A survey of variants and extensions of the resource-constrained project scheduling problem <s> Objectives Based on the Net Present Value <s> A zero-one integer programming formulation of the project scheduling problem is reported which maximizes the discounted value of cash flows in a project when progress payments and cash outflows are made upon the completion of certain activities. While this problem has been treated previously, it has not been addressed in a manner which allows for the type of constraints treated in this paper. And while cash can be treated as any other resource in the noncash flow, resource-constrained version of this problem, the effects of cash inflows and outflows and the time sequencing alternatives for them allow for a different treatment of cash resources than has been previously reported. <s> BIB001 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Objectives Based on the Net Present Value <s> This paper considers an activity scheduling problem for a project where cash inflows and outflows are given and availability restrictions are imposed on capital and renewable resources. The objective of the problem is to maximize the net present value of the project. Resource-duration interactions are considered in analyzing the problem. A two-phase heuristic solution algorithm is exploited and tested with various numerical problems for its effectiveness and efficiency. <s> BIB002 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Objectives Based on the Net Present Value <s> In this paper we present three new models for the resource constrained project scheduling problem, using two objective functions: minimizing the makespan and maximizing the net present value of the cash flows of the activities. In these models, compared to the models in the literature, we relax one of the most constraining assumptions about the integrality of the activity durations and the project time line. Furthermore resources are assumed to be partially-renewable. These models have multiple uses in Operations Management in that they can be used to solve resource constrained MRP problems. The models are found to be amenable to solution using the Optimization Subroutine Library (OSL). Extensive computational experiments with OSL on a total of 1400 problems with different parameter settings are reported. Detailed analysis is done to determine the factors that affect the computational efficiency of the code and the models. In general, results demonstrate that OSL is a powerful tool for solving the models with either objective function. Optimal solutions are obtained with reasonable computational effort. A discussion of the application of several special features of OSL is also given. <s> BIB003 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Objectives Based on the Net Present Value <s> Many of the recent studies on Project Scheduling focus on maximizing the Net Present Value (NPV) of cash flows that occur during the implementation of a project in the presence of precedence and resource constraints. The literature contains several integer programming formulations, heuristic procedures and their implementations. In most of these formulations, activity durations are assumed to be fixed. In this paper, we study a practical extension of the problem where the activity durations can be reduced from their normal durations by allocating more resources. Costs associated with such reductions are referred to as crashing costs. The problem under consideration involves determining the timing and duration of activities such that the NPV of all cash flows is maximized in the presence of precedence and resource constraints. We suggest a heuristic procedure for obtaining “good” feasible solutions. We compare the performance of three priority rules embedded in the proposed procedure. Two of these priority rules have been shown to be the most effective in the literature, the third is a new rule developed in this paper. Extensive computational experiments with the heuristic procedure on a total of 380 test problems are reported. The heuristic solutions are compared to benchmark solutions obtained from the best of 50-randomly generated solutions for each test problem. In addition, upper bounds on the optimal values are obtained by performing Subgradient optimization on a Lagrangian Relaxation of the problem. These upper bounds are used to assess the quality of the heuristic solutions. In general, none of the priority rules produced consistently better results for all the test problems. However, all three priority rules provide “good” quality solutions with modest computational effort. The Lagrangian approach is found to provide tight bounds on the optimal values. <s> BIB004 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Objectives Based on the Net Present Value <s> Management of projects is complicated by the scarcity of resources required to execute them. Limited resources usually extend the project completion times beyond those determined by CPM/PERT. Several solution procedures have been developed for solving the resource constrained project scheduling problem. One objective commonly used for these problems is to complete the project as early as possible (minimize makespan). The problem considered in this paper is a resource constrained project scheduling problem, with the added features that there are cash flows associated with the project activities, and the objective is to schedule the project activities in such a way that the net present value of cash flows is maximized. With these features the problem becomes financially motivated and more realistic. We introduce a branch and bound procedure to solve the resource constrained project scheduling problem with discounted cash flows. Our