cost tradeoff project scheduling problem with discounted cash flows

Abstract Many of the recent studies on Project Scheduling focus on maximizing the Net Present Value (NPV) of cash flows tlrat occur during the implementation of a project in the presence of precedence artd resource constraints. The literature contains several integer programming formulations, heuristic procedures and their implementations. In most of these formulatic%s, activity durations are assumed to be fixed. In this paper, we study a practical extension of the problem where the activity &nations can be reduced from their normal durations by al!ocnting rr,ore resources. Costs associated with such reditctions are referred to as crashing costs. The problem under consideration involves determining the timing and duration of activities such that the NPV of all cash flows 1s !:iaximized in the presence of precedence and resource constraints. We suggest a heuristic procedure for obtaining “good” feasible solutions. We compare the performance of three priority rules embedded in the proposed procedure. Two of these priority rules have been shown to be the most effective in the literature, the third is a new rule developed in this paper. Extensive computational experiments with the heuristtc procedure on a total of 380 test problems are reported. The heuristic solutions are compared to benchmark solutions obtained from the best of 50-rmdmly generated solutions for each test problem. in addition, upper bounds on the optimal values are obtained by performing Subgradient optimization on a Lagrangian Relaxation of the problem. These upper bounds are used to assess the quality of the heuristic solutions. In general. none of the priority rules produced consistently better results for all the test problems. However, all three priority rules provide “good” quality solutions with modest computa?onal effort. The Lagrangian approach is found to provide tight bounds on the optimal values.

1. Introduction In this paper, we introduce the Resource Constrained Time/Cost Tradeoff Project Scheduling Problem with Discounted Cash Flows, PI. This prob lem is applicable to situations in which the project manager has the option of reducing the duration of certain activities by allocating more resources to

* Corresponding author.

them. Some of these resources are available cnly in limrted quantities. The project manager’s objective is to schedule the project activities in such a way that the net present value of cash flows is maximized. Reductions of activity durations (crashing) may in-crease cash outflow, but if the correct activities are crashed, gain in net present value may also result. Thus, the project manager faces a scheduling problem Which involves resource limitations, variable activity durations and precedence relations. Despite its practicality, research on this problem is sparse. This is mainly because of the difficulty

0272-6963/96/915.00 @ 1996 Hsevier Science R.V. All rights reserved SSDf 0272-6963(95)00025-9

invoived in providing an efficient mathematical formulation and in developing an op!imal sohnion procedure for the problem. However. there are several approaches (both heuristic and exact) devclopcd for relaxations of this problem. One relaxation of this problcn is the Resource Constrained Project Scheduiing Problem with Discounted Cash Flows, P2. P2 involves scheduling the project activities with cash flows, in such a way that the resource and the precedence constrai. s are met and the net present value of the cash iiows is maximized. In P2. the project activities are dururion-dri~wz; activity durations remain constant regardless of the amount of resources allocated. Another relaxation of the probtem is the Time/Cost Tradeoff Project Scheduling P oblem with Discounted Cash Flows, P3. P3 involvcs determining a duration and a completion time for each activity of a project so that the net present va!~ of ail transactions that occur during the execution cf the project is m~xi,:m~d. In this problem, activities are effort-driven. Activity durations can be shortened by allocating more resources at an increased direct cost. In this relaxa!ion, it is assumed that resources are availabie in unlimited qnantities. Clearly, P1 is a combination of P2 and P3. A solution technique developed for PI will also provide solutions for both of these relaxations. in this paper, we present a multi-pass heuristic procedure and a bounding algorithm for solving PI. Heuristic sequencing rules that were found to be most effective in the P2 literature and a new priority rule, the Net Marginal Gain Index, are embedded in the procedure. Extensive computational experiments are reported to compare the performance of these rules. Results are presented for 380 problems that invo!ve different parameter settings. Furthermore, to test the effectiveness of the heuristic procedure, the test results are compared with the best of SO-random solutions. Finally, a Lagrangian Relaxation algorithm is introduced to obtain bounds on the optimal solutions. The plan of the paper is as follows. A summary of the project scheduling literature that is directly related to Pi will be presented in the following section, In Section 3, the problem formulation will be given and its managerial insights will be explained with the help of an example. In Scztion 4, the heuristic solution procedure ‘is presented. Furthermore, in Sec-

tion 4, a Subgradicnt Algorithm is developed to obtain upper bounds on tke optimal values and the computational results are summarized. Section 5 is devoted to conciusions.

2. LitekaPure review M19th research has been done to suggest solution procedures for problem P2. Models usually have different assumptions about the way they handle the cash flows but the fixed activity duration assumption is common. The tirst efforts were made by Doersch and Patterson (1977) and by Bey et al. (1981). Doersch and Patterson (1977) formulated the problem as a O-i ILP and attempted to solve the problem with a general purpose ILP code. Only small-size probterns were solved. Almost a decade later, Russell (198f! tested the performance of the heuristic rules which were found most effective for the makespan minimization problems, on P2. In general, no one heuristic rule outperformed the others consistently. Recently, Padman and Smith-Daniels (1993) compared the performance of several heuristic rules embedded in a single-pass Optimization Guided Aigorithm. Their early release heuristics provided superior results. Dayanand and Padman (1993b) intruduced several models for problem P2 where they relax the assumption that the amount and the timing of the progress payments are known. They discuss the features of each model and the characteristics of its optimal schedule. The reader is referred to (Dayanand and Padmsn, 1993a, 1993b; Eimaghraby and Herroelen, 1990; Icmeii and Erenguc, 1994a; Icmeii et al., 1993; Smith-Danieis and Smith-Danieis, 1987; Yang et al., 1992) for a complete survey of the studies on P2. In general, the Time/Cost Tradeoff models suggest allocating more resources to those activities which are less expensive to crash. However, in an inflationary environment with high interest rates this strqtegy may worsen the net present value of the project. Accordingly, recent studies are directed to combine the time/cost tradeoff concept with the net present value maximization objective. The reader is referred to (Icmeli et al., 1993) for a review of P3 and the related literature.

The first effort to solve the resource constrained time/cost tradeoff problem is by Talbot (19X2). He introduced a model in which the duration of each activity might vary according to the resources aiiocated to that activity and the objective is to minimize the makespan. He suggested an integer programming approach which is sufficiently general to soive the problem with different objective functions. Another related work was done by Leachman et al. (19901. They also considered the resource constrained, makespan minimization problem where the speed of an activity is directly related with the amount of resources allocated to it. A heuristic procedure is provided to solve; the problem. In their F 4% they do not include cr?shing costs, assuming that they never exceed the resource availatJities (no subcontracting or overtime). As it is clear from the review, no research has been done that suggests solution procedures for the resource constrained time/cost tradeoff problem with the objective of maximizing the Net Present Value of the cash flows although in many profit seeking organizations this is a viable objective. Although :he inclusion of the NPV objective makes the problem more difficult to solve, the modei becomes practically more appealing.

3. The problem formulation The resource constrained time/cost tradeoff project scheduling problem with discounted cash flows involves scheduling, and determining the duration of project activities with cash inflows and outflows, in such a way that the resource and precedence constraints are met, and the net present value (NPV) of the cash flows is maximized. This can be represented by the following integer program:

-

i

(t-due)XI,,PeWa’,

sL&ject to ;

<” = 1 Vi.

k= 1

5 X,, = 1 Vi,

Y,” E fO,l)

Vi,k,

(5)

X,, E {O,I)

Vi-t.

