planning and replanning in project and production scheduling

7/28/2019 Planning and Replanning in Project and Production Scheduling

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Omega 30 (2002) 155170

www.elsevier.com/locate/dsw

Planning and re-planning in project and productionscheduling

Kevin M. Calhouna, Richard F. Deckrob;, James T. Mooreb, James W. Chrissisb,John C. Van Hovec

aAnalysis Division, ACC=DRYM, 204 Dodd Blvd., Ste. 226, Langley AFB, VA 23665, USAbDepartment of Operational Sciences, Air Force Institute of Technology, AFIT=ENS, Bldg. 640, 2950 P Street,

Wright-Patterson Air Force Base, OH 45433-7765, USAcOperations Analysis Division, Air Force Wargaming Institute, 401 Chennault Circle, Maxwell AFB, AL 36112-6428, USA

Received 10 January 2001; accepted 14 December 2001

Abstract

Scheduling is a time and labor-intensive task. In addition, problems arise due to the lag time between the publication of the

schedule and the start of the scheduled work period. During this time, conditions in the dynamic environment can, and often

do, change (equipment breaks, orders are cancelled or increased, work force levels do not meet expectations, or a task simply

takes longer than planned, for example). While the scheduling literature is broad, very little of it addresses re-planning and

re-scheduling.

This research explores the use of tabu search (TS) seeded with a robust initial solution to determine good solutions for a

posted schedule. It considers preemptive tasking priorities. The procedure then adapts the TS to focus on generating re-planning

and re-scheduling options in a project or production setting. The TS procedure was implemented in Java to be portable and tomake the objects available for reuse and adaptation elsewhere in the planning hierarchy. Published by Elsevier Science Ltd.

Keywords: Scheduling; Planning; Re-scheduling; Project management

1. Introduction

Scheduling is a central function of all organizations. A

great deal of literature exists for scheduling projects and

job shops. Unfortunately, little of this literature deals withre-planning or re-scheduling activities. What literature does

exist generally focuses on dispatching rules and questions

The views expressed in this paper are those of the authors and

do not reect the ocial policy or position of the United States

Air Force, the Department of Defense or the United States govern-

ment. (Approved for Public Release, Distribution Unlimited) Pre-

publication draft. Please do not quote without the authors written

permission. Corresponding author. Tel.: +1-937-255-6565=4325; fax: +1-

937-656-4943.

E-mail address: [email protected] (R.F. Deckro).

of local changes vs. schedule regeneration. It is a commonly

held belief among operational schedulers, however, that the

schedule begins to be out of date from the moment opera-

tions begin (if it was even initially accurate).

Robust scheduling procedures have long been a goal ofresearchers and practitioners alike. While optimization ap-

proaches to scheduling are essential for eciently building

schedules and planning resource usage, the imprecision of

the existing data, the dynamic operational environment, and

the eects of uncontrollable events have always meant that

the schedule is a suggested plan. Without the ability to

re-plan and re-schedule, the schedule can become an in-

appropriate baseline for gauging performance.

This research focuses on schedule planning, re-planning,

and re-scheduling. A distinction is made between

re-planning and re-scheduling. Re-planning, as dened in

this paper, refers to that aspect of operations concerned with

0305-0483/02/$ - see front matter Published by Elsevier Science Ltd.

P I I : S 0 3 0 5 - 0 4 8 3 ( 0 2 ) 0 0 0 2 4 - 5


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156 K.M. Calhoun et al. / Omega 30 (2002) 155 170

xing the schedule before the start of the work period.

Re-scheduling refers to re-assigning tasks and resources, as

circumstances warrant, during the work period.

The overall problem is formulated as a project schedul-

ing problem (PSP). In order to obtain an initial solution

to the problem, a heuristic is developed based on key con-

cepts from scheduling theory. A tabu search (TS) procedurethen uses the result from the initial solution heuristic and

improves upon it. The TS procedure was tested by solving

PSPs with known optimal solutions.

The TS procedure presented herein takes advantage of

the structure of the planning, re-planning, and re-scheduling

problems. The research extends the eort of previous re-

search by capturing task priorities through the use of a goal

programming (GP) model.

A TS procedure for re-planning is developed. The study

has two re-planning objectives. The rst objective concen-

trates on the re-planning that must be done prior to the

start of the workday. It suggests new plans based on theresources, requirements and personnel currently available.

In many respects, the TS for re-planning is similar to that

used to generate the initial plan. The second objective of the

study was to modify the TS to perform quick re-scheduling

of activities during the workday. For both objectives, the ap-

proach applied is a lexicographic GP formulation that max-

imizes the number of high priority tasks that are scheduled

and then minimizes the number of tasks and resources to be

re-scheduled.

The TS generates a list of options for the planner to re-

view. The planner may then select the most desirable op-

tion for the current operational situation from the list. This

process uses the options developed by the heuristic, cou-pled with the decision makers experience and knowledge,

to develop a plan. It is acknowledged that the methodology

developed here is a tool to be used by, not a replacement

for, experienced planners and schedulers. The goal of this

research was to rapidly provide improved options to support

the decision maker.

To best exploit the advantages of object-oriented pro-

gramming, the heuristic procedures developed in this study

are written in Java. This makes the procedure portable

so that it can run on most platforms. The procedures

make Structured Query Language (SQL) calls to an Or-

acle or Access database. The use of ANSI-compliantSQL augments the portability of these procedures. These

queries pull only the attributes from dierent tables in

the database that are necessary to solve the planning

problem.