procedure exploits the known fact that potential resource violations can be eliminated by introducing additional precedence relations between certain project activities. Specifically, we use the "minimal delaying alternatives" concept to resolve resource conflicts. The bounds in the branch and bound procedure are computed by solving Payment Scheduling Problems, which can be formulated as linear programs and by that are well-solvable. We test our procedure computationally on a set of 90 test problems from the literature and compare it with the only other exact procedure we know of. <s> BIB005 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Objectives Based on the Net Present Value <s> Abstract The movement to product and process development projects that involve joint ventures among strategic partners, as well as the increasing prevalence of projects within organizations has led to increased implementation of project scheduling methods. It is frequently the case that a capital constraint is placed on a project, thus limiting the number and value of activities that can be scheduled to occur simultaneously. However, the quantity of capital available to schedule activities can increase as additional cash is received as progress payments for completed activities. Since the project manager'so objective is to maximize project Net Present Value (NPV), it is important for the manager to develop a schedule that balances the early receipt of progress payments (which improve NPV and increase the capital balance available), with the delay of particularly large expenditures. Due to the intractability of optimal methods, the use of heuristic methods is required to solve problems of practical size. This paper presents the first test of heuristic methods for solving this problem. We use information from a relaxed optimization-guided model that employs information from the unconstrained NPV-optimal problem in heuristic procedure for solving the capital constrained problem. An experimental design is employed to test the heuristics that includes multiple factor levels for a number of project characteristics, including capital utilization, frequency of progress payments, and project network structure. The results indicate very good relative performance for the optimization-guided procedures as compared to two benchmark heuristics. <s> BIB006 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Objectives Based on the Net Present Value <s> Abstract The vast majority of the project scheduling methodologies presented in the literature have been developed with the objective of minimizing the project duration subject to precedence and other constraints. In doing so, the financial aspects of project management are largely ignored. Recent efforts have taken into account discounted cash flows and have focused on the maximization of the net present value ( npv ) of the project as the more appropriate objective. In this paper we offer a guided tour through the important recent developments in the expanding field of research on deterministic and stochastic project network models with discounted cash flows. Subsequent to a close examination of the rationale behind the npv objective, we offer a taxonomy of the problems studied in the literature and critically review the major contributions. Proper attention is given to npv maximization models for the unconstrained scheduling problem with known cash flows, optimal and suboptimal scheduling procedures with various types of resource constraints, and the problem of determining both the timing and amount of payments. <s> BIB007 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Objectives Based on the Net Present Value <s> Projects are often evaluated on financial performance. The net present value (NPV) has long been established as a highly relevant criterion in the management and control of projects. Research in this area has emphasised scheduling of activities in projects to maximise the contractor's NPV assuming that cash flows (both positive and negative) that occur over the duration of the project are known. In practice, however, the contractor usually knows the expenses associated with project activities. He can use this information combined with the knowledge of other project parameters such as activity durations to negotiate the payments received for completed work so that the project achieves the maximum level of financial returns. Extending previous research, this paper examines the problem of simultaneously determining the amount, location and timing of progress payments. We present several models and their solutions for an example problem. Based on the assumptions made, the models are applicable in different project environments. The problem and models presented here can be used by the contractor and client for managing cash flows, setting milestones, and negotiating critical contract parameters, in addition to identifying the location, timing, and amount of progress payments. They also contribute to the body of research on formulating and solving difficult combinatorial optimisation problems. <s> BIB008 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Objectives Based on the Net Present Value <s> Abstract The problem of scheduling activities in a project to maximize its Net Present Value (NPV) has been solved for the case where net cash flow magnitudes are independent of the time of realization. This paper models a more realistic version of this problem — because of incentive payments and penalties for early and late event occurrences, respectively, and because of changing costs of resources over time, net cash flow magnitudes are dependent on the time of realization. We formulate an optimization program for this more general problem and present