1. INTRODUCTION

Project management is a key activity in software development organizations. One of the key problems in software project planning is the scheduling, which involves resource allocation and scheduling activities on time, optimizing the cost and/or duration of the project. The scheduling is a complex constrained optimization problem, because for the optimization of resources, time, and cost, it is necessary to consider a combination of variables, rules, and restrictions that cause the problem to be NP-hard search-based software engineering (SBSE) is an emerging area focused on solving software engineering problems by using search-based optimization algorithms. SBSE has been applied to several problems which arise during the software life cycle. The software project scheduling problem (SPSP) consists in finding a suitable assignment of employees to tasks, such that the project can be delivered in the shortest possible time and with the minimum cost [1,2].

In recent years, a considerable amount of research has been conducted around this topic. The studies that have been performed have focused on different aspects, for instance, on proposing models to solve the problem of allocating employees to tasks so that the project’s cost and duration are minimized, on introducing or applying more efficient optimization algorithms, and on studying the performance of different metaheuristics to determine which may be more appropriate to solve the problem.

We summarize generalizations of the activity concept, the precedence relations and of the resource constraints. Alternative objectives and approaches for scheduling multiple projects are discussed as well. In addition to popular variants and extensions such as multiple modes, minimal and maximal time lags, and net present value-based objectives in Table 1 [1–6].

As we can see in Table 1, the RCPSP can be classified by preemptive scheduling, resource requests varying with time, setup times, multiple modes, tradeoff problems, minimal time lags, maximal time lags, release dates and deadlines, time-switch constraints, etc. The RCPSP is a general problem in scheduling that has a wide variety of applications in manufacturing, production planning, project management, and various other areas. In this study, we aim to identify the key aspects of procurement in the software project management context and their relation to project success [2].

Project management is the core process of most current business activities, which include projects that have the goal of creating products, services, or original results. Project management tasks include project scope management, quality, project schedule, budget, resources, and risks. Still, there are more internal problems affecting the project which is out of the management of risk management parts such as the conflicts between the project partners or the conflicts in risk management itself. Therefore, detecting and analyzing these problems brings a necessary supplement for the project management tasks, thus ensuring all matters arising to be controlled and also enhancing the quality and chances of success of the project [1,2]. Based on various factors such as the source, cause, role, or function of conflicts in the project, conflicts can be classified into several categories as follows:

• By source of conflict, conflicts are classified as (i) conflict from plan scheduling, (ii) conflict from determining the priority in performing project task, (iii) conflict from the power sources, (iv) conflict from technical problems, (v) conflict from administrative procedure, (vi) conflict from the private issue, and (vii) conflict from the expenditure.

• By cause of conflict, conflicts are classified as (i) conflict from different goals, (ii) conflict from resource disparity, (iii) conflict by other people’s obstruction, (iv) conflict due to stress and psychological pressure from many people, (v) conflict due to ambiguity of jurisdiction, and (vi) conflict due to misleading communication.

• By role, conflicts are classified as (i) positive conflict and (ii) negative conflict.

• By function, conflicts are classified as (i) functional conflict and (ii) dysfunctional conflict.

Procurement is viewed as a strategic function to improve an organization’s profitability. In recent years, it has been shown that there are many challenges and difficulties in procurement, especially in multi-round procurement, where the result of selecting bidders in one round affects bidding behavior in subsequent rounds [7].

• The first challenge is the difficulty in choosing the right bidders to maximize profits while also ensuring project completion, which becomes more and more challenging when it is divided into many stages with several auction-goers. The material depends on each company and their pricing policies, which must be considered in order to choose the most reasonable ones (Quality and Cost-based Selection). However, quality cannot be assured afterward [8]. This implies many risks, as there are no measures or evidence to prove that the chosen bidder is the best one. Moreover, a win-win relationship requires a fair auctioning environment, which is ideal when all bidders are granted the highest profits.

• Secondly, dividing the project into many stages affects the project’s total cost, which is related to time. Good sustainability strategies require lowering the cost and, hence, increasing the profits and reducing the predicted price of each package; otherwise, it can lead to package over-budget, which causes incompletion and risks for the project.

In summary, a scientific and applicable method is necessary to make the final decision of the auction [8–13]. Apart from that, the main questions which must be considered for the problem are as follows:

1. When is the best time to open a bid?

2. Who are the most suitable auction-goers?

3. What are the most suitable quality and price at each stage?

4. How can we solve this?

The term sustainability strategy can be explained as a prioritized set of actions which provides an agreed framework in a cooperation situation. In other words, there are countless benefits from a good sustainability strategy, in terms of cooperation in a project, making the jobs of participants easier because long-term cooperation can be maintained by a rational plan of action. Rationality makes sustainability more effective and easier when it comes to project integration. Rationality can be obtained by forming a win-win situation for all sides.