As noted earlier, the TS procedure results are compared

for accuracy to the lower bounds of a test set of schedul-

ing problems. CPU processing times for the solutions are

reported for each problem to demonstrate how processing

times grow with the size of the problem. Inter-procedure

comparison of processing times is of limited value because

of computer platform dierences. Once the TS procedure

was validated against these test cases, it was modied

and applied to sample project planning, re-planning and

re-scheduling problems.

In the remainder of the paper, a brief literature review

is presented, followed by details of the development of the

TS procedure used to solve the GP models for the planning,

re-planning and re-scheduling problems. Computational re-

sults and a brief illustrative example are then presented. Thepaper concludes with a summary of the research, a review

of its contributions, and recommendations for future work.

2. Background

Scheduling concerns the allocation of limited resources

to tasks over time. It is a decision-making process that has

as a goal the optimization of one or more objectives [1,

p. 1]. The scheduling process exists in virtually all opera-

tional settings. It is a key to eective resource utilization by

world-class organizations.

A project is a systematic enterprise designed to accom-

plish some specic non-routine or low-volume task [2, p.

1]. For the purpose of this research, a project is a nite set

of activities with limited resources that must be scheduled

in agreement with certain precedence requirements between

activity pairs.

Precedence constraints dene timing requirements be-

tween activity pairs within projects. The most common type

of precedence constraints is of the nishstart variety and

is used to specify that a predecessor activity must end be-

fore its successor activity may start. Other common types

of precedence constraints are startnish, startstart, and

nishnish [2, p. 321]. Lag times are another importantconsideration. Certain activities must lag others by a partic-

ular interval. This gives rise to a minimum or maximum lag

time that may constrain the start time for successor activities.

The formulation of these generalized precedence

constraints can become complex with the addition of

multi-modal activities. Taken together, the minimum and

maximum lag times make up a time window for the suc-

cessor activity. The approach developed in this research

incorporates lag times to model staggered activities and

suggests start times.

2.1. Resource constraints

The simple PSP model does not account for limited re-

sources that are used by the activities. Hence, an optimal

solution to a PSP may be infeasible if resource constraints

are present. An expansion of the PSP model developed to

handle limited resources is the Resource Constrained Project

Scheduling Problem (RCPSP).

2.2. Heuristics

The RCPSP integer linear programming (ILP) model

yields an optimal solution if one exists. However,


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K.M. Calhoun et al. / Omega 30 (2002) 155 170 157

depending on the number of variables involved, it can be

time-consuming to solve. Therefore, decision makers are of-

ten interested in priority rulestechniques that yield good,

but not necessarily optimal, solutions quickly. A priority rule

frequently used when jobs are subject to arbitrary precedence

constraints and arbitrary job processing times is the

largest number of successors (LNS) rule. The job thathas the largest number of total successors in the prece-

dence constraints graph has the highest priority and would

be scheduled rst, subject to any additional constraints

[1, p. 71].

When job j can only be processed on a subset of the

available machines, the least exible job rst (LFJ) rule is

sometimes used. Every time a machine is freed, the LFJ rule

selects the job that can be processed on the smallest subset

of machines, with ties broken arbitrarily.

When multiple machines are freed simultaneously, the

LFJ rule does not specify which machine to consider rst.

If a number of machines are free at the same time, it may beadvantageous to consider the least exible machine (LFM)

rule. This rule assigns a job to the machine that can process

the smallest subset of remaining jobs [1, p. 71].

A combination of the preceding two rules gives priority

to the least exible jobs on the LFM. That is, for each time,

t, the LFM would be assigned the least exible feasible

job. This heuristic (dispatching rule) is abbreviated as the

LFMLFJ rule [1, p. 72]. The heuristics discussed in this

section were developed with the objective of minimizing the

makespan of the project.

2.3. The multi-modal resource constrained project

scheduling problem (MMRCSPSP)

The standard resource constrained project scheduling for-

mulation [3] or [4] may be extended to include the situation

where the activities can be completed in one of a number of

possible execution modes (MMRCPSP). The amount and

type of resources consumed and the processing time depend

on the mode selected [5, p. 350].

2.4. The generalized MMRCPSP

Generalized precedent constraints may be used to enforce

any timing requirement called for in an operational scenario

and are therefore required. The generalized MMRCPSP has

the abbreviation, MMGRCPSP.

2.5. Doubly constrained resources

In the formulation in Fig. 1 (following the work of

Sprecher and Sprecher et al. [6,7]), let K be the set of

Fig. 1. Van Hoves complete MMGRCPSP model formulation [8, p. 49]


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resource types in the problem and kK. The renewable

aspect of k is bounded at Rk, the per period availability of

renewable resource k[8, p. 43]. The non-renewable aspect of

resource kis the limit dictated byNk=rimk [8, p. 43]. Each unit

in the model is a doubly constrained resource. Therefore,

the resource types are both renewable and non-renewable.

2.6. MMGRCPSP formulation

In the formulation of the MMGRCPSP problem (Fig. 1),

the objective function, (1), minimizes the makespan of the

schedule. Eq. (2) enforces the non-renewable resource con-

straint, while Eq. (3) enforces the renewable resource con-

straint. The binary decision variable, ximt, is equal to one if

activity i starts in period t and is executed in mode m. Eq.

