TECHNISCHE UNIVERSITÄT DRESDEN Fakultät Wirtschaftswissenschaften

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1 TECHNISCHE UNIVERSITÄT DRESDEN Fakultät Wirtschaftswissenschaften Dresdner Beiträge zur Betriebswirtschaftslehre Nr. 168/12 A survey of recent methods for solving project scheduling problems Markus Rehm, Josefine Thiede Herausgeber: Die Professoren der Fachgruppe Betriebswirtschaftslehre ISSN

2 A survey of recent methods for solving project scheduling problems Markus Rehm, Josefine Thiede Institute of Material Handling and Industrial Engineering, Faculty of Mechanical Engineering, Dresden University of Technology, Germany Keywords: resource-constrained project scheduling, solution methods and algorithms Abstract This paper analyses the current state of research regarding solution methods dealing with resourceconstrained project scheduling problems. The intention is to present a concentrated survey and brief scientific overview on models, their decision variables and constraints as well as current solution methods in the field of project scheduling. The allocation of scarce resources among multiple projects with different, conflicting decision variables is a highly difficult problem in order to achieve an optimal schedule which meets all (usually different) of the projects objectives. Those projects, e.g. the assembly of complex machinery and goods, consume many renewable, e.g. workforce/staff, and non-renewable, e.g. project budget, resources. Each single process within these projects can often be performed in different ways so called execution modes can help to make a schedule feasible. On the other hand the number of potential solutions increases dramatically through this fact. Additional constraints, e.g. min/max time lags, preemption or specific precedence relations of activities, lead to highly complex problems which are NP-hard in the strong sense.

3 1 Introduction Multi-project management is seen by the majority of managers as the task of managing a list of individual projects rather than managing complex operations with a specific workload and demand of capacity [1]. However, project management can cover a wide variety of different kinds of projects, e.g. investment projects, product development projects or organisational projects. Those kinds of projects and the related strategic management will not be the focus of this paper but rather operative management. Therefore, the project management dealt with in this paper refers to operations scheduling, i.e. the process orientated project management or simply project scheduling. Operation scheduling on the other hand can cover many domains, e.g. the housing industry, the management in hospitals, schools or for railways. The primary focus of this paper is the domain of manufacturing. Hereby, solution methods were analysed regarding production requirements. Firstly, there are some general requirements for production including two main constraints. As resources usually have limited capacities, capacity constraints need to be considered whilst scheduling. Concerning multi-project scheduling, it is also important to pay attention to the fact that several projects can have access to the same limited resources, e.g. a common resource pool. In addition, the different kinds of resources to be found in production as well as the different kinds of objectives, i.e. time-oriented, cost-oriented and resource-oriented objectives, need to be taken into account. Furthermore, real-life production processes often demand several resources, several objectives and consist of a high amount of activities. Due to technological requirements, the other constraint concerns the precedence relations between activities. For these, activities can only start when all of its predecessors are completed. This means in general that obeying machine sequences is a fundamental requirement. Those precedence constraints can enforce minimum and maximum time lags. Additionally, some production processes allow different ways of performing activities, i.e. activities can have multiple execution modes. Each mode has a different effect on the duration of the activity, its resource requirements and the related costs and therefore trade-offs occur. Another characteristic of production is the possibility of activity splitting. Activities can either be interrupted once they have been started, which is the preemptive case or they cannot, namely the non-preemptive case [2]. Secondly, many other particular real-life characteristics exist, for example resource availabilities and requirements varying in time, due dates, activity ready times, activity overlaps [2], fixed starting times, set-up times, shelf-life constraints, releases or minimum and maximum overlaps of projects or activities that need to begin or end simultaneously [3]. Time-varying resource requests mean that activities can, in certain production processes, require different amounts of resources varying in time. The release date refers to the earliest time for an activity to begin, whereas the due date refers to the latest point-in-time for an activity to end. Those deadlines and related penalties are important for planning multiple projects simultaneously.[4] Set-up times can be sequence-independent, sequencedependent and schedule-dependent. Additionally, in certain production facilities manufacturing is not feasible at weekends and the workers need to take regular breaks. Therefore an activity can start only in a certain work window but not in a so-called rest window. This kind of constraint is called timeswitch constraint.[5] Another special case is covered by cumulative, i.e. multi-capacity resource scheduling problems, which are matched to the demands of many practical scheduling environments [6] as the resource capacity is not exceeded at any point and resources can perform several activities in parallel.[7] In section 2, an overview on several published solution methods will be given, as well as a classification of the scheduling problems dealt with. However, there is a high number of real-life 1