Despite the fact that, in some special types of problems, Nash equilibrium may not perform as well as other models, such as competitive equilibrium (i.e., when dividing the channels into two categories: high-quality and low-quality), it is still the most suitable model to find the appropriate benefit for all players by determining a win-win relationship [14–17].

The multi-round procurement problem was being an NP-hard problem, solution methods are primarily meta-heuristics. The above questions have already been answered by some researchers using genetic algorithms (GAs) and the Nash equilibrium [14,18]. However, when the threshold is changed, the performance of each algorithm is different.

Particle swarm optimization (PSO) and evolutionary algorithms such as GA have been used to solve multi-objective problems; they have many similar behaviors but also have differences in performance across certain problem areas. They both have the same mechanism, in that the system is initialized by generating a population randomly and searching for optima by refining this population. Nonetheless, PSO does not use techniques such as cross-over and mutation for updating and, so, may prematurely converge to a local optimum. The authors prove that PSO and GA have the same effectiveness in solution quality, but that PSO has superior computational efficiency over GA. The advantages of PSO are that it uses fewer parameters and evaluation functions than GA [19].

The main idea of GA is combining father and mother singles to generate many descendants, in order to create a better generation. In contrast, the PSO Algorithm focuses on developing its own members to better themselves. Therefore, instead of using GA and generating 100 particles, we can use PSO with fewer individuals and let them improve themselves [20,21]. For these reasons, we propose a new approach using PSO and Nash equilibrium to identify a sustainability strategy in a multi-round procurement problem which has higher accuracy and better speed. The following sections explain further details about the problem under consideration.

Our main contributions is presented a novel approach for solving a multi-round procurement. In this approach, by combining knowledge from the fields of game theory, Nash equilibrium, and PSO algorithm, we presented a game theory model as an optimal cooperative solution for the multi-round procurement problem. The parameter setting strategy is selected in an attempt to fine-tune the PSO algorithm for this problem. The Nash equilibrium point can be considered as the best solution for the conflict. The Nash equilibrium point can be solved by multi-objectives evolutionary algorithm. Compared with another multi-objectives evolutionary algorithms, the result suggested that PSO algorithm was significantly better than others with shorter runtime and higher quality for finding Nash equilibrium candidate in large-scale dataset.

2. RELATED WORKS

The software project management was being an NP-hard problem, solution methods are primarily meta-heuristics. We classify and present the state-of-the-art hybrid meta-heuristics for software project management and multi-round procurement problems in Table 2 [20–29]. These methods can be categorized based on local search meta-heuristics, evolutionary and population-based, learning meta-heuristics, and our proposed algorithms

In [12], the authors designed a new approach to find a winner in a multiple-attribute auction. The new method could help to address a mechanism for decision-making in dealing with multiple-attribute and multiple-sourcing procurement. Multiple-sourcing is usually proposed to prevent a variety of procurement risk and uncertainty. In [22], a model was developed to assist participants in multiple-attribute procurement in inferring their preference, based on the difficulty of elicitation.

In [23], the ability to use Bayesian equilibrium to predict human behavior in sealed-bid split-award procurement auctions was analyzed. The auction type was a multi-object extension, in which bidders may require more than one unit. The authors applied the Bayes–Nash equilibrium in the expanded game to forecast the strategies of participants. Then, they experimented with the accuracy of the equilibrium with computerized bidders. The results showed that the complexity of the multi-object does not have much effect on the result of procurement; rather, the risk aversion has a more pronounced effect.

In game theory, Nash equilibrium has been used to analyze the outcome of strategic interaction and also how conflict may be mitigated in competitive environments. A Nash solution is a mixed strategy profile with the property that no single player can obtain a higher value of expected utility by deviating unilaterally from this profile. It means all players obtain more revenue under cooperation, and thereby promoting sustainability. The above questions are already answered by some researchers using the GA and the Nash equilibrium. However, when the number of a threshold is changed, the performance of each algorithm is different. Furthermore, there are many meta-heuristic algorithms were proposed, such as GDE3 [29], ϵ-MOEA [31], ϵ-NSGA-II [33], NSGA-III [32], SPEA2 [36], ACO [25], Water drops [28], ANGEL [38], HEA [39], (1+1) EA and RS [40], EHH [41], Hybrid ABC-PSO [42], PSO Parallel (PPSO) [46], ACO Parallel (PACO) [48].

In [49], we proposed a GA model to infer the Nash equilibrium model for a multi-round procurement problem, where the buyer is not only focused on the revenue under the first-price bidder strategy, but also considers a variety of properties for bidder values (e.g., the relationship between investor and bidder, offer for future promotion, and the business credit score of the bidder). The chromosome in the GA represents the Nash equilibrium point, which includes all characteristics of the solution for the problem. An iterative GA is said to converge when, as the iterations proceed, the fitness value gets closer to specific values; we found a balance point for all sides in procurement.

