Solving Multi-Mode Time-Cost-Quality Trade-off Problem in Uncertainty Condition Using a Novel Genetic Algorithm

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1 International Journal o Management and Fuzzy Systems 2017; 3(3): doi: /j.ijms Solving Multi-Mode Time-Cost-Quality Trade-o Problem in Uncertainty Condition Using a Novel Genetic Algorithm Hasan Hosseini-Nasab 1, *, Masour Pourkheradmand 1, Naser Shahsavaripour 2 1 Industrial Engineering Department, Yazd University, Yazd, Iran 2 Department o Industrial Management, Vali-e-Asr University, Rasanjan, Iran address: hosseininasab@gmail.com (H. Hosseini-Nasab), mansour_kheradmand@yahoo.com (M. Pourkheradmand), shahsavari.dr@gmail.com (N. Shahsavaripour) * Corresponding author To cite this article: Hasan Hosseini-Nasab, Masour Pourkheradmand, Naser Shahsavaripour. Solving Multi-Mode Time-Cost-Quality Trade-o Problem in Uncertainty Condition Using a Novel Genetic Algorithm. International Journal o Management and Fuzzy Systems. Vol. 3, No. 3, 2017, pp doi: /j.ijms Received: May 13, 2017; Accepted: June 14, 2017; Published: July 21, 2017 Abstract: In this paper a Fuzzy Discrete Time-Cost-Quality Trade-o Problem (FDTCQTP), is presented. All o three main actors o a project are considered in uncertainty condition using uzzy theory. Time, cost and quality are considered as uzzy trapezoidal numbers and a novel Genetic Algorithm; Super Genetic Algorithm (SGA) is introduced to solve the problem. Project network paths are calculated via a new algorithm which it can be very useul or complex project networks and in order to comparing the uzzy numbers, a new Fuzzy Number Ranking (FNR) method is introduced. The proposed algorithm is compared with classic GA by ANOVA, and the results demonstrate its eiciency. An applied example is used to more details. Keywords: Project Scheduling, Time-Cost-Quality Trade-o Problem, Metaheuristics, Genetic Algorithm, Fuzzy Theory, Fuzzy Number Ranking, CPM, ANOVA 1. Introduction According to American national standard institute, project management is: The application o knowledge, skills, tools and techniques to project activities to meet project requirements [1]. In project management, it is sometimes required to complete the project beore the normal due time.naturally, in such cases; the duration o perorg some o the activities must be decreased. Decreasing the time leads to increasing resources consug, and costs. Since resources, execution methods and technology types in real world projects are represented by discrete values, or real world applications Discrete Time-Cost Trade-o Problem (DTCTP) is applied [2] The DTCTP is known as an NP-hard problem [3], and no exact solution method can be ound to have the required eiciency or solving DTCTP. However, there are some approaches based on heuristic algorithms or solving DTCTP [4, 5]. Quality is another important actor in executing the project which requires considerable attention. Clearly, reduction in time or changing execution methods lead to changes in quality o project execution. So a DTCTP can be generalized to a Discrete Time-Cost-Quality trade-o problem (DTCQTP). A project consists o dierent parts o an activity. These activities have dierent execution modes in which each mode has its own time-cost-quality execution. Babu and Suresh were the irst who suggested that the quality o a completed project may be aected by project crashing [6]. El-Rayes and Kandil [7], presented a model that is designed to transorm the traditional two-dimensional time-cost trade o analysis to an advanced three-dimensional time-cost-quality trade-o analysis. Iranmanesh and Skandari [9], ound Pareto- optimal ront o time, cost and quality o a project, whose activities could be done in dierent discrete modes and each mode has a dierent cost, time and quality. They proposed Fast PGA algorithm that was meta-heuristic and was developed based on a version o GA. In addition to quality, uncertainty is also another important actor that should be paid attention to in executing the project. In the real world there is nothing exact and everything is uncertain. In order to considering the uncertainty in calculations, uzzy theory is used as a popular method. Pour