(4) enforces the lag times between predecessor=successor

activities. The binary decision variable xjnt selectively en-

forces the applicable constraint and relaxes the redundantconstraints. As the delta (ijmn) can be positive or nega-

tive, both minimum and maximum precedence constraints

can be expressed by using this constraint twice, once for

precedence pair (i; j) and once for precedence pair (j; i) [9].

While the model generates an optimal schedule in terms of

makespan, it does not model task priority classications, nor

does it model maximum lag times or re-planning. For any

realistically-sized problem, schedule generation can be ex-

tremely time-consuming.

2.7. Tabu search

TS is a meta-heuristic procedure for solving large com-

binatorial optimization problems. TS has proven to be suc-

cessful in solving to optimality or near-optimality a variety

of classical and practical problems [10, p. 74].

Scheduling provides a fruitful area for heuristic tech-

niques in general and for TS in particular [11, p. 127].

Nowicki and Smutnicki [12] tested their TS application

for job shop scheduling against a benchmark set of

problems and compared their results to those of a shifting

bottleneck heuristic and simulated annealing. Their pro-

cedure outperformed the other methods with respect to

(adjusted) CPU time and the quality of the generatedmakespans [12, p. 808].

3. Methodology

This section provides details of the approach used to solve

the problem. A GP model and formulation for the plan-

ning and re-planning problems are presented. The section

concludes with a detailed look at the TS application devel-

oped to provide planners and managers with scheduling and

re-scheduling options.

3.1. Goal program

For the examples tested in this research, three priority

categories were used although the procedure is not limited

to three. Priority classes would be developed for an or-

ganization reecting rm-specic requirements. Dierential

weighting within classes is possible.For this GP model, the three priorities are assumed to be

preemptive. Tasks were classied as either Priority 1, 2, or

3. Classication would be pursuant to operational or con-

tractual considerations. This classication could be based

on utilization of key resources or personnel, political sen-

sitivity or visibility, precedence, or simply cost, for exam-

ple. Goal i is to schedule tasks in Priority class i with the

available resources and within the generalized precedence

constraints. The minimization of makespan was used as the

lowest priority goal (goal 4 in this case).

In Figure 2, Eqs. (5)(14) give the constraints on the de-

viations, di. It is possible that all of the ximt variables equalto zero for activity i, making the dxi variable equal to one.

This condition would occur for tasks that have no resources

assigned to them. The di variables are therefore constrained

to be equal to the number of priority i tasks that do not have

jobs scheduled against them in the optimal solution. The rest

of the model is identical to Van Hoves MMGRCPSP for-

mulation (Fig. 2) except that objective function (1) becomes

the constraint in Eq. (9) that denes Goal 4, minimizing the

makespan of the generated schedule (Table 1).

3.2. Initial solution heuristic

The objective in project or job shop scheduling is often to

minimize the makespan subject to precedence and resource

constraints. This suggests a heuristic to generate an initial

solution for tasking allocated resources to activities. The

heuristic is a combination of the LNS rule and the LFMLFJ

rule (referred to as the LNSLFMLFJ rule). The LNS

LFMLFJ rule provides a sound initial solution.

If an activity with many successors is assigned a late

start time, then a condition known as a bottleneck might

occur. That is, all of the successor activities would be on

hold until this key activity is suciently completed. To

avoid this, the initial solution heuristic assigns resources tofeasible tasks (predecessors met, resources available) with

many successors and shortest duration times. Finally, to be

a good initial solution for the GP TS, the heuristic strives

for a solution that satises the goal priorities of the GP.

Therefore, the heuristic given below considers the activities

in priority order.

Step 1. Starting with the highest priority unscheduled fea-

sible activities, order the appropriate modes for each activity

by the LNS rule.

Step 2. Within priorities order the modes rst by LFM

(least exible machine or mode) then by LFJ (least exible

activity or resource).


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Fig. 2. The MMGRCPSP goal programming model.

Table 1

Parameters and variables for the MMGRCPSP GP formulation

ParametersA1 the set of all priority one activities

A2 the set of all priority two activities

A3 the set of all priority three activities

A the set of all activities (A1 + A2 + A3)

d the index of the terminal activity

eim the earliest completion time for activity i in mode m

lim the latest completion time for activity i in mode m

Mi the set of all execution modes for activity i

im the duration of activity i in mode m

Si the set of generalized successors of activity i

ijmn the minimum lag between the start time of activity i

in mode m and the start time of activity j Si in

mode n

K the set of all double constrained resourcesrimk the amount of resource k required by activity i when

being executed in mode m

Rk the per period availability of resource k

Nk the total amount of resource k available

g the deadline for the project under consideration

Variables

ximt =1 if activity i starts in period t and is executed in

mode m, 0 otherwise

Pl The weight of goal, with P P +1dl # of Priority activities not covered, =1; 2; 3

d4 The makespan of the schedule

Step 3. Further sort the modes by ascending activity du-

ration times. This ensures that the modes with the quickest

completion times are paired up with the activities that have

the most successors, reducing bottlenecks.

Step 4. Schedule each activity in the resulting order from

Steps 1 to 3.

Step 5. After scheduling the highest priority unscheduled

activities, repeat Steps 14 for the next highest priority ac-

tivities. Continue until all activities are scheduled or until

there are no more resources available.

This heuristic is a pseudo-GP as it schedules each ac-

tivity by priority (goal) in descending order. It provides a

feasible, good starting solution for the TS application of the

GP model. While it does not explicitly consider makespan,

makespan is implicitly captured via the LFMLFJ rule. The

sorting of activity duration times was done to attempt to

eliminate situations where activities are waiting for longperiods for predecessors after a bottleneck. Fig. 3 presents

the owchart for the initial solution heuristic.