In our latest paper [50–54], we proposed a unified game-based model based on the concept of game theory and Nash equilibrium. This scientific model can help the decision-maker to recognize and define all characteristics of the conflict problems in project management. The model can be solved by a variety of multi-objective evolutionary algorithms. We introduced some important conflicts in project management and proposed a mathematical model addressing these problems. Our experimental results showed that the unified game-based model is useful to address these problems and effective in generating a near-optimal balance point.

3. MULTI-ROUND PROCUREMENT PROBLEM FORMULATION

4. A NOVEL PSO APPROACH TO SUPPORT DECISION-MAKING IN THE MULTI-ROUND OF AN AUCTION BY GAME THEORY

4.1. Nash Equilibrium Theory for Multi-Round Procurement Mixed Solution

Game theory is “The study of mathematical models of conflict and cooperation between intelligent rational decision-makers.” It has applications in a variety of fields of social science as well as in computer science [15,16]. In this research, we present a decision-supporting approach based on game theory. Multi-round procurement as a process relating to many participants requires a balance of harms and benefits of the decision. Using game theory, real-world conflicts for such situations as pricing competition and relationship negotiation can be laid out [17].

We study infinite games of perfect information played by multiple players, all players of a game pick and join to create a blended procedure, which is a combination of beliefs about probabilities over strategies and the choices of the other player. Technically, a Nash equilibrium is a set of mutual strategies that is considered as most beneficial by all parties and no player will defer from it. As mentioned in John Nash’s theory, the systems or joined moves are no mystery. The benefit value represents the quality of each player’s profile of action in terms of players cooperatively trying to reach an agreement [14].

Nash equilibrium is one of the most important tools to comprehend the contention between various players. For example, in the Prisoner’s dilemma, two detainees are scrutinized for the same crime. Provided that player A defects to the police and confesses that player B committed the crime, A gets his sentence removed and B gets full jail time. If both defect from each other, they get equally long prison sentences, and if both parties cooperate and remain silent, they each get reduced jail time. On the off chance that both parties blame each other, that implies a conundrum, both will be placed in prison for a long time (see Figure 1).

Figure 1

Solving conflict with Nash equilibrium.

Nash equilibrium is a widely utilized model to comprehend the contention between numerous players. For example, the two detainees question, if two detainees are scrutinized a similar interest that on the off chance that he concedes the other individual has made the wrongdoing, he can has ten years rebuff by detainment diminished or else both will be placed in prison for a long time. On the off chance that every one of the two blames the other, that implies a conundrum, both will be placed in prison for a long time. On the off chance that one denounces, the other stay silent, the tranquil individual will get longer discipline.

The conditions of the problem are shown in Table 3.

AB	Defect	Cooperate
Defect	2.2	0.3
Cooperate	3.0	1.1

Table 3

The problem circumstances.

By far the best strategy for the prisoners is cooperating to get a short jail term for them both. By contrast, receiving a long jail term is a bad strategy. As it has been pointed out, the problem draws a conflict for each individual in which they do not want to cooperate. It is a choice between competition and cooperation. The agreement to cooperate can guarantees that both win. By contrast, in a noncooperative situation, even a more attractive strategy can lead to worse results.

4.2. A Novel PSO Algorithm Applied for Multi-Round Procurement Problem

PSO [20,55–57] is an advanced computational technique which has been used previously, similar to GA [18], differential evolution (DE) [19], and ant colony optimization (ACO) [25,58,59]. The authors which proposed the algorithm were inspired by the idea of swarm intelligence, as typically seen in animal groups, such as birds, fish, and insects. PSO was first proposed by Russell Eberhart and James Kenedy in 1995 with the main idea of using individual self-experience combined with the social experiences of the entire population. However, the PSO algorithm is not the same as the GA algorithm, as it searches for optima by evaluating individual fitness values based on co-operation and competition among individuals. The PSO algorithm is the consequence of considering a herd of feathered creatures finding sustenance in a specific area. It is a metaheuristic algorithm based on the concept of swarm intelligence, which is useful in solving complex optimization problems [20].

In summary, the PSO algorithm implements a heuristic search through a technique that depends on joining every particle’s conduct and the entire swarm’s conduct by modeling an organic population. The distinction between the PSO and GA algorithms is extremely delicate. Both of them utilize the idea of ensuring the balance of the diversity of the population and similarly rely upon the information shared between their population to elevate their inquiry forms by joining probabilistic and deterministic standards. Therefore, they bring about similar arrangement quality. The idea which distinguishes PSO is its use of neighborhoods. Dissimilar to other heuristic search techniques, in PSO, every particle’s neighborhood is kept invariant. Experimental studies have shown that the PSO algorithm exhibits better robustness and converges much closer to the global optimum than the GA algorithm [20,27,28].