(61

where T is an upper bound on the completion time of the project, due is the due date on the completion time of the project, P is the penalty per period of delaying the project beyond its due date, N is the last activity in the project. qi is the cash flow that occurs at the completion time of activity i, K, is the number of possible durations for activity i, cl: is the kth possible duration of activity i, where d,’ conesponds to normal (longest) duration, r,:, is the number of units of resource u required per period by activity i with duration df? R, is the availability level of resource I’. u = ?, . . . .V, NF;” is the net cash flow at the comyle:ion time of activity i with duration dfi. NFs may be zero, negative or positive, d,X ,j= 1 f,: is the cash flow for activity i in period j, with duration df, j = 1, . . .,df, c” is the total crashing cost if duration df is chosen for activity i, cl = 0, I’”i = 1 if df; duration is chosen for activity i, Yi” = 0 otherwise, X,, = 1 if activity i is completed at time tt X,, = 0 otherwise, LYis the discount rate, ej is the eadicst completion time for activity i, and q is the latest completion time for activily i. The CPM method is used to determine the earliest and the Batestactivity compiietion times.

The first term in the objective function is the sum of the values of cash flows (including crashing cost) discounted to time zero. The second term is the discounted value of delay penalties which occur if the project duration exceeds its due date. Thus, the objective function maximizes the net present value of the cash flows that occur during the project life. The first set of constraints insures that only one duration alternative is chosen among the set of K, possible durations fo- each activity i. The second set of constraints requir-a that each activity is completed exactly once. The third set of constraints represents the resource constraints. At any time period t, the total resource consumption of the activities which v‘i: scheduled to be active at time t cannot exceed the resource availability at time t, for each resource type I,. Note that the resource requirement of an activity during a time period varies with the duration chosen for that acttvity. The fourth set of constraints represents the preceden:.:: eiations. If activity or immediately precedes activity H which has a duration di, then the completion time of activity m plus dt must be less than or equal to the completion time

of activity n. The fifth and sixth sets of constraints are the integrality requireme?rts on the variables Y,” and X,,, respectively. If the variable duration assumption is relaxed in PI, then problem P2 is obtained. Thus, the procedures that were used in ~Icmeli and Erenguc, 1994a) to determine a due -.,.te and the latest completion time T can be used to determine a due date and T for problem PI when activities are at their normal durations. In the model, the following assumptions are made: 1. Each activity i is assumed to - have K, possible integer durations, - require known levels of resources of each type, each of which increases as the duration of an activity shortens (effort-driven). - utilke constant level of each resource type over its total duration; - crashing cost per time unit is constant: - have no preemption. 2 The level of availability of each resource type is assumed to be constant throughout the schedule (renewable resources).

cashFlow w,flQo

Tiie(m0.) 4 8 15

-zE

NP\I=$548

ResourceLimits:

r,=3 r*=d

Titnetmo.) 4 8 14

Fig. I. (a) The optimal t,Russcll, I%%).

schedele

For the resource

constrained

problem.

(b) The optimal

schedule

with crashing.

The problem

is adopted

from

Note that the general assumption in P3 about the activities being effort-driven is adopted here. Since the activities that use renewable resources such as labor hours and machine hours are generally effortdriven, the assumptions listed above would not lessen the applicability of the mode: to real life situations. Note that in (Leachman et at., 1990i. it was assumed that unless resource usage exceeds the resource availabilities, no additional costs are incurred. However, .in this paper, we assume that ifan activity is performed in a shorter amount of time than its normal duralion, crashing costs are incurred. The resource availabilities represent the maximum amount of resources thr& can be allocated to activi!ies in a time period and cannot be exceeded (for z&le 24 machine (labor) hours per day). The following numerical example illustrates the relationship between problems PI and P2 dnd demonstrates why it should be beneficial to consider problem P1 rather than P2. Fig. 1 represents the activity on arc network oii a small project with five activities and four events. Tine numbers in the rectangles represent fixed activity durations the numbers in the brackets represent the resource usages (there are two resource types), the valses at the nodes represent the cash flows occurring upon rhe completion of all activities incident to a node and the numbers at the tail and head of each arc represent the optimal start and completion times, respectively, for the resource constrained problem with the NPV maximization objective (probiem P2). Now, consider the following scenario. The project manager can allocate more resources (machine hours) at a cost of $10 to reduce the duration of an activity by one unit. He is considering the possibility of increasing his profit without violating the precedence and resource constraints. The problem becomes a PI problem. Assuming a discount rate of 0.01 per period, the NPV is calculated to be $5dC when the project is scheduled as a P2 project. However. since he has the opportunity of shortening the activities by allocating more machine hours now, he would increase his profit by shortening for example the duration of activity 5 by one unit. The following computations show that he would gain $17 additional profit by solving the problem as PI. Overtime and hiring subcontractors would ahow the Project Manager to determine the best duration

f& a task based on tire effort of resrrurces assigned to that task. The reader is referred to (‘Leachman et al.. 1900; Leorch and Muckstadt, 1994) for additional examples.

4. A heuristic procedure for PI In this section, we present a heuristic procedure for the resource constrained time/cost tradeoff project scheduling problems with discounted cash flows. The procedure performs a forward pass through the network starting with the first activity, which is usually a dummy activity. At the completion time of an activity, ail the activities which can be scheduled according to their precedence relations are determined. Let L be the set which contains the schedulable activities. The best duration afternative is determined for each activity in L by computing an index ca!led the Net Marginal Gain (NMG). For each duration d:. k # 11 of each activity i the net marginal gain, HF, is computed. This gives the change in the net present value corresponding to a unit increase in the overall resource usage when the kth duration alternative, rather than df, is considered. Here, the overall resource usage is expressed in some aggregate units. For each activity in L the duration with the maximum NMG value is chosen. The procedure schedules the activities in L according to the priority rule selected until a resource conflict occurs. When a resource conflict occurs, unscheduled activities in L are delayed and the procedure moves to the next activity completion time. The procedure continues in this fashion until the last activity, which is usually a dummy activity, is scheduled. The foliowing priority rules are employed in the procedure: I. DCFW (Discounted Cash Flow Weight). For each activity in set L, the discounted value of future cash flows from all the successor activities is computed. Activities are scheduled in decreasing order of their DCFWs. This rule was found to be one of the most effective rules for problem P2 (Baroum and Patterson, 1993). 2. kX.3 (Oppanrtrtity Cm oj Scheduliq, Itmeditrte rz-slease).For each activity k in L, the opportu-

nity cost of scheduling activity k instead of other activities in the queue is determined. Activities are scheduled in. ascending order of their opportunity costs. c P,cd, -f [EC&,, k’EAkx

statement

k==2 ,.... K,,

of the procedwe

1. Start with scheduling the initial project activity usuaily a dummy activity, at t = 0. Step 2. At the completion time, t, of an activity, considering only the precedence relations, determine L, the set of all schedulable activities. step 3. IF L=@, THEN IF the last scheduled activity is N, the last activity of the network THEN return the schedule and the net present value and STOP; ELSE move to the next completion time and go to step 2. ELSE go to step 4.

Step

i E L. determine

j=I

- t)],

where A, = L\[kf is the set of remaining activities in L, P is the current time period, f,, is the cost of delaying the start of the activity in k’ E A, by one time period from its currently scheduled time of T,tklJ, EC, is the sum of the dual prices of the smn:ediate successors of k and d, is the duration of k To obtain Pk, and EC,, we solve the corresponding Payment Sckduling Problem (PSI’) (Padman and Smith-Daniels, 1993). Once the astivity durations are determined using Ihe Net Marginat Gain appmxh, resource tronstrainIs are rebaxcd to oWein the PSP. Grinold’s algorithm (Grinold, 1972) is used to solve the PSP and determine the activity dual prices. This rule was found to provide superior NPV results for problem P2 (Padman and Smith-Daniels, 1.993). 3. NMG (Net Margittal Gain). Let M be the sorted set of activities in L according to their NMG values. The procedure schedules the activitirs in I; in decreasing order of their NMGs. In the application of these priority rules, ties are broken by scheduling the lowest numbered task first.

