We prove that in the special caseof equal job durations, the algorithm terminates withan optimal crew assignment.
5.2. Dynamic Crew ReassignmentMotivated by yard operations where assignment ofstandard jobs is determined once in the beginning ofthe day and cannot be changed, our model for thecrew assignment is static. We now modify AlgorithmStoch-LPT such that standard jobs not yet started canbe reassigned later in the day as more informationabout the emergencies becomes available.
We assume that the standard jobs can be reassignedevery hour. We also assume that the arrival of emer-gencies follow a Poisson process with an arrival rate ã.Any arrival process can be used; however, we chose aPoisson process for the purpose of illustration. Recallthat the emergencies are found by dedicated companycrews that monitor for gas leaks in a shift of eight hours.Then there is a natural update rule for the belief onthe number of emergencies. Suppose there are s hoursremaining until company crews stop monitoring forleaks. Then the probability that n emergencies arefound within s hours is P 4L= n5= 44ãs5n/n!5eÉãs , forn= 011121 0 0 0 0
What we refer to next as the dynamic crew reassign-ment is the following. At the start of the day, determinethe standard job and emergency assignments accord-ing to Algorithm Stoch-LPT. We make a distinctionbetween Stoch-LPT’s assignments and the actual assign-ments that are based on the actual outcome of thenumber of emergencies. In the first hour, the numberof emergencies are realized according to the Poissondistribution. Make the actual emergency assignmentbased on Algorithm Stoch-LPT. For example, if there isone emergency, assign that emergency based on thealgorithm’s emergency assignment under the outcomewith one emergency. If a crew has an actual emergencyassignment, it starts work on that emergency in thecurrent hour. Otherwise, choose an actual standard jobassignment from the set of standard jobs that Stoch-LPTassigns to that crew. We choose the longest durationjob in the set.5 The crew starts work on the chosenstandard job (if any) in the current hour. Moving tothe next hour, we again apply Algorithm Stoch-LPT,but (i) with only the standard jobs not yet started,
5 It is possible to have a different rule for choosing the actual standardjob assignments. For instance, one can choose the shortest durationjob in the set.
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Figure 5 An Example of a Dynamic Assignment
t = 0 t = 2 t = 4 t = 6 t = 8 t = 10 t = 12
Crew A
Crew B
Crew C
Crew D
t = 0 t = 2 t = 4 t = 6 t = 8 t = 10 t = 12
Crew A
Crew B
Crew C
Crew D
(a) With no emergencies
(b) With two emergencies
Notes. The horizontal axis represents time. A rectangle represents a job;its duration is proportional to the length of the rectangle. Grey rectanglesrepresent standard jobs. Black rectangles represent emergency jobs.
(ii) with the current load of some crews reflecting thejob they started from the previous hour, and (iii) withan updated probability distribution of the number ofemergencies. Emergencies are realized for that houraccording to the Poisson distribution, and actual assign-ments of emergencies and standard jobs are determinedas before. Then continue the reassignment at the startof each hour until either there are no more standardjobs left or the end of eight hours is reached. At theend of the last hour, the LPT rule is applied for theremaining standard jobs (if any).
Figure 5 is an example of how the crew assignmentevolves as the emergencies arrive. The example usesfour crews, six jobs, and an emergency arrival rateof 0.2 per hour. Grey rectangles represent standardjobs. Black rectangles represent emergency jobs. Fig-ure 5(a) shows the job assignment when there are noemergencies. Figure 5(b) shows the job assignmentwhen two emergencies arrive at t = 3 and t = 5. Whitespace between jobs shows that the crew is idle dur-ing that period. Note that, depending on the arrivalof emergencies, the standard jobs assignments aredifferent.We compare the dynamic reassignment solution to
the static solution using simulation experiments. Con-sider a yard with 17 crews and 21 standard jobs withdurations varying from three to nine hours. An emer-gency has a duration of eight hours. Emergency arrivalsfollow a Poisson process with rate 0.352 per hour. Emer-gencies can only arrive in an eight-hour period, duringwhich there is an expected number of 2.8 emergencies.We simulate 100 sequences of emergency arrivals. For
Figure 6 Crew Work Hours Saved by Dynamic Reassignment
–3
–2
–1
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2
3
4
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6
7
8
0 1 2 3 4 5 6 7 8 9 10
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No. of emergencies
Note. Each data point corresponds to a different sequence of emergencyarrivals.
