Multi-objective multi-mode resource constrained project scheduling problem using Pareto-based algorithms

Abstract

This study addresses the multi-objective multi-mode resource-constrained project scheduling problem with payment planning where the activities can be done through one of the possible modes and the objectives are to maximize the net present value and minimize the completion time concurrently. Moreover, renewable resources including manpower, machinery, and equipment as well as non-renewable ones such as consumable resources and budget are considered to make the model closer to the real-world. To this end, a non-linear programming model is proposed to formulate the problem based on the suggested assumptions. To validate the model, several random instances are designed and solved by GAMS-BARON solver applying the ε-constraint method. For the high NP-hardness of the problem, we develop two metaheuristics of non-dominated sorting genetic algorithm II and multi-objective simulated annealing algorithm to solve the problem. Finally, the performances of the proposed solution techniques are evaluated using some well-known efficient criteria.

1 Introduction

Resource investment is one of the most important issues in the resource-constrained project scheduling problem (RCPSP) which tries to reduce the resource employment costs of a project. In other words, many project activities are allowed to have a delay and in this situation, completion costs of projects, as well as the level of investment in resources, are targeted in the planning stage before starting the project. So decision-makers seek to create a desired balance between time and cost in projects. For example, in capabilities planning project of military industries with the aims of achieving defined military capabilities according to the investment constraints, optimizing the amount of resources invested in the project with respect to the two factors of time and cost is very critical [4].

In RCPSP, minimization of the cost related resource is done by determining the desirable level of resources required for project activities, mainly at the planning stage. In resource investment, considering the resources needed to carry out activities as well as the precedence relationships, the desired level of access to resources is considered as a decision variable and all activities are scheduled according to determined levels. The resource investment problem was originally introduced by Möhring [30]. He tried to minimize required resources cost while considering the due date of projects. According to Möhring [30], project scheduling problem (PSP) is divided into two categories with respect to the completion time and available resources:

• (1) RCPSP: the access level to all kinds of resources is constrained and the goal is to achieve the shortest possible completion time of the project.

• (2) Time-constrained project scheduling problem (TCPSP): the total time available to complete the project is constrained and the goal is to determine the optimal level of resources and minimize the utilization cost of resources, assuming that the required resources are unlimited.

The first category is studied in this research with a wide range of applications in the real world. In other words, RCPSP is the most comprehensive subject of scheduling problems which even includes job shop, flow shop and open shop scheduling problems. Generally, RCPSP seeks to find a suitable sequence for performing the activities of a project in such a way that the precedence relationships of the project network and different types of resource constraints on the project are satisfied simultaneously. All of these limitations are in line with a particular measure like time or cost which should be optimized.

RCPSP is one of the problems with rich literature in the field of project management. So far numerous articles and books have been published due to two main reasons. Firstly, this problem is various in terms of the objective function, the characteristics of the activities, resources and precedence relationships, according to practical and industrial conditions and secondly, due to the NP-hardness of these problems, researchers have always sought to develop more efficient solution techniques. Some of the most applicable fields of the project scheduling is product design [8], software design and engineering [7, 29], military capability planning [4, 48] as well as research and development projects for goods and services [13].

Normally, the duration of an activity is fixed and cannot be changed, but in some cases, this time is changeable by incurring additional costs. In fact, the duration of the project activities depends on its importance for the project manager where it can be reduced by considering more expenses. In the multi-mode RCPSP (MMRCPSP), there are a set of acceptable execution modes for each activity such that each one has its own specific and unique duration and consumption level of resources [39]. The time and cost of performing each activity are characterized by the selection of its execution modes [19]. The aim is to determine the optimal execution mode and the start time of each activity to minimize the total completion time of the project. Brucker et al. [5] proved that if the number of non-renewable resources and execution modes of each activity is more than two, MMRCPSP is an NP-Complete problem.

Project management utilizes the knowledge, skills, tools, and techniques required to manage the execution of activities in order to meet the requirements and expectations of the project managers. It employs three powerful tools including required data collection, project planning and project scheduling [38].