3.3. Initial solution evaluation

To judge how well the initial solution schedules the activ-

ities, the number of unscheduled activities for each priority

level is totaled. The makespan (last goal in the planning

problem) is determined by the nish time of the last task

completed. In a problem with four goals, an array of four

integers is used to represent the quality of the solution.

For instance, if the initial solution covered all of the P1


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Terminate

InitialSolutionMethod

Invoked,time = 0

Resourcesand tasks

uncovered?

Yes

No

Get nextmode

Task

Covered?

Yes

Resourcesavailable?

Yes

No

No

Find time window for

the activity. Check forsuccessor violation.

SuccessorViolations?

YesAssign activity.Performcalculations.

NoResourcesavailable?

Yes

No

Fig. 3. Initial solution heuristic owchart.

activities, all but one of the P2 activities, and all but ve

of the P3 activities, while achieving a makespan of 550

time units, then the objective function vector would be

[0,1,5,550].

3.4. Tabu search for the project planning problem

The initial solution heuristic provides a robust schedule

that can be used on its own but is intended as a good starting

point for the Tabu Search for the Project Planning Problem

(TS3P). Parameters for TS3P such as tabu tenure and total

number of iterations must rst be specied. The aspiration

criterion is the best objective function value found. The ini-

tial solution is passed to the TS engine and TS3P starts.

3.5. Move manager

The move type chosen for this research is the toggle.

Toggling consists of taking one of the elements in the binary

solution vector and switching it to the opposite position; this

may be the easiest type of move to implement. Employing

a toggle allows activities that were not initially scheduled to

rotate into the plan.

It is easy to see that there are n neighbors for any solu-

tion of an n-variable problem. While the toggle move may

seem simplistic, successive toggles achieve results similar

to other move types. The rst consideration for TS3P is the

move neighborhood. A decision must be made concerning

which of the activities in the current solution are eligible for

toggling; the eligible set of activities is called the candidate

list.The Move Manager selects the candidate list. This

method considers activities associated with only one specic

priority category at a time for inclusion on the candidate

list. The activity priority category under consideration alter-

nates at each iteration; only P1 activities may be considered

at iteration 1, only P2 tasks are considered at iteration 2,

and only P3 activities are considered at iteration 3. Then,

the cycle repeats. The reasoning behind this strategy is: if

activities from all categories were included on the candidate

list and an unimproving move was necessary (an activity

must be turned o), then only tasks associated with P3

activities would be selected. The TS would never select a


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task assigned to a P2 activity or P1 activity to toggle o if a

resource assigned to a P3 activity was available. In essence,

the candidate list would then only consist of P3 activities

and only those P1 and P2 activities that have not yet been

scheduled. This would result in little or no inter-modal swaps

and an overly restrictive neighborhood.

The Move Manager selects P1 tasks for inclusion on thecandidate list if one of the following criteria is met: either no

resources are currently assigned to the associated task or, the

activity under consideration is assigned a specic mode with

an activity completion time equal to the makespan of the

schedule. A move that toggles o resources to a P1 activity is

the worst kind of unimproving move (the deviation variable

for goal one would increase by one). Therefore, including

this type of move in the candidate list requires a very good

rationale. We allow an assigned P1 task on the candidate list

only if it is associated with the makespan. This decreases

the size of the candidate list while allowing a decrease in

the makespan.Selecting move candidates among P2 and P3 tasks is less

complex. An activity is placed on the candidate list if either

the task has no resources assigned to it or if the task is an

assigned task and its current mode of operation is set to

zero.

3.6. Move

Each element of the candidate list is operated upon by

the Move method. First, the mode selected to execute the

task is toggled (turned o if it was previously on and vice

versa). Next, the Solution Modier method is called.

The Solution Modier method performs quickly because

it has no loops. It only calculates the marginal change in

makespan resulting from the move under consideration.

Consider a potential move where a mode is toggled o; its

corresponding element in the solution vector is changed to

zero and the marginal change in makespan is determined.

When the mode was toggled on, the method checks to

see if there are enough resources left to execute that task.

If not, the method skips to the next task on the list. If there

are enough resources, an attempt is made to locate a time

window for the activity in a fashion similar to that of the

Initial Solution method. The dierence during the TS isthat even if successor violations exist, the time window is

assigned to the activity, an objective function penalty is

incurred, and the next move is evaluated. This keeps the

search moving quickly.

If a move is chosen that does violate any successor con-

straints, the successors that are in violation are the only

moves allowed on the candidate list until all violations are

reconciled. For example, say elements 5, 12, and 21 of the

solution code array are successor activities and all three are

violating precedence constraints. On the next three moves,

only these three elements will be on the candidate list. Once

these are all dropped from the schedule, TS3P can proceed

choosing its candidates as before. Fig. 4 is a owchart that

illustrates this process.

3.7. Move evaluation

The appropriate element of the objective function vector

is decremented or incremented by one depending on whether

a task was toggled on or o. The makespan of the current

move is also evaluated. After the current move is evaluated,

it needs to be inspected to determine if it is an allowable

move. If not, then the aspiration criteria are checked. If the

move is admissible, the index of the move and its objective

function value are stored until all the moves on the candidate

list are considered and evaluated. When this is accomplished,

the best move is selected and a new iteration is begun.

Fig. 5 is a owchart of the move evaluation method.