each sequence, we apply both the static Stoch-LPT andthe dynamic Stoch-LPT. Figure 6 shows the number ofwork hours saved by the dynamic reassignment plottedagainst the number of emergencies in the sequence.The static assignment, which is decided at the start ofthe day, is conservative. It does not adjust its solutionand maintains some idle crews even toward the end ofthe day if no or few emergencies arrive. Therefore, someof the hours logged by crews are actually idle timewhile waiting for emergencies. For these emergencysequences, the dynamic reassignment converts someof the idle time into time actually working, therebyreducing the total number of work hours. However, insequences with many emergencies, dynamic reassign-ment often results in more work hours compared to thestatic assignment. A disadvantage of the dynamic reas-signment is that the resource planner has to coordinatechanging the crew assignments around and callingsupervisors to update them about the new assignments.Therefore, in the yard setting, the utility has chosen toimplement a dynamic model, but with reassignmentonly occurring once midday.
6. Business Analytics for aUtility’s Gas Business
In this section, we describe how the research aboveapplies to the scheduling of operations at the gasbusiness of a large multistate utility. This is basedon a joint project between the research team and thecompany that gave rise to the results of this paper.Refer to §2 for the details of the utility’s operations.We discuss how we used the optimization models andheuristics described in this paper so that the companycould develop better strategies to create flexibility inits resources to handle emergencies.
6.1. Overview of the ProjectAt the onset of the project, our team analyzed sourcesof inefficiency in yard operations by mapping in detail
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the existing yard processes. We visited several companyyards and interviewed a number of resource planners,supervisors, and crew leaders as well as membersof the Resource Management Department. Our teamshadowed crews from multiple yards performing dif-ferent types of jobs and documented the range ofprocesses followed. We also constructed historical jobschedules based on data gathered from the company’sjob database.Our project with the utility had three main objec-
tives. The first was to develop a tool that can be usedwith ease in the company’s daily resource allocation.Based on the models and heuristics we discuss inthis paper, we created a tool—the Resource Allocationand Planning Tool (RAPT)—to efficiently schedulejobs and to assign them to crews while providingflexibility for sudden arrival of emergencies. RAPT hasaccess to the job and time-sheet databases and uses thisinformation to estimate the distribution of the numberof emergency jobs and their durations. The resourceplanner can view a webpage showing RAPT’s outputof the weekly schedule for each crew and detailedplans under different emergency outcomes. This tool isbeing piloted in one of the company’s yards.The second objective was to create and improve
processes related to daily resource allocation so that thetool could be easily embedded into daily schedulingprocess. We observed that a lot of the data in thedatabase were either missing, inappropriately gatheredor not vetted before entry into the system. Havingmissing or inaccurate data makes it difficult to apply adata-driven tool such as RAPT. Processes were created
Figure 7 Hypothetical Scenario: Crew Hours Worked if Optimization Model Is Used to Schedule Jobs
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to ensure that when new jobs were added to thedatabase, they had the right database fields set in aconsistent manner across all jobs and yards.
The third objective was to analyze the impact of keyprocess and management drivers on operating costsand the ability to meet deadlines using the optimizationmodel we developed. Results from this analysis willhelp the company deploy the optimization model withall the necessary process and management changes tocapture the potential benefits outlined in this paper.This analysis of process improvement is discussedin §6.2. For yard-level process improvements, RAPTalso has an option for resource planners to analyzethe impact of different process changes to the yard’skey performance indicators using simulation, therebyenabling better informed decisions.