4. /. A formal

Step 4. For each a&vity

wherl: CFtk = NFtk - cf, and H,” is the net marginal gain of activity i with duration ~1,“. t," is the completion time of an activity i with duration (1:. ti is the completion time of activity i with normai duration, y,“, are the r,:s expressed in some aggregate unit such as dollar value, y,: indicates the nnrfual resource consumption in some aggregate unit. Srep 5. Determine the duration with the highest net marginal gain for each activity i in L, ;f,“‘=max(H,“:k=

l....K,).

Let dr * be the duration chosen for activity i. Sfep 6. At aime f schedule the aciivities in t by using a priority rule until a resource conflict ocCUTS.

IF no resource conflict occurs THEN schedule all the activities in L and go to Step 7, ELSE if a resource conflict occurs during the scheduling of activity j E L, then activity i and all the succeeding elements of L are delayed, go to Step 7. Step 7. Go to step 2. One of the notable features of the procedure is that the duration of each activity is determined based on the time the activity is considered for scheduling. That is, the NMG values are updated each time a new L is formed. The procedure might choose to crash an activity if it is considered for scheduling at time t and might choose to perform the activity at its normal duration at time r’ depending on the NMG values of the activity at these two different time periods. For each duration alternative of each activity, the project manager needs to estimate the cash flow CF,k, the crashing cost cf and the aggregated value of resource usage -p;. The cash flow value should include the lump sum cash to be received or paid at

the completion of an activity plus the discounted value of all transactions to occur during the implementation of the activity. The ys can be expressed in some aggregate unit such as dollars or hours. As an example consider reducing an activity duration by one anit which causes machine-l to be used 2 additional hours and machine-Z to be used 4 additional hours. Ii the efficiency of machine-l is 20% higher than that of machine-2, then machine-2 is actually used an addttiona13.2 machine,.1 hours. Thus, a total of 5.2 additional machine-l hours are used. This number can be used as the denominator in the N&E formula. Finally, note tnat this procedure will always produce a feasible solution to Pi regardless of the values of the cash flows. This is mainly becaust be procedure has a front-load scheduling scheme. That is, the activities in L zre delayed only if there is a resource conffict and the deIayed activities will be eligible for scheduling at the next time period where an :xtivity is scheduled to complete. In contrast lo previous research (Russell, 19861, there is no need to ensure a positive NPV by assign:ng a large positive cash inflow at the completion of tbe project. This attribute allows the heuristic procedure to be used for government and defense/aerospace projects where the least costty project is selected and profit is not a concern, As will be explained in the next section, Baroum and Patterson’s (Baroum and Patterson, 1993) right and left shift routines are apphed to the schedule obtained by the front-loading scheme to improve the solution, if possible. 4.2. Computational experience The heuristic procedure was coded in FORTRAN and tested on two data sets. ‘The first data set contains 50 test problems constructed from the Patterson data set (Yang et al., 1992). These problems consist of projects with activities ranging from 7 to 51 and with up to and mostly 3 resource types The original Patterson data set, however, does not contain data on cash flows for the activities, crashing costs, crash durations and the increase in resource consumptions in case of activity crashing. Thus, additional data were generated to modify the original data set. For each activity three durations are considered: minimum, average and normal. The minimum and aver-

age durations are obtained by reducing the normaI activity durations by 60% and 4O%, respectively. However, if an activity has a normal duration of 1 time unit, then minimum and average dnratjo~s are set equal to the normal duration, Tne crashing costs per unit time are determined as follows: Cash received at the compietion time of the activities, qis, are generated randomly and uniformly between [- 5OO,lOOO]. For each activity the Crashing Cost (CC) per unit time was generated from a uniform distribution on [l,b], where b is either S or 10 or 20. Without toss of generality, the cash flows, &, are assumed to be zero. The resource requirements for the activities at their average and minimum durations are tied to the crashing costs incurred. The percent increase in crashing costs determines the percent increase in the amount of resources allocated when crashing occurs. In the computational testing. r$ was set equal to r:. for al? values of i, i and k. Mso, if the normal duration resource requirement of an activity is zero, then tie average and the ~~~~irn~r~~ duration resource requirements are set to 1 and 2, re5pectiveIy. The second data set is a larger data set and prepared by Patterson et al. (1990). There are 115 test problems each of which has six resource types and the number of activities range from IO to 100, Crashing costs and cash flows were generated and added to this data set in the same way as was done for the first data set. .. . The Average Utthzation cL actar (AUF) has been used as a measure of resource constrainedness (Barotum and Patterson, 1993). For a given project schedule, the AUF for each resource type is computed as the ratio of the total usage of that resource to the tOta amount Of the same that iS available over the unconstrained critical path duration. AUFs were observed to determine if priority rules were impacted by the problem constrainedness. For the first problem set the following combinations of discount factor and b, upper limit of the uniform distribution for generating crashing costs, were used: CO.O1,5), @LOI,lO~, (0.02,20>. For the larger problem set these combinations were (0.01,20), (O.O5,20). Each problem was solved with the three prior&y rules: NMG, DCFW and ICES. Computational results are summarized in Tables l-6. Tables 4-6 pet”ain to the second data set. In each table, the

262

R

24

iI

2

5

i 22

22.23

27

35

51

with

DCFW

IO.21

I .0x 0.79 0.19

RLLIml

O.RS 0 53 0.85

Avenge and the range of number of activites crashed to minimum dunrtton

1.50 1.00 !.OO

'5.50

IS.50

17.30 17.30

10.299.62

3K.OQ 23.80 23.30

24.00

24.90

II.58

9.14 x.42 8.42

Average of the total numtxr of periods crashed

---,

[0.$5,0.76] 10 !iJ,O.85]

lO.6~,0.~21IO.n7.0.8~~(O.G6,~~.X51

(0.55,0.76] ]0.54.0,62]

[0.57] [0.571

[0.53,0,7(r) ]0.61,0.77]

I&57]

[0.59*0.761~5.58,0.771[0.58,0.77]

]0.63.0.X6]]0.61,0.87]]0.61,I~.87] ]0.64,0,80] [0.61,0.79] [0.6l.C.Hl]

~0.41,1.041I0.~11,1.031I0.41,1.1)31 [0.62,0.83] 10,48,0.&Y] (0.4X,0.84]

[0.5S,1.14][0.55,1.11][0.5j,1.12] [o.4s,l.l2](o.4~.l.otil(O.Jf,l.U9j

10.50‘1 .OOl [O.SO, I.001 [S.SO. I.001 10.43,0.561]0.4?,0.57] [0.43.0.37] [~.6.s,O.?2](0.6S,0.72][O,6S,0.72]

Range of h’.JF

0.29

0.13

0.1 I

0.10

0.06 ‘1

SCCOlldS

AWX!@ CPlJ time in

.. -

increw with

0.01 I fl.0l00.021

0.030 0.034 O.Odh

0.0150.0120.026

0.01 I 0.009 0.019

0.#100.0100.021

multipass

Avcragc in NPV

I-

0.053

0.021 0 130

0.03 I 0.073

0.0 IS 0.070

0,066 0,038

0.066

0.044

0.030

14.4c 9.20 X.60

9.50 5.50 20.00

1.72 4.63 4.72

6.08 4.00 5.08

3.57 3.57

3.a

0.021 0.017

0.058

Average drcrcase in project completion time with crashing Average increase in NPV crashing YerSUS no crashing

and the third entries correspond to results with u = 0.01 and CC C. (1.5). LX= U-0 I ;md CC & 1I, IO], and w 3~.0.02 and CC E a1.2OJ. rcspcctivcly. in the discount rates and crashing costs did not change lhe nvernge CPlJ times ccrnsutned.

