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Proceedings of the Fifth International Conference on Machine Learning and Cybernetics, Dalian, 13-16 August 2006

1-4244-0060-0/06/$20.00 ©2006 IEEE 176

ANT COLONY ALGORITHM FOR SCHEDULING RESOURCE CONSTRAINED PROJECTS WITH DISCOUNTED CASH FLOWS

YONG-YI SHOU

School of Management, Zhejiang University, Hangzhou 310058, China E-MAIL: [email protected]

Abstract: For long-term projects with considerable cash flows, the

project managers aim to maximize the net present values instead of to minimize project durations. An ant colony algorithm is proposed to coordinate the allocation of scarce resources to improve the net present values. The algorithm adaptively adjusts resource allocation according to the pheromone generated by artificial ants employed to search for suitable schedules. The crossover operation, inverse mutation and elitist strategy are applied to accelerate the searching, and a backward scheduling technique is adopted to postpone cash outflows. An experimental testing indicates that the proposed algorithm helps to improve the net present values of resource constrained projects.

Keywords: Project scheduling; ant colony optimization; discounted

cash flow; resource constraint

1. Introduction

The classic resource-constrained project scheduling problem (RCPSP) is a typical scheduling problem which allocates scarce resource over time to perform a set of activities in order to minimize the project duration. The problem is well known to be strongly NP-hard. Many optimization methods have been proposed for the problem in the literature, such as implicit enumeration methods, zero-one programming, dynamic programming, and etc. Nevertheless, due to the inherent complexity of the RCPSP, it is no surprise that the majority of current algorithms are heuristic in nature. Comprehensive reviews of the state-of-the-art of the RCPSP could be found in the literature [1] and [2].

Presently, most researches on resource-constrained projects focus on analysis of minimizing project duration. However, for many large scale engineering projects, especially those with long time horizons, the management of cash flows is a significant problem, since the variance of net present values (NPVs) could be tremendous with different resource allocation mechanisms. Therefore, the

resource-constrained project scheduling problem with discounted cash flows (RCPSP-DCF) has attracted increasing attentions recently. The RCPSP-DCF has a so-called non-regular objective function, since the net present value of a project may not be optimized even if the project duration is minimized [3]. Hence, many effective methods for the RCPSP may not be suitable for the RCPSP-DCF. Due to computational complexity of the RCPSP-DCF, few exact methods have been suggested in the literature [3], and various heuristics are proposed [4~8]. In this paper, an ant colony algorithm is suggested for the RCPSP-DCF.

2. Problem Description

A classic RCPSP can be represented by an oriented and acyclic activity-on-node (AON) network: (1) A finite set of J activities or jobs, and each activity is indexed so that the preceding activity must have a smaller index and the J-th activity must be the only end activity; the processing time of the j-th activity is pj, its start time STj, its completion time CTj, CTj = STj + pj, and its associated net cash flow NCFj at the completion time. (2) A finite set of precedence constraints: S = {Sj: j = 1,…,J} where Sj is the set of activities immediately succeeding activity j. (3) A finite set of renewable resources with limited capacities, denoted by R. The capacity of resource r is Kr, and kjr units of resource r is required by activity j for each period activity j is in process.

For an RCPSP-DCF, an optimal project schedule is one that conforms the above-mentioned constraints and maximizes the net present value of the project. Let It be the set of in-process activities at time t for a given schedule and ω be the discount rate, the problem can be formulated as:

∑ −= jCT

jjeNCFNPV ωmax (1)

s.t. jhj ShSTCT ∈∀≤ , (2)

∑∈

∀≤tIj

rjr trKk ,, (3)


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3. Basic Ant Colony Algorithm

The so-called schedule generation scheme (SGS) is a basic procedure for many project scheduling heuristics. An SGS determines how a schedule is constructed gradually, building a feasible full schedule for all activities by augmenting a partial schedule covering only a subset of activities in a stage-wise manner. Two schemes are usually distinguished. In the serial SGS, in each stage, an eligible activity is selected and scheduled as soon as possible without violating the constraints. In the parallel SGS, a schedule proceeds by considering the time periods in chronological order and in each period all eligible activities are attempted to start at that time if resource availability allows. The serial SGS is more convenient for ant colony algorithms and hence is adopted in this paper.

The serial SGS divides the set of activities into three disjoint subsets: scheduled, eligible, and ineligible. An activity that is already in the partial schedule is considered as scheduled. Otherwise, an activity is called eligible if all its predecessors are scheduled, and ineligible otherwise. The subsets of eligible and ineligible activities form the subset of unscheduled activities. The scheme proceeds in N = J stages, indexed by n. On the n-th stage, the subset of scheduled activities is denoted as Sn and the subset of eligible activities as decision set Dn. On each stage, if more than one activity is eligible, one activity j from Dn is selected and scheduled to begin at its earliest feasible start time. Then activity j is moved from Dn to Sn which may render some ineligible activities eligible if now all their predecessors are scheduled. The scheme terminates on stage N when all activities are scheduled.