a simulated annealing solution approach. We test different implementation strategies for this algorithm and suggest a method for choosing neighborhood moves. We compare the NPVs of the solutions obtained from our formulation with the NPVs of early start schedules and with late start schedules for 168 different problems. These computational results show that the simulated annealing approach consistently produces substantially better solutions than the early start or late start schedules. Even poor simulated annealing neighborhood moves give improved solutions for most problems studied. <s> BIB009 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Objectives Based on the Net Present Value <s> Considers the resource‐constrained project scheduling problem where cash inflows and outflows are tied to the occurrence of events. The objective is the maximization of the project net present value (NPV) as well as the minimization of project tardiness in the presence of a project due date. Develops hybrid scheduling rules with both NPV and tardiness considerations to enhance both objectives. Experiments extensively with a set of benchmark problems originally designed for the objective of minimizing the project duration. Demonstrates that thje hybrid rules developed here are superior in performance with respect to both objectives when compared with well known rules which are developed for the two objective of minimizing the project duration. Demonstrates that the hybrid rules developed here are superior in performance with respect to both objectives when compared with well‐known rules which are developed for the two objectives taken individually. Furthermore, the iterative algorithm improves the performance of all tested rules significantly. <s> BIB010 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Objectives Based on the Net Present Value <s> Abstract Project scheduling is concerned with single-item or small batch production where scarce resources have to be allocated to dependent activities over time. Applications can be found in diverse industries such as construction engineering, software development, etc. Also, project scheduling is increasingly important for make-to-order companies where the capacities have been cut down in order to meet lean management concepts. Likewise, project scheduling is very attractive for researchers, because the models in this area are rich and, hence, difficult to solve. For instance, the resource-constrained project scheduling problem contains the job shop scheduling problem as a special case. So far, no classification scheme exists which is compatible with what is commonly accepted in machine scheduling. Also, a variety of symbols are used by project scheduling researchers in order to denote one and the same subject. Hence, there is a gap between machine scheduling on the one hand and project scheduling on the other with respect to both, viz. a common notation and a classification scheme. As a matter of fact, in project scheduling, an ever growing number of papers is going to be published and it becomes more and more difficult for the scientific community to keep track of what is really new and relevant. One purpose of our paper is to close this gap. That is, we provide a classification scheme, i.e. a description of the resource environment, the activity characteristics, and the objective function, respectively, which is compatible with machine scheduling and which allows to classify the most important models dealt with so far. Also, we propose a unifying notation. The second purpose of this paper is to review some of the recent developments. More specifically, we review exact and heuristic algorithms for the single-mode and the multi-mode case, for the time–cost tradeoff problem, for problems with minimum and maximum time lags, for problems with other objectives than makespan minimization and, last but not least, for problems with stochastic activity durations. <s> BIB011 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Objectives Based on the Net Present Value <s> Abstract This paper deals with resource-constrained project scheduling problems with nonregular objective functions where general temporal constraints given by minimum and maximum time lags between activities are prescribed. In particular, resource leveling and net present value problems are studied. We present different heuristic and exact procedures for (approximately) solving those problems. A detailed experimental performance analysis shows that these procedures also solve large problem instances in reasonable computing time. <s> BIB012 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Objectives Based on the Net Present Value <s> Abstract This study reports on a new approach to the payment scheduling problem. In this approach, the amount and timing of the payments made by the client and received by the contractor are determined so as to achieve an equitable solution. An equitable solution is defined as one where both the contractor and the client deviate from their respective ideal solutions by an equal percentage. The ideal solutions for the contractor and the client result from having a lump sum payment at the start and end of the project, respectively. A double-loop genetic algorithm (GA) is proposed to solve for an equitable solution. The outer loop represents the client and the inner loop the contractor. The inner loop corresponds to a multi-mode resource-constrained project scheduling problem (RCPSP) with the objective of maximizing the contractor's net present value (NPV) for a given payment distribution. When searching for an equitable solution, information flows between the outer and inner loops regarding the payment distribution over the event nodes and the