A BALANCED ALLOCATION METHOD FOR MINIMUM SPANNING TREE PROBLEM Su Yeon Cho1, Young Hae Lee†2, Dong Won Cho3 1-3
Department of Industrial and Management Engineering Hanyang University 55 Hanyangdeahak-ro, Sangnok-gu Ansan Kyeonggi-do, 426-791, South Korea Corresponding author’s e-mail:
[email protected]†2
Abstract: In today’s customer-centric services society, provide fast services to customers can be a competitive advantage. For this purpose, we propose the problem of allocating a set of customers to multiple nodes and reducing delay. Also this paper proposes a mathematical model and an efficient solution procedure for the balanced allocation minimum spanning tree (BA-MST) problem which consider total weighted cost with a balanced allocation and the total delay of minimum spanning tree problem. The BA-MST is a more realistic representation of the practical problem in the real world. The proposed solution procedure utilizes a genetic algorithm that is designed to find Pareto optimal solutions for this problem. The experiment results show that the proposed GA algorithm produces reasonable solutions.
1. INTRODUCTION The purpose of this paper provides a method for fast services to customers can be a competitive advantage in today’s customer-centric services society. For example, there are data communication, transportation and supply chain management. Data communication specially must provide fast services to customers to get an advantage in the competition to business. For this purpose, we propose a balanced allocation method and minimum spanning tree problem in wireless network for data communication as a case study. Recently, with the rapid growth of the smart phone users, location-based services and augmented reality has become major issues in wireless network. Location-based services are a software application defined as a general class of computer program-level services used to include specific controls for location and time data for an IP-capable mobile device that requires knowledge about where the mobile device is located. Current location-based services use information about current locations of users to provide services, such as nearest features of interest, they request (Vu et al., 2009). Augmented reality, which create an artificial environment through the combination of real-world and computer-generated data, is a new form of the man–machine interaction (Dangelmaier et al., 2005). Computer-generated information is shown in the user’s real field of view. Therefore, it is necessary to provide customers with location-based services and augmented reality in wireless network nowadays. However, the introduction of these services has significantly increased the load on wireless network. More significantly, location-based services and augmented reality generally place considerably more strain on the network compared to the original transmission services, since the network with these services contain many access points overloaded with a lot of customers at once and the significantly increased delay time in transmission services. As a result, not only are wireless network with these services and heavier loaded, but both the many overloaded access points and the increased delay time makes providing good service extremely difficult. Figure 1 shows the connection between multiple access points (APs) and the customers in wireless network for data communication. Some improvements haven’t been made at the wireless network with location-based services and augmented reality in order to increase the quality of service (QoS) through reducing delay time as well as make an efficient allocation of the customers to multiple APs. However, an improvement in customer allocation to multiple APs and delay time are necessary to maximize user satisfaction. Both the proper allocation of customers to multiple APs and reduced delay time has the great impact on increased user satisfaction since they allow the network to adapt to widely varying customer needs as well as to provide the differentiation in QoS. Thus, the main drive to improve user satisfaction in the wireless network is enhancement of the efficient allocation of the customers to multiple APs and the high QoS through a significant reduction in delay time. To deal with this problem, we explore the possibility of noteworthy compressing delay time as well as efficiently allocating a group of customers to multiple APs. The two objective performance criteria that we evaluate are total weighted communication cost with the balanced allocation and the total delay time, since these criteria represent main components of cost and QoS in wireless network, respectively. In this paper, we define this particular problem as the balanced allocation minimum spanning tree (BA-MST) problem. In the BA-MST problem, a capacitated balanced allocation problem (CBAP) is applied for the balanced allocation and minimum spanning tree (MST) is applied for the high QoS through reducing delay time. Therefore, the BA-MST problem is conjunct to the well-known generalized assignment problem and minimum spanning tree problem. As a result, it is NP-hard
in the strong sense due to its balanced allocating element along with its interaction with minimum delay decisions. To solve the BA-MST problem, we adopt a genetic algorithm (GA) approach based on Excel solver by FrontLine Systems.