The swarm is comprehended as an accumulation containing various particles which impact the calculation productivity. Be that as it may, particles can be picked haphazardly by and large. A population of particles is initialized with arbitrary speeds and positions, and the wellness capacity is assessed by utilizing a particle’s location and direction. The process initiates the population of particles (N) with some random values for position X and velocity V.

• Position: Xi=xi1,xi2,..,xiN∈RN.

• Velocity: Vi=(νi1,νi2,..,νiN)∈RN

Each particle maintains knowledge of its best previous evaluated position, represented as Pi=(pi1,pi2,..,pi,N)∈RN, and also has knowledge of the single global best solution found so far. Then, is it decided whether the optimization function F(opt) needs to be minimized or maximized. The two best values for each particle, pBesti=f(Pi) and gBest=f(Pg), Pg∈RN, are calculated, where pBest is its own best solution in its history (fitness) and gBest is the best value collected out of all particles in the population. The comparison of fitness values is done with the pBest value, and the gBest value is replaced when the current fitness value is found to be better than the existing gBest value; considering this, the particle with the best fitness value among the whole population is selected. This is not quite the same as GA, which utilizes various particles for each generation created.

4.2.1. Initialize particles

The representation step plays an important role in the successful design of PSO-MRP algorithm, which aims at finding an appropriate mapping between problem solution and PSO-MRP particle. Assuming that N particles form a population, each particle in the initial solution for the problem was generated randomly. A chromosome is the set of strategies selected for each generation. The best global chromosome found can be considered as the Nash equilibrium point, and the chromosomes in each generation are candidates for the Nash equilibrium point.

The chromosome should bring useful information to represent the solution for the multi-round procurement problem. We propose the format of the Nash equilibrium chromosome=[4,2,4,0,3,1] as follows (see Figure 2):

Figure 2

Nash equilibrium’s format.

A generation includes the number of rounds in procurement, where each round contains bidding and player information. In the first generation, the estimation of the fitness value of every player is random.

4.2.2. Parameters selection

In each new generation, the algorithm performs the following steps:

• Step 1: Call the fitness function and provide fitness values to adjust the priority of particles.

• Step 2: Adjust the optimal solution (i.e., global optimum).

• Step 3: Update the speed vectors.

Parameters selection of PSO-MRP will play an important role in performance and efficiency of the algorithm. Based on the analysis of the problem, this paper propose an PSO-MRP with the following parameters as shown in Table 4.

Parameters Name	Variable	Value
Number of particles	N	100
Number of iterations	Max	1000
Inertia weight	wmin and wmax	0.4 and 0.9
Cognitive coefficient	c1	0.5
Social component	c2	0.5
Stopping criteria	StopCondition	1000

Table 4

Selected control parameters in PSO-MRP algorithm for multi-round procurement problem.

4.2.3. Fitness function

Multi-stage procurement has a lot of cases and possibilities which can happen, considering the extra conditions given by bidders and investors in order to gain satisfactory benefits. Hence, the constraints considered were as follows:

• The whole project is divided into many stages, where the total time of each round equals the project length and each round lasts at least 1 month;

• Bidders joining must provide sufficient materials and participate until the end of the project. Moreover, they can join at any stage they want, within their capability.

The constants used are the values that maintain the balance of adaptive function factors. These constants have the role of adjusting the fitness function, in order to direct all of the particles to convergence while satisfying the criteria to get the best result. This function is used to ensure the benefit of both sides in obtaining the maximum profits. Therefore, a solution to optimize the objective function is required.

Particles are evaluated through a fitness function, the fitness value represents the quality of the solution that the particle expresses and also shows its influence in the population. In this problem, we must bring the Fitness function to a minimum, the fitness value is calculated by

Fitness=A∑i=1p∑i=1n∑j=1mxjciIije−r.tj∑i=1n∑j=1mxjciCije−r.tj−BA.e−r.tFinish∑i=1n∑j=1m(xjPj)ie−r.tj−1+SFC2(14)

or

Fitness=|AFc−BF0|+SFC2→Min(15)

where

• A,B are constants

• Fc is the total profit of bidders

• Fo is the benefit of project owner

• SFC2 is the variance benefit of bidders

4.2.4. Personal best, global best, and particle update

In each generation, particles are updated by comparing their fitness value to that of their best position pBest in all previous iterations. The pBest value is replaced by a new one, if the current fitness value is better. In the next step, the fitness value of each particle is compared with the global best of the population (gBest), where gBest is replaced if a better fitness value is obtained (see Figure 3).

Figure 3

Outstanding particles.

The model to change nodes is if the gBest member has a lower cost, higher quality, and higher association with the venture proprietor than that of particle i. These criteria help to pick the best possible closeout member with the best cost while ensuring the quality of the items delivered. These models are evaluated as a point. If the criteria are met, the point is raised up; otherwise, they are dropped down by one unit. In the event that the fact is sure, the node will be replaced.