In the same vein, another important issue is the project payment planning (PPP), which determines the time and the amount of payment flows and can be done through one of the four existing models [45]:

• Lump-Sum Payment (LSP): the total cost is paid to the contractor when the project is successfully completed.

• Payments at Event Occurrence (PEO): the payments are made during the occurrence of the events.

• Payments at activities’ completion times (PAC): the payments are made when each activity is finished.

• Progress Payments (PP): the payments are made at regular intervals and the last payment is made at the completion of the project.

In this research, a bi-objective MMRCPSP with discounted cash flows and PEO-based payments is studied, which is an extension of the research done by He et al. [19]. In the proposed problem, renewable resources (including manpower, machinery, and equipment), as well as non-renewable resources (including consumption and money), are also taken into account to be assigned to the activities during the project. The aim of considering the discounted cash flow is to calculate and maximize the NPV of the project. Moreover, the second objective is to minimize the completion time of the project based on the occurrence of the last event. In other words, the main goals are to determine and allocate the amount of payments and to schedule the activities considering the appropriate execution modes with respect to the time–cost trade-off. To validate the proposed model, the ε-constraint method is applied to cope with the bi-objectiveness of the model, then it is implemented by CPLEX solver of GAMS software. Furthermore, two efficient Pareto-based metaheuristics including non-dominated sorting genetic algorithm II (NSGA-II) and multi-objective simulated annealing (MOSA) algorithm are developed to solve the problem approximately.

The remaining sections of the paper are organized as follows. Section 2 represents the literature review of the research, then the proposed problem and mathematical model are described in Sect. 3. Section 4 introduces the applied ε-constraint method, NSGA-II and MOSA, and the computational results are presented in Sect. 5. Finally, the conclusions and outlook of the research are described in Sect. 6. Moreover, the proposed flow graph of the research is shown in Fig. 1.

Fig. 1

The proposed flow graph of the research

2 Literature review

The objective of this section is to investigate the shortcomings of the studies on RSPSPs. During the last decades, RCPSP was widely investigated by researchers [20, 23]. Although the main focus is on the minimization of completion time of project [17] while some other objective functions implicate on maximizing the net present value (NPV) of the project [19].

Since the discussion of cash flows in PSP was introduced by Russell [36], the problem of scheduling the activities of a project aimed at maximizing the NPV has attracted much attention in the literature. The efforts of many research works have led to the examination of models and methods for solving a large variety, along with the representation of critical path method (CPM), cash payment patterns, resource constraints, and time–cost exchanges. In the context of a review on the PSPs with regard to PPP with the aim of maximizing NPV, the studies conducted by Icmeli and Erenguc [22], Özdamar et al. [34] and Hartmann and Briskorn [17] can be referred.

Dayanand and Padman [9] presented several deterministic models for maximizing contractor’s NPV where the delivery time and the amount of pre-determined payments are determined for the project. In the next study, they determined a set of payments regarding the scheduling, and then, in the second step, re-scheduled them to improve the NPV [10]. Szmerekovsky [41] presented a Branch and Bound (B&B) method to solve the project payment scheduling problem (PPSP), which provides the scheduling of the project’s payments on the customer request, and the contractor can defend his interests by selecting activities scheduling and rejecting payment schedules.

Ulusoy and Cebelli [44] developed a genetic algorithm (GA) to solve the PPSP considering a timely payment plan that provides the benefits of both customer and contractor. He and Xu [18] examined the effect of the incentive-fining policy on the payment schedules and found that the existence of such a structure would improve the flexibility of a payment schedule. He et al. [19] developed two metaheuristics of simulated annealing (SA) algorithm and Tabu Search (TS) algorithm for multi-mode PPSP (MMPPSP) and examined their performance by generating and solving random samples.

On the other hand, other problems were studied as capital-constrained PSP (CCPPSP). A number of studies have been done on CCPPSP by Özdamar and Dündar [33], Ulusoy et al. [45], Mika et al. [28], and many others who examined the CCPPSP problem in a single-mode and multi-mode context. Chen and Zhang [6] examined the RCPSP to maximize the NPV of the project under uncertain conditions. They considered random stochastic time and cost for each activity and solved the problem using ant colony optimization (ACO) algorithm.