4 requirements which need to be met and many methods do not. Section 3 will state several concluding remarks. 2 Presentation of results The resource-constrained project scheduling problem (RCPSP), which is a generalization of the popular Job-Shop-scheduling problem, covers not only capacity constraints, i.e. deals with the issue of allocating resources available in limited amounts but also deals with precedence of activities. Additionally, activities have a certain duration as well as a certain demand of limited resources. The RCPSP is an optimisation problem, which is NP-hard in the strong sense. The objective of the RCPSP planning is to determine a valid schedule, i.e. a schedule accounting for precedence and capacity constraints, which optimises a given target criterion. Due to the fact, that the RCPSP covers only renewable resources and that its activities can only be executed in one mode, a more realistic problem needs to be focused on the MRCPSP, the multimode resource-constrained project scheduling problem. This is a well-known variant of the RCPSP, in which non-renewable resources can be taken into consideration as well. Even doubly-constrained or partial renewable resources may be taken into account. In addition, activities can be executed in one out of a number of different modes. Each mode stands for a different way in which alternative levels of resource demands are combined with their specific duration. Due to the multiple modes several types of trade-offs may occur, e.g. between the duration of an activity and its resource use, the socalled time-resource trade-off or between several resources used by the activity regarding their quantity and combination, so-called resource-resource trade-off or between the duration of an activity and the related costs, the time-cost trade-off. Except for the similarity of the RCPSP and the MRCPSP being NP-hard, both problems can be preemptive or non-preemptive. Buddhakulsomsiri and Kim [8] have shown that for the MRCPSP preemption helps to improve the optimal project duration in case that resource absenteeism as well as occasional resource unavailability occur. Also, preemptive problems are more closely linked to real scheduling problems, especially when involving human resources. Moreover, solution methods can either deal with scheduling problems which consist of either only one objective or multiple objectives. The latter case needs to be divided into two instances: The case that different objectives can be inserted and the case that different objectives can be pursued at the same time, the so called multi-objective scheduling problem. This problem is more difficult than the one with a single objective as not only one but several optimal solutions can be found because there is hardly an approach that solves all objectives simultaneously [9]. Another categorization is the division of the MRCPSP into problems which consider a single or multiple resources. Reddy, Kumanan and Chetty [10] name the latter case as the multi-mode multiresource-constrained problem (MMRCPSP), Kim/Gen/Kim [11] term it in their work as multiresource-constrained project scheduling problem with the multiple modes (mcpsp-mm). Since only [10] and [11] uses special terms, the problem is simply referred to as multi-mode resource-constrained project scheduling problem (MRCPSP) in the following. An opposing example is the paper of Basnet, Tang and Yamaguchi [12] who presented a beam search heuristic for the multi-mode single-resourceconstrained project scheduling problem in which only a single renewable resource is considered. 2

5 A similar case is the division into single- and multiple-category resource-constraints scheduling problems, introduced by Słowiński, Soniewicki and Węglarz [13]. They present a decision support system for multi-objective project scheduling under multiple-category resource constraints with multiple performing modes (MMCRCPSP). The solution method they propose can account for renewable, non-renewable and doubly constrained resources and deals with conflicting time and cost type objectives [13]. The general scheme for the single-mode problem with multiple-category resource constraints was already proposed by [14] in 1989 but the paper on the MMCRCPSP by [13] remains the only one which uses the term mentioned despite other authors also proposing solution methods for the MRCPSP with two or three resource categories. Consequently, the problem is in the following also simply called MRCPSP. Various heuristic and exact approaches have been suggested in past years to find solutions for the MRCPSP. Following Słowiński [15], who applied linear programming to solve a MRCPSP first in 1981, Talbot [16] presented his solution method for MRCPSP but only with time-resource trade-offs in More than a decade had to pass before the stochastic scheduling method by Drexl and Gruenewald [17] was proposed. Exact solution methods were then proposed by Sprecher in 1994 [18] as well as by [19], [20] and [18]. The latter three are all branch-and-bound algorithms, which deal with time-orientated objectives and renewable and non-renewable resources. Multi-objective solution methods were presented by [15], [13] and [9] who used simulated annealing and tabu search to find a metaheuristic solution for the MRCPSP. A quite recent metaheuristic solution method is the adaptive large neighbourhood search algorithm by Muller [21]. Figure 1 shows the number of different solution methods, summarising them according to their date of publication. Figure 1: Number of published general solution methods per year Until 1992 all solution methods dealt only with the MRCPSP, but in 1993 [22] presented an integer model for solving the multi-mode resource-constrained multi-project scheduling problem (MRCMPSP), a generalized case of the RCPSP. In this problem the activities of each project can be performed in one out of several modes in compliance with the given precedence and resource constraints. Only little research has been done on this problem. Artigues and Roubellat [23] proposed a polynomial activity insertion algorithm to deal with rescheduling caused by unexpected activities that repeatedly question the effectiveness of the schedule previously generated. Solution methods dealing with the complete scheduling problem were introduced by [22] and [24]. Tseng (2008) even proposed two heuristic algorithms to find a solution for the MRCMPSP. [25] and [23] term the multi- 3