timing of these payments. An example problem is solved and analyzed. A set of 93 problems from the literature are solved and some computational results are reported. <s> BIB013 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Objectives Based on the Net Present Value <s> Resource-constrained project scheduling under a net present value objective attracts growing interest. Because this is an NP-hard problem, it is unlikely that optimum solutions can be computed for large instances within reasonable computation time. Thus, heuristics have become a popular research field. Up to now, however, upper bounds are not well researched. Therefore, most researchers evaluate their heuristics on the basis of a best known lower bound, but it is unclear how good the performance really is. With this contribution we close this gap and derive tight upper bounds on the basis of a Lagrangian relaxation of the resource constraints. We also use this approach as a basis for a heuristic and show that our heuristic as well as the cash flow weight heuristic proposed by Baroum and Patterson yield solutions very close to the optimum result. Furthermore, we discuss the proper choice of a test-bed and emphasize that discount rates must be carefully chosen to give realistic instances. <s> BIB014 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Objectives Based on the Net Present Value <s> In this paper we study the resource-constrained project-scheduling problem with discounted cash flows. Each activity of this resource-constrained project-scheduling problem has certain resource requirements and a known deterministic cash flow that can be either positive or negative. Deterministic cash flows are assumed to occur over the duration of the activities. Progress payments and cash outflows occur at the completion of activities. The objective is to schedule the activities subject to a fixed deadline to maximize the net present value subject to the precedence and resource constraints. With these features the financial aspects of project management are taken into account.We introduce a depth-first branch-and-bound algorithm that makes use of extra precedence relations to resolve a number of resource conflicts and a fast recursive search algorithm for the max- npv problem to compute upper bounds. The recursive search algorithm exploits the idea that positive cash flows should be scheduled as early as possible while negative cash flows should be scheduled as late as possible within the precedence constraints. The procedure has been coded in Visual C++, Version 4.0 under Windows NT, and has been validated on two problem sets. <s> BIB015 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Objectives Based on the Net Present Value <s> The paper studies the unconstrained project-scheduling problem with discounted cash flows where the cash flow functions are assumed to be linear-dependent on the completion times of the corresponding activities. Each activity of this unconstrained project-scheduling problem has a known deterministic cash flow function that is linear and non-increasing in time. Progress payments and cash outflows occur at the completion times of activities. The objective is to schedule the activities in order to maximize the net present value (npv) subject to the precedence constraints and a fixed deadline. Despite the growing amount of research concerning the financial aspects in project scheduling, little research has been done on the problem with time-dependent cash flow functions. Nevertheless, this problem gives an incentive to solve more realistic versions of project-scheduling problems with financial objectives. We introduce an extension of an exact recursive algorithm that has been used in solving the max-npv probl... <s> BIB016 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Objectives Based on the Net Present Value <s> The main purposes of this study are to incorporate both the project delay penalty and early completion bonus into the objective function of the resource-constrained multi-project scheduling problem with discounted cash flows (RCMPSP-DCF) and to develop an efficient heuristic search scheduling rule. The effectiveness of the proposed heuristic rule is evaluated by comparing it with the optimal solution obtained by the optimal model for 42 small-scale problems. The result indicates that the solution obtained using the proposed heuristic rule is very close to the optimal solution, and that the proposed heuristic rule provides significant savings in computation time. Moreover, the proposed heuristic rule is also compared with four existing heuristic rules based on an experiment involving the single-project and multi-project scheduling problems. The results indicate that the proposed heuristic rule is superior to the four existing rules under the performance criteria of the average total project net present value and the average total project delay. In addition, the number of times that the proposed heuristic rule can obtain the best solution is also far more than that of any other rule considered in this paper under the two aforementioned performance criteria. Furthermore, this study has found that the performance of a heuristic scheduling rule is significantly affected by the number of projects. A rule that can lower the total project delay to the minimum is also able to maxi5 mize the total project net present value. <s> BIB017 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Objectives Based on the Net Present Value <s> Abstract In this paper we study the unconstrained project scheduling problem with discounted cash flows where the net cash flows are assumed to be dependent on the completion times of the corresponding activities. Cash outflows occur when an activity is