Aboutalebi et al. [1] presented a bi-objective mathematical model to solve a multi-mode resource-constrained project scheduling problem with discounted cash flows (MMRCPSP-DCF) with the objectives of minimizing the completion time and maximizing the NPV of projects. They applied NSGA-II and a multi-objective particle swarm optimization (MOPSO) algorithm to solve the proposed problem. Hosseini et al. [21] presented a mathematical model to solve MMRCPSP with positive and negative cash flows aimed at maximizing the NPV of the project. They designed an efficient GA to solve the problem.

Leyman and Vanhoucke [26] presented a single-objective model for MMRCPSP with renewable and non-renewable resources and positive and negative cash flows to minimize the NPV of the project. They studied three different payments of PAC, PEO and PP and applied a GA to solve the problem. Sebt et al. [37] proposed a hybrid metaheuristic algorithm including GA and particle swarm optimization (PSO) algorithm to solve MMRCPSP with the aim of the completion time minimization. Geiger [15] developed an iterated variable neighborhood search (IVNS) to solve a multi-project, multi-mode resource-constrained project scheduling problem (MPMMRCPSP). They tested the performance of their proposed algorithm on some benchmark instances.

Oztemel and Selam [35] designed a bee colony optimization (BCO) algorithm for MMRCPSP in a molding industry. They demonstrated that their proposed algorithm could generate suitable schedules for the projects with a high number of activities and limited resources.

Nabipoor Afruzi et al. [31] studied a robust multi-project resource-constrained scheduling problem (MPRCSP) with uncertainty in activity duration. The main goal of this study was to maximize the total weighted tardiness of the projects. Küçüksayacıgil and Ulusoy [25] studied a bi-objective MPMMRCPSP. Their proposed objectives were to minimize the completion time of projects and the mean of the flow times for individual projects as well as maximizing the NPV of all projects. They implemented a hybrid GA to solve the problem.

In Table 1, briefly, important studies are presented, taking into account the payment method and modeling according to the objective functions.

Table 1 Literature overview

According to the literature review, the main contributions are described as follows:

• Developing a novel bi-objective mathematical model to consider the two practical objectives of NPV maximization and completion time minimization which is the extension of the proposed model by He et al. [19],

• Integrating the decisions of optimal allocation of payments to the events, optimal modes of activities, calculation of the income and expenses of the project on each event,

• Considering the real-world conditions of activities demand for renewable and non-renewable resources and possible multi-mode of activities which leads to the time and cost trade-off,

• Developing three efficient solution techniques the of ε-constraint method, NSGA-II and MOSA,

• Applying different comparing measures to test the performance of the solution methods.

3 Problem definition

Assume that there are several activities which have several required resources and several execution modes that each mode has a specific duration, cost and the amount of the required resources. The objectives of the proposed problem are to maximize the NPV and minimize the completion time of the project concurrently. The main questions of the problem are:

• 1. When and how much payment should be assigned to each event?

• 2. Which execution mode should be considered for each activity?

• 3. When is each event completed?

• 4. How much is the cost of each event?

• 5. How much is the income of each event?

Consider a project that its contractor’s initial capital is equal to ICA. Duration, cost, and demand of activity i for rth resource under mode j are defined by $dur_{ij}$, $\cos t_{ij}$, $dem_{ijr} ,$ respectively, where i = 1, 2, …, n; r = 1, 2, …, R and j = 1, 2, …, J. The available resources are also defined as $a_{r}$ that can be either renewable or non-renewable. The cost of each event m (m = 1, 2, …, M) is defined as follows:

$$ e_{m} = \sum\limits_{{i \in S_{m}^{start} }}^{{}} {\left[ {\zeta_{i} \,\sum\limits_{j = 1}^{J} {\text{cost}_{ij} y_{ij} } } \right] + } \sum\limits_{{i \in S_{m}^{end} }}^{{}} {\left[ {(1 - \zeta_{i} )\,\sum\limits_{j = 1}^{J} {\text{cost}_{ij} y_{ij} } } \right]} $$