Figure 1. The structure of wireless network for data communication
In the literature review, the successful application of genetic algorithms to generalized assignment problems and multiple objective network design problems is given (Deb, 1999; Gen et al., 1998a; Gen and Cheng, 2000; Horn, 1997). Ever since the GA approach was introduced by Holland (1975) to solve combinatorial problems, it has known as one of the most efficient heuristic algorithms for solving various network design problems in communication network, logistics and supply chains (Zhou et al., 2003). In general, a genetic algorithm (GA) is referred to as a stochastic search optimization technique whose solution search process imitates natural evolutionary phenomena: genetic inheritance and Darwinian strife for survival (Gen and Cheng, 1997; Michalewicz, 1999). The GA typically consists of five components (Gen and Cheng, 2000; Kamrani et al., 2001; Michalewicz, 1999): 1. A chromosome representation of potential solutions to the problem. 2. A way to construct an initial population (a set of potential solutions). 3. An evaluation of the objective function that measures the fitness of solutions to see whether they will survive. 4. An application of the genetic operators that alter the genetic composition of offspring. These operators include reproduction, crossover and mutation. Reproduction is a process in which individuals (solutions) are copied through the selection of the more fit individuals. Fitter chromosomes have higher probabilities of being selected. Crossover combines the features of two parent chromosomes to produce a new generation of individuals (offspring) by exchanging corresponding attributes of the parents. Mutation randomly alters one or more genes of a selected solution to obtain extra variability. 5. A setting of parameter values that determine population size to form a new population, crossover rate, and mutation rate. Although GA pursues a goal to produce global optimal solutions efficiently and effectively, its population size have no choice but to be finite. Thus, its final solution can be partial due to limited sampling of potential solutions. Nevertheless, GA was successfully applied to some challenging multiple objective network design problems. These problems include: logistics network problems (Chan and Chung, 2004; Erol and Ferrell Jr, 2004; Gen, et al., 1998b; Guillen et al., 2005; Altiparmak et al., 2006, 2009; Zhou et al., 2003), communication network problems (Elbaum and Sidi, 1996; Gen, et al., 1998; Kim and Gen, 2000; Lo and Chang, 2000), and advanced planning and scheduling problems (Gen et al., 2008; Moon and Seo, 2005). On the other hand, the application of GA to multiple objective communication network problems with the BA-MST problem is still rare, despite the fact that GA can produce Pareto-optimal or non-dominated solutions to the multiple objective problems by using the notion of fitness sharing. Some of the pioneering attempts to utilize GA for multiple objective communication network problems include Elbaum and Sidi (1996) for solving a selection of optimal local area networks (LAN) configuration and Kim and Gen (2000) for solving bicriteria communication network topology design. The selection of optimal LAN configuration is an NP-hard problem. Therefore, heuristic methods can be employed to solve the problem. Elbaum and Sidi (1996) used GA based on Huffman tree for topological designing of LANs. In particular, Gen et al. (1998a) proposed a GA for solving bicriteria network design problems considering connection cost and average message delay. It demonstrated that GA could find Pareto optimal (or non-dominated) solutions to the multiple objective problems much quicker than traditional weighted-sum approaches by keeping dominated solutions out of consideration (Bagchi, 1999).
Given the proven success of GA to multiple objective communication network design problems, we believe that GA is suitable for solving the BA-MST problem. To solve the BA-MST problem we propose a genetic algorithm (GA) approach to spreadsheet using Excel solver developed by FrontLine Systems. In contrast to the traditional multiple criteria programming techniques such as goal programming that allows the decision maker to arbitrarily assign weighting coefficients and/or preferences on multiple criteria and consequently presents a dominated solution, the proposed GA is designed to produce a wide range of non-dominated solutions without the arbitrary determination of weights. In other words, we proposed a GA approach to generate non-dominated solutions for assigning customers to a capacitated multiple APs and transmitting data using cost and delay time, respectively. This paper addresses modeling issues, solution approaches and potential benefits for the BA-MST problem. The main contributions are in two areas: 1. This study, to the best of our knowledge, is the first for the BA-MST problem that includes the two objective performance criteria that include total weighted communication cost with the balanced allocation and the total delay time. 2. To solve the BA-MST problem we propose a genetic algorithm (GA) approach to spreadsheet using Excel solver developed by FrontLine Systems. The computational results show that GA approach can produce reasonable solutions. The paper is organized as follows: In Section 2, we describe the mathematical model with problem description. In Section 3, In Section 2, we describe the mathematical model with problem description. In Section 3, Genetic algorithm approach is given to solve the BA-MST problem. Section 4 shows results of numerical experiments. Finally, in Section 5 the study is concluded.