When the kth particle moves to new position Xk, if a greater fitness value is achieved, its personal best pBestk is replaced with Xk. If fitness value of any pBest is greater than current gBest, gBest is replaced with that pBest. Particle i is updated, based on gBest (see Figure 4).

Figure 4

Example of update process.

The velocities of particles are updated regularly based on its best experience pBest and the leader gBest. The below equation indicates how kth particle updates its velocity.

Vk(t+1)=w.Vkt+c1r1(pBestkt−Xkt)+c2r2(gBestkt−Xkt)(16)

in which

• c1 and c2 are positive acceleration constants, which control the influence of pBest and gBest on the search process.

• r1 and r2 follow uniform distributions in the range of [0,1].

• w is inertia weight, which usually decreases from a large value (wmax=0.9) in the beginning to a small value (wmin=0.4) in the last iteration to control the exploration and exploitation abilities of the swarm.

The update of the velocity has been considered an important step since the first generation of development of the PSO-MRP algorithm. Velocity regulation is carried out according to the number of particles that exceed the feasible area during the optimization process. The velocity vector for a particle i is updated according to three other vectors: (i) the personal best position of the ith particle; (ii) the best position found over the whole swarm; and (iii) the current position of the particle. It is necessary to reduce the velocities of the particles by speeding up the attenuation rate of the inertia weight and reducing the maximum allowed velocity (see Figure 5).

Figure 5

Velocity update process.

The new position is updated by

Xk(t+1)=Xkt+maxi=1..NVkt+1(17)

4.2.5. Stop conditions

For efficient implementation of the PSO-MRP algorithm, we have to determine when to stop iterating. The stop condition of the algorithm can be classified into two techniques:

• Firstly, in real-world problems in which a desirable solution range may be available, the stop condition of the algorithm is based on the closeness of the potential solution to this range;

• Secondly, the iteration can be stopped when a specific number of generations has been reached.

4.2.6. Implementation of PSO-MRP algorithm for multi-round procurement problem

The PSO-MRP algorithm is described in Algorithm 1. Details of the implementation of the PSO-MRP algorithm for multi-round procurement can be found in the supplementary material files.

5. EXPERIMENTS AND RESULTS

5.2. Testing Result

5.2.1. Testing environment

Our experiments were conducted on a computer configured as shown in Table 10.

Parameter	Configuration
CPU	Intel Core i5 6200U (Skylake), 2.7GHz
RAM	8 GB
Operating System	Windows 10 Pro
Programming language	Java 11, json-simple-1.1.1.jar

Table 10

Testing environment.

5.2.2. Case study 1

In the first experiment, we installed and compared the performance of six algorithms, GDE3 (The third evolution step of generalized DE) [29], Multi-Objective Evolutionary Algorithm(ϵ-MOEA) [32], Strength Pareto Evolutionary Algorithm 2(PESA2) [33], NSGA-III [35], Nondominated Sorting Genetic Algorithm(ϵ-NSGA-II) [36], and our PSO-MRP algorithm on data set 1. We compared two criteria: the payoff value of the solution and the runtime. The payoff of the solution was calculated by summing the “goodness” for each corresponding weighting goal. Thus, the payoff value was within [0,1]; the closer to 0, the better the solution.

After testing the program 10 times, the populations were randomly initialized, including 100 individuals; the maximum number of individuals created for each runtime was 10,000. The comparison of the payoff parameter and the runtime on data set 1 is given in Table 11. We compare the performance of our algorithm and other algorithms, in terms of best, worst, and average, in Figure 6.

	NSGA-III [35]		ϵ-MOEA [32]		GDE3 [29]		PESA2 [33]		ϵ-NSGAII [36]		PSO-MRP
Ord	Payoff	Time	Payoff	Time	Payoff	Time	Payoff	Time	Payoff	Time	Payoff	Time
1	0.2961	4.197	0.2755	6.602	0.2805	5.981	0.2741	3.630	0.2744	8.954	0.2805	4.300
2	0.3322	4.178	0.2714	6.633	0.2799	5.992	0.2743	3.051	0.2765	8.921	0.2811	4.369
3	0.2911	4.209	0.2711	6.590	0.2799	5.974	0.2747	3.555	0.2801	8.933	0.2816	4.321
4	0.2966	4.166	0.2706	6.602	0.2811	6.000	0.2751	3.525	0.2809	9.102	0.2796	4.256
5	0.2979	4.178	0.2754	6.613	0.2821	5.981	0.2714	3.617	0.2712	8.990	0.2801	4.273
6	0.3025	4.184	0.2755	6.595	0.2797	6.112	0.2777	3.504	0.2708	9.051	0.2823	4.291
7	0.3105	4.189	0.2723	6.621	0.2797	5.969	0.2750	3.550	0.2714	8.915	0.2841	4.315
8	0.2923	4.171	0.2746	6.608	0.2810	5.980	0.2707	3.544	0.2721	8.956	0.2800	4.324
9	0.2940	4.190	0.2704	6.615	0.2807	5.987	0.2721	3.601	0.2725	8.937	0.2799	4.210
10	0.3025	4.178	0.2705	6.601	0.2805	5.994	0.2733	3.539	0.2708	8.929	0.2796	4.356