(1)

where $S_{m}^{start}$ is the set of activities starting from mth event, $S_{m}^{end}$ is the set of activities that end at event m and $\zeta_{i}$ ($0 \le \zeta_{i} \le 1$) is the cost ratio distribution of the ith activity during its execution. The income amount of event m is calculated by $v_{m} = \sum\nolimits_{{i \in S_{m}^{end} }} {w_{i} } ,$ where $w_{i}$ is amount of the gained income from activity i. The final payment is done at the last event of the project which is $p_{K} = U - \sum\nolimits_{k = 1}^{K - 1} {p_{k} }$. Here, U, D and α are the contract cost, the considered completion time of the project and the return rate for each period, respectively where $U = \sum\nolimits_{i = 1}^{n} {w_{i} } .$

To describe the proposed network of the project, activity on arc (AOA) network is employed so that arcs show the activities and nodes represent the events. The suggested example is depicted in Fig. 2.

Fig. 2

General scheme of proposed AOA network [19]

3.1 Mathematical model

Indices and sets

i :

Index of activities

j :

Index of activity modes

k :

Index of payments

m :

Index of events

r :

Index of resources

t :

Index of time period

I :

Set of activities

J :

Set of activity modes

M :

Set of events

R :

Set of resources including renewable and non-renewable

K :

Set of payments

T :

Set of time periods

$S_{m}^{start}$ :

Set of activities that begin at event m

$S_{m}^{end}$ :

Set of activities that end at event m

Parameters

ICA :

Initial capital of the contractor amount

$dur_{ij}$ :

Duration of activity i under mode j

$\cos t_{ij}$ :

Execution cost of activity i under mode j

$dem_{ijr}$ :

Demand of activity i for resource r under mode j

$a_{r}$ :

Available amount of resource r

$E_{m}$ :

Earliest occurrence time of event m

$L_{m}$ :

Latest occurrence time of event m

$\zeta_{i}$ :

Cost distribution ratio of activity i during the start and end of the activities

$w_{i}$ :

Income amount of activity i

U :

Contract price of the project

D :

Project delivery time

α :

Rate of return in each period

Decision variables

$x_{km}$ :

Binary variable; it is equal to 1 if payment k is assigned to event m, otherwise, it is 0

$y_{ij}$ :

Binary variable; it is equal to 1 if activity i is done in mode j, otherwise, it is 0

$z_{mt}$ :

Binary variable; it is equal to 1 if the event m is completed in period t is equal to 1, otherwise, it is 0

$e_{m}$ :

Cost of event m

$v_{m}$ :

Income of event m

$p_{k}$ :

Amount of payment k

Now, the proposed bi-objective model for our MMRCPSP-DCF is defined as follows:

$$ {\text{Maximize}}\;{\text{NPV}} = \sum\limits_{k = 1}^{K} {\left\{ {p_{k} \sum\limits_{m = 1}^{M} {\left[ {x_{km} \sum\limits_{{t = E_{m} }}^{{L_{m} }} {(\exp ( - \alpha t)z_{mt} )} } \right]} } \right\}} - \sum\limits_{m = 1}^{M} {\left\{ {e_{m} \sum\limits_{{t = E_{m} }}^{{L_{m} }} {(\exp ( - \alpha t)z_{mt} )} } \right\}} $$

(2)

$$ \begin{aligned} & {\text{Minimize}}\;{\text{makespan}} = \sum\limits_{{t = E_{M} }}^{{L_{M} }} {t.z_{Mt} } \\ & {\text{subject to}} \\ \end{aligned} $$