2. MATHEMATICAL MODEL The BA-MST problem introduced in this paper is formulated based on the models described in Min et al. (2005) and minimum spanning tree problem. Min et al. (2005) proposed integer programming model concerned with the CBAP. In the remaining part of the paper, we use the following notation: Indices i=1, 2, ..., n j=1, 2, ..., m Parameters |V|={1, 2, ..., n, n+1, ..., n+m} e=|E| vi qj wij∈W dij∈D Decision variables xij
index of customer index of AP number of nodes number of edges (a connection between APs and their customers) each customer demand each AP capacity weight of communication cost between APs and their customers delay of edge (i, j) 0-1 decision variable (if customer i allocates to AP j, then xij=1, else xij=0)
The objective minimizes the total weighted the communication cost and the total delay time. The mathematical model is given as follows: n
min max f1 ( x) = å vi wij xij j =1,..., m
(1)
i =1
n+m n+m
f 2 ( x) = å å dij xij
min
(2)
i =1 j =1
Subject to n
åv x
i ij
i =1
£ q j , j = 1, 2,..., m
(3)
m
åx
ij
= 1, i = 1, 2,..., n
(4)
j =1
n+m n+m
ååx
ij
= n + m -1
(5)
£ S - 1 for any set S of nodes
(6)
i =1 j =1
n+m n+m
ååx
ij
i =1 j =1
xij = 0 or 1, "i, j
(7)
In the above formulation, objective function (1) minimizes the total weighted communication cost sum between customers and their serving APs, while balancing data traffic at each AP as equitably as possible. Objective function (2) minimizes the total delay time for wireless network QoS. Constraint (3) ensures that the total demand of customers should not exceed the given capacity of AP serving them. Constraint (4) assures that every customer be served by only one AP. Constraint (5) is a cardinality constraint implying that we choose exactly n+m−1 edges, and the packing constraint (6) implies that the set of chosen edges contain no cycles (if the chosen solution contained a cycle, and S were the set of nodes on a chosen cycle, the solution would violate this constraint). Constraint (7) is the domain constraint. The wireless network is modeled using an edge-weighted undirected graph G=(V, E) with n+m nodes and e edges. Figure 2 shows the wireless network with 12 nodes.
Figure 2. The wireless network with 12 nodes
3. SOLUTION PROCEDURE 3.1 Genetic algorithm based on spreadsheets program To solve the BA-MST problem we apply a spreadsheets program using Excel solver, not those program in source codes. To design of minimum spanning tree is not easy method in a spreadsheets program. Nonetheless, we developed a spreadsheets program of the BA-MST to easily use and modify for users. Our approach is guaranteed reasonable solutions based on a spreadsheets program using Excel solver. Also, a spreadsheets program has advantage to apply for broad problems without limit a specific problem. A spreadsheets program tested in this paper is Excel solver developed by FrontLine Systems. Frontline's Excel solver has three methods or algorithms: GRG nonlinear for solving smooth nonlinear optimization problems; Simplex LP for solving linear problems; and Evolutionary for nonsmooth problems (Frontline Systems, 2013). The procedure of evolutionary solver is particularly useful for tackling optimization models containing nonsmooth functions (Baker, 2011). Also, Baker (2011) introduced the advantage that the evolutionary solver is not handicapped by the presence of nonsmooth functions, as would be the case for the linear and nonlinear solvers. Therefore, we adopt the evolutionary solver by FrontLine Systems. Baker (2011) is explained the evolutionary solver principle through the following four step of the solution procedure. First,
the evolutionary solver works with a population of solutions. This step means that an offspring solution should combine traits from each of two parent solutions. Second, there are occasional mutations, which are offspring solutions with some random characteristics that do not come from each of two parent solutions. Third, over the course of the procedure, the population is governed by a fitness criterion (based on the objective function) that removes the poorer solutions and keeps the better ones. This process of selection drives the population toward better levels of fitness (better values of the objective function). Finally, if there is evidence that the population is no longer improving, or if one of the user-designated stopping conditions is met, then the procedure stops. When it stops, Solver displays the best member of the final population as the solution.
