Appendix A Test Instances For testing the algorithms developed in this work, we have employed sev­ eral standard project instance sets which have been used in many studies reported in the literature. Section A.l describes the classical Patterson set of single-mode project instances. Subsequently, Section A.2 summarizes the so­ called ProGen instance sets, which contain both single-mode and multi-mode instances. A.I Patterson Instance Set For a comparison of algorithms for the RCPSP in 1984, Patterson [162] col­ lected several project instances that had been used for testing procedures before. This benchmark test set then became known as the Patterson in­ stance set. It contains 110 single-mode instances with up to 51 activities and up to 3 resources. For all of these test problems, the optimal solutions are known, see, e.g., Demeulemeester and Herroelen [48]. As it was subsequently used by many researchers in their studies, the Pat­ terson set became a standard for testing both heuristic and exact algorithms for the RCPSP (cf., e.g., Bell and Han [15], Cho and Kim [30], Demeule­ meester and Herroelen [48, 51], Hartmann [92], Lee and Kim [133], Leon and Ramamoorthy [134], Klein [115], Kolisch [120, 121], Mingozzi et al. [145], Ozdamar and Ulusoy [156, 157], Sampson and Weiss [173], and Thomas and Salhi [201]). The Patterson instances are available from the internet based project scheduling problem library PSPLIB at the University of Kiel (Germany). For further information, we refer to Kolisch and Sprecher [129] and Kolisch et al. [131]. 182 APPENDIX A. TEST INSTANCES A.2 Instance Sets Generated by ProGen The Patterson instance set described in the previous section was among the first of widely used standard test sets to evaluate exact and heuristic al­ gorithms for resource-constrained project scheduling. There were, however, several points of attack: First, as a collection of instances, the Patterson set was not systematically generated in terms of adjustable problem parameters. Hence, it does not allow a detailed investigation of the impact of project characteristics on the behavior of solution procedures. Second, with contin­ uous progress in project scheduling research as well as in the development of faster computers, the Patterson instances became too easy to solve for mod­ ern algorithms in the early nineties. Therefore, appropriate computational tests required more challenging problem instances. Third, the Patterson set considers only the single-mode RCPSP but not any of its extensions. To overcome some or all of these shortcomings, several researchers de­ veloped project instan<;e generators which allowed to generate instances on the basis of controllable problem parameters (cf. Agrawal et al. [1], Demeule­ meester et al. [47], and Kolisch et al. [130)). In this work, we consider the project instance generator ProGen introduced by Kolisch et al. [130]. We have selected ProGen because it allows to generate both single- and multi­ mode instances on the basis of parameters which have been shown to have a high impact on the behavior of solution procedures. Moreover, several sets of instances have already been generated and used in many previous stud­ ies such that we were able to use existing standard instance sets instead of generating new ones. In order to allow a convenient access to ProGen and the standard ProGen instance sets, the internet based project scheduling problem library PSPLIB has been set up at the University of Kiel (Germany). For further information on the instance sets and on recent developments of the project scheduling problem library PSPLIB, the reader is referred to Kolisch and Sprecher [129] and Kolisch et al. [131]. Motivated by the broad acceptance of ProGen and the related instance sets in the project scheduling community, several researchers have extended ProGen in order to cover further general project scheduling models. A ProGen based generator for networks with minimal and maximal time-lags (cf. Subsection 2.2.2) called ProGen/max has been developed by Schwindt [180]. Drexl et al. [64] introduced ProGen/1Tx which extends ProGen by par­ tially renewable resources (cf. Subsection 2.2.3), the mode identity concept (cf. Subsection 2.2.1), and some other new modeling features. The remainder of this section summarizes the main characteristics of those ProGen instance sets that have been used throughout this work. Subsection A.2.1 describes the single-mode instances while Subsection A.2.2 reports on the multi-mode instances. A.2. INSTANCE SETS GENERATED BY PROGEN 183 A.2.1 Single-Mode Instance Sets We use three standard ProGen instance sets which have been introduced in Kolisch et al. [130], Kolisch and Sprecher [129], and Kolisch et al. [131]' respectively. Each set is characterized by the number