Forward-backward Improvement for Genetic Algorithm Based Optimization of Resource Constrained Scheduling Problem

Size: px

Start display at page:

Download "Forward-backward Improvement for Genetic Algorithm Based Optimization of Resource Constrained Scheduling Problem"

Cecil Jack Edwards
5 years ago
Views:

1 2017 2nd International Conference on Advances in Management Engineering and Information Technology (AMEIT 2017) ISBN: Forward-backward Improvement for Genetic Algorithm Based Optimization of Resource Constrained Scheduling Problem Jian-cheng WANG 1,* and Wen-ran LIU 2,3 1 Department of Equipment Command, Equipment Academy, Beiing , China 2 Department of Graduate Management, Equipment Academy, Beiing , China Troops, China *Corresponding author Keywords: Resource constrained, Proect duration optimization, Equipment support, Forwardbackward improvement, Genetic algorithm, Network planning, Activity on arc network, Scheduling generation scheme. Abstract. Due to the development of management idea and the scarcity of some resources, the lean management has become the necessary request to implement effective control of resource constrained proect. Resource constrained proect scheduling is the significant guarantee to attain the lean management. Genetic algorithm is one of the basic heuristics to solve the resource constrained proect scheduling problem (RCPSP), the precedence relations of which is described by an activity-on-arrow (AOA) network. The chromosomes are encoded as the extended priority value list (EPL) and decoded by the parallel schedule generation scheme (PSGS). To enhance the exploitation ability, the iterative forward-backward improvement as the local search procedure is applied upon all generated solutions using PSGS. The chromosomes supplied by the genetic algorithm are then adusted to reflect the solutions obtained by the improvement procedure. The overall framework of the GA with forward-backward improvement for RCPSP is developed and the algorithm of FBI is schematically designed. Comparative computational experiments demonstrate the effectiveness of the proposed algorithm in solution to a medium-sized benchmark RCPSP with its precedence relation of activities being diagramed as an AOA network. Introduction Resources are important foundation to fulfill a task. In the field of equipment support, however, some kinds of the resources are usually limited. On account of the limited resource units that can be assigned to activities, the predicted task completion time will most likely be longer than that without taking into account the resource usage limit. How to properly schedule the activities in a proect under constraint of limited resources belongs to the resource constrained proect schedule problem (RCPSP) [1-3]. To assist task managers or decision-maker in well organizing and carrying out the task, it necessary to develop efficient solution method for solving medium-sized or even large-sized RCPSP. The genetic algorithm (GA) is one of the usually adopted meta-heuristics. To the best of the authors knowledge, RCPSP with precedence relationship expressed in an activity-on-node (AON) format is well studied, but little efforts are given to solution of RCPSP with precedence relationship expressed in an activity-on-arc (AOA) format. This work is focused on the theories and computation experiments upon genetic algorithm for solution to the RCPSP with precedence relationship expressed in an activity-on-arc (AOA) format to assist the resource constrained proect scheduling. To further increase the efficiency of the algorithm, a forward-backward improvement (FBI) for enhancing the exploitation ability are incorporated into the framework of the traditional GA, and a chromosome representation of extended priority value list (EPL) are utilized for fast and easy decoding of the chromosome. 349

