Fuzzy Least-Squares Linear Regression Approach to Ascertain Stochastic Demand in the Vehicle Routing Problem

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1 Applied Mathematics, 0,, doi:0.436/am Published Online Januar 0 ( Fuzz Least-Squares Linear Regression Approach to Ascertain Stochastic Demand in the Vehicle Routing Problem Abstract Fatemeh orfi,, Reza Zanjirani Farahani 3, Iraj Mahdavi Department of Industrial Engineering, Islamic Azad Universit, Semnan Branch, Semnan, Iran Department of Industrial Engineering, Mazandaran Universit of Science and echnolog, Babol, Iran 3 Department of Industrial Engineering, Amirkabir Universit of echnolog, ehran, Iran irajarash@rediffmail.com, f_torfi000@ahoo.com Received September 5, 00; revised November 3, 00; accepted November 7, 00 Estimation of stochastic demand in phsical distribution in general and efficient transport routs management in particular is emerging as a crucial factor in urban planning domain. It is particularl important in some municipalities such as ehran where a sound demand management calls for a realistic analsis of the routing sstem. he methodolog involved criticall investigating a fuzz least-squares linear regression approach (FLLR s ) to estimate the stochastic demands in the vehicle routing problem (VRP) bearing in mind the customer's preferences order. A FLLR method is proposed in solving the VRP with stochastic demands: approximate-distance fuzz least-squares (ADFL) estimator ADFL estimator is applied to original data taken from a case stud. he SSR values of the ADFL estimator and real demand are obtained and then compared to SSR values of the nominal demand and real demand. Empirical results showed that the proposed method can be viable in solving problems under circumstances of having vague and imprecise performance ratings. he results further proved that application of the ADFL was realistic and efficient estimator to face the stochastic demand challenges in vehicle routing sstem management and solve relevant problems. Kewords: Fuzz Least-Squares, Stochastic, Location, Routing Problems. Introduction he problem within distribution management of scheduling vehicles from one or more fixed positions (depots) to service a given set of locations (customers) is called the vehicle routing problem (VRP) []. Vehicle routing problems are important and well-known combinatorial optimization problems occurring in man transport logistics and distribution sstems of considerable economic significance vehicle routing problem with stochastic demand (VRPSD) has recentl received a lot of attention in the literature []. his is mainl because of the wide applicabilit of stochastic demand in real-world cases. In the routing problem with stochastic demands (RPSD) a vehicle has to serve a set of customers whose exact demand is known onl upon arrival at the customer s location. he objective in these problems is to find a permutation of the customer's demands that the penalties for losing a customer are minimized [3]. he actual demand of each customer depends on common assumptions on mathematical programming where all problem data are known in advance. In most cases, however, decisions have to be made before the realizations of random variables are known. A classical approach is to work with estimations of random data and to solve the stochastic problem similar to the deterministic cases. Moreover, it is often preferable to explicitl incorporate uncertaint in the models. Fuzz heor is a powerful tool, for decision making in fuzz environment. Crisp methods work onl with exact and ordinar data, so there is no place for fuzz and vagueness data. orfi et al. [4] proposed a Fuzz approach to evaluate the alternative options in respect to the user's preference orders in a fuzz environment. Human has a good abilit for qualitative data processing, which helps him or her to make decisions in fuzz environment. In man practical cases, decisions are uncertain and the are reluctant or unable to make numerical input and output data. orfi et al. [5] applied a