(3)

$$ \sum\limits_{m = 1}^{M - 1} {x_{km} } = 1\quad k = 1,2, \ldots ,K - 1 $$

(4)

$$ x_{KM} = 1 $$

(5)

$$ \sum\limits_{k = 1}^{K} {x_{km} } \le 1\quad m = 1,2, \ldots ,M $$

(6)

$$ e_{m} = \sum\limits_{{i \in S_{m}^{start} }}^{{}} {\left[ {\zeta_{i} \,\sum\limits_{j = 1}^{J} {\text{cost}_{ij} y_{ij} } } \right] + } \sum\limits_{{i \in S_{m}^{end} }}^{{}} {\left[ {(1 - \zeta_{i} )\,\sum\limits_{j = 1}^{J} {\text{cost}_{ij} y_{ij} } } \right]} \quad m = 1,2, \ldots ,M $$

(7)

$$ \sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{J} {dem_{ijr} \,y_{ij} } } \le a_{r} \quad r = 1,2, \ldots ,R $$

(8)

$$ \sum\limits_{{t = E_{m} }}^{{L_{m} }} {z_{mt} } = 1\quad m = 1,2, \ldots ,M $$

(9)

$$ \sum\limits_{{t = E_{{s_{i} }} }}^{{L_{{s_{i} }} }} {\left( {z_{{s_{i} t}} .t} \right)} + \sum\limits_{j = 1}^{J} {\left( {dur_{ij} y_{ij} } \right)} \le \,\sum\limits_{{t = E_{{e_{i} }} }}^{{L_{{e_{i} }} }} {\left( {z_{{e_{i} t}} .t} \right)} \quad i = 1,2, \ldots ,n $$

(10)

$$ \sum\limits_{m = 1}^{M} {\left( {e_{m} \sum\limits_{t = 0}^{T} {z_{mt} } } \right)} \le ICA + \sum\limits_{k = 1}^{K} {\left[ {p_{k} \sum\limits_{m = 1}^{M} {\left( {x_{km} \sum\limits_{t = 0}^{T} {z_{mt} } } \right)} } \right]} \quad T = 1,2, \ldots ,D $$

(11)

$$ \sum\limits_{k = 1}^{K} {p_{k} } = U\quad k = 1,2, \ldots ,K $$

(12)

$$ \sum\limits_{j = 1}^{J} {y_{ij} } = 1\quad i = 1,2, \ldots ,\text{n} $$

(13)

$$ \sum\limits_{{t = E_{M} }}^{{L_{M} }} {\left( {z_{Mt} .t} \right)} \le D $$

(14)

$$ x_{km} ,\,z_{mt} ,\,y_{ij} \in \{ 0,1\} $$

(15)

The objective function (2) represents the amount of contractor’s NPV which is equal to the current value of payments minus all costs associated with the project. The objective function (3) represents the minimization of the total completion time of the project, which is equal to minimizing the final event of the project. Equation (4) denotes the assignment of payment k (k = 1, 2, …, K − 1) to a particular event. Equation (5) ensures that the last payment K should be assigned to the last event M. Equation (6) represents that only one payment occurs in a particular event. Equation (7) calculates total costs of an event. Equation (8) indicates the amount of available resources for activities under each mode. Equation (9) shows the occurrence time of event m in the possible time window $[E_{m} ,\,L_{m} ]$. Equation (10) denotes the precedence relationships. Here, $E_{{s_{i} }}$ and $L_{{s_{i} }}$ are the earliest and lateness occurrence time of the event that the activity i begins at it. Furthermore, $E_{{e_{i} }}$ and $L_{{e_{i} }}$ denote the earliest and lateness occurrence time of the event that the activity i ends at it. Equation (11) ensures that the sum of the contractor’s output financial flows should not exceed its initial capital plus the amount of the input financial flows. Equation (12) ensures that the sum of all payments is equal to the contract price of the project. Equation (13) ensures that each activity should be performed only by one execution mode. Equation (14) also ensures that the occurrence time of the final event should not exceed the project delivery time. Equation (15) defines types of the variables.

4 Solution methods

The proposed model is a mixed integer non-linear programming (MINLP) and due to its high complexity, it is NP-Complete [19]. Therefore, implementing heuristic and meta-heuristic approaches is necessary to solve the problem. On the other hand, in order to validate the proposed mathematical model, small-sized samples are solved by GAMS software-BARON solver using the ε-constraint method to cope with the bi-objectiveness of the model.

