Constrained multi-project plannings problems: A Lagrangean decomposition approach

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH

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European Journal of Operational Research 78 (1994) 267-275

Constrained multi-project planning problems: A Lagrangean decomposition approach Carlo Vercellis Dipartimento di Economia e Produzione, Politecnico di Milano, I20133 Milan, Italy

Abstract

This paper describes a Lagrangean decomposition technique for solving multi-project planning problems with resource constraints and alternative modes of performing each activity in the projects. The decomposition can be useful in several ways: from one side, it provides bounds on the optimum, so that the quality of approximate solutions can be evaluated. Furthermore, in the context of branch-and-bound algorithms, it can be used for more effective fathoming of the tree nodes. Finally, in the modelling perspective, the Lagrangean optimal multipliers can provide insights to project managers as prices for assigning the resources to different projects.

Keywords: Multi-project planning; Lagrangean decomposition; Lagrangean relaxation; Project management

I. Introduction

Among the different stages in which the decision processes associated to project management can be hierarchically decomposed, multi-project planning appears to be one of the most critical steps to be undertaken, at least for companies operating within the construction building and engineering industries. Indeed, at this tactical planning level, managers are faced with crucial decisions such as allocating resources among different projects, establishing due dates and other milestones for bidding proposals, determining the optimal trade-off between the absorption of resources, the time duration and the costs associated to alternative 'modes' of performing each activity. It should be noticed that such decisions have a great impact over the whole productivity performance of a company, and that they can even influence its competitive strength, by determining the cash-flow profiles and the delivery dates specified for bidding proposals. The time horizon over which this planning analysis is conducted is generally medium to long, and is only based on an aggregate level of knowledge of the different activities composing the set of projects. Despite its practical relevance, the multi-project tactical planning problem has not received adequate attention in the literature. An early mathematical programming formulation of the multi-project problem was presented in (Pritsker et al., 1969). The problem was then solved by means of a general purpose 0-1 implicit enumeration code. The papers by Kurtulus and Narula (1985) and Kurtulus and Davis (1982) considered the categorisation of heuristics based upon priority rules. Both papers were aimed at scheduling problems at the operational level. A 'macro' formulation, oriented to a tactical analysis and 0377-2217/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0 3 7 7 - 2 2 1 7 ( 9 4 ) 0 0 0 8 7 - S

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based upon continuity assumptions, was given in Tavares (1987). The number of authors who devoted their analyses to the case in which activities can be performed in several alternative modes is also very limited, two exceptions being represented by Talbot (1982) and Patterson et al. (1989), who however considered the single project environment. A third contribution in this direction was due to Speranza and Vercellis (1993), who proposed a multi-objective approach to the multi-project planning problem, with different goals prevailing at either the tactical or the operational levels. Finally, it should be mentioned that a few authors have considered the objective of maximising the net present value of the project, instead of the more traditional makespan minimisation. More or less explicitly, these authors, such as Doersch and Patterson (1977), Russell (1970), Russell (1986), Speranza and Vercellis (1993), seem to have placed their analyses at the tactical level. In this paper, we propose a model-based decomposition approach to the tactical planning analysis of the multi-project problem. The model developed is aimed at the maximisation of the total net present value of the set of projects, in consideration of investments and operating costs, revenues and penalties. To allow for the analysis of the trade-off between resources usage, time and cost, each activity has an associated set of alternative modes in which it can be performed, corresponding to different combinations of the resources. A Lagrangean decomposition technique for solving th~ resulting optimisation problem is then introduced. Lagrangean techniques for deriving hierarchical problem decomposition have proven useful in many contexts (see, for instance, other recent applications of this methodology to other problems in Fumero and Vercellis, 1993a,b, 1994). In fact, there have been previous attempts to use Lagrangean relaxation also in connection to project scheduling problems (Christofides, 1987), for the case of a single project with a single mode of performance for each activity. In this latter case, the reported results did not appear particularly promising for the test instances considered by the authors. However, the Lagrangean decomposition scheme we propose is quite different from the one adopted in the quoted paper, since we do relax a set of constraints which is disjoint from the one considered by Christofides et al. (1987). Therefore the resulting sub problems in our case are also of a different nature, since they lead to a set of separable multiple choice knapsack problems. The proposed decomposition can be useful in several ways: from one side, it provides bounds on the optimum value, so that the quality of approximate solutions produced by means of heuristic algorithms can be evaluated a posteriori also for large instances of the multi-project problem, for which the true value of the optimum is not known. Furthermore, in the context of branch and bound algorithms, the Lagrangean relaxation can be used for obtaining better upper bounds on the best value which can be achieved from the descendants of a given node, leading therefore to a more effective fathoming of the tree nodes. In particular, we have incorporated the Lagrangean bounds within an improved version of the branch and bound algorithm first proposed in Speranza and Vercellis (1993), conducting a number of tests indicating an improved performance of the algorithm. Finally, in the modelling perspective which may help decision makers, the Lagrangean optimal multipliers can provide insights to project managers as prices for assigning the resources to different projects. Finally, we notice that the proposed models and algorithms are well suited to be framed within a decision support system aimed at assisting project managers in deeper understanding of the interrelations among the allocation of resources to the projects, the timing of the milestones and the cash flows.