2 F. ORFI E AL. 65 new fuzz decision making model to determine the weights of multiple objectives in combinational optimization problems. In this paper, we appl their basic approximation operations in fuzz least-squares estimator. his paper considers a stochastic routing problem in which a set of customers is given, each of which will require service after the a priori decision is made. Uncertaint is modeled b using a vector of dependent variables which are intervallic and similar to the demand vector. Each of these in turn depends on independent variables which are also intervallic. However, in man practical cases, the customer's demands are uncertain and those who demand service are reluctant or unable to make a numerical permutation of the customer's demands. Fuzz least squares linear regression was assumed to be a powerful tool for decision-making in fuzz environment. Fuzz regression analsis is a fuzz (or possibilit) tpe of classical regression analsis. It is applied under circumstances where evaluation of the functional relationship between the dependent and independent variables in a fuzz environment is necessar. anaka et al. [6] initiated a stud in fuzz linear regression analsis that considered the parameter estimation of models as linear programming problems. Based on the findings of anaka et al., further investigations were made, which took two approaches: the linear-programming-based methods [6-9] and fuzz least-squares methods [0-]. Most of these fuzz regression models are analticall considered with fuzz outputs and fuzz parameters but non-fuzz (crisp) inputs. his paper aims to stud fuzz linear regression model with fuzz outputs, fuzz parameters, and fuzz inputs. Sakawa and Yano [9] proposed a fuzz parameter estimation model for the fuzz linear regression (FLR) model as follows: Yi A0 AX j AkX jk, j,,, n. Where both input data X j, X j,, X jk and output data Y j are fuzz. hree tpes of multi-objective programming problems were further formulated for the parameter estimation of FLR models along with a linear-programming-based approach. his multicriterial analsis of FLR models provided an appropriate method of parameter estimation b using the vagueness of the model via some indices of inclusion relations. Alternativel, a fuzz least-squares approach directl uses information included in the input-output data set and considers the measure of best fitting based on distance under fuzz consideration. Fuzz least-squares are fuzz extensions of ordinar least-squares. In this paper, one tpe of fuzz leastsquares is proposed as the parameter estimation for the FLR model is proposed as follows: Yi A0 AX j AkX jk, j,,, n. Yang and Lin [] used two approaches to evaluate the functional relationship between the dependent and independent variables in a fuzz environment. heir analticcal framework involved fuzz linear regression models with fuzz outputs, fuzz inputs, and fuzz parameters, but the fuzz numbers the considered in their model were of LR-tpe. In this paper, attempt is made to appl an extension of one of their approaches with triangular fuzz numbers. he proposed methodolog presents the extension of approximate-distance fuzz least-squares (ADFL) estimator. he proposed method is assumed to be appropriate alternative approach to estimate the stochastic demands in the routing problem. he remainder of this paper is outlined as follows: Sections introduce the method used to compute the stochastic demands. hen, the routing problem with stochastic demands is presented in Section 3. Section 4 presents the results of computational experiments to assess the value of the proposed approach and reports a comparative performance analsis to alternate method. Finall, in Section 5 conclusions and future researches are drawn.. Fuzz Least-Squares Linear Regression he rationale for the Fuzz heor is briefl reviewed before developing fuzz Least-squares Linear Regression as follows:.. Fuzz Arithmetic Definition.. A Fuzz set M in a universe of discourse X is characterized b a membership function M x which associates with each element x in X, a real number in the interval [0,]. he function value M x is termed the grade of membership of x in M [3]. he present stud uses triangular Fuzz numbers. A triangular Fuzz number, M, can be defined b a triplet M,,. Its conceptual schema and mathematical form are shown b Equation (). 0 x x x a x () x x x Definition.. Let M,, and N,, be two triangular Fuzz numbers, then the vertex method is defined to calculate the distance between them, as Eq-