Finally, we set out to determine the potential impactof the RAPT tool to the company. Refer to Figure 2,which shows the actual crew hours worked in April2011 for one of the company’s average-sized yards(with 25 weekday crews and 4 weekend crews). Fig-ure 7 shows the profile for the same set of jobs if RAPTis used to schedule jobs and assign them to crews.The result is a 55% decrease in overtime crew hours forthe month. Clearly, the schedule and crew assignmentsproduced by RAPT are superior to those producedpreviously by the resource planner. However, even ifcompared to the best possible schedule where uncer-tainty is removed, the decisions produced by RAPTcompare favorably. The “perfect hindsight” schedul-ing and assignment decisions are based on completeknowledge of the outcomes of emergencies that occur
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in the month. The “perfect hindsight” model resultsin the maximum possible reduction in overtime sincethe yard can plan completely for emergencies. Eventhough the RAPT model assumes a random numberof emergencies, it still is able to capture 98.6% of themaximum possible overtime reduction by “perfecthindsight.”
6.2. Using the Model to RecommendProcess Improvements
Using the models from this paper, we conducted astudy to understand the impact of changes in yardprocesses on yard productivity. Based on past studiesthe company had conducted, the company understandsthat yard productivity is driven by process settingssuch as the size of work queues (i.e., jobs availablefor scheduling), effective supervision, incentives, andcultural factors. Our team analyzed three drivers ofproductivity: work queue level, use of crew-specificproductivity data, and the degree of field supervision.The analyses were used to recommend specific processchanges to management and their resulting productivityand cost improvements.
6.2.1. Optimal Work Queue Level. Jobs need to bein a “workable” state before crews can begin executingthem. For instance, one of the steps in getting a jobto a workable state is to apply for a permit with thecity of jurisdiction. Jobs in a “workable jobs queue”are jobs ready to be scheduled by RAPT. A queueis maintained since “workable” jobs are subject toexpiration and require maintenance to remain in aworkable state (e.g., permits need to be kept up to date).We observed some yards kept only a few workablejobs in the queue at times. The short workable jobsqueue adversely impacted the RAPT output by notfully utilizing the tool’s potential. The team decidedto run simulations to determine a strategic level ofworkable jobs in the queue to maximize the impact ofRAPT while minimizing the efforts to maintain theworkable jobs queue.
In what follows, we describe simulations usingdata from one of the company’s yards. Five crews,each with eight-hour shifts, are available to work ineach simulated day. There are 10 types of jobs to beexecuted (9 standard job types and 1 emergency jobtype). Table EC.17 in the electronic companion showsthe average durations of the job types. On each day,the Resource Management Department announces aminimum quota of the number of jobs required tobe done for each standard job type. These quotas arerandom and depend on various factors beyond theyard’s control. Based on historical quotas, we estimatethe probability distribution of daily quotas for eachjob type. Table EC.17 gives the probability distributionof quotas for the standard job types. To meet thesequotas, the yard maintains a workable jobs queue for
each standard job type. For example, consider the CMPjob type in Table EC.17 with an average duration of5.85 hours. Suppose today the quota for CMP jobs isthree, but there is only one job in the workable CMPjob queue. Then today the yard will execute one CMPjob and will carry over the remaining two CMP jobs asa backlog for the next day.
Getting jobs to a workable state (and into the queue)is done by the yard’s resource planner, who processesthe paperwork and applies for city permits. Sinceresource planners have many responsibilities in the yard(including scheduling, dispatching, and monitoringcrews), they only process paperwork for jobs in batchesand only when the jobs in the queue reach a lownumber. Hence, the workable jobs queue is replenishedthrough a continuous review policy specified by areorder point and an order quantity. Each time the totalworkable jobs (both in the queue and in the pipeline)drops below the reorder point, the yard requests newworkable jobs. The size of the request is equal to theorder quantity. The request is added to the pipelineand arrives after a lead time of three days. This leadtime includes time used for administrative work toapply for a permit. Suppose the yard chooses a reorderpoint of two and an order quantity of five for the CMPworkable jobs queue. Then each time the total CMPworkable jobs drops below two, the yard places anadditional request for five workable CMP jobs.