Note: NSGA, non-dominated sorting genetic algorithm; MOEA, multi-objective evolutionary algorithm; GDE3 3rd evolution step of generalized DE; PESA2, strength pareto evolutionary algorithm 2; PSO, particle swarm optimization; MRP, multi-round procurement.
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Table 11

Comparison of the payoff parameter and runtime (seconds) on data set 1.

Figure 6 shows the difference in convergence and stability of the considered algorithms. The proposed PSO-MRP algorithm met both criteria, showing good convergence in the fitness function and fast execution.

Figure 6

Comparison of payoff and runtime (second) in the best, average, and worst results in data set 1.

5.2.3. Case study 2

In the second experiment, we performed the same experiment as in case study 1 on data set 2. The comparison of the payoff parameter and the runtime on data set 2 is given in Table 12. We also compare the performance of our algorithm and other algorithms, in terms of the best, worst, and average, in Figure 7.

	NSGA-III [35]		ϵ-MOEA [32]		GDE3 [29]		PESA2 [33]		ϵ-NSGAII [36]		PSO-MRP
Ord	Payoff	Time	Payoff	Time	Payoff	Time	Payoff	Time	Payoff	Time	Payoff	Time
1	0.2674	5.6390	0.2590	7.6990	0.2555	7.5150	0.2671	5.6690	0.2555	12.4990	0.2569	5.4960
2	0.2654	5.6360	0.2596	7.6960	0.2545	7.5170	0.2702	5.6470	0.2569	12.5010	0.2555	5.4750
3	0.2666	5.6330	0.2610	7.7210	0.2569	7.4860	0.2666	5.6210	0.2518	12.4200	0.2547	5.5020
4	0.2668	5.6230	0.2591	7.7100	0.2518	7.4970	0.2697	5.5870	0.2525	12.4250	0.2603	5.5140
5	0.2697	5.6990	0.2610	7.6550	0.2581	7.5250	0.2653	5.5920	0.2522	12.3870	0.2522	5.4770
6	0.2701	5.5870	0.2586	7.6470	0.2525	7.5290	0.2666	5.6210	0.2557	12.4650	0.2518	5.4210
7	0.2670	5.6210	0.2589	7.6960	0.2533	7.5330	0.2701	5.5880	0.2525	12.4690	0.2555	5.4330
8	0.2655	5.6520	0.2587	7.7100	0.2529	7.5290	0.2655	5.6490	0.2551	12.4700	0.2569	5.4350
9	0.2654	5.6270	0.2590	7.6990	0.2519	7.6000	0.2654	5.6760	0.2543	12.4340	0.2566	5.4150
10	0.2680	5.6190	0.2599	7.6870	0.2520	7.5550	0.2680	5.6960	0.2569	12.4530	0.2589	5.4500

Note: NSGA, non-dominated sorting genetic algorithm; MOEA, multi-objective evolutionary algorithm; GDE3 3rd evolution step of generalized DE; PESA2, strength pareto evolutionary algorithm 2; PSO, particle swarm optimization; MRP, multi-round procurement.

Table 12

Comparision of payoff parameter and runtime (seconds) on data set 2.

Figure 7 shows the differences in convergence and stability of the considered algorithms. The proposed algorithm had the best runtime.

Figure 7

The comparison of payoff and runtime (second) in the best, average, and worst results in data set 2.

5.2.4. Case study 3

After testing the program 10 times, the result of the solution is listed below:

1 Nash Equilibrium found as follows:

2 Number of particles: 10

3 The Nash solution:

4 Chromosome:

5 [4 0 4 2 1 0 ]

6

7 Fitness: 30642322

8 The benefit of bidders: 23624700

9 Bidder0: 128295300

10 Bidder1: 65719300

11 Bidder2: 3999400

12 Bidder3: 0

13 Bidder4: 38233000

14

15 The benefit of project owner: 14472956

16 Investor payment: 13181444

17 Estimated price: 27654400

18

19 Successfully obtained after 0 hours 6 minutes 5 seconds

The comparison of fitness and runtime (minutes) of our algorithm on data set 3 is shown in Table 13. Our proposed algorithm demonstrated stability and convergence of estimated cost and payment values in different experiments.