2. The multi-project management planning problem The multi-project management planning problem (MPMP) is defined as follows. A project s consists of a given set V~ of activities, on which a partial order relation is assigned. We say that i precedes k whenever activity i must be terminated before starting activity k. The set V~ includes two dummy

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269

activities, a s and z s, representing the start and the end of the project, respectively. The precedence among the activities can be represented by an oriented acyclic graph G s = (V~, Ps)- An arc (i, k) in Ps exists whenever i precedes k. In order to be performed, each activity requires the usage of a given amount of each of a set of resources. Denote by R the indices of the resources, and let W~ be the availability of resource r ~ R over each time interval. The usage cost of a unit of resource r ~ R during a unit of time will be denoted by Cru, while to each unused unit of a resource r ~ R will be inputted a cost Crv. It is assumed that each activity i can be carried out in several modes, each corresponding to a different absorption of resources and to different processing times. Let M i be the set of modes of accomplishment for activity i. Hence, it follows that different modes correspond also to different costs for each activity. Specifically, if activity i is carried out in mode i an amount Wijrt of resource r is consumed at time t where the time index is computed relative to the starting time of activity i. The duration of activity i carried out according to mode j will be denoted by dij. A single mode is defined for each of the dummy activities, which does not require any resource, and whose duration is null, i.e. das 1 = d z s I = 0 . A feasibility interval [ei, li] is associated to each activity i, where e i and I i represent the earliest and latest starting time of activity i, respectively. Each project is part of a program, composed by a set of projects, whose indices are denoted by S. The projects are interrelated and influence each other in at least two ways: at a general level, they are in competition for the use of the limited resources; moreover, they can be related by a further partial order relation. We say that project s 1 precedes project s 2 whenever z~l precedes zs2. The program can be represented by an oriented acyclic graph G = (V, P). The set of vertices V = U s ~ s ~ u a u z includes two dummy vertices a and z which represent the start and the end of the whole program. Again, these dummy activities have a single mode, and time durations das ~ = dzs ~ = 0. The set of arcs P = U s~sPs UPs is given by the precedence arcs Ps for each project s ~ S, and by additional arcs Ps which are introduced for two reasons. First, there are arcs expressing precedence relationships between projects. Such an arc (zsl, a s 2 ) ~ P s exists whenever project S 1 precedes project s 2. Then, there are arcs connecting the beginning and the end of some projects to the dummy nodes a and z, respectively: an arc (a, a s) ~ Ps exists whenever project s is not preceded by any other project and analogously an arc (z~, z) ~ Ps exists whenever project S does not precede any other project. Analogously to the elementary activities, a feasibility interval [e~, I~] is associated to each project s, with e~ and I~ representing the earliest and latest starting time of the project respectively. The amount of the investment for project s will be denoted by c si and is independent from the modes of accomplishment of the activities of the project. It is assumed that such amount has to be paid at the beginning of the project. A revenue fs will give rise to a positive cash flow at the completion time of the project. Moreover, a due date h s can be associated to each project s: if a project s is not completed within h~, a penalty Ps will be paid at time h~. The solution of the project planning and scheduling problem corresponds to determining the mode of accomplishment and the starting time for each activity. In order to formulate the MPMP as a mathematical programming problem, it is convenient to introduce the following binary decision variables, for i ~ V, i E M i , t = e i ..... li:

Xijt

=

0 1

if activity i starts at time t in mode j, otherwise.