To solve the problems with large sizes, NSGA-II and MOSA algorithm are designed to solve the problem as the main methods and are evaluated in comparison with each other and the ε-constraint method. NSGA-II is one of the most common and powerful algorithms available to solve multi-objective optimization problems so that its effectiveness in solving various problems has been proven [11]. MOSA algorithm is a fast local search algorithm that has the ability to escape local optimum solutions [16]. This algorithm has much efficiency in solving problems with a discrete or on-convex solution space.

4.1 ε-Constraint method

The ε-constraint method is one of the well-known approaches to deal with multi-objective problems which can generate Pareto solutions [14]. The formulation of the ε-constraint method is as follows:

$$ \begin{aligned} & {\text{Minimize}}\;f_{ 1} (x) \\ & {\text{subject}}\;{\text{to}} \\ & f_{ 2} (x) \le \varepsilon_{2} , \\ & \ldots \\ & f_{n} (x) \le \varepsilon_{n} , \\ & x \in X. \\ \end{aligned} $$

(16)

The ε-constraint method steps are as follows:

• Step 1: Choose one of the objective functions to be introduced as the main objective function.

• Step 2: Solve the problem according to each single objective function, then obtain the optimal values of each objective function and the other obtained values for remaining objective functions. If we have n objective functions, we should solve the single-objective model for n times with all objectives. In each single-objective model, n values are determined for all n objective functions.

• Step 3: Find the two best values for each sub-objective functions. Divide the difference between these two values to a given number (the number of breakpoints) and create a table of values for $\varepsilon_{2} , \ldots ,\varepsilon_{\text{n}}$.

• Step 4: Now, solve the single-objective model with the main objective function for each value of $\varepsilon_{2} , \ldots ,\varepsilon_{\text{n}}$.

• Step 5: Report Pareto solutions findings.

In our proposed model, the first objective is considered as the main objective and the second objective is the sub-objective with 10 breakpoints. So the formulation is presented as follows:

$$ \begin{aligned} & {\text{Maximize}}\;f_{1} (x) \\ & {\text{subject}}\;{\text{to}} \\ & f_{ 2} (x) \le \varepsilon_{2} , \\ & x \in X. \\ \end{aligned} $$

(17)

4.2 NSGA-II

GA is an efficient algorithm which has been extensively applied to different optimization problems [27, 42, 47].

Srinivas and Deb [40] proposed the non-dominated sorting genetic algorithm for the first time; it divided the evolutionary group into several levels based on a dominance relation for selection and solution. Deb et al. [12] optimized an operational NSGA scale, such that elite mechanism was used instead of sharing coefficient of density function; this algorithm is known as NSGA-II. The proposed pseudo-code of the applied NSGA-II in this research is shown in Fig. 3.

Fig. 3

Pseudo-code of NSGA-II [2]

The main information about the mechanism of the proposed NSGA-II is as follows: two-point crossover operator and one-point mutation operator are used for crossover and mutation, respectively. Furthermore, the stopping condition of this algorithm is met when there is no improvement in 50 consecutive iterations. Moreover, the values of the parameters are determined by trial and error method which is described in Table 2.

Table 2 Optimal levels of the parameters for the proposed NSGA-II

4.3 MOSA algorithm

SA is another well-known fast metaheuristic algorithm has been studied to improve the solutions generated initially in different optimization problems [3, 43]. On the other hand, MOSA attempts to generate non-dominated solutions by using a simple probability function that tries to generate solutions on the Pareto optimal front. The probability function is varied in such a way that the total space of objective is covered uniformly obtaining as many possible non-dominated and well-dispersed solutions [46]. These features have made MOSA a fast efficient algorithm compared to the other existing multi-objective algorithms. Figure 4 illustrates the pseudo-code of MOSA algorithm. Furthermore, the mechanism of the suggested MOSA is adopted from that proposed by Kubotani and Yoshimura [24]. Note that MOSA is designed on the basis of SA considering the non-dominance concept, which is implemented by NSGA-II.