of activities within a project. In the three sets, we have J = 30, J = 60, and J = 120 non­ dummy activities, respectively. As displayed in Table A.l, all instances were generated with certain fixed parameter ranges, leading to activity durations between one and ten periods and four renewable resources for each project. Parameter min max Pi 1 10 IKPI 4 4 Table A.l: ProGen single-mode instances: Fixed parameter levels Kolisch et al. [130] identified three parameters which have a strong impact on the performance of solution procedures, namely the network complexity, the resource factor, and the resource strength. The network complexity NC reflects the average number of immediate successors of an activity. The re­ newable resource factor RFP is a measure of the average number of resources requested per job. The renewable resource strength RSP describes the scarce­ ness of the resource capacities. If the latter is high (Le., close to 1), the availability is high, which leads to a smaller solution space and hence easier problems. On the other hand, a low resource strength (Le., close to 0) implies scarce resources and more difficult instances. The instance sets with J = 30 and J = 60 were generated by a full facto­ rial design obtained from three network complexity levels, four resource factor levels, and four resource strength levels. For each of the resulting 3·4·4 = 48 parameter combinations, 10 instance were randomly generated, leading to 480 instances in each of the two sets. The instance set with J = 120 was gen­ erated similarly, with the exception that five levels for the resource strength were chosen. Again, 10 instances for each parameter combination were ran­ domly constructed, yielding 600 instances. An overview of the systematically varied parameter settings within the three sets is given in Table A.2. The set with 30 non-dummy activities currently is the hardest standard set of RCPSP-instances for which all optimal objective function values are known (cf. Demeulemeester and Herroelen [51]). For the other two sets, lower bounds on the project's makespan can be easily derived using forward recursion (cf. Subsection 2.1.2). Clearly, the earliest precedence feasible start time ESJ+l of the dummy sink activity is a lower bound on the makespan, the so-called critical path based lower bound (cf. also Stinson et al. [195]). Further lower bounds have been developed by, e.g., Baar et al. [8], Brucker and Knust 184 APPENDIX A. TEST INSTANCES J parameter levels 30 60 RFP 0.25 0.50 0.75 1.00 RSP 0.20 0.50 0.70 1.00 NC 1.50 1.80 2.10 120 RFP 0.25 0.50 0.75 1.00 RSP 0.10 0.20 0.30 0.40 0.50 NC 1.50 1.80 2.10 Table A.2: ProGen single-mode instances: Variable parameter levels [27], Heilmann and Schwindt [99], Klein and Scholl [117], Mingozzi et al. [145], and Stinson et al. [195]. The library PSPLIB which is frequently updated contains the currently best lower and upper bounds for these instances. Some or all of the three instance sets considered here have been widely used by researchers, making them a standard for evaluating and comparing solution algorithms. We refer to the studies of Baar et al. [8], Bouleimen and Lecocq [25], Brucker et al. [29], Demeulemeester and Herroelen [51], Hartmann [92], Hartmann and Kolisch [95], Klein [115], Klein and Scholl [116], Kohlmorgen et al. [118], Kolisch [120, 121, 122], Kolisch and Hartmann [126], Mingozzi et al. [145], Schirmer [174], Schirmer and Riesenberg [176, 177], and Sprecher [191]. A.2.2 Multi-Mode Instance Sets We consider six standard ProGen instance sets for the MRCPSP. As for the single-mode case, each set is characterized by the number of activities within a project. We have J = 10, J = 12, J = 14, J = 16, J = 18, and J = 20. Table A.3 shows the fixed parameter ranges which were used to generate all of these multi-mode instances. We have three modes for each non-dummy activity and two renewable as well as two nonrenewable resources. Observe that, for the multi-mode case, the network complexity NC (cf. Subsection A.2.1) has been fixed to 1.8 arcs per activity. Parameter min max Pj 1 10 M j 3 3 NC 1.8 1.8 IKPI 2 2 IKvl 2 2 Table A.3: ProGen multi-mode instances: Fixed parameter levels A.2. INSTANCE SETS GENERATED BY PROGEN 185 In order to obtain the multi-mode instance sets including nonrenewable resources, the following four ProGen parameters were systematically varied: The resource strengths for the renewable and nonrenewable resources, RSP and RSI!, were treated separately, as well as the resource factors for the re­ newable and nonrenewable resources, RFP and RFI!. With the two resource factor levels and the four resource strength levels listed in Table A.4, a full factorial design with 10 instances for each parameter level resulted in 640 instance for each project size.