4. Re-planning

Re-planning often occurs during the operational time

frame. Plans are often developed days, weeks or months

before they are to be executed. Should any changes occur

to the plan during this time, the activities aected by the

changes may have to adjust. The impact of any change

should be minimized. In the following section, we propose a

methodology for minimizing the impact of changes to a plan.

4.1. GP model for re-planning

A slightly modied GP model is appropriate for

re-planning. Consider the GP for the planning example. It

consisted of four goals:

Goal 1 Maximize P1 activity coverage



Goal 4 Minimize makespan of the schedule

These goals, however, are not sucient for the re-planning

application. An additional goal has been inserted betweengoals 3 and 4 (although it could be inserted wherever the

project manager or planner feels it is appropriate). In order

to minimize the impact of a change to the planned schedule,

the number of activities aected should be minimized.




Goal 4 Minimize number of changed activities

Goal 5 Minimize makespan of the schedule


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Initiate TS3P

Move Manager

generatescandidate list

Solution Modifierunassigns activities

updates

timeWindow, and

resources

Mode

on?

Move methodtoggles next

mode on list

Solution Modifierassigns activity,

updates timeWindow

and resourcesremaining, calculates

activity duration times

Move Lag

enforces lag

times for mode

Move Evaluation

calculates objectivefunction value for

current move

Determine ifmove is

allowable

No

Yes

Is this mode

last on the

candidate list?

Select bestcandidate, iteration

complete

Max iterations

reached?

No

Yes

Yes No

Terminate TS3P, report results, store

solution in database

Fig. 4. TS3P ow chart.

4.2. Tabu search for re-planning

Since the theoretical formulation of the GP model for

re-planning is similar to that in planning, the toggle move

is used for the re-planning TS. Likewise the tabu tenure

and aspiration criterion are identical. To model the goal of

minimizing the number of activities to be re-planned, the

only change that needed to be made to the TS3P application

to obtain the Tabu Search for Activity Re-planning (TSAR)

was to incorporate an additional deviational variable and its

corresponding constraint.

When TS3P was used to solve each planning problem

instance, a copy of the best solution obtained was saved to

the database. For TSAR, an initial solution based on the

current scenario situation is obtained using the LNSLFM

LFJ heuristic. This initial solution is evaluated just as in

TS3P, but with one dierence. The solution vectors from

the TS3P best solution and the TSAR initial solution are

compared to one another. Each time the value of element

i in one vector diers from the value of element i in the

other vector, the value of the deviational variable associated

with re-planning activities is incremented by one. At each


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Initiate the EvaluateMethod

Is currentmode

assigned?

Increment assignedtargets for

appropriate targetory

Get next mode

Is this the

last mode?

Subtract assignedmodes from number oftargets for each priority

class

Calculate latest

job completiontime

Report objective functionvector and terminate method

No

No

Yes

Yes

categ

Fig. 5. Flow chart for evaluate method.

iteration, every neighbor to the current solution is likewise

compared to the TS3P best solution; the TSAR attempts to

minimize this value as well as the values for the other goals.

This approach allows for a rapid comparison of the

re-planning solution with the original plan. By minimizingthe value of the deviational variable associated with the

re-planning goal, the operational impact of a change to

the schedule is decreased. If this number is small, most

of the activities in the original plan can be executed as

scheduled.

5. Re-scheduling

Re-scheduling is dened in this research as re-assigning

activities, as circumstances warrant, while the current work

package or planning horizon is already being executed. It

diers from re-planning by the amount of time planners

have available to nd a solution and, subsequently, by the

amount of time that work crews have available to retool (if

necessary), develop new detailed work plans and times, and

review operational instructions associated with a new task.This means that an application for re-scheduling must work

rapidly. The solution obtained should, if possible, impact

the scheduled plan to an even lesser extent than that for

re-planning.

The re-scheduling application developed for this re-

search reduces operational impact by liberally using a

feature that allows locking in activities. While a plan-

ner may selectively lock in any specic activities, for the

examples discussed below all unchanged tasks assigned

to priority one and two were locked, thus allowing only

unchanged tasks currently assigned to priority three to be

considered for re-assignment. Should the magnitude of the


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re-scheduling situation be such that reassigning all of the

activities assigned to priority three is not sucient to cover

all priority one tasks, some or all of the priority two tasks

could be unlocked. This restricted approach allows for rapid

re-scheduling.

The re-scheduling problem is solved by modifying TSAR

to create Tabu Search for Activity Re-scheduling (TSARS).By locking in a quantity of activities, the value of the devi-

ational variable corresponding to the change goal is already

quite small, presenting a good starting solution for the TS.

In addition, the candidate list becomes quite small, thus al-

lowing the TS to execute rapidly. This extra speed may be

necessary in a dynamic environment. Fig. 6 is a owchart

of the overall process.

6. Results

This section presents the outcome of applying TS3P to

several MMGRCPSP problems having known optimal or

lower bound solutions. In addition, a case study is described

and the results of applying TS3P and TSAR are provided.

6.1. Problem sets

The problem sets used for testing TS3P were generated

using the ProGen problem generator [4,13]. A total of ve

10-job problems, three 30-job problems, three 60-job prob-

lems, and two 90-job problems were used for testing.

Each of the 10-job problems could be executed in three

dierent modes; each mode uses dierent quantities and

types of resources and has dierent processing times. The

30- 60- and 90-job problems consisted of jobs that may be

executed from a choice of one, two, or three possible modes.