[93J11[6+71f6,71 Il.21 19.20 12.00 It.80 1.60 1.00 1.00 1~2.261~9,171~9,161[1,71[0,31[0,~1

10.00 6.50 6.50

11,31[1,31[0,?1

i8~131[6.12115,121

1.10

2.00

9.80 7.20 7.20

I.10

UAI01ml KM1 [O,Sl[O,31 Ku]

4.04 3.87 3.87

3.42 a 3.00 3.00 ~2SlI2.41 k41

Average and the range of number of activitcs crashed to average duration

results

“, In eacfl cdl, the first, the second For each problem class, changes

Numbf!r of problems

Number of activities

Table 2 Computational

-

with IOCS

l.l20.81

[0,31 [O,.?l IO,31

5.08 4.37 3.62

[0.10110,81[2,81

24

II

2

5

22.23

27

35

51

[5,131~6,81~5,121

38.1023.8023.30

24.00 14.00 IS.50

24.00 8.95 16.90

13.40 14.90 9.04

9.14 8.42 8.42

AW.Wge of the total number of ~riods crashed

(0.65.0.721

[OS. I. IO] [0.48.0.991 [O-49, I. 121 [0,45.I.O6] [0.66.t.O81[0.45,0.94) [0.~i.1.041[0.61,c;,~l1[0.4l,1.i~] [0.62,0.831 [0.5S.1,091[~.48.0.841 [0.64.1.l21[0.45,0.941[0.65,1.091 [0.64.0.80][0.4l.1.0?1[0.51,0.81] [0.59,0.761 [O.S8.0.7S1[3.58,0.771 (0.571 [O.S81 IO.571 [~.S~.O.771[O.SS,O.76][0.55.0.761 [0.56.0.77] [C.S4.0,62] [O.S4,O.SS] [O.fiS.O.82] [0.67,0.841[0.6(~,O.x5j

[O.65,0.721[0.65,0.72]

-[O.fO. l.OO] [0.50,1.00] [o.so.r.oo] [0.43.0.521[0.43.0.571[0.43.0.571

Range of AIJF

I__

19.40

18.45

3.66

1.09

I. I5 ”

Average i’PlJ time in : .xonds

0.065 0.030 0.053

0.066 0.005 0.067

0.042 0.017 0.0’73

0.022 0.013 0.071

0.032 0.038 o.osx

-. Averngc increase in NPV crashing versus no ctashing

and u .= 0.02 and CC E [I.20

0.01 I 0.013 0.021

0.020 3.043 0,086

0.015 0.013 0.028

0.015 0.008 0.016

0.0130.0120.021

Average incrcnse in NPV with multipass

a In each cell, the first, the second and tlte third entries correspond IO results with a = 0,OI and CC E [l,S], B - 0.01 aad CC E [l,lOj. ’ For each problem class, changes in the discount rates and crashing costs did not change the average CPU times consumed.

10.00 7.50 6.50 I.50 0.00 I .oo [9,111[7,81[6.71 [I.21 19.00 12.00 11.80 1.80 1.00 1.00 112,261 [9.171[9.161[1,41 [OJl (031

1.90 0.62 1.09

[1.41[0.31tO.31

9.63 6.45 7.09

0.75

10.21 r0.21 ra21

0.85 0.85 0.85

3.42 ’ 3.00 3.00 [2.51[2.41[2.41

8

<22

Average and the range of number of activitrs crashed to minimum duortion

Numher of probkms

Average and the range of number of activites crashed to average duration

results

Num---ber of activities

Table 3 Computational

vspectively,

14.20 5.40 8.60

9.50 I .OO 5.50

6.63 O.?O 5.09

4.47 3.65 4. I2

4.57 4.51 3.57

Avenge decrease in prqec, completion time with crashing

z

m

0

8 >

5.50 6.00

37.80 39.20 [33/W] j37.421

17.80 18.60 [ 11.27) [ 13,241

25

25

5

10

111 20

IV 30

v IQ0

vi 50

3.40 3.68

a In each cell, the first and the second b For each set, changes in the discount

entries correspond rates and crashing

I3,81[4,91

2.32 2.40 lo.61 KU1

I .76 2.08

IO,41[DA

7.04 7.16

[2,111[4,~~1

k51 I~71

43.90 46.70

75.40 79.60

26.36 26.28

16.?6 18.20

6.30 8.20

7.88 8.12

Avenge of the total number of periods crashed

tMLF---

~0.l~,1.23110.l1,~.2~1

[0.37, I !06] lO.36, I .05] (0.38,1 .Q41 io.50, I .051 lO.34, I LIO] [0.36,1.03] (0,39,0,991 IO.37,1.201 (0.57, I. I91 IO*5.5,I. 191 [0.57,lr0sl[0*59,1.0~3 2.82 [0.49,0!85] [0.51,0.85] [0.66,0!901 [O&LO.941 I0.51,0.,R91[0.51,0.91 I [0.64,1.05] [0.65,1.041 [0.8l,l.22][0.86.l.26] [0.56,1.52] [0.56,1.441 0.47 [0.39,0.:12][0.40,0.811 [0.57.4~.')0][0.58.0.90~ [0.52,0.w]10.49,l.041 [0.60, I .:!O] [OS’), I .27] 10.60, R. ! 7][0.62,1.23]

~0.4S.I.~2~~~0.44,1.19~ 0.15

[0.16~0.96j [0.16,0.96] [0.19$.961 iO.19.0.981 [0.26,(\).741[0,26,0.74] [0.23.i\.00] [0.23,0.96] [O.22,l’.Ol](O.29,l.O3] [0.2g,O~.6l][O.ll.l.l7] 0.06 [0,24,0;.68] [0.16,0.87] jO.74,0;90] (0.19, I. 181 ~0.26,01.60] [0.26,0.68] [0.30.0:.78] [0.26,0.91] I0.29,0,731[0.29,1.071 [0.40,1.05] L0.40,1.02] 0.04 [0.28.0:.74] [0.28,0,97] to.3 1,0,.89] IO.3 1,0.89] [0.29,0!.78] f0.32,0.82] [0.38.0.94] [0.50,0.94] [0,59, I .iool lO.45, I .021

AWagc CPU time in i,zconds

Range of AUF

to results with u = 0.01 and o’ = 0.05, respectively. costs did not change the average CPU times consumed.