In the simulations, the order quantity is set for eachjob type queue so that, on average, new requests aremade every week. The reorder point is determinedfrom a service level the yard chooses, where the servicelevel is the probability that there is enough jobs in theworkable jobs queue to meet new quotas during thelead time period (i.e., while waiting for new work-able jobs to arrive). For each simulated day, quotasare randomly generated and met to the maximumextent possible from the workable jobs queue. The jobsare assigned to the five crews using the RAPT crewassignment model. Figure 8 shows the evolution ofthe workable jobs queue in one simulated month for a50% service level. The net inventory level correspondsto the total number of workable jobs currently in thequeue. When the net inventory is negative, then thereis a backlog of workable jobs for that job type (i.e.,there are not enough workable jobs to meet the quotas).
Table 2 summarizes the results of the simulation fordifferent service levels. Note that increasing the servicelevel increases the average size of the workable jobsqueue. With 50% service level, the average inventoryper day in the workable jobs queue is 28.8. However,seven quotas have not been met in time. Increasing theservice level to 75% requires increasing the averageinventory per day to 35.6, resulting in eliminating anybacklogged jobs. Increasing the service level furtherto 90% or 99% results in higher average inventories
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Figure 8 Workable Jobs Queue over One Simulated Month with 50% Service Level
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inve
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y
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PVIP
LKTPDP
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CUSTREQ
MPP
LEAK2A
CMP
Note. Each line is the net inventory of workable jobs for a job type in Table EC.17 of the electronic companion.
of workable jobs, but with essentially the same effecton backlogged jobs as the smaller service level 75%.A backlog of quotas also affects the daily crew utiliza-tion. If there is an infinite supply of workable jobs,crews will be working on average the same amount ofhours. Having a finite supply of workable jobs meansthat crew utilization is variable from day to day (crewswork less during backlogged days but work more inensuing days). Note that with a 50% service level, onaverage a three-member crew is logging 10.8 hoursmore overtime per month than with the other servicelevel choices.We have embedded in RAPT an option to conduct
an analysis of the workable queue inventory policyfor the yard. This analysis includes a workable queuesimulation and a summary of the service level’s effecton crew utilization and on the yard’s ability to meetquotas. A yard’s resource planner can use RAPT tochoose an appropriate policy for the yard.
6.2.2. Using Crew Productivity Data. At present,detailed crew productivity data are not available in the
Table 2 Effect of Service Levels on Average Workable JobsInventory, Backlogged Jobs, and Overtime Crew Hours forOne Simulated Month
Service level 50% 75% 90% 99%
Average inventory per day 2808 3506 3706 5006Total backlogged jobs 7 0 0 0Average overtime per day (crew hours) 14067 14055 14055 14055
company’s database. As such, it is not possible to makecrew assignments to take advantage of the inherentjob-specific productivity differences between crews inthe crew assignment phase. We used simulations to aidcompany management in understanding the impact ofusing job-specific productivity data in crew assignmentversus assigning jobs based on average productivity.This analysis was used to motivate upper managementto provide the appropriate resources to keep track ofcrew productivity.In the simulations, a yard has five crews that are
each “experts” in one of the job types. We let É 2 60115be an expertise factor, which is the percentage reductionin a job’s duration if an expert works on it. Largervalues of É mean that experts are more productiverelative to regular crews. In our experiments, we letÉ = 0% (base case), 5%, and 10%. We run the simulationfor 30 days. In each day, the work that has to beassigned is randomly generated. (The distribution ofquotas is given in §EC.13 of the electronic companion.)We observe that the assignment model assigns mostjobs to crews that have expertise in them. Table 3shows the total expected overtime crew hours over aone-month period. By having expert crews who work
Table 3 Total Expected Overtime Crew Hours for DifferentExpertise Factors
Base case É = 5% É = 10%
Total expected overtime crew hours 34004 32904 30206% improvement over base case — 3023 1101
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Table 4 Simulation Results for Increasing Supervisor Presence in the Field
Base case 5% reduction in job durations 10% reduction in job durations 25% reduction in job durations
Average overtime per day per crew 3009 2081 2048 1051(crew hours)
% improvement over base case — 902 1907 5101
with 5% reduced durations, overall overtime hourscan decrease by 3.23%. The decrease in overtime hoursis nonlinear since if expert crews can work with 10%reduced durations, the total overtime hours in onemonth are reduced by as much as 11.1%.