Although the previous set of binary variables contains all the information required to the decision maker, it is convenient to introduce a set of additional dependent variables, which will be useful in the subsequent developments of Section 3. Let T~ be a variable representing the starting time of activity i. Let also D i be the duration of activity i. With Wrs we denote a variable representing the maximum

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270

amount of resource r ~ R which will be allocated to project s. For each project s, the quantity Wrs is taken from the total available amount Wr of resource r ~ R. As already noticed, the Net Present Value (NPV) appears to be the most natural objective function to maximise in the context of multi-project planning analysis, over a medium to long term time horizon. Hence, letting a denote the discount rate, we can express the cash flows at time t determined by the revenues C~, the investments C/, the penalties Cte and the cost of the resources CtU, as follows:

C R = ~_, f~Xzs,l,,

(1)

sGS

c/ = E 'c,x. At

(2)

sES

ce=

E

Ps

sES:hs=t

E

Xz~,lo

(3)

O=max(ei,t-diy+ l )

C? = E E E E Wijr(t-O+I)(CUr--CN)Xijo+CNWr • r ~ R i~ V j ~ M i O=max(ei,t +dij+ 1)

(4)

The NPV can be defined in terms of the previous cash flows as

c?- c / - c,

NPV= E ,=0

c?

(5)

(1 + , ~ ) t

By substituting the expressions (1), (2), (3) and (4) in (5), one may see that the NPV can be rewritten as a linear function of the variables xijt, in which the coefficients cij t are conveniently defined in terms of the original cost coefficients of the problem. Therefore, the multi-project planning problem (MPMP) is formulated as follows: MPPM lz

Z* =

max

NPV = E

E E CijtXijt t=0 i ~ V j E M i

x,T,D,W li

s.t.

Ti = E

E txijt,

i ~ V,

(6)

j ~ M i t=e i

Di = E

li dij E xijt,

j~M i

i ~ V,

(7)

t=e i

Tk-Ti>Di,

(i,k)~P~,

(8)

TR-Ts>D,,

(s,p) ePs,

(9)

min(li,t -dij+ 1) E E E i~ Vs j ~ g i "r= max(ei,t - d i j + E Wrs~-~Wr, s~S

Wijr(t_.r+l)Xij,r~_~Wrs,

reR,

t~T

(10)

1)

reR,

(11)

li

E Exij, =1, i e v ,

(12)

j ~ M i t=e i

XijtE{0,1},

i~V,

j~Mi,

t = e i . . . . . I i.

(13)

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271

Methods for the exact solution of the integer linear programming problem MPMP can be devised. For instance, in (Speranza and Vercellis, 1993) a branch-and-bound procedure for solving a variant of MPMP has been proposed and analysed. However, due to the inherent complexity of problem MPMP, such methods inevitably lead to prohibitively high computing times even for instances of the problem of moderate size. Thus, for solving practical problems, one has to rely upon heuristic methods, aimed at achieving a 'good' approximate solution of MPMP. In Speranza and Vercellis (1993) it was proposed to solve a variant of MPMP by means of a hierarchical decomposition, which led to three classes of sub problems, easier to solve than the original one, and corresponding also to significant layers of the managerial decision making process. The next section presents a different decomposition scheme, based upon Lagrangean relaxation techniques, which may prove helpful in obtaining exact as well as approximate solutions to the MPMP problem.

3. A Lagrangean decomposition scheme If one considers the formulation of the MPMP problem as given in Section 2, it appears that there are only two groups of constraints which interrelate among themselves the projects s ~ S. The first group is represented by the set of constraints (9), expressing the precedence relations among the set of projects, while the second group is given by the set of constraints (11), which partition the available resources among the different projects. Hence, if we relax these two groups of constraints by introducing two sets of multipliers, denoted respectively as A and ~r, we obtain the following Lagrangean relaxation of the MPMP, denoted as LMPMP: LMPMP lz

Z*(A,~r)=

max

L(A,~r)=~

x,T,D,W

E CijtXijt-~-

~

t=O i~V j~M i

E

Apq(Tq-Tp-Dp)

(P,q)~Ps

lz

- E E

rs,(Wr- WrA

t=O r~R Ii

s.t.

