scatter search for project scheduling with resource availability cost jia-xian zhu

Scatter search for project scheduling with resource availability cost

Jia-Xian Zhu

Introduction

Project scheduling– resource constrained project scheduling problem (RCPSP).– resource availability cost problem (RACP). (Mohring, 1984)– The both problem is NP-hard. (Blazewicz et al., 1983), (Moh

ring, 1984) Difference between the RCPSP and the RACP

– The RACP time for completing the project is limited and the resources are unlimited at a non-decreasing discrete cost function.

– The RCPSP involves minimizing the cost of the resources required to complete the project by a specified deadline.

This paper addresses the RACP.

Introduction

The objective of this paper is to develop solution procedures based on the scatter search methodology (Glover, 1977, 1988; Laguna and Marti’, 2003) for the RACP.

Computational experiments– For medium and larger size instances– For small problems

The performance of the proposed procedures is tested against optimal solutions generated by cutting plane procedure from Demeulemeester (1995), and upper and lower bounds from Drexl and Kimms (2001).

References review

The RCPSP recent reviews about exact methods and heuristics can be found in Herroelen et al. (1998), brucker et al. (1999), Kolisch and Hartmann (1998) and Hartmann and Kolisch (2000).

The authors are not aware of heuristics developed for the RACP in the literature.

The resource availability cost problem

di duration of activity i. rik units of resource type k. At the set of activities in progress

during the time interval (t-1,t]. D the project deadline. Ck(ak) a discrete non-decreasing cost

function associated with the availability ak of resource type k.

ak resource availability values. fi the finish times of the activities.


12 activitiesm = 3 resource typesD = 18

Improvement heuristic for the RACP

The improvement heuristic is composed of two phases:– Feasibility (Phase 1)– improvement (Phase 2)

RSBF is hybrid multi-pass method that combines random sampling within a backward-forward scheduling method. (Tormos and Lova, 2003)


Phase 1 is applied– The amount of available resources is increased.

Phase 2 consists of decreasing the amount of available resources, one type at a time, by one unit.

Slake elimination procedure (SEP)– MaxResk < a_solk => Then the resource availability of resou

rce k is reduced to MaxResk (the maximum per period requirement of resource k).


For this project, a_sol = (2,2,2) is likely infeasible. Phase 1 of the improvement heuristic is applied. (

Phase 1 procedure)– Since activities 6, 9, and 11 have the largest value of Si - LS

i, we randomly select activity 9. (go to table)– Activity 9 is r9,1=2, r9,2=2, r9,3=1.– The resource availabilities are increased to a’=(2+2,2+2,2+

1)=(4,4,3). The resource slack of type 2 is eliminated with the S

EP. – generation a new solution a_sol = a’ =(4,2,3).


This solution is feasible and Phase 2 is started. (Phase 2 procedure)– Iteration 1: current solution: a_sol=(4,2,3)

a’=(4-1,2,3)=(3,2,3) － RSBF generates a feasible schedule with start times shown in the last row of Table.

Apply the SEP => a_sol=(2,2,3)

– Iteration 2: current solution: a_sol=(2,2,3) a’=(2-1,2,3)=(1,2,3) － infeasible schedule a’=(2,2-1,3)=(2,1,3) － infeasible schedule a’=(2,2,3-1)=(2,2,2) － infeasible schedule


Pseudocode of the improvement heuristic.

Scatter search

Scatter search is an evolutionary method proposed by Glover (1977,1998).

– Hard combinatorial.– Non-linear optimization problems.

The following parameters are used in the remaining of our discussion:– PSize size of the set of diverse solutions generated by the

Diversification Generation Method.– b size of the reference set (RefSet)– b1 size of the high-quality subset of RefSet– b2 size of the diverse subset of RefSet– MaxSol maximum number of solution evaluations– λ,g parameters used in the Diversification Generation Method.

Scatter search

Scatter search method

Scatter search

Diversification generation method

Let [Zk0,Zk

g) be an interval of integers containing the possible values of resource availability of type k.

Suppose this interval is divided in g sub-intervals [Zk0,Zk

1), …, [Zki-1,Zk

i), …, [Zkg-1,Zk

g). Frequency matrix stores the number of times the a_solk variable a

ssumes a value in the sub-interval [Zki,Zk

i+1). A sub–interval is then selected with probability inversely proportio

nal to its frequency in M.

niikk

ikni

k

mkru

mkrl

,...,1

,...,1

.,...,1},{

,,...,1},{max

Scatter search

Procedure to generate diverse solutions.