6.2.3. Increasing Supervision over Crews. A priorstudy conducted by the company observed that crewproductivity is directly related to field supervision.More time spent overseeing crews in the field results inmore productive crews. We used simulations to validateand measure the appropriate level of supervision tomaximize productivity since field supervision hasa cost.
In these simulations, we compare the effect of havingan increased supervisor presence in the field to theaverage expected overtime incurred by crews. Con-sider the work types given in §EC.13 of the electroniccompanion. Assume that by having increased super-visor presence, the durations of work types can bedecreased. We will compare different cases: the basecase (no reduction), 5% reduction, 10% reduction, and25% reduction. We simulate a yard with five crewsand with the daily quotas randomly generated basedon Table EC.17. We assume that there is an infinitesupply of permitted work (so the inventory policy isnot a factor). Each day, we assign the work to the fivecrews using RAPT and note the total expected overtimeincurred by the five crews during that day. For thedifferent cases, we run this simulation for 30 days andcalculate the total expected overtime averaged over30 days.Table 4 reports the result of the simulation for the
different cases. We can infer that each 5% decreasein job duration (by increasing supervisor presence)results in a reduction of 1.6 overtime crew hours eachday for the five-crew yard. Therefore, assuming thatthere are three members in a crew, a 5% increase inproductivity results in reducing a total of 143 overtimehours charged for the yard in one month.
6.2.4. Projected Financial Impact from Changes. Inour project, we used the previous analyses to computethe projected financial impact of implementing processchanges in the utility company. We demonstrate thiswith a hypothetical utility that has an operating profitof $3.5 billion per year. The hypothetical utility employs10,000 field personnel. The straight-time hours perperson per year are 2,000, with an additional 500overtime hours per person per year. The average
wage of a field personnel is $50 per hour. Overtimeis paid out at $75 per hour. The hypothetical utilityspends $1 billion in straight-time labor costs (20 millionhours), with an additional $375 million in overtimelabor costs (5 million hours). We estimate the financialimpact to this hypothetical utility of introducing thebusiness process changes described earlier in thissection. The percentage savings in overtime costs areall based on the analyses in §§6.2.2 and 6.2.3.
If the utility were to keep crew-specific productivitydata as described in §6.2.2 (assuming expert crewsare 5% more productive), we would anticipate annualsavings of about $12 million, which represents 0.3% ofthe utility’s annual operating profit. Suppose the com-pany were to increase crew supervision as describedin §6.2.3. Based on previous company studies, increasedsupervisor presence reduces job durations by at least10%. This results in annual savings of $74 million (or2% of the annual operating profit). If the company isable to implement both changes, this has a cumulativesavings of about $84 million per year, which represents2.4% of the annual operating profit.
7. ConclusionsIn many industries, a common problem is how to allo-cate a limited set of resources to perform a specific setof tasks or jobs. However, sometimes these resourcesare also used to perform emergencies that randomlyarrive in the future. For example, in hospitals, operatingrooms are used both for elective surgeries (that areknown in advance) and emergency surgeries (whichneed to be performed soon after they arrive). Anotherexample, which motivated this paper, is schedulingmaintenance crews in a gas utility company. Mainte-nance crews have to execute standard jobs (pipelineconstruction, pipe replacement, customer service) andrespond to reports of emergency gas leaks. Emergenciesarrive randomly throughout the day. With randomlyarriving emergencies, the problem becomes more com-plicated since the resources need to be allocated beforerealizing the number of emergencies that have to beperformed. Thus, a schedule needs to be flexible inthat there must be resources available to perform thesefuture emergencies.