Fig. 4

MOSA pseudo-code [32]

The values of MOSA parameters are determined by trial and error method which is represented in Table 3.

Table 3 Optimal levels of the parameters for the proposed MOSA algorithm

5 Numerical results

In this section, model validation, sample problems generation and solution methods evaluation are presented and described.

5.1 Sample problems generation

To validate the proposed model and evaluate the developed solution techniques, 12 sample problems are designed in different scales where the input parameters are generated randomly. The required information about these samples is provided in Table 4. Moreover, Fig. 2 represents AOA network of problem 1, and Figs. 5, 6 and 7 depict AOA networks of problems 2–4, respectively. The other samples are so large to show their related AOA networks.

Table 4 Input information about samples

Fig. 5

AOA network of sample 2

Fig. 6

AOA network of sample 3

Fig. 7

AOA network of sample 4

It has been concluded that the ε-constraint method cannot solve the samples 5–12 by considering the run time constraint of 12,000 s. Thus, these samples are considered as large-sized problems.

Table 5 represents the obtained results by the ε-constraint method. It’s worth mentioning that problems are run on a laptop with specs (Intel Core i7-RAM 8 GB) by GAMS software and BARON MINLP Solver.

Table 5 Optimal values of objective functions for sample problems

In the following, 10 values of epsilons (breakpoints) are considered based on the third step of the ε-constraint method and then 10 Pareto points are obtained after implementing steps 4 and 5. These results are presented in Table 6 for the first sample in comparison with the other solution techniques.

Table 6 Pareto solutions obtained by NSGA-II, MOSA and ε-constraint method

5.2 Solution representation

In meta-heuristic algorithms, a solution string including binary matrices of K * M, I * J and M * T is used to display a feasible solution. Figure 8 shows the defined string related to solution representation.

Fig. 8

Chromosome-based solution representation

In the first part, if the payment k is assigned to the event m, the corresponding cell takes the value of 1; otherwise it takes 0. In the second part, if activity i is executed under mode j, the corresponding cell takes the value of 1; otherwise, it takes 0, and finally in the last one, if the event m occurs at period t, the corresponding cell takes the value of 1; otherwise, it takes 0.

5.3 Solution results

In this section, the numerical results obtained by the proposed solution methods are analyzed. First, in small and medium-sized problems, the results of NSGA-II and MOSA algorithms are compared with the results of the ε-constrained method. Since the ε-constrained method is not possible to solve the larger problems within 12,000 run time limitation, these problems are solved by the proposed metaheuristic algorithms and the obtained results are compared using different measures. It should be noted that the proposed algorithms are coded in MATLAB programming language.

Table 6 represents the obtained Pareto solutions by these three solution methods for sample 1. As is clear, NSGA-II could find 8 Pareto solutions and MOSA algorithm has found 6 Pareto solutions for the first sample.

According to Fig. 9, it is clear that the Pareto frontiers obtained by MOSA and NSGA-II are close to one obtained by the ε-constraint method. MOSA algorithm has a better and closer performance than NSGA-II for solving the sample 1. However, for more accurate evaluation of the proposed algorithms, some well-known measures including mean ideal distance (MID) measure, spacing metric (SM), diversification metric (DM) and run time are used. The definition of these measures is explained in Zitzler et al. [49]. The obtained results based on these measures are reported in Table 7 and also depicted in for the proposed methods.

Fig. 9

Pareto frontiers obtained by different solution methods for the first sample

Table 7 Comparison of the proposed solution methods in the small and medium sized samples

As can be seen in Fig. 10, the proposed meta-heuristic algorithms perform close to the ε-constraint method. MOSA has been better in the measure of MID and NSGA-II could obtain the superior results in terms of DM. However, they are very similar in terms of SM where each one may be better in different small and medium-sized samples. According to the obtained results in these samples, we can conclude that MOSA algorithm is slightly better than NSGA-II, but the important point is that the output of both algorithms is reliable and valid in comparison with the ε-constraint method. Therefore, they can be used as a suitable solution for large-sized problems.