6 mode resource-constrained project scheduling problem in their work as MMRCMPSP. In this paper the term MRCMPSP is preferred because MMRCMPSP can also stand for multi-mode multi-resource constrained project scheduling problem. In 1997 a parallel randomized solution approach for a generalisation of the MRCPSP, the modeidentity and resource-constrained project scheduling problem (MIRCPSP) was presented by Salewski, Schirmer and Drexl [26]. Here, the set of all activities is divided into several disjoint subsets, in which all the activities are performed in the same mode. The cost and time incurred, while such a subset is being processed, depend on the resources assigned to it [27]. There are hardly any methods available to solve this problem. [26] were the first to present an approach that deals with this problem, namely a parallel regret-based biased random sampling approach called RAMSES, into which diverse priority rules can be included. In 2010 [28] proposed an exact solution method, in particular a branch-and-bound procedure. The resource-constrained project scheduling problem with multiple crashable modes (RCPSPMCM) was introduced by Ahn and Erenguc [29] in 1998, when they combined the MRCPSP with the timecost trade-off problem. Crashable modes mean that the duration of a certain mode can be decreased at some cost. In order to reduce the duration (crashing), additional shifts can be used or more resources can be assigned, which might be easily achieved by incurring more money. Besides the difference that the RCPSPMCM and not the MRCPSP considers time-cost trade-offs within the activity, the objective is to minimise the sum of all activity costs and the costs of tardiness instead of minimising the makespan [30]. To solve this problem, [29] suggested an exact solution procedure as well as a heuristic procedure. Another exact solution method was introduced by [30] in 2000 but after that, no other author dealt with the RCPSPMCM ever again. Several real-life projects, for example in civil engineering for chemical or food industries, can be modelled using an extension of the MRCPSP, namely the multi-mode resource-constrained project scheduling problem with minimum and maximum time lags (MRCPSP/max) [31], also known as MRCPSP with generalized precedence relations (MRCPSP-GPR). A solution method dealing with this problem was introduced by [2] in The generalized precedence relations, or minimal and maximal time lags between the start of the activities and their completion, can be divided into four categories which are start-finish (SF), finish-start (FS), start-start (SS) and finish-finish (FF). Minimal time lags specify that a certain activity can only begin or end if the previous activity has already begun or ended for a specific amount of time.[2] With the help of this, many real-life situations can be modelled, making the related solution methods very applicable and, therefore, well researched. Sabzehparvar and Seyed-Hosseini [31] proposed a mathematical model to find an exact solution for MRCPSP with mode-dependent time lags. Six other authors introduced heuristic solution procedures to solve the MRCPSP/max. Those heuristic methods are able to account renewable and non-renewable resources whereas the exact algorithm considers only renewable resources. In order to be able to deal with financial aspects when using the MRCPSP, Ulusoy, Sivrikaya- Şerifoğlu and Şahin [32] introduced in 2001 the multi-mode resource-constrained project scheduling problem with discounted cash flows (MRCPSPDCF). The objective of this model is to maximise the net present value of every cash flow. Time value of money is considered by discounting those cash flows. In addition, activities and/or events are associated with the in- and out-flows of cash. Cash outflows occur, for example, at the beginning of each activity. To solve this problem [32] proposed a genetic algorithm approach that investigates four different payment models, whereas Chen et al. [33] presented an ant colony optimisation approach to solve the problem. 4