6.80 7.60 [4,111 [X121

12.00 12.00 I9,171[9,161

25

II 10

0.30 0.56 [0,21[0,21

0.48 0.56 [0,2!IO,21

4.04 ’ 3.56

[Ml i2.61

25

of

Number

I IO

Set and thenumber of activities in each set

problems

with DCFW Average and the range of number of activites crashed to minimum duartion

results

Average and the range of number of activites crashed to average duration

Table 5 Computational

0.012 0.037

0.01 S 0.046

o.ot I 0.039

0.007 0.033

0.003 0.022

0.005 0.022

-, Avciage increase in NW with multipass

0.02 I

0.032

- 0.036

0.03 I 0. L113

0.035 0.058

0.002 0.038

0.003

Average increase in NPV crashing versus no crashing

3.64 1 I 20

2“14.00

1 I .OR 10.60

5.40 s.52

0.50 2.28

1.16 1.24

Average decrease in project complefion time without crashing

-

25

25

25

5

10

I IO

II 10

iI1 20

1v 30

v 100

VI 50

IOCS

18.30 18.30 113,251113,251

38.~0 39.80 121,411139,421

k1 JlIfa II

7.04 6.80

3.64 3.92 [1,51Il.71

[W1 hV51

3.64 a 3.68

4.60 s.90 [3,71 i&31

6.80 760 [3,~0~[4,12~

2.32 2.40 RI.71 IO,41

0.52 0.68 IO,21 (0.21

0.48 0.56 IO,21 [CL21

40.20 45.40

77.40 80.80

24.36 27.00

17.36 lb.20

8.68 9.24

7.88 8.12

Average of the total number of periods crashed

[0.16.0.961[0.16,0.961 [O, I9,O.S,6] [O. 19,0.98] [0.26,O.b9] [0.26,0.771 (0.23, I .Fx)I IO.23S1.961 lO.22, I .(I1 3 10.29, I.O3] [0,2R,I.~r41iY).11,1.171 [O. I6,Q.R3[ [0.16,0.87) [0.l9,1,10][0.24,1~1~] [0.26,0.68] [0.26,0.68j [0.26,O.S6][0.26,0.961 [f~.22,1,~131[0.29,1.07j (0.40,1.05110.40,1 *n21 [0.28,0.751[0.28,0.75[ [o.3i,o.921[0.31,~.39~ [0.32,0.813 Kk32.0.821 lOAl .0.94110.42,0.941 [0.45,1 .O’Ol I&45, I .OZl [0.44,1.22][a44,1.19] [0.36,1.07[ @36,l .OSj [0.30,1.071[0.30, I .OSl [0,36,1.01] [0.36,1.03] [0,37.1.20] [0.37,1.20] [0.55,1,1111[0.55.1,181 [0.53, I .Q:i] 10‘5S, I.051 [0.50,0.831 lO.Sl,O.R31 10.67,0.931[0.70,0.931 10.50,0.8!)1[0,52,0.8P1 [0.59,l.0~][0.66,1.0l[ [0.83,1.2:1][O.L13,1.27] [0.~4,l.5~~1[0.~7.1.4~1 [0.40,0.81 I [0.40,0.76] 10*53,0.901 Ia64Lhsll [0.47,I.Ottj [0*48,1,06f (0.57, I .x*1 [0.56, I Xl [0.60,1.15] [O.6O,l,ZSi

Range of AUF

15.79

120.50

?,28

1.22

0.48

Average CPU time in seconds

to results with a = 0.01 and LY :‘= &OS, respectively. costs did not change: the averagr: CPU times consumed.

Average and the range of number of activites crashed to minimum duartion

entries correspond rates and crashing

Average and the range of number of aetivites crashed to average duration

with

a In each cell, the first and the second b For each set, changes in the discount

25

Set and the number of activities in each set

results

Number of problems

Table 6 Computational

-

Average increase in NW with multipass

0.045

0.028

0.005

0.064

0.084

0.022 0,056

O.Olii

O.oos, 0.023

Average increaw in NW clashing WhSII($ no crashing

IO.00 8.70

^_d Average decrease in project completion time with crashing

number of problems solved and the number of activitics in each problem are given in thy First two columns. Columns 3 and 4 give the average and the range of the total: number of activities crashed to their avcragc and to their minimum durations, rcspectiveIy. Column 5 gives the average of the total number of periods projects in each category were crashed. The minimum and the maximum values of AUF for each category and for each resource type are reported in column 6. Note that AUF values change with different parameter settings. The total resource amount required by all the activities changes zs :Sresult of a change in the activity durations. Each schedule generated with a different parameter setting mig: :, leave different duration alternatives chosen for activities. Column 7 gives the computational effort in CPU seconds on an IBM3!190 computer. Column 8 gives the increase in hJPV as i: resr;It of enhanckg the heuristic proccdurc by applying the right and left shift routines to the schedule obtained by the procedure. The heuristic procedure was also used to solve the test problems without the crashing option. The gap between the NPV of the schedules obtained hy noi aliowing crashing and the NPV of the schedules obtained with crashing is reported in column 9. The last column gives the percentage decrease in the project completion time when crashing is allowed. Tn all the problems, the total number of activities crashed to their average durations is greater than or equal to the number of activities crashed to their minimum durations. With the given parameter values, it turns out that for most of the activities it is not profitable to crash the activity durations all the way to their minimum values. As it can be observed from the tables, as the crashing cost per unit time increases, the total number of periods crashed decreases. This decrease is steep when the crashing cost is increased from [ 1S] to [l,lO], however, it slows down when it is increased from [ 1, IO] to [ 1,20] together with an increase in the discount rate (from 0.01 FQ0.02). The AUF values are between 0.1 I and 1.54 which indicates that in most of the schedules, the resource constrainedness is moderate. In the literature, AUF values of less than 0.5 are generally accepted as low resource constrainedness, between 0.5 and 1.5 as moderate to tight and higher than 1.5 as very tight. (Baroum and Patterson, 1993; Padman and Smith-Daniels. 1993).

In the larger data set, the crashing costs arc kept the same but thr discount rates considered are 0.01 and 0.05. In general, more activities are crashed to their ~lini~~ur~~durations when the discount rate is increased to 0.05, a,.j an increase in the average of the total number of activities crashed is observed. As one wouEd suspect, the quality of the solution is very sensiL/e to the discount rate. As discount rate increases. the impact of it on the NPV becomes more pronounced. Thus. the cost of schcdulinp an activity earlier or later than its optimal completion time increases considerably. Addition of left and right-shift routines (multipass) to the heuristic procedure almost always guaranteed an improvement in the NPV values. This improvement is the highest when the DCFW rule is used {around 2%) and is the lowest when the IOCS rule is used (around 0.8%). Due to tightness of the resource cunstraints, this improvement in the NPVs is considerably smaller for the problems in the larger data set. But since the increase in she computational effort with the addition of multi-pass to the heuristic prz;cc&re is minimal. it is advisable to enhance the procedure with the left-shift and right-shift routines, regardless of the priority rule employed. In almost all of the problems. reductions in the activity durations rest&cd in better NPVs (column 9). The only exception occurred with DCFW in Table 5 where the NPVs worsened with crashing (the negative number in column 9). The last column in each table indicates that thcrc are, almost always, reductions in the prqiect completion times as a result of crashing the activities. although shorter project life does not guarantee better NPV values. Note that crashing activities guarantees better solutions for prob!cms with the objective of minimizing the project completion time or guarantees better NPVs for problems with a very large positive cash flow on the last nctivfity of a project. There is only one instance where the project duration actually increased with crashing. This occurred with the NMG rule and discount rate of 0.05 (Table 4). This is mainly because the heuristic procedure delayed the activities with negative cash Bows in order to improve the NPV values. In order to compare the relative performances of the three priority rules, the following additional computations were undertaken. We generated 5O-rut&~n2

(RAN) for each problem and selected the best of the 50 as a benchmark solution, To generate the Xl-random solutions, Step 6 of the procedure given in Section 4.1 was modified. The modification consisted of selecting the next activity to be scheduled from the set t randomly, rather than using a priority rule. In Tables 7 and 8 we compare the performance of the priority rules to that RAN and ta one another. In Table:,, 7 and 8, the three priority rules and RAN are compared. The three entries in each line of a cell compare the column rule with tE. cow rule in the following m::;nner. If we denote each entry by the triple (~,h.c), where b idcnti.%s the rule which produced more solutions with strictly higher :“.:ective function values, u is the number of problems in which the priority rule denoted by b produced better solutions, and c is the nizmber of problems in which the two priority rules produced the same objective function vrtlue. For exampk. the first entry in Table 8 indicates !hat for the I IS test problems solved, salhms

Table 8 Relative performances problerns~

of the priority

NMG

DCFFV ----“________ 61. DCW,I:! i

DCFW

Qfl. DCFW, -

rots

-

IS ”

rules.