			Investor (thousand VND)			Bidder (thousand VND)
Ord	Time	Fitness	Estimated cost	Payment	Benefit	0	1	2	3	4
1	6:05	33476600	27654400	13181444	14472956	2211446	143314	79090	0	157556
2	6:08	29213000	27654400	13181444	14472956	1458516	143314	183402	0	224774
3	5:55	32380516	27654400	13181444	14472956	2211446	283406	39994	0	157556
4	5:51	29300604	27654400	13181444	14472956	1542919	138075	183402	0	157556
5	5:51	28376784	27654400	13181444	14472956	1204395	283406	18340	0	224774
6	6:13	31539874	27654400	13181444	14472956	940112	657193	39994	0	690008
7	6:08	27531258	27654400	13181444	14472956	940112	423862	183402	0	233302
8	6:10	29300604	27654400	13181444	14472956	1542919	138075	183402	0	157556
9	6:28	31750880	27654400	13181444	14472956	1960984	240098	79090	0	224774
10	6:16	28376784	27654400	13181444	14472956	1204395	283406	183402	0	224774

Table 13

Comparision of fitness and runtime (minutes) of our algorithm on data set 3.

Our proposed algorithm shows the stability and convergence of values estimated cost and payment in different experiments. Through experiments shows that the values of estimated cost (27,654,400), investor payment (13,181,444), and benefit of project owner (14,472,956) are not changed. However, the distribution of the bidder values will fluctuate, resulting in a fluctuation in the fitness function (see Table 14)

	Min	Max	Mean	std
Time	5.51	6.28	5.945	0.29789
Fitness optimal	27531258	33476600	30124690.4	1999915.156
Bidder 0	940112	2211446	1521724.4	474601.161
Bidder 4	138075	657193	273414.9	163915.868
Bidder 2	18340	183402	117351.8	71889.476
Bidder 4	157556	690008	245263	159996.683

Table 14

Analysis and statistical evaluation of time, fitness, bidders’ value of PSO-MRP algorithm in data set 3.

5.2.5. Case study 4

To evaluate the performance and convergence of our algorithm in terms of both the fitness and time functions, we compared the performance between the different meta-heuristic algorithms on a large-scale project in data set 4, which had a large number of bidders, resources, items, proposals, and budgets. The experiment results are listed in Tables 15 and 16, which show that our algorithm had the best performance and quality of Nash equilibrium point when the fitness values were lowest.

Ord	NSGA-III [35]	ϵ-MOEA [32]	GDE3 [29]	PESA2 [33]	ϵ-NSGAII [36]	PSO-MRP
1	1,307,737	1,251,764	1,028,268	1,269,256	1,240,694	1,055,274
2	1,275,423	1,240,213	1,087,182	1,284,698	1,259,906	953,342
3	1,280,605	1,440,048	1,048,430	1,289,464	1,165,777	934,703
4	1,369,824	1,113,870	1,118,879	1,137,605	1,142,129	998,258
5	1,251,295	1,175,737	1,171,253	1,225,694	1,268,246	1,064,634
6	1,297,487	1,343,591	1,186,463	1,020,719	1,209,126	953,237
7	1,390,135	1,208,916	1,148,664	1,294,479	1,146,635	1,037,284
8	1,362,089	1,299,300	1,006,452	1,046,804	1,164,376	935,569
9	1,334,285	1,123,940	1,186,586	1,272,498	1,145,861	998,714
10	1,246,259	1,368,942	1,164,761	1,273,764	1,156,935	1,090,015

NSGA, non-dominated sorting genetic algorithm; MOEA, multi-objective evolutionary algorithm; GDE3 3rd evolution step of generalized DE; PESA2, strength pareto evolutionary algorithm 2; PSO, particle swarm optimization; MRP, multi-round procurement.

Table 15

Comparison of the fitness values of different algorithms on data set 4.

Ord	NSGA-III [35]	ϵ-MOEA [32]	GDE3 [29]	PESA2 [33]	ϵ-NSGAII [36]	PSO-MRP
1	10m 20s	15m 43s	13m24s	12m 59s	9m 12s	7m 07s
2	10m 42s	15m 48s	13m37s	14m 36s	10m 07s	6m 23s
3	11m 6s	15m 29s	14m 02s	15m 13s	11m 01s	6m 11s
4	9m 53s	14m 55s	14m 22s	14m 23s	9m 46s	7m 25s
5	9m 48s	15m 17s	13m 45s	12m 15s	9m 22s	6m 48s
6	11m 27s	14m48s	13m 51s	12m 23s	10m 56s	6m 29s
7	11m 1s	14m 35s	13m 11s	14m 53s	8m 54s	7m 16s
8	9m 46s	16m 01s	12m 35s	13m 49s	9m 36s	7m 13s
9	10m 31s	15m 24s	12m 41s	13m 08s	9m 57s	6m 49s
10	10m 33s	15m 47s	14m 48s	12m 52s	10m 24s	6m 27s

NSGA, non-dominated sorting genetic algorithm; MOEA, multi-objective evolutionary algorithm; GDE3 3rd evolution step of generalized DE; PESA2, strength pareto evolutionary algorithm 2; PSO, particle swarm optimization; MRP, multi-round procurement.