Again, each mode uses dierent resources and processing

times. The doubly constrained resources each carried vary-

ing quantities of renewable and non-renewable assets. The

problems within each set also diered in their network con-

guration and complexity. The number of successors and

YesYes

Initiate

Application

Re-plan?

Invoke Initial

Solution

Method

Evaluate

Initial

Solution

Re-plan?

Start TS3P

Re-schedule?

Start TSAR

Start TSARS

End

Get Current

Solution

RetrieveInitial Data

Yes

No

No No

Fig. 6. Flow chart for overall application.


11/16


predecessors for each job were varied among the problems

to give an array of network structures.

The lag times between a predecessor=successor pair de-

pends on the mode combination being used. For example,

predecessor job J1 executed in mode A followed by suc-

cessor job J2 executed in mode A would have a dierent

lag time than predecessor job J1 executed in mode A fol-lowed by successor job J2 executed in mode B. Random

maximum (negative value) or minimum (positive value) lag

times were generated for each predecessor=successor mode

combination. The lag times were uniformly distributed, ran-

domly generated integers restricted to be in the range from

negative one-half the duration of the predecessor activity to

the duration of the predecessor activity. For example, if the

predecessor activity had a duration of 10, then the lag times

associated with the successor activity would be in the inter-

val [ 5; 10].

Optimal solutions were available for four of the ve

10-job problems. However, relaxed optima, using only thenon-renewable resource constraint, were obtained for the

remaining 10-job problem and for one of the 30-job prob-

lems. The true optima for the doubly constrained problems

cannot be less than the relaxed solution; the relaxed optima

are lower bounds on the fully constrained problems. If any

of the heuristics, which do satisfy all the constraints, nds

a solution that has the same objective function value as the

relaxed optima, that solution is optimal. For the rst phase

of testing of the initial solution heuristic and the general TS

heuristic, goals were not considered. The objective in these

tests was to minimize makespan. However, other goals are

considered in the case study.

The remaining 30-, 60-, and 90-job problems had noobtainable lower bounds within the limited time and

resources available. Results of TS3P for these jobs are

reported in this research for two primary reasons: rst,

to determine how problem size aects solution time and

second, to measure the performance of the initial solution

heuristic vs. the TS.

7. TS3P results

Barr et al. [14] suggest the following experimentation

steps for heuristics [14, p. 1128]:

1. Dene the goal of the experiment.

2. Choose measures of performance and factors to explore.

3. Design and execute the experiment.

4. Analyze the data and draw conclusions.

5. Report the experiments results.

The goal of this initial testing was to show that the initial

solution heuristic and the TS perform quickly and provide

reasonable solutions when compared to a lower bound on

the makespan for the problem.

The relaxed (no renewable resource constraint) integer

programs were solved using HyperLingoJ Version 5.0

on a Pentium II with a clock speed of 400 MHz and 256

megabytes of memory. TS3P was run on a Pentium II with

a clock speed of 300 MHz and 128 megabytes of memory.

All times reported for TS3P were for 1000 iterations and a

tabu tenure of 13. As there was potential value inherent in

the initial solution heuristic itself, solution times and values

were recorded for the initial solution heuristic as well as forTS3P.

The results are presented in Tables 25. The statistics

for the 10-job problems are reported in Table 2 where the

delta column is the raw dierence between the solution

obtained via the specied heuristic approach and the lower

bound. The % delta column is the percentage away from

the lower bound for the specied heuristic (i.e. (delta=lower

bound)%). Table 5 contains data including minimum, max-

imum, and mean solution times as well as the standard

deviations and condence intervals for each size problem.

For the 10-job problems, the relaxed TS3P achieved

optimality three times, while the initial solution heuristicattained the relaxed optimum twice. On one occasion, the

solution found by TS3P for the fully-constrained problem

equaled the relaxed optimum and therefore is known to

be optimal for the fully-constrained problem. TS3P was

able to improve the solution obtained by the initial solution

heuristic three out of ve times (on the other two occasions,

the initial solution heuristic found the relaxed optimum).

TS3P improved the fully-constrained initial solution for all

but one of the ve 10-job test problems.

Only one lower bound was available for a 30-job prob-

lem. For this one problem, TS3P for the relaxed problem

was within 8% of the optimum. In all but two of the larger

problems, TS3P was able to improve the initial solution. Forthe cases where the initial solution heuristic failed to sched-

ule all of the jobs, TS3P was able to nd a solution where all

jobs were scheduled, sometimes with a subsequent longer

makespan, but since makespan is a lower priority goal, these

solutions are still an improvement. All solutions were fea-

sible. Table 6 summarizes solution times.

Solution times for the test problems naturally increased

as the problem size increased. Graphs of the TS3P (Figs. 7

and 8) and the initial solution heuristic times show that they

follow similar, although dierently scaled curves. The initial

solution heuristic is very fast for the problem sizes tested.

Regression analysis indicated an exponential relationshipbetween the number of jobs in the problem and solution

time.

Overall, the initial solution heuristic and the TS both gave

rapid, feasible answers to the job shop models on which

they were applied. The initial solution heuristic gives fea-

sible answers in a very short time. The TS generally gives

solutions with better objective function values, taking some-

what longer than the initial solution heuristic, but it is sub-

stantially faster than optimization. Since the solution times

for both methods grow as the number of jobs increase, test-

ing the methods against larger problems needs to be accom-

plished.