2 Problem Formulation The relationship among activities of a proect can be described either as an AOA network or an AON network. The latter one is usually utilized in most research papers, but the former one is especially adopted here for purpose of comparison study. The AOA network is a directed acyclic diagram (DAG) G(V, A) where V is the set of nodes and A is the set of activities between which a finish-start precedence relationship with time lag 0 exists. Each activity can be distinguished either by number index or by node pair (p, q). In normal case, the number index expression of an activity in AOA network is adopted for simplicity and ease of description. The node pair expression of an activity is used where necessary. Sets of all immediate predecessor and successor activities of an activity are denoted by P and S, respectively. The AOA network of the network in AON format in reference [4] is shown in Fig. 1. The non-virtual activities correspondence between AOA network and AON network is shown in Table Figure 1. AOA network for the instance example with 12 nodes and 15 activities including 4 dummy ones. Table 1. Non-virtual activities correspondence between AOA network and AON network. Activity No AOA network AON network The RCPSP is as follows: a proect consists of N activities in all including the dummy ones. The activities are interrelated by two types of constraints. The first one is a precedence relation between and i P which forces activity not to be started before any one of its immediate predecessor activities, i P, has been finished. This type of constraint is determined by the corresponding AOA network of the proect and the activity durations. The constraint of the second type is related to the resource requirements. The renewable resources are considered here. The set of renewable resource types is K. While being processed, activity requires r k units of resource type k K, during each time period of its non-preemptable duration d. Resource type k has a limited capacity of R k, constant over the proect execution, which cannot be violated at any time period, i.e., the sum of resource usage of all ongoing activities A t in time instance t should not exceed R k units of resource type k K. The parameters d, r, and R k are assumed to be nonnegative, deterministic, and integer. k The obective for RCPSP is to find a schedule such that the proect duration is minimized subect to precedence and resource constraints. The problem is formulated as an optimization problem as in (1), (2), (3) and (4).The decision variable is the actual start time of activity, t. AS T min p = max{ t AS + d A}. (1) s.t. t AS AS ti + di, A \{1 }, i P, (2) 350

3 k k r t R t T k K ( ) π, = 1,,, A t t, (3) AS t 0, A. (4) In which, A t is the set of ongoing activities in time instant t, as defined in (5): AS AS At = { A, d 0, t t t + d 1}. (5) k An upper bound on the proect s makespan is denoted as T = = d 1, and π R t is the as-yet-unused resource units of type k at time instant t. The obective function (1) minimizes the makespan of the proect. Constraints (2) take into consideration the precedence relations between activities i and, where i immediately precedes. Finally, constraint set (3) limits the total resource usage within each time period to the maximum available amount. Solution Using Genetic Algorithm with FBI Being based on GA, genetic algorithm with forward-backward improvement is also an iteration procedure that operates on population of chromosomes generation after generation, which resulting in solution to the decision variables of RCPSP. The overall framework, chromosome expression, forward-backward improvement [5], and chromosome adustment are given below. For other description of basic components of the heuristics, the reader is referred to [1, 6]. Overall Framework The overall framework of genetic algorithm with FBI is developed as in Algorithm 1. It is actually constructed as an iteration process. Initialization is done first, and then enters the outermost loop. Within the loop, the population is evaluated. As contrast to the traditional GA, the evaluation is immediately succeeded by FBI. The best individual of current generation and the best individual so far are updated according to the evaluation results. Generation gen is increased by 1. Terminating condition is now checked. If the terminating condition is not satisfied, selection, crossover and mutation operators as well as the elitist strategy are executed to generate the next population, which is the beginning of the next iteration; otherwise, exit outermost loop and necessary quantities are output. Algorithm 1. Framework of Genetic Algorithm with FBI /* initialization */ set apopulation.pl=rand(popsize, N) /* main loop */ set gen = 1 repeat if gen>1 then /* generate next population */ roulette wheel selection parametrized uniform crossover of apopulation.pl mutation of apopulation.pl /* perform elitist strategy: replace priority value of a individual by that of current best one */ ip=unidrnd(popsize) apopulation.pl(ip, :)= currentbestindividual.pl /* evaluation loop */ for i = 1 to popsize do set PL i = apopulation.pl(i, :) decode PL i by using PSS to get AL i, SL i and FL i of individual i set apopulation.al(i, :) = AL i set apopulation.sl(i, :) = SL i set apopulation.fl(i, :) = FL i 351 N