3 66 F. ORFI E AL. uation ():, x x x d X Y () he basic operations on Fuzz triangular numbers are as follows [4]: For approximation of multiplication [5]:,,,,,, (3) For addition:,,,,,, (4) Given the above-mentioned Fuzz theor, the proposed Fuzz Least-squares Linear Regression Approach is then defined as follows:.. Developed Version of the Approximate-Distance Fuzz Least-Squares his is basicall an extension of and improvement on the model applied b Yang and Lin [6] above which is expressed b the FLR model as follows: FLR : Y A A X A X, j,,, n, (5) 0 i j k jk,,, inputs Where outputs Y j Y j Y j Yj X ji X,, ji X ji X ji Aj a,, j a j aj and parameters i,,, k, j,,, n so that the notion M,, is triangular fuzz number. he difficult in treating model (5) of fuzz inputoutput data is that A i X ji ma not be of triangular fuzz number. Although the product of two triangular fuzz numbers ma not be a triangular fuzz number, Dubois and Prade [7] presented an approximation form. Based on this analtical framework, Yang and Ko [] further developed the model presented b Dubois and Prade and suggested an approximation tpe of fuzz least-squares. What follows here is the application of approximation to present an algorithm for parameter estimation of the FLR model (5). B assuming M and N,,,, to be two triangular Fuzz numbers; therefore, b using the basic operations on Fuzz triangular numbers, it will be possible to express an approximation of multiplication and addition as follows: A A X A X where 0,, j k jk j j j * k j a0 ap xjp p k j a * 0 ap xjp p k j a * 0 ap xjp p Since A0 AX j AkXjk is of approximate triangular fuzz number, the distance d is defined on two triangular fuzz numbers. hus, the following object- tive function is considered: n 0,,..., k j, 0 j k jk U A A A d Y A A X A X j n j j j j j j j 3 0,,, k over A i subject to 0a, 0, 0, 0,,,, i a i a i k is i called the developed version of the approximate-distance fuzz least-squares method. he minimization of U A A A.3. Fuzz Membership Function he existing precise values is deliberatel transformed here to five levels ranking order of Fuzz linguistic variables: ver low (VL), low (L), medium (M), high (H) and ver high (VH). he commonl used Fuzz numbers applied for the triangular fuzz numbers are likel to be appropriate for the trapezoidal ones due to their simplicit in modeling interpretations. Both triangular and trapezoidal fuzz numbers are applicable to the present stud. he triangular fuzz number applied here can adequatel present an analtical framework within sevenlevel Fuzz linguistic variables, for the present stud. hese linguistic variables can be expressed in triangular numbers as ables and [3]. able. Linguistic variables for the importance weight of each criterion. Rank Ver low (VL) Low (L) Medium (M) High (H) Ver high (VH) Criteria grade Membership function (0.00,0.0,0.5) (0.5,0.30,0.45) (0.35,0.50,0.65) (0.55,0.70,0.85) (0.75,0.90,.00) able. Linguistic variables for the ratings. Rank Ver poor (VP) Poor (P) Medium (MP) Good (G) Ver good (VG) Criteria grade Membership function (0,,.5) (.5, 3, 4.5) (3.5, 5, 6.5) (5.5, 7, 8.5) (7.5, 9, 0)

4 F. ORFI E AL Statement of the Problem In this stochastic routing problem, capacities on the plants are given in terms of number of customers whom each plant can actuall serve, and the capacit of vehicles emploed to provide services are available. he aim of this stud is to allocate customers to the plants in question. It is assumed that there are circumstances when a plant is overloaded, (i.e., the number of customers on its route requesting service exceeds its capacit), and a number of customers are left without service, which incur an additional costs to the plant. his additional cost can be interpreted as the penalt for losing a customer, or as the cost of acquiring external resources to provide the service. Here, the demand made b the set of customer, which the plant has to satisf, is stochastic. he stochastic nature of the demand is a good reason to take the fuzz approach. he demand made b each node is a function of factors such as the demand time, age of the vehicles emploed, qualit improvement of product and distance of a particular node to the plant, etc. he goal in this RPSD is to minimize the penalt for losing a customer, defined as the sum of the expected penalties incurred for the customers who did not receive service. he uncertaint in demand is modeled b using a vector of dependent variables, which in turn the linear combination of the dependent variables generates independent variables. However, in man practical cases, the customer s demands are uncertain and the are reluctant or unable to make a numerical permutation of the customer's demands. Fuzz least