2 33 Hasan Hosseini-Nasab et al.: Time-Cost-Quality Trade-o Problem in Uncertainty Condition Using a Novel Genetic Algorithm Modarres et al [2], used uzzy theory to manage the uncertainty o quality, and considered the two other actors as crisp. They also used a novel genetic algorithm to solve the problem. Zhang and Xing [8], presented a uzzy-multi objective particle swarm optimization to solve the problem, which time, cost and quality described by uzzy numbers. SantoshMungle and LyesBenyouce proposed a uzzy clustering-based genetic algorithm (FCGA) approach, and provided a case study o highway construction to demonstrate the applicability o the proposed approach. In this study, time-cost and quality o project network activities detered by some experts through uzzy theory and linguistic variables. These quantities are presented as uzzy trapezoidal numbers. The aim o a Fuzzy Discrete Time-Cost-Quality Trade-o Problem (FDTCQTP) is to select a set o activities or crashing and an appropriate execution method or each activity such that the cost and time o the project is imized while the project quality is imized. In this paper, a novel Genetic Algorithm; Super Genetic Algorithm (SGA) is introduced to solve the problem. A new algorithm is proposed to calculating the project network paths which is very useul or complex project networks, and a new method is introduced to ranking uzzy numbers. This paper is organized as ollows: In section 2, the problem deinition and ormulation is provided. Section 3, discusses about solution procedure. In section 4, an experimental example is provided. Finally, section 5, concludes the paper. 2. Problem Deinition and Formulation The overall completion time, cost and quality o a construction project are detered by the duration, cost and quality or each activity involved in the project [8]. Activities are demonstrated by arcs in the project network. In order to perorg the activities, several modes are available. For example, to perorg the activity i, there are several methods and dierent technologies (modes) which each mode has its own duration, cost and quality. Also, in some cases, resources are not suicient or technical knowledge is inadequate. In this way, subcontracting should be implemented. Considering that subcontractors perorm the activities with dierent times, costs and qualities, a mathematical modeling is required to achieve the optimum solution. The aim o this study is imizing the time and cost and imizing the quality o the project. Notations o the problem are as ollows: Parameters: M ij Set o available execution modes or activities i and j, where (i, j) E Cɶ ijk Fuzzy direct cost o activities i and j perormed by execution mode k; ɶ Fuzzy duration o activities i and j perormed by T ijk execution mode k; qɶ ijk Fuzzy quality o activities i and j perormed by execution mode k; w Quality weight o activities i and j in the project, ij Decision variables: 1 i mode k is assigned toactivities i and j y ijk 0 otherwise xɶ i Early event time i, (i={1,2,...,m}) The model o FDTCQTP is as ollow: Max Qɶ wij. qijk. y = ɶ ij E k M ijk (1) IJ Min C ɶ Cijk. y = ɶ ij E k M ijk (2) IJ Min T ɶ = x ɶ x ɶ (3) n 1 = 1, ij E (4) ijk y k M ijk ij = 1 (5) w ij E ij x 0 i { 0,1}, i V (6) y, ij E, k Mij (7) Equation (1) imizes the total project quality and Equations. (2), (3), imize the total project cost and time respectively. Equation (4), guarantees that one and only one execution mode is assigned to each activity. In Equation (5) summation o activities important weight actor should be equal to Solution Procedure To solve the problem it is necessary to calculate the time, cost and quality o the project. To calculating the duration o perorg the project, critical path method (CPM) is used. To calculating the CPM, it is irst necessary that the project network paths be identiied Calculating the Project Network Paths In this paper, to calculate dierent paths o a project network, a new algorithm is presented. Let m be the number o nodes and Am m = aij be the node-node adjacency matrix with the ollowing notations, 1 i node i starts arc j a ij = (8) 0 otherwise The procedure o paths detecting is explained in Figure 1. Input o the algorithm is the matrix A, and an arbitrary number m that indicates the number o detected paths.