The idea of ant colony optimization is utilized in the process of a serial schedule generation scheme. In each generation of project schedules, several ants are employed to search a feasible schedule according to some simple heuristics and a dynamic pheromone matrix. For a certain ant, it selects an activity from the eligible set in each stage of schedule generation scheme. The factors that influence the ant’s choice include (1) the problem dependent heuristic information ηij, acted as visibility of the next candidate activity for ants, (2) the pheromone information τij, as the pheromone intensity. Here, the heuristic information ηij is independent to the searching history of the ant colony, while the pheromone information τij is history dependent.

The latest finish time (LFT) heuristic is used in the paper since it is reported to be an outstanding heuristic for evolutionary algorithms. The heuristic information of activity j is calculated by

εη +−=∈

jhDh

ij LFTLFTmax (4)

where ε is added to assure that the activity with lowest priority still has a small possibility to be selected by the ants.

In the i-th stage of a serial SGS, the possibility of activity j being selected from the eligible set is proportional to the value of βα ητ ijij ⋅ , where α is the heuristic desirability

of selecting activity j on the i-th stage of scheduling, and β is a parameter which determines the influence of the heuristic information. Hence, the possibility of selecting activity j is:

∑∈

⋅

⋅=

Dhihih

ijijijp βα

βα

ητ

ητ (5)

An alternative way to determine the possibility of selecting activities is as follows:

∑ ∑

∑

∈ =

=

⋅

⋅=

Dhih

i

kkh

ij

i

kkj

ijpβα

βα

ητ

ητ

)(

)(

1

1 (6)

It is argued that this summation evaluation method helps avoid schedule certain important activities too late [9], otherwise the project would have a poor performance. The following experimental tests confirmed this argument, and the summation evaluation method is adopted in this paper.

After one generation of schedules are constructed by the artificial ants, the pheromone matrix is updated using the following classic formula:

*1 1)1(

NPVgij

gij ρτρτ +−= − (7)

where NPV* is the NPV of the best schedule ever found, so */1 NPVg

ij =∆τ is the pheromone augmented by the ants in the g-th generation. In case the pheromone is too small so that some activities in the eligible set would be neglected, a lower limit ε is set to assure every activity in the eligible set has a possibility to be selected.

The schedule generation scheme terminates if at least one of the following criteria is satisfied: (1) a certain number of generations, (2) the average quality of solutions found by the ants fails to improve in several generations.

4. Improvement of the Algorithm

4.1. Crossover

The 2-opt strategy is widely adopted in ant colony algorithms for the traveling salesman problem (TSP), but is cumbersome for the RCPSP-DCF since it is time consuming and a schedule applied 2-opt strategy normally


178

becomes infeasible. Hence, other local optimization strategies are necessary. The crossover operation adapted from genetic algorithms appears appropriate for this purpose.

In a certain generation of scheduling, a set of schedules are constructed by several ants. Two different schedules are selected from this set randomly. For a given schedule, it can be represented by an order list of activities. Hence, the two selected schedules could be represented by two lists.

A single point crossover operator could be used to generate two children from these two parent lists. First, two base points are determined according to the parent lists. Due to the precedence relations, the sequence of anterior activities is normally fixed. If a crossover point is selected in this sequence, the child list is more likely to be identical to its parent. So does the case of the order list of completing activities. Therefore, a lower base point and an upper base point are necessary to assure the sequences before and after the base points are not identical. Then, a single crossover point within the range from the lower base point to the upper base point is selected randomly. For the first child, it keeps the partial list in the first parent list before the crossover point, and constructs its remaining partial list according to the order in the second parent list. The same procedure is applied to generate the second child list. And the child list with a better net present value is selected to upgrade the program. The crossover operation is illustrated in Figure 1. Since the order in each parent list is precedence feasible, the order in the generated child list is also feasible and the corresponding schedule is also feasible. This is an appealing feature, especially compared with the 2-opt strategy which generates infeasible schedules. 1 2 3 4 5 6 7 8 9

1 2 4 5 3 7 6 8 9

1 2 3 4 5 7 6 8 9

1 2 4 3 5 6 7 8 9 Figure 1. An illustration of crossover of two parent lists

4.2. Global optimization strategy

On a certain stage of schedule generation, the best solution found so far could have great influence on succeeding ants’ choice. This may lead to a bad situation of local optimization. It is suggested that for a small possibility pu>0 the best solution in the current generation replaces the best solution so far to avoid local optimization [10]. However, it is noticed that if solutions in later generations are all local optimal solutions this strategy fails to jump out local optimization since the ant colony algorithm is effective in positive feedback learning.