T/=

E

Etxijt,

(14)

i~V,

jEM i t=e i li

Di = E

dij E xijt,

j~M i

(15)

i ~ V,

t=e i

(i,k) sP ,

(16)

min(l/,t - dij + 1) E E E Wijr(t-'r+l)Xij'r~-Wrs, i ' ~ V s j ~ M i r = m a x ( e i , t - d l j + l)

r~R,

t~T

(17)

li

E

EXijt =1 ,

(18)

i~V,

j~M i t=e i

xijt~{O, 1},

i~V,

J~Mi,

t = e i . . . . . Ii.

(19)

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272

Notice also that problem LMPMP is decomposable into s separate subproblems, one for each project s ~ S. We have therefore: Iz

Z*(A,o')=

EZ*(A,o')-

E

s~S

Aa,(Ta+Da)+

(a,p)Ep s

E

A,zTz+ E

(p,z)EP s

Ears,W~,

t = 0 r~R

where LMPPM(s) lz

Z*(A, ~r) =

max

L s ( ) [ , Or)= E E E CijtXijt-E Aasp(Tas+ Oas) x,T ,D ,W t=O i~Vs J~M i (as,P)~Ps ls

+

E l~p,zsTz s - E E OrrstWrst (p,Zs)~p s t = 0 r~R

li

s.t.

Ti=

2 2txijt, j ~ M i t=e i li

O i= }~ d i j ~ , x i j t , jEM i

(20)

i~V~,

i~V~,

(21)

s,

(22)

t=e i

Tk-Ti>-Oi,

(i,k)~P min(li ,t - diy + 1)

E E E Wijr(t-'r+ 1)Xij'r ~-~ Wrs, iEl/s j E M i ,r=max(ei,t-du+ l)

r~R,

t~T

(23)

li

E

E x i j t =1,

ieV~,

(24)

j E M i t=e i

XijtE{O, 1},

i~V~,

j~Mi,

t = e i , . . . , l i.

(25)

An upper bound D* to the optimum value Z* can be obtained by minimising the Lagrangean function Z*(A, or), as in the following problem DMPMP: DMPMP min Z*(A, ~r) = • Z*(A, ~r). A,o,> 0

s~S

Each of the sub problems LMPMP(s), s ~ S, is easier to solve than the original one LMPMP, since it includes far less variables and constraints. Furthermore, for those cases in which the size of some of the sub problems LMPMP(s) still remains beyond reach for the exact methods, in the next section it will be presented an approximate solution, again based on the application of Lagrangean relaxation.

4. Algorithms and computational results

In this section we will discuss the methods proposed for the solution of the Lagrangean sub problems LMPMP(s), s ~ S and the dual problem DMPMP, as well as the original problem MPMP. Consider first

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the Lagrangean sub problems LMPMP(s), s ~ S. In the experimental tests conducted we have used a variant of the branch-and-bound algorithm proposed in Speranza and Vercellis (1993) for solving each of the LMPMP(s), s ~ S, at least for those projects s ~ S of moderate size: this means projects with up to 30 activities, with 3 modes of execution and 3 resources for each activity. As an alternative to exact methods, when dealing with larger instances of LMPMP(s), we have relied upon the following Lagrangean relaxation of the problem. Introduce a set of multipliers /z associated to the precedence constraints (22), and define the following relaxation MK(s) of LMPMP(s): MK(s) lz

K*(A,o-,/z)=

max

~

]~

~

x,T,D,W t=0 i~V s j~M i

cijtxij t -

~

Aaw(Tos+Das)+

(as,P)~P s

~]

Ap,zT~

(p,Zs)~p ~

lz -- E

E

OrrstWrst -

t=o r~R

E ~ i k ( T k -- Ti - D i ) (i,k)~P~

li

-s.t=

T/= ~_ ~ txi)t,

i ~ V~,

(26)

.i~34 i t = e i li

D i=

~_, d i j Y ' . x i j t, jt~M i

i~Vs,

(27)

t=e i

min(li ,t - dij + 1) E E Z Wijr(t_.r+l)Xijr~_~Wrs, i ~ v~ j ~ M i .c = max(e i ,t - d q + 1)

r~R,

t~T

(28)

li

~.,xijt=l,

i~V~,

(29)

j~M i t=e i

XijtE{O, 1},

i ~ V s,

j~Mi,

t = e i .... ,1 i.