3 International Journal o Management and Fuzzy Systems 2017; 3(3): Ater detecting the paths, uzzy time o the existing activities in these paths are added up and the biggest uzzy number is identiied using FNR algorithm Fuzzy Number Ranking Method FNR method is described using the ollowing algorithm. The uzzy number o A ɶ is set in [0, 1] by µ (x) Aɶ membership unction (Figure 2). The trapezoidal uzzy A ɶ = a, a, a, β which its number o A ɶ is shown as ( ) membership unction is deined by: 1 2 Figure 1. Flowchart o detecting the network project paths. A A Suppose that and B = ( b1, b2, a B, βb ) are two uzzy numbers, whose summation and multiplying is computed by: A + B =, ( a + b, a + b, + β + β ) a A a B A B (10) A B =, I 0 ( a b, a b, a A + β a + β ) (11) B B A = (12) K ; K(.) A ( Ka1, Ka2, Ka A, KβA ) I 0 K ; K(.) A = ( Ka, Ka1, K A, Ka A ) 2 β (13) a1 x 1 a1 α x a1 α µ 1 a Aɶ = 1 x a2 (9) x a 2 1 a2 x a2 + β β ɶ 1 2 A A. Figure 2. Deinition o A = ( a, a, a, β ) To comparing the uzzy numbers A ɶ and B ɶ, x 2A, x2b are calculated or dierent values o Y, and at each iteration, a pre-deined superiority is attributed to the bigger number. The superiority becomes more when y getting close to 1. Y=1/ƞ (14) Ƞ is an arbitrary number which deteres the number o intervals in [0, 1]. For example i Ƞ=10, then y, at each iteration, has values o 0, 0.1, 0.2,, 0.9, 1. x 2A, x 2B are calculated as ollows: For the uzzy number A ɶ : ( ) x = a + 1 y β (15) 2A 2 A

4 35 Hasan Hosseini-Nasab et al.: Time-Cost-Quality Trade-o Problem in Uncertainty Condition Using a Novel Genetic Algorithm x = a + (y 1) α (16) 1A 1 A x b ( ) or the uzzy number B ɶ : = + 1 y β 2B 2 B (17) x = b + (y 1) α (18) 1B 1 B Finally the number with the higher superiority is the bigger one. (Figure 3). In case o equality these steps will repeat or x 1A, x 1B. Figure 3. Fuzzy ranking method. The FNR algorithm is shown in Figure 4. Figure 4. FNR algorithm.

5 International Journal o Management and Fuzzy Systems 2017; 3(3): Introducing a Novel GA; Super GA As it mentioned in part 1, FDTCQTP is known as NP-hard problem. So it is required to use a heuristic or metaheuiristic algorithm to solve it. In this paper SGA is presented which it has higher perormance with respect the classic GA. It is explored using ANOVA in section (5). In classic GA, selection probability o chromosomes represents by; Pi ( ) = N 1 (19) Which (i) denotes the itness value o chromosome(i) in current generation. It can be say that this itness value is not enough strong to be cause o a good chromosome to be selected. Also direct selection methods can be used, but some other deects still exist. In direct selection methods, there is no chance or a bad chromosome which has a great potential to become a very good chromosome using partial disturbance. So, the new itness value, new is introduce as ollow: new Table 1. Comparing o new and classic itness unctions. = e µ σ (20) new is the new itness unction, and µ and σ, are the mean and standard deviation o the all itness unctions at the same iteration. See table 1 or more details. P = new N 1 new (21) Equation (21), calculates the new selection probability o each chromosome. The selection probability o chromosome (5) is equal to 0.24 ( (5) =0.24), which it has not distinct dierent with chromosomes (4) ( (4) =0.22). However, using the new itness value, selection probability o chromosome (5) increased to 0.52 while the other itness values are not zero and they can be selected with the lower probability. Using this technique, a chromosome, with the small superiority, has a good chance to be selected. This change improved classic GA so as it can be say it is better to use SGA instead o classic GA, always. No. Fitness unction value classic Selection probability (-µ)/σ e (-µ)/σ New Selection probability sum 22.5 µ 4.5 σ Proposed Algorithm Implementation In order to solve the problem using the presented algorithm, it is necessary irst to deine the chromosome. Each chromosome has n genes which n is the number o project network activities. Each gene contains a random number between 1 and imum execution modes or each activity (M ij ). For example, the ollowing chromosome shows that activity 1 will execute with mode ith, activity 2 with mode j th and activity n with mode k th. (Figure 5) Calculating Fitness Function To calculate itness unction or each chromosome three parameters o time, cost and quality should be calculated. For each chromosome, time, cost and the quality o project should be calculated. C ; is the Total cost o project. T ; is the total project time o project which is calculated by CPM method. Q ; is the quality o project which is obtained by Equation 20. ( i ) Q = n 1 w i. q ij (22) Figure 5. Chromosome o the problem. i, j, k (1, imum number o available modes o each activity). F The itness unction (F (i) ) o chromosome i is calculated by ollowing equation: T T + γ C C + γ Q Qi ( ) + γ = wt. + wc. + wq. (23) T T + γ C C + γ Q Q + γ Where, w c, w t and w q are the planner-speciied weight. They indicate the relative importance o project cost, duration and quality respectively. And w t + w c + w q =1.0. C, C, T, T, Q and Q are the