4 set apopulation.obvalue(i) = max(fl i ) /* begining of the FBI*/ compute bal i, bsl i and bfl i using backward schedule compute fal i, fsl i and ffl i using forward schedule /* end of the FBI*/ /* chromosome adustment */ set TP i = max(fl i ) set TP' i = max(ffl i ) if TP' i < TP i then compute the priority value, adpl i, which will have to result in fal i, or fsl i when using PSS obtain other lists adal i, adsl i, and adfl i of the chromosome with priority value adpl i using PSS /* update PL i, AL i, SL i and FL i of individual i with new ones */ set apopulation.pl(i, :) = adpl i set apopulation.al(i, :) = adal i set apopulation.sl(i, :) = adsl i set apopulation.fl(i, :) = adfl i set apopulation.obvalue(i) = max(adfl i ) end for /* find the best individual */ set bestindividual = apopulation(1) set bestindex = 1 for i = 1 to popsize do if apopulation.obvalue(i)<bestindividual.obvalue then set bestindividual = apopulation(i) set bestindex = i end for /* find out the best individual so far */ if gen=1 then currentbestindividual = bestindividual else if bestindividual.obvalue<currentbestindividual.obvalue then currentbestindividual = bestindividual set gen = gen + 1 until currentbestindividual.obvalue is good enough or gen>maxgen /* output results */ output currentbestindividual Chromosome Expression In general, there are two forms of chromosome representations: permutation-based representation and priority-based representation [7,8]. A single permutation-based or priority-based chromosome expression should be extended to enhance the performance of GA-FBI. The latter is adopted here for its remarkable merit that the activity list decoded by an SGS from the priority value of a chromosome after crossover and mutation is still precedence feasible, and no repairing operator is needed to resolve the precedence conflict, which would appear if the former expression of a chromosome is used. In the GA-FBI, a list with the start (finish) time of every activity will be helpful when the backward (forward) schedule is performed. The extended chromosome representation is proposed and termed as extended priority list (EPL), which contains four parts: (1) a priority value list (PL) {ρ 1, ρ 2,..., ρ N }, the th element of which stands for the priority of activity, which will be used by the chromosome decoding. For activity, it would compete with this priority among its competitors in current decision set to be scheduled first; (2) an activity list (AL) {π 1, π 2,..., π N }, which is formed by decoding the priority value list; (3) a list with the start time of every activity (SL) {σ 1, σ 2,..., σ N }; and (4) a list with the finish time of every activity (FL) {φ 1, φ 2,..., φ N }. 352

5 Consider the example in Fig. 1, the EPL representation for a feasible schedule of the example is given in Fig. 2, where the priority value, start time and finish time of each activity can be seen clearly from the EPL representation. For example, for activity 8, the priority value is 1.4, the start time is 13, and the finish time is 16. Note that an EPL is the intact representation of a feasible schedule because the necessary information of all activities, including dummy activities, can be obtained from it. Forward-backward Improvement After an initial solution is obtained by the PSS, the forward- backward improvement (FBI) [5] is activated to possibly reduce the makespan of the proect. The FBI procedure iteratively schedules the problem by alternating between backward and forward scheduling. The FL of an initial schedule or a forward schedule determines the activity priorities for the backward schedule of the FBI. Similarly, the SL of a backward schedule determines the activity priorities for the forward schedule. In Fig. 3, a single iteration is used to illustrate procedure of the FBI process. The resource usage profiles of an initial solution scheduled using PSS is shown on the top of Fig. 3(a). As illustrated in the second and third tables of Fig. 3(b), the backward improvement is implemented by shifting each activity (in the descent order of activity finish time) to right as much as possible. In this way, we obtain a schedule with makespan of 20, which can be also read from the middle plot of Fig. 3(a). Forward schedule is used to try to further improve it by shifting activities to left as much as possible. As expected, no further reduction of the proect duration is obtained in forward schedule, as shown in the last two tables of Fig. 3(b) and the plot at the bottom of Fig. 3(a), because the exact solution has been searched in the previous pass. As a whole, it is obvious that the FBI is of certain ability to improve the quality of the individual. (a) A feasible schedule of the example in Figure 1 PL AL SL FL Chromosome Adustment (b) EPL representation for the schedule shown in (a) Figure 2. A feasible schedule and its EPL representation. Since the GA has no knowledge of the chromosome changes that occur in the final solution, the heuristic adusts the chromosome to reflect these changes. To make the chromosome supplied by the GA agree with the local updated solution supplied by FBI, the heuristic adusts the order of priorities of the genes according to the starting times. This chromosome adustment improves not only the quality of the solutions but also decreases the number of iterations necessary to obtain the best values. 353