Fig. 10

Comparison of the proposed solution techniques based on different measures in small and medium-sized samples

For further investigation, these algorithms are tested on large-sized samples too (Table 8 and Fig. 11).

Table 8 The comparison results of NSGA-II and MOSA algorithm in large-sized samples

Fig. 11

Comparison of NSGA-II and MOSA algorithm based on different measures in large-sized samples

According to Table 8 and Fig. 10, MOSA algorithm has a better performance just in MID measure while NSGA-II outperforms it in terms of SM and DM measures. Therefore, NSGA-II has a better overall performance and can be introduced as the most efficient algorithm to solve large-sized problems.

Finally, the optimal Pareto frontier obtained by these algorithms for a large-sized sample (No. 5) is depicted in Fig. 12.

Fig. 12

Pareto frontier of the NSGA-II and MOSA algorithm for the 5th sample

As it is clear in Fig. 12, NSGA-II can find more Pareto solutions than MOSA algorithm. In addition, the distance between two successive Pareto solutions is lower in NSGA-II. But Pareto frontier quality created by MOSA algorithm is much better because this algorithm is able to generate Pareto solutions closer to the ideal point (origin coordinates). Now, as a final comparison of the proposed solution methods, they are compared in terms of run time. Figure 13a, b illustrates these comparisons for all the proposed solution methods and two suggested algorithms, respectively.

Fig. 13

Run time comparison of the proposed solution techniques in all samples

As it is obvious, the required run time to solve the samples 1–4 by the ε-constraint method is growing exponentially to the extent that it cannot solve the samples 5–12 within the considered time limitation of 12,000 s. However, meta-heuristic algorithms could solve them in a much shorter time. Based on Fig. 13b, NSGA-II requires less run time to find its Pareto frontiers, and it can be the advantage of this algorithm against MOSA.

Consequently, the proposed NSGA-II and MOSA algorithms in this study can be regarded as effective tools for solving large-sized problems in reasonable time. As an important managerial insight, we can see that the time–cost trade-offs would generate solutions with negative NPV, which needs to be analyzed for different situations and goals. Accordingly, the management needs to investigate whether more resources should be provided or not, which can be done by the proposed methodology of this research.

6 Conclusions and suggestions

In this research, a multi-mode resource-constrained project scheduling problem with discounted cash flow (MMRCPSP-DCF) is proposed and formulated based on the real-world conditions considering renewable resources (including manpower, machinery and equipment) as well as non-renewable resources (including consumption and money). Furthermore, payments’ planning is studied in the problem which is realized by payments at event occurrence (PEO) model. Thus a bi-objective mixed-integer non-linear programming (MINLP) model is developed with the aims of NPV maximization and completion time minimization of the project. In order to solve and investigate the validity of the proposed model, different random samples are generated and solved by the ε-constraint method to cope with the bi-objectiveness of the model. On the other hand, two efficient metaheuristic algorithms include non-dominated sorting genetic algorithm II (NSGA-II) and multi-objective simulated annealing (MOSA) algorithm are also developed and implemented to solve the large-sized problems and generate optimal Pareto solutions. Finally, the performance of these solution methodologies is analyzed in terms of mean ideal distance (MID) measure, spacing metric (SM), diversification metric (DM) and run time. The obtained results demonstrate that MOSA algorithm has better efficiency in small-sized problems and NSGA-II outperforms MOSA algorithm in large-sized problems.

The outlook of the research is listed as follows:

• (1) Studying uncertainty in the problem, especially in activities demand, cost, or execution time and develop the model using efficient techniques such as stochastic programming, robust optimization, etc.,

• (2) Designing and testing other meta-heuristic algorithms such as multi-objective particle swarm optimization (MOPSO) and multi-objective variable neighborhood search (MOVNS),

• (3) Considering energy minimization in the project besides the other objectives so that there are a different interval of energy consumptions in each time period,

• (4) Investigating different levels of the required skills for each activity under each mode.