6 resource

IOCS 66, 53. 5*. _b IO.

of problems

in which

i I I5

RAN

IOCS, NklG. IOCS. IOCS,

I:! I6 8 ICB

h: In cornping rhc row rule 2nd the column stricriy better objective fun&on rslues in more a. Number solution.

types

rule

“I?“

T3, RAN, 0 77 > R-MN ‘ *0 75, RAN, 0 80, RAN, 73, RAN, 81,RAN.0

ruie. “I?“’ problems. xoduced

0 0

found a betier

l-able7 Reiative performances pFOhlCIIlSf

of the priotity

Lx”Fw

rules.

3 resource

IOCS

types

(50

RAN

NMG

33. NMG,J

’-

29. NMG.

2 3

DCFW

34. NMG.5 30. NMG.6 -

iocs

-

35, NMG, 3 27, NMG, 6 17. DCFW. IS 21. DCFW, 7 I. IOCS, 49 -

2

35. RAN.

0

38. 36. 38. 41, 39,

0 5 0 0 0

RAN, RAN. R 4N. RAN. RAY.

41, R‘AIZ. 0 40, RAN. 9 36, RAN, (i

.-

1: Results obtained range of [ 1,s:.

wilh

001

diacolinr

rtlte and

;L crashing

cost

2: Results obtained range offl,lO]. 3: Results obtained

with

0.01

discount

rate and

a crashing

cost

with

0.02

discount

rate and

a crashing

cost

mngc of [I 201. 4: Results obtained 5: Results obtained

with with

the 0.01 discount the 0.05 discount

rate (large rate (larger

data set). data set).

- (a.b,c). b: In comparing the row rule and the column rule, “h” found strictly better objrcrive function values in more problems. a: Number of problems in which rule “b” produced a better solution. c: Number of problems gave the same objective

in which function

the row value.

rule and the column

rule

compared to the NMG rule, the DCFW rule produced strictly higher NPV values for 64 problems and in 12 problems the NNG and the DCFW rules obtained she same objective function value. The following observations were made: None of the priority rules and the 50-random solutions performed c.:nsistently better on ail the test problems. However, RAN produced the best solution most often. All the priority rules produced soiu?it;t~ within 10% of the sotutions obtained by RAN. Furthermore, the performances of all the rules are close in that the gap between any two heuristic solutions never exceeds 5%. The performance of the heuristic procedure with different priority rules is affected by the resource constrainedness, discount rates and the crashing costs. The number of activities in a problem ii found to be a significant factor affecting the computational effort, For the first data set, DCFW and NMG rules produced solutions within 0.29 seconds, whereas IQCS required 19.40 seconds. This is due to the fact that IOCS needs to repeatedly solve Paymeti Scheduling Problems to produce the required opportunity costs. Similarly, for the larger set,

IOCS required the most computational effort. Probiems with 100 activities consumed 120 seconds on the average with this rule. The computational effort required by RAN was 11 times the computational effor: required by NMG and DCFW for small problems (10 activities) and this factor went up to 26 for larger problems CIO0 activities). Interestingly, ICKS needed more CPU time than RAN. Also nate that changing the values of the parameters had no effect on the computational effort. In general, it was found that, for problems l.vith moderate resource constrainedness. the NMG rule yielded better results. IOCS produced better solutio.rs for high discount rates and tight resource conl.:rainedness with significantly higher computational effon. Alihough a benchmark such as best of tl-random sn!utions CUE he used to judge the quality of the heuristic solutions. a bc~tel measure wouTd br ob. tained by comparing the heuristic solutions to *‘tight” upper bounds. In the next section, we develop a subgradient procedure for obtaining an upper bound on the solution value of Pl.

The Lagrangian Relaxation of Pl that was used and the details of the Subgradient algorithm are given ira Appendix A. The relaxation is obtained by relaxing the resource constraints and putting this set of constraints in the objective function with the corresponding Lagrangian multiplier vector, A. It is also shown that the relaxed problem can be written as a Time/cost Tradeoff Project Scheduling Problem with Discounted Cash flows (P3). It is further shown that, following the work of (Erenguc et al., 1993), the Generalized Benders Decomposition technique can be used to solve the relaxed problem. In this section, a discussion of the steps of the Subgradient algorithm is given and computational experiments are reported. The solutions produced by the heuristic procedures are compared to the upper bounds obtained by the algorithm. Let PI be written in the following simplified form: (S) Maximize NPV( ,s, .v) ~

sub.+

to

tr( .r,?.) G a, where s={X,,: (y,“;

i=

1 .-~‘7 I..

g(

.X.y)

<

i-

b.

1.2, . . . . N:

xEX,yEY,

I==

I,2 ,....

Tj.

y==

, IV;

k = 1,2,. . .A!,}, tz is the set of precedence relations (constraints 3). and p is the set of rcsourcc availability constraints (constraints 3). The set X is defined by constraints (2) and (6) and the s-1 Y is defined by constraints (1) and (5). Then, the Lagrangian dual of (S) is (LS) Minimize a( A).

subject to A 2 0,

where H(A) = max ,,,.(NPV(~,J) + A(b - g(s,y)): h(1.y) G R, x E x. ?’ E Y}. The following subgradient procedure is tised to solve (LS) and in turn (S). Srep 0. Set iteration counter, ITER, to zero. Select A > 0. Determine a lower bound, f?. SIP,CJ I. Solve the following subproblem by using GBD: @A) Max,,,, (NPV(x,y) + A(D - g(x,yj): h(x,~),(a, XEX~ KEY). Let x”.?;’ be the optimal solution. SW/~3. Determine B(A) = NPV(.x _,y * ) + A(b g(x’,y-)j. IF 0(A) is close enough to the lower bound THEN stop, return x * , .v * . ELSE, IF a predetermined maximum number of ITER is reached THEN return the best solution found so far and STOP. ELSE go to step 3. Step 4. Let A=msx(O, A+t(g(.r*,~*)-b)}, where t=

EC@(A) - 0) llgll?