3.2 Pareto solution approach We combined two objective functions into a single scalar objective function as following equation (Collette and Siarry, 2003):
f ( x) = w1 × f1 ( x) + w2 × f 2 ( x)
(8)
where w1 and w2 are constants representing weights for f1(x) and f2(x), respectively. To evaluate all the individuals (candidate solutions), we adopt a strategy that Zhou et al. (2003) propose the entire solution space in order to avoid local optima and thus gives a uniform chance to search all possible Pareto solutions along the Pareto frontier. The fitness value of an individual in each generation is calculated according to the following formula (Zhou et al., 2003):
Eval( f ( x)) = w1 × f1¢( x) + w2 × f 2¢( x) Ꞌ
(9)
Ꞌ
where f1 (x) and f2 (x) are the normalized values of f1(x) and f2(x), respectively. To rescale the objective functions of f1(x) and f2(x), Zhou et al. (2003) normalized those functions by dividing each coefficient of the objective function by the norm of f1(x) and f2(x), f1Ꞌ(x) and f2Ꞌ(x) are derived from the following equations within the range [0, 1]:
f1¢( x) =
f1 ( x) - f1,min ( x) f1,max ( x) - f1,min ( x)
, f 2¢( x) =
f 2 ( x) - f 2,min ( x) f 2,max ( x) - f 2,min ( x)
.
(10)
Also, w1 and w2 denote the weights that are determined as follows (Zhou et al., 2003):
w1 =
w1 w2 , w2 = w1 + w2 w1 + w2
(11)
where
¢ ( x), w2 = f 2¢( x) - f 2,min ¢ ( x) w1 = f1¢( x) - f1,min Ꞌ
Ꞌ
Ꞌ
(12) f2Ꞌ(x)
where f1 ,min (x) and f2 ,min (x) are the minimum of f1 (x) and among the current population, respectively. In other words, f1Ꞌ,min (x) and f2Ꞌ,min (x) represent an ideal point in the objective space whose value has been normalized.
4. NUMERICAL EXPERIMENT The experiment was performed to investigate the accuracy of the proposed approach. The experiment in this paper is implemented in a Microsoft Excel Solver platform developed by Frontline Systems and executed on Intel® Core™ i7 CPU 3.40 GHz with 8GB of RAM. The Microsoft Excel solver has extraordinary capabilities to handle very large models and is suitable for many truly industrial strength applications (Frontline Systems, 2013). The evolutionary solver parameter is set as follows: Population size = 100; Mutation rate = 7.5%; Convergence: 0.0001. The customers demand randomly generated 3 types of data traffic. The complete network structures of 11 nodes (#of nodes n=11). The delay time in the experiment was referenced by OR-Library (Beasley, 1990). The numerical experiment for Pareto solutions is summarized in Fig. 3.
5. CONCLUSION In this paper, we formulated multi-objective optimization model for the BA-MST problem in wireless network for data communication as a case study. And we GA based on spreadsheets program through numerical experiment. The experiment result shows that the proposed GA based on spreadsheets program produces reasonable solutions through Pareto solutions. In the future research, this research can be applied to network structure similar to this particular problem as a case study.
Total delay time
Total communication cost (×102) Figure 3. Pareto solutions for the numerical experiment
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