6 37 Hasan Hosseini-Nasab et al.: Time-Cost-Quality Trade-o Problem in Uncertainty Condition Using a Novel Genetic Algorithm imal and imal values o cost, duration and quality in the current population respectively. γ is a very small positive number in order to prevent dividing by zero in the itness unction. In this paper, two -point crossover and uniorm crossover o Hasançebi and Erbatur [10] with pre -speciied weights are used. Also two-point mutation is used. Two random chromosomes are selected to show the operators. Parent 1= [2,1,5,3,2,4,1,5,3,6,1,7] Parent 2= [4,3,2,5,4,1,3,2,5,2,4,6] Applying the operators, the ospring produced rom the parents are as ollows: Two-cut-point crossover with random points (e 1 =4, e 2 =7). Ospring 1= [2,1,5,5,4,1,3,5,3,6,1,7] Ospring 2= [4,3,2,3,2,4,1,2,5, 2,4,6] Uniorm crossover with a random mask chromosome [1,1,0,1,0,0,1,0,0,1,1,0]. Ospring 1= [4,3,5,5,2,4,3,5,3,2,4,7] Ospring 2= [2,1,2,3,4,1,1,2,5,6,1,6] Two point mutation with random points (e 1 =6, e 2 =12). Ospring 1= [2,1,5,3,2,7,1,5,3,6,1,1] The steps o our proposed SGA are summarized in Figure 6. Figure 6. Flowchart o proposed algorithm. 4. Numerical Example To show the eiciency o the suggested algorithm, an example is presented here. This example deals with a project network (Figure 7). This project network consists o 12 activities. There are several modes or executing each activity. Each mode has its own time, cost and quality that are shown in terms o trapezoidal numbers in table 2. The eect o weight o each activity in total quality (w e ) is also considered in Table 2. Figure 7. The Project Network.

7 International Journal o Management and Fuzzy Systems 2017; 3(3): Table 2. Activities executions modes. e1 e2 e3 e4 e5 e6 t c q t c q t c q t c q t c q t c q t7 c7 q7 wq e7 e8 e9 e10 e11 e The model is programmed in Microsot Excel using Visual Basic Application (VBA).In this model the initial chromosomes includes 12 genes in which each gene shows the number related to a random execution mode. In order to calculate itness unction or each chromosome, computing time, cost and quality o that chromosome is required. To ind the time o each chromosome, all the available paths in the project network are identiied and time o the activities on that path are added up. Then these numbers that are in orm o trapezoidal uzzy numbers are compared with each other through FNR algorithm. The longest time shows critical path o that chromosome. Following network matrix was used to ind the network paths (Table 3.). Table 3. Network matrix was used to ind the network paths In this example, there are 18,500,000 solutions. To obtain the optimal solution, the proposed model was used. To solve this model the parameters o the problem are as ollows: w t =.3, w c =.4, w q =.3, initial population=200, population