(30)

It is not hard to see that, in the formulation of MK(s), the only significant decisions within the optimisation problem reduce to a multi-constrained multiple choice {0, 1} knapsack problem. Indeed, expressions (26) and (27) may be substituted into the objective function of MK(s), obtaining the multiple choice knapsack problem in the variables xij k. The resulting knapsack problem, in turn, may be solved by either exact methods based on dynamic programming, in the case of instances of moderate size, or by approximation heuristics for a higher number of variables. In particular, we solved MK(s) by using a generalisation of the approximation algorithm proposed and analysed in Rinnooy Kan et al. (1993). From a solution to MK(s), one may derive an upper bound to the optimum value of LMPMP(s), by solving the following minimisation problem, for each s ~ S. DMK(s)

min K*(A, or,/z).

A,o-,p~_>0

The dual problem DMPMP, as well as problem DMK(s), s ~ S, were solved by means of sub-gradient optimisation (see, for instance, Shapiro 1979). It has been implemented using the multipliers updating formula suggested in Held et al. (1974): U n + l ~ - U n "IV "l~ng n,

(31)

274

C. Vercellis / European Journal of Operational Research 78 (1994) 267-275

where u n is the generic multipliers vector at step n, On is the updating step length and g n the sub-gradient of the dual objective function at step n. The formula needs to be modified since the multipliers are constrained and need therefore to be projected onto the feasible region: we have therefore chosen at each step the maximum between 0 and the value given by the updating formula (31). The step length On has been computed according to the expression (Held et al., 1974) On = tn[W -- w ( u n ) ] / / 1 1

en II z,

where ~ is a known upper bound to the dual problem, iteratively updated, and w(u ~) is the optimal value of the dual objective function at step n. The parameter t n has been set equal to 2 at the first step and then halved every time there was no improvement over a prefixed number of iterations. Although convergence is not guaranteed, since no sufficient condition is met; nevertheless this choice performed quite well in all computational experiences. The Lagrangean relaxation procedure described was framed within a branch-and-bound algorithm for solving the original problem MPMP, according to the general scheme described in Speranza and Vercellis (1993). In order to validate the whole methodology, the algorithms proposed for solving MPMP, DMPMP, LMPMP(s), DMK(s) and DMK(s), for s ~ S, have been implemented and tested on a set of problem instances drawn from a real world application to a company operating in the construction building industry. The computer programs for the tests have been coded using the C + + language, and the tests were conducted on a PC equipped with a fast 80486 CPU. The problem instances for the tests had a number of projects ranging from 5 to 10, while the number of activities per project ranged from 10 to 20, over a planning horizon of 24 months. Two or three modes per activity were generally available. Up to 6 renewable resources were consumed by the different activities. The computational experience indicated that the running times were all in the order of minutes. Moreover, the branch-and-bound incorporating the Lagrangean bounds showed a reduction in both the number of nodes explicitly considered and the computing time. In particular, the number of nodes visited by the new algorithm was generally less than 20% than those visited by the original version described in Speranza and Vercellis (1993).

5. Conclusions

It has been argued that the multi-project planning is one of the most critical steps to be undertaken in the context of project management. The tactical planning level deals with crucial decisions such as allocating resources among different projects, optimal trade-off between the absorption of resources, the time duration and the costs associated to alternative 'modes' of performing each activity. These features make inadequate the support of most traditional scheduling optimisation models for project management, which are generally oriented toward short time horizon. The methodology proposed in this paper is based upon Lagrangean decomposition, which leads to an explicit co-ordination scheme among the different projects. We notice that the methodology proposed may be useful in the design of a decision support system for multi project planning. Hence, the whole approach may assist the decision makers in allocating scarce resources among projects; evaluating different cash flows in light of the net present value maximisation and identifying the most appropriate modes of accomplishment of the various activities. In particular, the Lagrangean decomposition can be used to evaluate the allocation of the resources among the various projects. Indeed, the Lagrangean multipliers tr can be interpreted as marginal prices for the allocation of the resources to each project in each time period. A further way of applying the proposed Lagrangean relaxation is for the posterior evaluation of the quality of approximate algorithms for MPMP.

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