7 In 2002 [34] introduced a solution method for the resource-constrained multi-project scheduling problem (RCMPSP), although the first approach was presented by Pritsker, Watters and Wolfe already in 1969 (see [25]). This is quite applicable to real-life problems as many companies need to manage various projects simultaneously, which all share the same pool of resources (all assumed renewable) [34]. A common objective is to find a resource and precedence feasible completion time for every activity so that a minimum duration for the multi-project is achieved [35]. The precedence constraint of multi projects mean that the project cannot be changed while the RCMPSP is being finished, once it has been initiated in a certain project. The precedence constraints of activities, on the other hand, imply that the starting time of an activity is determined by the completion time of other activities [36]. Although the RCMPSP is seen as not unimportant, there are only few studies which deal with this problem, for example [35], who give a particle swarm optimization, and Gonçalves, Mendes and Resende [37], who presented a genetic algorithm. Most of the analyzed solution methods deal with the MRCPSP, see figure 2. The MIRCPSP, which is strongly NP-hard [26], as well as the RCPSPMCM and the MRCPSPDCF are hardly ever dealt with, as each of them covers only 3% of the studies total. With 11 percent for the MRCPSP/max and 6 per cent for the RCMPSP, these problems are also rather popular. Figure 2: Percentage of types of problems considered by the solution methods Figure 3: Number of solution methods depending on classification (multiple answers possible) Regarding the classification of methods the authors applied to solve the different models, the genetic algorithm, which primarily deals with time-orientated problems, was used most frequently, followed by the branch-and-bound algorithm which was used in 6.5 % of the cases, see figure 3. Five out of the six exact solution methods are based on branch-and-bound algorithms. Simulated annealing is frequently applied with half of the multi-objective problems being solved with this approach. Examples of solution methods using those algorithms have been described earlier on in the text as well as the ant colony optimisation and also a beam search approach. This is due to the fact that searchbased algorithms are also rather common. The total of all different types, like tabu search, local search or hybrid scatter search covers 7.2 % of the applied types of algorithms. The application of an artificial immune system, Petri Nets or a population learning algorithm, however, was infrequent. As far as the latter case is concerned, this might be due to the fact that the computational time required was too long. In conclusion, it can be stated that there is no ideal method which perfectly suits and serves every area or covers all real-life demands of production, as this is a highly complex issue. The mentioned requirements include the ability to deal with a large problem size, different constraints, np-hardness, different objectives, preemption, different kinds and numbers of resources and multiple modes. Only 5

8 few of the introduced solution methods are exact procedures, and often they take only a single resource or a single objective into account. Usually, the more activities the solution methods can deal with, the less objectives or resources it can handle and vice versa. However, there are many methods which closely resemble real-life production problems, meet many requirements and offer good solutions within a reasonable amount of time. 3 Concluding remarks In this paper several solution methods for different project scheduling problems, including MRCPSP, MIRCPSP, MRCMPSP, MRCPSP/max, MRCPSPDCF, RCMPSP and the RCPSPMCM, were gathered and evaluated facing requirements of production processes and multi-project scheduling. The analysis showed that since 1981 solution methods for the introduced project scheduling problems have been proposed, and since 1993, an average of three new solution methods have been published each year. Another finding is that most solution methods deal with MRCPSP. Other project scheduling problems, which consider additional specific requirements, e.g. multiple crashable modes, mode identity constraints or even multi projects are less frequent. Most solution methods can consider up to 51 activities, are heuristic and use a genetic algorithm. All solution methods deal with renewable resources, and 70% also consider non-renewable resources. Regarding their objectives, most solution methods consider time-orientated objectives, whereas the resource-orientated objective is not at all considered. There is no solution method that covers all the high real-life demands of production; however, viable solution methods have been recommended for specific areas of application in this paper. Furthermore, a need for a precise and standardised rating system for all kinds of project scheduling problems and domains as well as standardised definition of terms becomes apparent. Due to the fact that there are still many aspects which need further research, it can be assumed that the keen interest in this important field will persist. During the recent years, a growing number of applicable solution methods has been published an output likely to continue in the years to come. Acknowledgments Very special thanks to Mr. André Gräning, Dresden University of Technology, for his constructive and valuable support. 6

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