Feasible: Zkg=λ*a_solk

Infeasible: a_solk>Zkg

Zkg=λ*a_solk

Scatter search

Reference set update– The subset of quality solutions (RefSet1) contains the b1 best solutions, as evaluated

by the objective function.– The subset of diverse solutions (RefSet2) consists of b2 diverse solutions.

The initial RefSet– The b1 best solutions that belong to P.– The b2 elements from P that maximize the minimum distance to RefSet.

There are two main aspects that must be considered when updating the RefSet.– The timing of the update.

Static update Dynamic update

– Choosing the criteria for adding to and deleting elements from RefSet. Quality and diversity (QD) Quality (Q)

The subset generation methods proposed in the template by Glover (1998).– Four types of subset are generated: ST1, ST2, ST3 and ST4.

Scatter search

Combination with path relinking

mk

jk

ik

ji aaaad,...,1

),(

Scatter search

Scatter search

Procedure to generate neighbors.

Scatter search

Multi-start heuristic

The main steps of the multi-start heuristic are:– Step1. Generate resource values, a_solk,k=1,…,m.– Step2. Apply the improvement heuristic.– Step3. Repeat steps 1 and 2 until the number of evaluated

solutions reaches MaxSol. Two versions of the multi-start heuristic were develo

ped.– Random multi-start (RMS)

Step1. from a discrete uniform distribution U[lk,uk].– Frequency-based multi-start (FMS)

Step1. from the diversification generation method.

Generation of problem instances

Instance with four resources and 30, 60, 90 and 120 activities are directly obtained from the PSPLIB. The remaining instances were generated with Progen (Kolisch et al., 1995).

Two important parameters for Progen– the network complexity (NC) reflects the average number of

immediate successors of an activity.– Resource factor (RF) varies between 0 and 1, reflecting the

density of the different resource types needed by an activity. If RF=1, Each activity requires all m types of resources. RF=0 indicates that activities do not require any type of resour

ces.


D the deadline for the project. DF the deadline factor. EFi the earliest finishing time of activity i. D = DF * max EFi

Following parameters values for RF and NC:– RF: 0.25, 0.5, 0.75 and 1.0.– NC: 1.5, 1.8 and 2.1.– DF: 1.1, 1.2 and 1.3.

The stopping criterion MaxSol for the scatter search and multi-start method is the maximum number of evaluated solutions.

– MaxSol = m * n2 * 25


Run 144 instances (3 RF, 3 DF, 3 NC, 4 activities). Twelve scatter search versions were generated by c

ombining the following factors:– Three values for (b1,b2): (5,5), (7,3) and (3,7)– RefSet update: Static or Dynamic– RefSet update criterion: QD or Q.

The scatter search version (7,3)/Dynamic/QD, with parameters PSize=10, g=10 and λ=1.9, dominates all other versions in terms of mean gap and maximum gap.

– gap=100(SSV-Best)/(Best).

Computational results for set 1

To compare the performance of the best scatter search version (SSD － scatter search with dynamic update) with two multi-start heuristics: FMS and RMS.

Run another 144 instance (4 RF, 1 DF, 3 NC, 4 activities, 3 resources).


Performance of SSD and FMS with respect to the number of resources.


Average computational time (in seconds).


Classification of best solutions found with SSD.


The performance of the scatter search version SSD is compared to upper bounds, lower bounds and exact solutions.

Optimal solutions are generated with a cutting plane algorithm proposed by Demeulemeester (1995), with time limit of 600 seconds.

– If an optimal solution cannot be found within this time limit then the resulting infeasible solution represents a lower bound for the problem.

Lower and upper bounds for the RACP are also obtained by a Lagrangean relaxation and column generation methods proposed by Drexl and Kimms (2001).


Performance of scatter search in the 30-activity and 20-activity sets.


Average computational time for 30-activity and 20-activity sets. (in seconds)


Distribution of instances by interval gaps for 30-activtity and 20-activity sets.

Conclusions

Scatter search obtained better or equal results than the multi-start heuristics.

The results of the extensive experimentation show that this method is capable of providing high-quality solutions to fairly large instances of the RACP in reasonable computational time.


Applying the improvement heuristic to the example.

scatter search for project scheduling with resource availability cost jia-xian zhu

Documents

racp phase

resource availabilities

improvement phase

resource availability

improvement heuristic

racp time

resource slack of type

resource types d