completed whereas cash inflows are incurred as progress payments at the end of some time period. The objective is to schedule the activities in order to maximize the net present value of the project subject to the precedence constraints and a fixed deadline. This paper extends the ever-growing amount of research concerning the financial aspects in project scheduling in which cash flows are time-dependent. We introduce a branch-and-bound algorithm which computes upper bounds by making piecewise linear overestimations. In doing so, the algorithm transforms the problem into a weighted earliness–tardiness project scheduling problem. The algorithm is extended with two new rules in order to reduce the size of the branch-and-bound tree. Computational results are reported. <s> BIB018 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Objectives Based on the Net Present Value <s> In this paper the multi-mode resource-constrained project scheduling problem with discounted cash flows is considered. A project is represented by an activity-on-node (AoN) network. A positive cash flow is associated with each activity. Four different payment models are considered: lump-sum payment at the completion of the project, payments at activities' completion times, payments at equal time intervals and progress payments. The objective is to maximize the net present value of all cash flows of the project. Local search metaheuristics: simulated annealing and tabu search are proposed to solve this strongly NP-hard problem. A comprehensive computational experiment is described, performed on a set of instances based on standard test problems constructed by the ProGen project generator, where, additionally, the activities' cash flows are generated randomly with the uniform distribution. The metaheuristics are computationally compared, the results are analyzed and discussed and some conclusions are given. <s> BIB019 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Objectives Based on the Net Present Value <s> The need to develop schedules for projects with resource constraints and cash flows arises in organizational settings ranging from construction planning to research and development. Given the intractable nature of the problem, a variety of knowledge sources relevant to the project scheduling task have been identified in the Operations Management literature. These include a large number of heuristic procedures that can be used to generate feasible project schedules as well as recent neural network-based approaches that can select appropriate heuristic procedures to apply to a specific instance of the project scheduling problem. While integrated application of these knowledge sources is required to effectively support scheduling, previous work has focussed on developing and implementing them in isolation. The problem space computational model presented in this paper addresses this shortcoming by integrating these various knowledge sources, thus enabling the development of decision support systems for resource constrained project scheduling. More generally, the modeling approach used in this paper can be applied to create systems to assist knowledge intensive tasks that arise in many organizational settings. <s> BIB020 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Objectives Based on the Net Present Value <s> A resource investment problem with discounted cash flows is a project scheduling problem in which the availability levels of the resources are considered decision variables and the goal is to find a schedule and resource requirement levels such that the net present value of the project cash flows optimizes. In this paper, we present a genetic algorithm to solve this problem. We explain the elements of the algorithm such as chromosome structure, fitness function, crossover, mutation, and local improvement operations and solve more than 220 problems with known optimal solutions to evaluate the performance of the proposed algorithm. The results of the experimentation are quite satisfactory. <s> BIB021 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Objectives Based on the Net Present Value <s> To stay ahead of their competition, pharmaceutical firms must make effective use of their new product development (NPD) capabilities by efficiently allocating its analytical, clinical testing and manufacturing resources across various drug development projects. The resulting project scheduling problems involve coordinating hundreds of testing and manufacturing activities over a period of several quarters. Most conventional integer programming approaches are computationally impractical for problems of this size, while priority rule-driven heuristics seldom provide consistent solution quality. We propose a Lagrangian decomposition (LD) heuristic that exploits the special structure of these problems. Some resources (typically manpower) are shared across all on-going projects while others (typically equipment) are specific to individual project categories. Our objective function is a weighted discounted cost expressed in terms of activity completion times. The LD heuristics were subjected to a comprehensive experimental study based on typical operational instances. While the conventional "Reward---Risk" priority rule heuristic generates duality gaps between 47---58%, the best LD heuristic achieves duality gaps between 10---20%. The LD heuristics also yield makespan reductions of over 30% over the Reward---Risk priority rule. <s> BIB022 </s> A survey of variants and extensions of the resource-constrained project scheduling problem <s> Objectives Based on the Net Present Value <s> In this paper discrete-continuous project scheduling problems with discounted cash flows are considered. In discrete-continuous