8 39 Hasan Hosseini-Nasab et al.: Time-Cost-Quality Trade-o Problem in Uncertainty Condition Using a Novel Genetic Algorithm size=100, generation number=500, betta=8, ala=4, two point cross over rate=0.5, uniorm cross over rate=0.3, mutation rate=0.2. These algorithms are coded and implemented in Excel 2007 by the VBA and are run on a PC with CPU Core i7, 2.0 GHz and 4 GB o RAM memory which chromosome Table 4. Final outputs o the NHGAIII. [4,1,3,2,3,4,3,2,4,4,2,4] and its corresponding time, cost and quality, T=[94,104,11,18], C=[127,140,19,26], Q=[90,93,5,4] are obtained as output. Dierent execution modes o each activity associated with its time, cost and quality are presented in table 4. W t W c W q time Cost Quality Solution chromosome Experimental Evaluation This section evaluates the perormance o SGA and the classical GA. The Weighted Relative Deviation (WRD) is used as a common perormance measure to compare these algorithms that are computed by: Ta lg T Ca lg C Qa lg Q WRD = wt. + wc. + wq. Where, T a lg, C alg and Q ɶ a lg are the total project cost T T C C Q Q and time and quality or a given algorithm, respectively. Q, Q, C, C, T and T are calculated among inal answers o each time the program is run. The proposed algorithm, SGA and classic GA are implemented with same parameters or ity times. Their results are analyzed via the analysis o variance (ANOVA) method. Main hypotheses, containing normality, homogeneity o variance and independence o residuals, are checked and ound no bias or questioning the validity o the experiment. The means plot and least signiicant dierent (LSD) interval or the SGA and classic GA are shown in Figure 8. It demonstrates that the SGA gives very better outputs than classic GA statistically. Figure 8. Means plot and LSD intervals or the Super GA and classic GA. 6. Conclusion This study investigated multi-mode time-cost-quality trade-o problem in uncertainty condition. Due to the big solving space o the algorithm, a new metaheuristics algorithm, Super Genetic Algorithm (SGA), was presented. The eiciency o this algorithm with respect to classic GA was perormed by ANOVA. In order to calculate critical path o project network a new method was presented. Time, cost and quality were estimated in terms o trapezoidal uzzy numbers by experts and regarding the need or comparison o uzzy numbers, a new method was suggested or such a comparison. The algorithm was coded in VBA and an example was provided to illustrate the capability o algorithm. Considering the SGA s eiciency, it is recommended to use it instead o classic GA or NP-Hard problems. For uture researches solving a problem

9 International Journal o Management and Fuzzy Systems 2017; 3(3): considering the budget, resources and time constraints and achieving a necessary quality o project in uncertainty condition is suggested. Reerences [1] Institute, P. M. (2004). A Guide to the Project Management Body o Knowledge, Third Edition (PMBOK Guides), PMI Publisher. [2] N.S. Pour, M. Modarres, R. T. Moghaddam (2012). "Time- Cost-Quality Trade-o in Project Scheduling with Linguistic Variables." World Applied Sciences Journal 18(3): [3] SantoshMungle and LyesBenyouce, (2013). "A uzzy clustering-based genetic algorithm approach or time cost quality trade-o problems: A case study o highway construction project." Engineering Applications o Artiicial Intelligence 26: [4] Zheng, D. X. M., et al. (2004). "Applying a Genetic Algorithm-Based Multiobjective Approach or Time-Cost Optimization." Journal o Construction Engineering and Management 130(2): [5] Azaron, A. and T. R (2007). "Multi-objective time cost tradeo in dynamic PERT networks using an interactive approach." European Journal o Operational Research 180(3): [6] Babu, A. J. G. and N. Sureshb (1996). "Project management with time-cost and quality considerations." European Journal o Operational Research 88(2): [7] El-Rayes, K. and A. Kandil (2005). "Time-cost-quality tradeo analysis or highway construction." Journal o Construction. Engineering Management 131(4): [8] Zhang, H. and F. Xing (2010). "Fuzzy-multi-objective particle swarm optimization or time cost quality tradeo in construction." Automation in Construction 19: [9] Iranmanesh, H., et al. (2008). "Finding Pareto Optimal Front or the Multi- Mode Time, Cost Quality Trade-o in Project Scheduling." International Journal o Computer, Inormation & Systems Science 2(2): [10] Hasançebi, O. and F. Erbatur (2000). "Evaluation o crossover techniques in genetic algorithm based optimum structural design." Computers & Structures 38(1-3):