Table 16

Comparison of the runtimes of different algorithms on data set 4.

Figure 8 shows the differences in convergence and stability of the considered algorithms. The proposed algorithm had both the best runtime and best fitness.

Figure 8

Comparison of fitness and runtime (second) in the best, average, and worst results in data set 4.

5.2.6. Case study 5

In this experiment, the PSO-MRP algorithm was executed 100 times with randomly initialized populations including 100 individuals, where the maximum number of individuals created for each runtime was 10000. We evaluated the stability of the algorithm based on certain parameters, such as the minimum (Min), maximum (Max), mean, variance, and standard deviation (Std) of fitness, as well as time cost, in Tables 17 and 18.

	Payoff value					Time cost (second)
Dataset	Min	Max	Mean	Variance	Std	Min	Max	Mean	Variance	Std
1	0.280	0.284	0.281	2.095E-06	0.0014476	4.210	4.369	4.302	0.0022183	0.0470992
2	0.252	0.260	0.256	7.017E-06	0.0026489	5.415	5.514	5.462	0.0012731	0.0356801

Table 17

Analysis and statistical evaluation of convergence of the proposed PSO-MRP algorithm in data sets 1 and 2.

	Fitness optimal				Time cost (second)
Dataset	Min	Max	Mean	Std	Min	Max	Mean	Std
3	27,531,258	33,476,600	28,915,472	2315.3072	5.51	6.28	5.945	0.297890733
4	934,703	1,090,015	975,453	57,217.74	371	445	409	25.667

Table 18

Analysis and statistical evaluation of convergence of proposed PSO-MRP algorithm in data sets 3 and 4.

5.2.7. Discussion

From the experimental results in Tables 4 and 5, the ϵ-NSGA-II algorithm gave the best results with 2 data sets; otherwise, PESA2, NSGA-III, and PSO-MRP had the fastest running times. In addition, we have some comments and evaluations, as follows:

• The order of runtime was similar in the two data sets.

• Both the ϵ-MOEA and ϵ-NSGA-II algorithms were the worst-performing algorithms, due to the ϵ-dominance calculations taking much more time than conventional ones.

• In both data sets, the NSGA-III algorithm gave the worst results, as it very easily became stuck at local solutions.

• The ϵ-NSGA-II algorithm had good results in both data sets; so, while using ϵ-dominance is often costly, in terms of time, it had significant efficiency.

• Both ϵ-MOEA and PESA2 divided the target space into hyperboxes and, so, the algorithm efficiency depends on the width of the target space; therefore, there were differences in the result order, compared to the other algorithms, on the two data sets.

• Similarly, PSO-MRP and GDE3 had in common that the selection was not carried out through a mating process but, instead, was based on an individual transformation (mutation, or change in rules); therefore, the resulting orders had similarities in the two data sets.

From overall observation, the authors drew some conclusions for case study 3:

• The result after each run converged and shared a similar solution.

• Contractors were likely to have similar interests.

• The choosing decision was based on many other criteria, such as the price, the relationship, and the quality of each bidder. However, more trustworthy contractors were more likely to be chosen; especially bidder 3, which did not have any relationship with the investor.

• The program took some time and depended on the size of the swarm population and the max running fold.

The proposed PSO-MRP algorithm met both criteria, demonstrating good convergence in the fitness function and fast execution. Our algorithm can, thus, provide the best performance in terms of payoff and time taken with limited iterations, as seen in case studies 4 and 5.

6. CONCLUSION

This paper at hand presents a novel approach on how to combine Game theory and PSO algorithm to support decision-making in the multi-round of an auction. In the first phase, we propose a game model which brings all needed information, criteria, and constraints of the problem. In the second phase, the PSO algorithm is utilized with a suitable set of parameters to figure out the Nash equilibrium point. We proved that conflict, such as the competition and cooperation in multi-round procurement, can be resolved by defining a sustainability strategy that brings all participants of the problem into a win-win solution. Additionally, the model was successfully applied in our experiments, demonstrating the usefulness of the Nash equilibrium point for the multi-round procurement issue.

However, restrictions have not vanished, as procurement is typically a secure project. Therefore, the data gathering process encountered many difficulties and information was ultimately lacking. Moreover, the program requires an optimized solution and may need to be run several times to get the best result. Some unexpected risks and issues during the project were not calculated. Therefore, it is still necessary to do more experiments and research to develop an optimized method.

For future research, the authors wish to improve the optimization of the algorithm, reduce the time cost to run the program, and complete the software development.

CONFLICTS OF INTEREST

The authors have declared no conflicts of interest.

ACKNOWLEDGMENTS

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 102.03-2019.10.