project scheduling activities require for their processing discrete and continuous resources. The processing rate of an activity depends on the amount of the continuous resource allotted to this activity at a time. A positive cash flow is associated with each activity. Two common payment models-lump-sum payment and payments at activities' completion times-are considered. The objective is the maximization of the net present value of all cash flows of the project. Some properties of optimal schedules are discussed, and the formulation of a mathematical programming problem for an optimal continuous resource allocation is presented. Applications of a local search metaheuristic-tabu search, as well as simple search methods-multi-start iterative improvement and random sampling are described. The algorithms are compared on the basis of a computational experiment, the results are analyzed and discussed. Some conclusions as well as directions for further research are given. <s> BIB023
Another important type of objective emerges if cash flows occur while the project is carried out. Cash outflows are induced by the execution of activities and the usage of resources. On the other hand, cash inflows result from payments due to the completion of specified parts of the project. Typically, discount rates are also included. Note that cash flows related to activity j might occur at several points in time during execution of j. However, they can easily be compounded to a single cash flow at the beginning or the end of j. These considerations result in problems with the objective to maximize the net present value (NPV) of the project subject to the standard RCPSP constraints. Recent papers include Kimms BIB014 , Mika et al. BIB019 , Padman and Zhu BIB020 , and Vanhoucke et al. BIB015 . In the notation of Brucker et al. BIB011 , the objective to optimize the NPV is represented by ∑ c F j β C j for γ, where c F j is the cash flow of activity j and β is the discount factor. Furthermore, NPV objectives have been investigated for the multi-mode RCPSP (see Ulusoy et al.  , Varma et al. BIB022 , and Waligóra BIB023 ), the resource investment problem (see Najafi and Niaki BIB021 ), and the RCPSP with minimal and maximal time lags (see Neumann and Zimmermann  BIB012 ). Icmeli and Rom BIB003 employ the NPV objective in a problem with continuous activity durations and time-dependent resource capacities. An overview of problem settings with NPV objectives is given by Herroelen et al. BIB007 . A comparison of problems with cash flow based objectives is provided by Dayanand and Padman BIB008 . Several variants of the NPV objective mentioned above arise in literature. Instead of fixed cash flows, Etgar et al. BIB009 consider individual cash flow functions of the completion time for each activity. In Vanhoucke et al. BIB018 , at specific points of time a cash inflow related to activity j occurs which amounts to the fraction of j being finished at this moment. Vanhoucke et al. BIB016 assume the cash flow of an activity to be a linear non-increasing function of the activity's completion time. Najafi and Niaki BIB021 consider a cash flow related to a subset of activities, that is, a cash flow is initiated when the last activity of the subset is finished. Furthermore, the cash flow associated with activity j might depend on the mode chosen for j as lined out in Icmeli and Erenguc BIB004 . Some authors have extended the NPV objective by additional payments upon project completion. Icmeli and Erenguc BIB005 as well asÖzdamar et al. BIB010 consider a penalty being charged for each period the project is finished after a given due date. Chiu and Tsai BIB017 , Doersch and Patterson BIB001 , and Sung and Lim BIB002 extend this by including also bonus payments for early completion. Smith-Daniels et al. BIB006 and Sung and Lim [170] propose the objective to maximize the discounted amount of cash available in each period. This amount is influenced by cash flows associated with each activity. In Smith-Daniels et al. BIB006 this is extended by a constraint: An activity j may not be started at a specific point in time if the cash outflow at the beginning of j exceeds the available cash. Cash can be seen as a renewable resource if the outflow equals the inflow for each activity. However, if this does not hold, the capacity of this resource depends on the point in time and the activities being already started (note that there is a similarity to the concept of cumulative resources, see Section 4.3). In Sung and Lim BIB002 there is no limit for the amount of cash available. In particular, the amount of available cash might be negative which can be interpreted as being indebted. Ulusoy and Cebelli BIB013 investigate the negotiation process to find the timing of payments and the amount of each specific payment between a client and a contractor. Obviously, the client seeks to minimize the NPV while the contractor aims at maximizing it. The objective in Ulusoy and Cebelli BIB013 is to find the payment structure which minimizes each party's loss in comparison to the respective ideal payment structure. Dayanand and Padman  treat a similar problem but restrict themselves to the client's point of view. The client might associate a specific value with each event (start or completion of a job). Cash outflows can be assigned to each event having a positive value. The problem is to find a project schedule and decide cash outflows to happen at a given number of events. The total outflow might exceed the total value of finished activities (minus a percentual retention) at no point of time. The objective is to find a solution such that discounted cash inflow (associated with finishing the project) minus total discounted cash outflow is maximized.