6 Computation Experiment An AOA version of a medium-sized benchmark instance [9] with 27 nodes and 39 activities including 14 dummy ones is used in the experiments. Computation program of the GA with FBI is developed in Matlab Language. Two computation experiments are executed to evaluate the performance of the proposed heuristic. The basic evolutionary parameters of the program include: popsize= 200, N=39, P c =0.80, Perc m = 75, T CP =0.7. The maximum generation, maxgen, is 30 and 120 for experiment 1 and 2, respectively. The first experiment is for the program run of GA with FBI while the second one is for that of GA without FBI. For both experiments, the programs are all randomly run 10 times. In both experiments, all runs have obtained the exact solution of the problem, but the generation in a run at which the exact one has been found for the first time, G *, is different. The computed proect duration, T p, and the corresponding G *, are listed in Table 2 (index to GA running is indicated by IX). The means of these generations in each experiment showed the apparent superiority of the GA with FBI over the GA without FBI in solving RCPSP. Table 2. Comparison between results of GA solution with FBI and without FBI. Results with FBI (maxgen=30) without FBI (maxgen=120) IX T p G * T p G * Conclusions This paper presents the forward-backward improvement for GA based optimization of RCPSP with precedence relations expressed in AOA format. The overall framework, the EPL representation of the chromosome, and the forward-backward improvement are studied in detail. The experiments show that FBI is an efficient local search technique which can be incorporated into the genetic algorithm for solution to medium or even large sized resource-constrained proect scheduling problem. References [1] S. Hartmann, R. Kolisch, Experimental evaluation of state-of-the-art heuristics for the resourceconstrained proect scheduling problem, Eur. J. Oper. Res. 127 (2000) [2] S. Kreter, J. Rieck, J. Zimmermann, Models and solution procedures for the resource-constrained proect scheduling problem with general temporal constraints and calendars, Eur. J. Oper. Res. 251 (2016) [3] J.-G. He, X.-D. Chen, X. Chen, A filter-and-fan approach with adaptive neighborhood switching for resource-constrained proect scheduling, Computers & Operations Research 71 (2016) [4] H. Wang, T.-L. Li, D. Lin, Efficient genetic algorithm for resource-constrained proect scheduling problem, Transactions of Tianin University 16 (2010) [5] K.Y. Li, R. J. Willis, An iterative scheduling technique for resource constrained proect scheduling, European Journal of Operational Research 56 (1992) [6] J.-C. Wang, Optimization and simulation of resource constrained scheduling problem using genetic algorithm, Science Journal of Business and Management 4 (2016)

7 [7] J. Gonçalves, M. Resende, J. Mendes, A biased random-key genetic algorithm with forwardbackward improvement for the resource constrained proect scheduling problem, Journal of Heuristics 17 (2011) [8] L. Wang, C. Fang, An effective shuffled frog-leaping algorithm for multi-mode resourceconstrained proect scheduling problem, Information Sciences 181 (2011) [9] J.-C. Wang, Resource constrained proect scheduling using particle swarm optimization, in Proceedings of 2016 International Conference on Applied Mathematics, Simulation and Modelling (AMSM 2016). Beiing: Atlantis Press, 2016,

PL 1.1 0.7 0.9 0.2 1.2 1.0 1.3 0.1 0.3 0.4 0.6 1.5 0.5 0.8 1.

according to FT increasingly AL 1 3 7 2 6 8 4 11 9 5 12 10 13 14 15 SL 0 0 2 0 4 2 4 5 8 12 15 15 19 21 24 FL 0 2 2 4 4 5 8 8 12 15 17 19

21 24 24 Sort activities according to ST increasingly AL 1 2 3 8 4 6 7 11 5 9 10 13 14 12 15 SL 4 4 4 6 8 9 9 9 12 15 15 19 21 22 24 FL 4 8

8 PL AL SL FL Sort activities according to FT increasingly AL SL FL Backword schedule AL SL FL Sort activities according to ST increasingly AL SL FL Forword schedule AL SL FL (a) Figure 3. A single iteration of the FBI (b) 356

PROPOSED METHODOLOGY FOR COMPARING SCHEDULE GENERATION SCHEMES IN CONSTRUCTION RESOURCE SCHEDULING. Jin-Lee Kim

Proceedings of the 009 Winter Simulation Conference M. D. Rossetti, R. R. Hill, B. Johansson, A. Dunkin and R. G. Ingalls, eds. PROPOSED METHODOLOGY FOR COMPARING SCHEDULE GENERATION SCHEMES IN CONSTRUCTION