12/16


Table 2

Solution statistics for 10-job problems

Job size Method CPU time Cmax Delta % Delta Jobs not

10 (s) (solutionLB) (delta=LB) scheduled

Problem 1 Lower bound (LB) 83.00 10 0

Relaxed initial soln. 0.03 10 0 0 0Relaxed TS 10.80 10 0 0 0

Fully const. init. soln. 0.03 13 3 30 0

Fully const. TS 14.90 11 1 10 0

Problem 2 Lower bound 64.00 11 0

Relaxed initial soln. 0.03 19 8 73 0

Relaxed TS 11.20 11 0 0 0


Fully const. TS 11.40 15 4 36 0



Relaxed TS 12.00 11 2 22 0


Fully const. TS 11.00 16 7 78 0



Relaxed TS 10.50 9 0 0 0


Fully const. TS 10.60 9 0 0 0



Relaxed TS 10.50 11 2 22 0


Fully const. TS 12.50 12 3 33 0

Delta = heuristic solution lower bound, %Delta = (delta=lower bound)%.

Table 3


Job size 30 Method CPU time Cmax Delta % Delta Jobs not

(s) scheduled

Lower bound 2734.00 26 0

Problem 1 Relaxed initial soln. 0.12 44 18 41 0

Relaxed TS 34.20 28 2 7.90 0Fully const. init. soln. 0.11 60 34 131 1

Fully const. TS 36.80 57 31 119 0

Problem 2 Relaxed initial soln. 0.12 21 1

Relaxed TS 27.00 23 0

Fully const. init. soln. 0.08 41 1

Fully const. TS 30.62 41 0






13/16


Table 4


Job size 60 Method CPU time Cmax Jobs not

(s) scheduled


Relaxed TS 110.10 30 0Fully const. init. soln. 0.67 40 0










Table 5


Job size 90 Method CPU time Cmax Jobs not

(s) scheduled





Problem 2 Relaxed initial soln. 0.62 33 0Relaxed TS 227.90 30 0



8. Case study

The case study for this research is the 100 activities, 4

machine types, 296 mode planning scenario described in

detail in Van Hove [8, Chapter 7]. It was adapted for thisresearch by designing and populating a database to store all

of the scenario data.

This problem is larger than the 90-job problems dis-

cussed in the previous section. In the 90-job problems,

each job could be executed in one mode from a selec-

tion of from one to three modes, resulting in roughly

530 predecessor=successor mode combinationsthe real

driving force behind the size of the problem. In Van Hoves

[8] scenario, each activity could be accomplished from

a selection of either two or four modes, resulting in 999

predecessor=successor mode combinationsalmost double

the size of the 90-job problems.

The objective function in Van Hoves model was to min-

imize makespan, while the objective of TS3P is threefold:

maximize activity coverage, minimize operational impact

(for re-planning=re-scheduling), and minimize makespan.

One of Van Hoves assumptions for his model was thatadequate resources were available to implement the plan.

During the early stages of the plan, that may be the case.

Optimization is valid in pre-operation planning to shape re-

source requirements. By the time the process reaches de-

tailed planning, new solutions may be needed. Therefore,

TS3P was applied to the scenario as Van Hove originally

designed it, but the scenario was also solved more than once

with varied amounts of assets.

The rst application of TS3P was against a scenario with

sucient renewable and non-renewable assets planned. This

was done to obtain a baseline case to see how the solutions

would be aected by varying the quantities of each asset,


14/16


Table 6

Solution time statistics

Initial solution TS3P

(1000 iterations)

10 Jobsa

Min 0.02 10.4Max 0.03 14.9

Mean 0.027 11.53

St. deviation 0.00483 1.360596

95% conf. int. (0.024, 0.029) (10.686, 12.373)

30 Jobsa

Min 0.08 27

Max 0.12 36.8

Mean 0.0983 31.65

St. deviation 0.0204 3.475

95% conf. int. (0.082, 0.115) (28.869, 34.431)

60 Jobsa

Min 0.26 110.2Max 0.861 137.4

Mean 0.429 118.2


95% conf. int. (0.214, 0.643) (108.38, 128.02)

90 Jobsa

Min 0.541 227.9

Max 1.642 264.3

Mean 1.0934 250.338


95% conf. int. (0.270, 1.917) (234.984, 265.691)

a

As there was no statistical dierence between the mean solutiontimes of relaxed vs. fully constrained problems within a job size,

the statistics are pooled for each job size.

in turn. Table 7 shows that tightening the non-renewable

resource constraint had little impact on the solution, while

tightening the renewable resource constraint had a consid-

erable impact on the solution. Interestingly, TS3P was

unable to improve the initial solution for the non-resource

constrained problem instance.

The levels of both types of resources were then set at

the threshold. TS3P was applied to the scenario at thislevel and then at resource levels of 95%, 90%, 85%, and

80% of the threshold level of the non-renewable resource.

At the 95% level, and each subsequent level, tasks were

uncovered, but in each case, TS3P was able to improve the

initial heuristic solution.

Two re-planning scenarios were run, one with the attrition

of four pieces of equipment from the rst work crew, and the

other with an entire crew unavailable. Recall that the fourth

goal in TSAR is to minimize the impact of any changes in

the overall plan. In both cases TSAR was able to reduce the

number of jobs re-planned vs. the solution obtained by the

initial solution heuristic. Results are summarized in Table 7.

TSACP-Time per 1000 Iterations

S

econds

0

50

100

150

200

250

300

10 Job 30 Job 60 Job 90 Job

Fig. 7. Graph of TS3P time for 1000 iterations.