The inverse mutation could be applied on the best solution so far to overcome this deficiency. For a small possibility pu>0, the best solution for the current generation is used for inverse mutation.

For a given schedule, its corresponding activity list is used for the inverse mutation. Two random points are selected. The sequence of the activity list between these two points is inversed. The new activity list is normally infeasible, so a conversion procedure is required. A simple serial SGS is adopted. For each stage of the SGS, an activity is selected from the eligible set according to the order in the newly generated list. This new schedule is then used to replace the best schedule so far and to update the pheromone matrix.

4.3. Elitist strategy

To speed up the convergence, an elitist strategy is adopted. In the scheduling process, if the best schedule of one generation is strictly superior to the best schedule so far, the pheromone augmented by the elitist ant is doubled. Similarly, if the schedule generated by a crossover or inverse mutation is superior to the best schedule so far, the elitist strategy is also applied so as to accelerate the searching speed.

4.4. Backward scheduling

The serial generation scheme promises to generate a left-justified schedule which tend to have a short duration but it does not promise to generate schedules with a larger NPV. Li and Willis [11] proved that backward scheduling helps to improve the NPV of a left-justified schedule. Hence, a backward scheduling procedure, as described in [11], is applied to the best schedule found by the ants to postpone the cash outflows so as to improve the NPV further.

5. Experimental Testing and Analysis

The well-known Patterson test problems [12] are adopted for algorithmic testing and analysis. The set includes 110 project scheduling problems, each problem with 7 to 50 activities and 1 to 3 renewable resources.

Since the original Patterson test problems do not concern with cash flows, the problems are modified to suit the requirements. Each resource demanded by projects has a unit cost per time, and the cash outflow occurs at its completion time. A certain amount of cash inflow occurs at the completion of the project. Several projects with negative or small net present values are discarded to assure


179

the accuracy of comparison. A total of 103 test problems are employed for testing and analysis.

For the ant colony algorithm suggested in this paper, the program terminates after 1000 generations, and 5 artificial ants are employed in each generation. All programs are run on a Pentium 4 computer with Intel 1.80GHz CPU and 256M RAM.

One project instance is selected to demonstrate the searching process of artificial ants, as shown in Figure 2. In the first generation, the five ants found a feasible schedule for the project with a net present value of 11.4. Then the crossover operation was applied and the schedule was improved to obtain a new NPV of 11.6. The ants kept searching for better schedules using updated pheromone. In the third generation, the ants improved the NPV to 12.2 and in the fourth generation to 12.7. Later, in the 12th generation, after an inverse mutation the net present value was improved to 13.5. In the succeeding generations, the ants do not improve the schedule further. The instance shows that the artificial ants are capable to search for better solutions in a small number of generations.

10

11

12

13

14

15

1 6 11 16 21 26 31

NPV

1

3

4

12

Figure 2. The searching history of artificial ants

A variety of methods are employed to solve the above-mentioned test problems. For example, the late start and short processing time (LSSPT) heuristic [4], the cumulative cash flow weights (CCFW) heuristic [5], the discounted cash flow of future activities at late-finish times (ΣDCFLF) heuristic [6], the weighted resource utilization ration and precedence (WRUP) priority rule [7], the composite regret-based random sampling (CRBRS) method [13], and the minimum slack (MINSLK) and the minimum latest finish time (MINLFT) priority rules [14].

The experimental results are listed in Table 1. Since the optimal solutions to these RCPSP-DCFs are unknown, the best solutions found by the above-mentioned heuristics are used as the basis for comparison. Obviously, the proposed ant colony algorithm presents significantly better solutions with an average deviation of 0.20% to the best solutions while the second best CRBRS method has an

average deviation of 5.59%. Moreover, the proposed ant colony algorithm (ACA) gives 89 best solutions out of total 103 problems.

6. Conclusions

An ant colony algorithm is suggested in this paper to improve the net present value of resource-constrained projects. The crossover operation and inverse mutation are adopted in the ant colony algorithm to enhance its capability of searching for better solutions. The backward scheduling technique is utilized to postpone the cash outflows. The experimental testing indicates that the proposed hybrid algorithm helps to improve the net present value of projects.

Acknowledgements

This paper is supported by the National Natural Science Foundation of China (Project 70401017), and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry of China.

References

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[4] Ghaleb Y. Abbasi, and Y.A. Arabiat, “A heuristic to maximize the net present value for resource-constrained project scheduling problems”, Project Management Journal, Vol 32, No. 2, pp. 17-24, 2001.

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