’

0<~~2.UpdateAforallr~,~~=I,...,Vandfor all k. ITER = ITER + 1 and go to step 1. Additional

computations

The Subgradient algorithm was programmed in FORTRAN and executed on an IBM 3090 computer. The GBD procedure coded by Erenguc et al. (1993) was used as a subroutine to solve problem @A). For the details of the subroutine refer to (Erenguc et al., 1993). Initially, the Lagrangian multipliers, A:, , are set to zero. The heuristic solutions found in the

previous section were used as the lower bounds (LB) on 9. A tolerance of 5% was used. That is, the algorithm terminates either when the sahnion obtained, the upper bound (UP), is within 5% of the lower bound or the number of iterations reaches a predetermined maximum. The maximum number of iterations was set at 50. The test problems from both data sets that have the data requirements of the Erenguc et al. (1993) subroutine were solved. If an upper bound repeated: itself at feast once within 10 iterations, hf,. values were perturbed by a random quantity. Computational results are summarized in Table 9. In the table, each row corresponds to results obtained by a different parameter setting. Ten of the 16 problems in the first category have 10 acm4es and are from the larger data set. These problems were solved with a discount rate of 0.01. Column three of the table gives tna average gap between the upper bounds obtained by the Subgradient algorithm and the solutions (that is. sclutions to P3) obtained Ejr the GBD procedure (with it = 0). The last two columns give the average gtip between the best heuristic solution (excluding RAN) and the upper bound obtained by the Subgradient procedure and the average CPU times consumed, respectively. In generaI, the gap between the NPV of the solutions obtained by the GBD procedure without the resource constraints and the NPV of rhe solutions

Table

obtained by the Subgradient algorithm is very sma2. In smail problems, as the crtlshing cost increases, ftwer activities are crashed which results in less resource usage. and therefore in loose resource constraints. As a result, Lagrangian algorithm produces solutions ctose to the GRIf solutions. IIowever, for larger problems (23 activities) we see an increase in the gap between the two as the crashing costs increase. In these instances, although higher crashing costs result in fewer activities to be crashed, thus loose resource constraints, the densities of the networks cf these problems are high enough to cause Subgrzdient algorithm to produce inferior solutions. A reduction in crashing costs and/c. a reduction in resource consumption reduce the optimahty gap considerabfy. For larger problems, for example, the average gap is 17% with crashing cost generated randomly between [I ,201 per unit time. This gap dropped to 9% when the crashing cost is r&uced to a range between 1 to 5 and discount rate to 0.01. The optimality gap increased as the interest rate increased. This increase in the gap is due to the fact that as the interest rate increases. completing an activity earlier or :ater than its optimal completion time becomes more costly. It should be noted that the gaps reported here are relative to upper bounds rather than the optimaa solutions. Accordingly, one would expect the true gaps to be smaller.

9

Compurntional Number activities

results

with

of

the Subgradient Number problems solved

of

Algorithm Avemgc between and P3

gap PI

Average gap between the best heuristic solution and the

0.00016 0.00013 0.00009 o.ctx%

’ 2 3

0.024 0.027 0.061 0.093

Average CPU time keconds)

upper bound < 22

16

22

8

0.0010 0.0-012 I: Results 2: Results 3: Results

obtained obtained obtained

with with with

0.01 discount 0.01 discount 0.02 discount

rate and a crashing rate and a crashing rate and ;L crashing

0.096 0.175 --cost range of [I,S]. cost range of !I. IO!. cost mange of Ii.201.

1.48

30.21

In this paper, we introduced and discussed in detait a heuristic procedure f5i obtaining “‘good” solutions to a practical project scheduting problem, the resource constrained NPV problem with time/cost tradeoffs. The problem is interesting in that it gives valuable insights about the tradeoff between the expediting (crashing) costs and the time value of money. The problem is one of determining the duration and timing of project activities so that the net present value of all the cash fLows that occur during the life of a project is maximized. The probEert is rather difficult in that even certain relaxations ~3’ it are very hard to solve. Consequently. one is motivated to del;elop well-performing heuristic procedures.

The heuristic procedure introduced in this paper was implemented with t;:r~ : different priority rules NMG, IGCS, DCFW. Solutions obtained by the heuristic procedure were improved through applying the Left and right-shift routines of Baroum and Patterson (1993.X Extensive computationat testing indicated that for smaller problems the multi-pass heuristic with the NMG rule produced better results than the other two. For large problems with higher discount rates and tighter resource constraints, IOCS gave better quality solutions. Wowever, IOCS was also the computationally most demanding of the three. In general, the computational requirements, especially for NMG and DCFW are minimal. Therefore the heuristic procedure can be effectively used for large networks to obtain “good” feasible solutions. We also generated 50 random solutions for each test problem and selected the best of them as a benchmark solution against which the heuristic solutions were compared. In addition, we introduced a Lagrangian Relaxation of the problem and a Subgradient algorithm for solvmg it. The subgradient solution provided an upper bound on the optimal NPV and this bound was used to asses the gap between the optimal solution and the solution provided by the heuristic procedure. Problems for which the Subgradient procedure was implemented had an optimality gap ranging from 2.4% to 17.5%. Of course, these are worst case gaps in that they are relative to upper bounds rather than to optimal solutions.

Gur study provides valua’;iie insights to practitioners. The hemistic procedure is easy to implement and can effectively be used for targe networks to obtain good feasible solutions. Furthermore, the procedure provides the opportunity to anafyze the solutions obtained by different priority rules in different environmen tat settings. Nowever. one should be careful about choosing the priority rule for the procedure. The rl,oice depends mainly on the characteristics of the problem. For example, in high inflationary environments where the cost of not being able to schedule the activities at their optimal times is much higher, it makes good sense to include the IOCS rule in the procedure. Our computational results clearly demonstrate the trade-off between the crashing Cost and the gain that ensues from receiving payments earlier. Crashing, which is undertaken at a cost, may in some cases result in a positive net benefit. For instance, an activity with a positive cash flow can be crashed to be able to receive the payment earlier. If the increase in NPV. as a consequence of receiving !he cash at an earher period, is higher than the cost of crashing, the result will be a net positive gain. The NMG Index provides this information. To our knowledge, this paper is the first one to research the problem under consideration. Our hope is that it will provide motivation for further research in this area.

Appendix A. A subgradient proach to PI

optimization

ap-

The Subgradient Optimization technique is commonly used for solving the L,agrangian dual problem of a nonlinear programming problem (Bazaraa and Shetty, 1979). The following definitions of Lagrangian duality will be needed to facilitate our presentation. The Lagrangian dual proi5lem

Let the primal problem (Pi’) be (PP) Maximizef(n).subjectto~(s)


Then, the Lagrangian Dua! problem of problem (PPj is (LD) Minimize tI( II). subject to II 2 0, where Ci(iif =mas(f(s>

+!r(h--i:(.r)):seXf.

The Lagrangian Dual problem is obtained by incorporating the constraint set g(.r) in the objective function with corresponding multipliers, II, ca!led the Lagrangian multipliers. A problem might have several different relaxations and accordingly there might be several different approaches for solving these relaxations. Tie subgradient optimization technique uses the idea that moving a smal? step i : 1 suhgmdiem direction gets us closer to an optimal dual solution. Under certain convexity assumptions, the optimal dual solution obtained is an optimal solutrc:<: to maximization problem (PP). In the procedure, it is presumed that relaxing the complicated constraints would make the resuiting problem easier to solve. Thus. the efficiency of the algorithm solely depends or the computational efliciency of the technique used to solve the subproblem at each iteration. Before moving to the next iteration, the Lagrangian muitipiiers are updated by choosing a suitable step size r. There are two commonly used step sizes (Polyak, 1967). Step size 1.

Let @R“1 be the problem obtained by applying Lagrangian relaxation to the resource constraints of problem (Pli and by including the penalty value of the objective function as a cash outfiow for the last activity of the network.

’ = countllg”lf ’ where count is the current iteration number. Step size 2.

t=

.E(l!?(u)- 6) llgll’

Note that in IPR’i’) there is no harm in letting the index t in the first term go from I to 7’. For fix-d A, h’ the objective function of (PI?) can be written as

-

where E is generated frcjm a uniform distribution on (0,2) and I) is a lower bound on 0(u) which can be obtained by applying a heuristic to the primal problem (PP). In general, empirical results indicate that step size 2 yields superior results compared to the results obtained with other step sizes even though it requires a lower bound which might be difficult to find for some problems. For details of step sizes and the applications of the subgradient optimization technique the reader is referred to (Bazaraa and Shetty, 1979; Fisher, 1973; Polyak, 1967).