0

0.2

0.4

0.6

0.8

1

1.2

10 Job 30 Job 60 Job 90 Job

Secon

ds

Solution Times for Initial Heuristic

Fig. 8. Graph of solution time for initial heuristic.

9. Conclusions and recommendations

Several important contributions were provided through

this research. The rst contribution is a fast heuristic ap-

proach for the project planning problem incorporating

activity priority classications and generalized precedence

constraints, as well as specialized machine and least ex-

ible activity. The initial solution heuristic developed here

considers an array of schedule attributes. This heuristic, as

well as the tabu search, reads from a database so as to more

closely replicate conditions in an operational environment.

Options allow specic jobs to be frozen and priority

classes of activities to be considered.

While the initial solution heuristic was designed to pro-

vide the tabu search with a good starting solution, as a

stand-alone application the initial solution heuristic workswell. Indeed, the quality of the solutions obtained by the ini-

tial solution heuristic was such that, on several of the test

problems, the tabu search was unable to improve upon its

starting solutions. The initial solution heuristic found the op-

timal solution for two of the test problems. Moreover, this

heuristic works very quickly, obtaining good solutions in a

fraction of a second for the smaller problems (up to job size

60) and in less than two seconds for the larger problems,

including the case study.

Another contribution of this research is the construction

of a tabu search that operates on a goal programming for-

mulation of the planning problem. The tabu search yielded


15/16


Table 7

Results of applying TS3P to the planning scenario

Renewable Non-renewable Method Times Goal one Goal two Goal three Goal four Goal ve

(S) deviation deviation deviation deviation deviation

Innite Innite (Makespan)Initial solution 1.70 0 0 0 NA 389

TS 1000 iters. 152.05 0 0 0 NA 389

Innite 330

Initial solution 1.64 0 0 0 NA 471

TS 1000 iters. 145.62 0 0 0 NA 389

120 Innite


TS 1000 iters. 139.73 0 0 0 NA 648

Threshold

120 330


TS 1000 iters. 179.6 0 0 0 NA 677

120 314


TS 1000 iters. 180.47 0 0 4 NA 568

120 298


TS 1000 iters. 197 0 3 0 NA 593

120 282


TS 1000 iters. 203.25 0 3 3 NA 552

120 266


TS 1000 iters. 211.1 0 5 4 NA 518

Re-plan Loss of 4 machines from rst unit116 324

Initial solution 1.98 0 0 1 10 748

One unit TS 1000 iters. 184.5 0 0 0 6 736

Lost

84 250

Initial solution 1.76 0 0 10 50 755

TS 1000 iters. 204.52 0 0 10 40 735

good solutions with consideration to task priority classica-

tion, and generalized precedence relationships. A list of the

best k solutions oers options for planners to use during theplanning process. The tabu search worked quickly, even on

the larger test problems and the case study, therefore demon-

strating its potential to operate within a planning tool.

The methods developed in this study were applied to the

problem of re-planning. In many situations, a great deal of

time, money, and personnel hours is spent up front gener-

ating a comprehensive plan of operation. Should the plan

break down due to unforeseen circumstances, planners fre-

quently have no option but to cobble together an ad hoc x,

especially when time is a critical factor. These quick xes

to the overall plan may mean extensive re-planning by the

managers of the individual jobs and tasks. By using TSAR,

planners can generate new solutions with regard to the cur-

rent situation while attempting to minimize the impact of

changes throughout the system.

References

[1] Pinedo, Michael. Scheduling theory, algorithms, and systems.

New Jersey: Prentice-Hall, 1995.

[2] Shtub A, Bard J, Globerson S. Project management

engineering, technology, and implementation. New Jersey:

Prentice-Hall, 1994. p. 1321.

[3] Pritsker A, Watters L, Wolfe P. Multi-project scheduling

with limited resources: a zero-one programming approach.

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[4] Kolisch R, Sprecher A, Drexl A. Characterization and

generation of a general class of resource-constrained project

scheduling problems. Manuskripte aus den Instituten fur

Betriebswirtschaftslehere, No. 301, University of Kiel,

Germany, December 1992.

[5] Boctor FF. A new and ecient heuristic for scheduling

projects with resource restrictions and multiple execution

modes. European Journal of Operational Research

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1998.

[9] Van Hove JC. Personal E-mail. Air Force Logistics

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[10] Glover F. Tabu search: a tutorial. Interfaces 1990;20(4):

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[13] Kolisch R, Sprecher A, Drexl A. Characterization and

generation of a general class of resource-constrained project

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1693703.

[14] Barr RS, Golden BL, Kelly JP, Resende MGC, Stewart WR,

Jr. Designing and reporting on computational experiments

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932.

Further reading

[15] Deckro, Richard, John Hebert. Polynomial goal programming:

a procedure for modeling preference trade-os. Journal of

Operations Management 1988;7(4):149 64.

[16] DellAmico M, Trubian M. Applying tabu search to the

job-shop scheduling problem. Annals of Operations Research

1993;41:23152.

[17] Glover F, Laguna M. Tabu search. Boston: Kluwer Academic

Publishers, 1997.

[18] Koewler, David A. An approach for tasking allocated combatresources to targets. Thesis, Air Force Institute of Technology,

Wright-Patterson Air Force Base, OH, March 1999.

[19] Laguna M, Barnes J, Glover F. Scheduling jobs with linear

delay penalties and sequence dependent setup costs using

tabu search. Research report, Department of Mechanical

Engineering, University of Texas-Austin, April 1989.

planning and replanning in project and production scheduling

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