We use stochastic optimization to model the problemfaced by the gas utility. The problem is decomposedinto two phases: a job scheduling phase and a crewassignment phase. The optimization problems result-ing from each phase are computationally intractable,
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but we provide tractable heuristics for solving eachof them. The job scheduling phase heuristic solvesa mixed integer program, for which we propose anLP-based heuristic. We are able to prove a data-drivenperformance guarantee for this heuristic. The crewassignment phase solves a two-stage stochastic mixedinteger program. Here, we propose an algorithm thatreplicates the structure of the optimal crew assign-ment. We demonstrate how the two heuristics can beimplemented in a rolling horizon for rescheduling andreassignment in response to the state of emergencies.
We used our models and algorithms to improve jobscheduling and crew assignment in the gas business ofa large multistate utility company that faced significantuncertainty in its daily operations. Our models werealso used to help the utility make strategic decisionsabout changes in its business and operations. In simula-tions using actual data and our models, we project theimpact of different process changes to crew utilizationand overtime labor costs.
7.1. Future DirectionsThere are several future directions that go beyond thescope of this paper and could be pursued.
In this paper, we focused on the job scheduling andcrew assignment problems assuming that there is notravel time between two jobs. In the real-world applica-tion, this simplifying assumption makes sense becauseof the small distances between jobs. However, a futuredirection might be considering geography in makingdecisions. For instance, the job scheduling model caninclude a penalty proportional to the distance betweenjobs scheduled for the same day.Another possible direction is to have emergencies
with random durations. This is related to literature onscheduling under stochastic job durations, where jobsneed to be processed on parallel machines withoutpreemption. The number of jobs is known (unlikeour setting), but the processing time of each job isan independent random variable. The objective isto minimize expected makespan (like our setting).It is known that the longest-expected-processing-time(LEPT) rule minimizes the expected makespan forexponential jobs or for remainders of i.i.d. decreasinghazard rate jobs (Pinedo and Weiss 1979, Weber 1982).In general, LEPT is a good but not optimal heuristic(Pinedo and Weiss 1979). For this reason, and based onpreliminary experiments, we believe that AlgorithmStoch-LPT would perform well under the case whereemergency durations are random.Another potential direction is an analytical perfor-
mance guarantee for the crew assignment heuristic,Stoch-LPT. We are able to prove that Stoch-LPT termi-nates with the optimal crew assignment under specialcases. However, establishing a guarantee for the generalcase is an interesting direction.
We demonstrated the potential impact of the resourceallocation planning tool in managing uncertainty inyard operations and decreasing labor costs. However,there is still some further work to be done for the yardsto achieve these results. These include gaining grassrootsupport from the workers’ union and continuing withstrong management leadership. Some new processesneed to be also introduced in all of the company’syards to ensure that the tool can be implementedsuccessfully. The purpose of these new processes is toensure integrity of the data fed into the model, andto create multiple levels of accountability for betteroversight and cost control.
AcknowledgmentsThe authors thank Noah Gans (special issue coeditor), theassociate editor, and the anonymous referees for their insight-ful comments, which have helped to significantly improvethis paper. The authors also thank the anonymous utility thatinspired and supported this work as well as the Leadersfor Global Operations Program for their support. GeorgiaPerakis and Joline Uichanco thank Retsef Levi, Melvyn Sim,and Chung-Piaw Teo for insightful comments on this paper.Georgia Perakis also acknowledges the National ScienceFoundation [Grants CMMI-1162034, CMMI-0824674, andCMMI-0758061].
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