Rearranging the terms in the objective function we okstain

In a recent study by Ere?guc et al. (1993). the time/cost tradeoff project scheduling problem with dkcounted cash flows is fomn~!ated as a mixed integer nonlinear program and was solved by using the Generalized BUlitCrS Decomposition (GBD) technique developed / ’ Geoffrion (1972). The probkm involves scheduling activities with cash Rows, where shorter activity durations can be obtained by incurring higher costs. The objective of the probiem is to maximize the net present value of cash flows. The following formulation proposed by the authors allowed the GBD technique to be used for solving P3. Kl, (PE) Max f Q, 4c 1 NF,: yi: e-“Tl j= 1 ! (r,J)EB(j) k= I I

Let min{t,dj) Aft = ---yy--A,, e

7

where A)‘, 2 0. Then

- f[( I - e-“y/(e” sub.ject to q-T,,<

Let the constant ierm in the bracket be NF,:. Note that NF,/: denotes the net future vatue of all cash flows associated with activity i if the kth duration is selected and the activity is completed at time period t. Now. the ei and z on the activity completion times can be included again to reduce the number of variables. Then (PR’) can be written as = Max 5

;

i=l

k=l

subject to

z NF,tX,,e-“‘Y,‘, I=e,

- I)] 1

K:, - Cd:,Y,:,

j=2

,..., M.(i,j)EB(j),

k=J

k=I

Y,tE(O,I}, q>0,

k= I ,..., K,,,(i,j)EA, iEN,

where NF,: is the net future value of activity (i,j) if off;/ duration is chosen for it, B(j) are all activities immediately leading into event j, I is the indirect cost per unit time paid at the end of each time period, Q, is the payment received with the realization of event j, A is the set of activities, K is the due date on the project completion time, N is the set of events, q is the occurrence time of event j, 1* is the nominal interest rate, Y,: = 1 if the kth duration is selected for activity (i,j), yIi = 0 otherwise, and M is the last activity of a network. The above formulation is rewritten in (Erenguc et al., 1993) in a form which is amenable to solution with the GBD technique, The details of the GBB technique and the applicability of GBD to (FE) can be found in (Erenguc et al., 1993).

Note that in (PR’). the cash Raw on the activities. iW,t, depend on their completion times. However. in (FE) the cash flows are indepcndcnt of activi1y complerron times. ft can be shown (Icmcli and Ercnpuc. 1994b) that if we use the following definition fcr the multipliers h)y, ikn PI?.’ can be written as (PE) of Erenguc et al., 19%): A;, +--cl, if A,, = 0 for all I, A:<.+- min,(min(r,d~}( X(.iJ,/e-“‘)

A,,/e-“‘):

otherwise min{r,dfj

> 0) forall a.

lcmcli, 0. and S.S. Erenguc. IY94a. “A Tabu search procedure for resource constrained project scheduling problems with discounted cash flows”, Computers and Opemtions Research, vol. 21. no. 8. pp. X41-853. Icmeli. 0. and S.S. Erenrguc, 199Jb. Solving the constrained project scheduling prohlcms with time/cost tradeoffs. ing lcmeli. ing and

NPV Work-

Paper, Cleveland State University. Cleveland. Ohio. O., S.S. Erenguc and J.C. Zappe, 1993. “Project schedulproblems: .4 survey”. lntems;ional Journal of Operations Production Management. vol. 13. no. 11.

Leachman. R.C., A. Dincetler and S. Kim, 1990. “Resource-constrained scheduling of projects with variable-intensity activities“. HE Trarisactions. vol. 22, no. 1; pp. 31-i?. Lrotch, A.G. and J.A. Muckstadt, 1994. “An approach to production pkming and scheduling in cyclically scheduled man&c-

References Baroum, SM. and J.H. Partrrsori 1993. .4 compararivr evaluation of cash flow weight heuristics for maximizing the net present value of a project, Working Pap,tr. indiana University. Bloomington, Indiana. Bazaraa. M.S. and CM. Shetty, 1979, Nonlinear Programming Theory and Algorithms. Wiley. Ne%, t ;,rk. Bey. R.P., R.H. Doersch and J.H. Patlerson, 1981. “The net present value criterion: Its impact on project scheduling”. Project Management Quarterly. vol. 12. no. 2. pp. 35-45. Dayanand. N. and R. Padman. i993a. The payment scheduling problem in prqject networks. Working Paper 93-31, The Heinz School of Public Policy and Management. Carnegie Meilon University, Pittsburgh, PA 15213, June. Dayanand, N. and R. Padman, 1993b. On modeling payments in project networks. Working Paper 93-36, The Heinz School of Public Policy and Management, Carnegie Mellon L;nivrrsity, Pittsburgh. Doersch, R.H. maximize approach”,

Fisher. M.L.. 19’73. “Op!m~al solution 3f schedulmg pmhiems usrng Lagrange multilthrrs: Fart i “. Oper. Res.. vol. 7, no. ;5, pp I ! id- 1127. Geoffrion. A.M.. 197’. “Gcnendi~ad Benders decomposition”, Joumat of Opttmizatlon Theory and Applicattons. voi. 10, pp. 260-771. Grinold. R.C.. i972. “Payment scheduling probfem”. Naval Research Logistrcs Quarterly. vol. 19, no. 1. pp. 123-136.

PA 15213. June. and J.H. Patterson. !977. “Scheduling a pmject to its net present value: A zero-one programming Management Science. vol. 23, no. 8. pp. 882~S83.

Elmaghraby, SE. and W.S. Herroelen, 1990. “The scheduling cf activities to maximize the nrt present value of projects”, European Journal of Oper&ions Research. vol. 49. pp. 35-49 Erenguc, S.S., S. Tufekci and C.J. Zappe, 1993. “Solving time/cost tradeoff problems with discounted cash tlows using generalized Benders decomposition”, Naval Research Logistics, vol. 40, pp. 20-25.

turing systems”, Intemationti Journal Product R~\e~ch. vol. 32, no. 4, pp. 85 1-87 I. Padmar. R. and D.E. Smith-Daniels, 1993. “Early tardy cosi trade-offs in resource constrained prqjects with cash flows: An optimization guided hemistic approach”, European Joumat of Operational Research, vol. 64, pp. 29.5-311. Patterson. J.H.. F.B. Taibot, R. Slowinski and J. Weglmz, 1990. “Computational experience with a backtracking algorithm for solving a general class of precedence and resource constrained scheduling problems”, European Journal of Operational Research. vol. Polyak. B.T.. problems”, Russell, R.A..

49. pp. 68-79. 1967. “A general method for solving extremal Sov. Math. Dokl., vol. 8, pp. 592-597. 1986. “A comparison of heuristics for scheduling

projects with cash flows and resource restrictions“, Manngement Science. vol. 32. no. 10. pp. 1291-1300. Smith-Da&%, DE. and V.L. Smith-Daniels, 1987. “Ma?rimizing tbc net present value of a project subject to material and capital constraints”, Journal of Operations Management, vol. 7. no. 1. pp. 33-45. Talbot. B.F., 1982. “Resource constrained project scheduling with time-resource tradeoffs: The nonpreemtive case”, Management Science. vol. 28, no. 10, pp. 1197-1210. Yang, K.K.. F.B. Talbot and J.H. Patterson, 1992. “Scheduling a project to maximize its net present value: An integer programming approach”, European Journal of Operational Research, vol. 64, pp. 188-195.