squares liner regression is a powerful tool for decision making in fuzz environments. Fuzz regression analsis is a fuzz (or possibilit) tpe of classical regression analsis. It is used in evaluating the functional relationship between the dependent and independent variables in a fuzz environ- ment. his paper uses estimation method along with a fuzz least-squares approach, which can be effectivel used to estimate the customer demand. he outcomes of the method are compared for problem solution. A fuzz regression model is used in evaluating the functional relationship between the dependent and independent variables in a fuzz environment. Most fuzz regression models are considered fuzz outputs and parameters but non-fuzz (crisp) inputs. In general, there are two approaches in the analsis of fuzz regression models: linear-programming based on fuzz least-square methods. Sakawa and Yano [9] considered fuzz linear regression models with fuzz outputs, fuzz parameters and fuzz inputs. he formulated multi-objective programming methods for the model estimation along with a linearprogramming-based approach. 4. Procedure Experiment In Sections, a fuzz least-squares method has been constructed for the estimation of a routing problem with stochastic demands and fuzz input-output data. Given the above-mentioned approach, the procedure experiment is then defined as follows: Step. he first step for constructing the linear regression model involved collection of available data about the problems to be solved. For the purpose of this stud, the most important and effective factors, which are perceived b the experts to influence the demand made b a particular node, will be determined. Since the influencing factors in this stud were merel the perceptions of the stakeholders involved, and as such, are considered as qualitative data (i.e., conditions of qualit improvement of product). his renders the data subjective, as different experts view the world realit from their own perspective. his in turn depends on their experience and professional qualifications. Four of the most effective factors for estimating the demand made b a specific node can be seen in able 3. Step. he second step involved measuring the real demand for the 90 nodes in a single period of one ear. he following factors were taken into consideration: hese factors, as independent variables, are used in the model to estimate the demand made b each node. For example, X is the variable for qualit improvement of product, which consists of four evaluative categories like in line with the Likkert scale model, which comprises good, average, rather poor and poor. Ver high VH, average H, rather poor M and poor L, smbolize the good emulative categor. hese are also used for other independent variables and the results are shown in able 4. Step 3. Functional objectives and the constraints associated with the model used in this stud are calculated from the method provided in., which make mathematical programming problem and model parameters, which are triangular fuzz numbers, are the answer to these problems. Numerical results of the problem under investigation using the Mathematica and Lingo8 software, which include coefficients of variables of the able 3. Effective factors for estimating the demand. NO 3 4 CRIERIA Distance of the plant to particular node Qualit improvement of product Demand time Age of the vehicles emploed SAUS Less than 5, Between 5~0, More than 0 Good-Average-Rather poor-poor Spring-summer-autumn-winter Less than 5, Between 5~0, More than 0

5 68 F. ORFI E AL. able 4. Results of real demand for the 90 nodes. variable variable variable node node node L M M H VH 3 H VH M H VH 6 L VH M VH VH L M M VH H 3 L M H VH VH 6 L M M VH H 3 L VH M VH VH 33 M VH M VH VH 63 H H H VH H 4 L M H VH M 34 M VH H H H 64 M VH H VH VH 5 M H M H H 35 VH H M VH VH 65 L M M H VH 6 L VH L M H 36 L H H M H 66 VH VH M L H 7 M H VH H H 37 L H H H VH 67 M H M M VH 8 H H M L H 38 L L M VH H 68 VH VH M H H 9 L H H VH VH 39 L M H H VH 69 M H H VH H 0 L M H H H 40 L VH H H VH 70 L VH M H H M H M H H 4 L H H H H 7 L M H VH VH L L H VH H 4 VL M H VH H 7 VH VH H H H 3 L H H H VH 43 M H M M H 73 L M H VH H 4 L H H VH VH 44 M M M H H 74 VH H M H VH 5 M H M H H 45 L H VH H H 75 M VH H H VH 6 L VH H H H 46 M H H VH VH 76 L H M H H 7 H H H VH H 47 L VH M L VH 77 L VH H VH VH 8 M H VH H VH 48 M H VH H VH 78 L H M VH H 9 L VH H VH VH 49 M M M VH VH 79 M VH H VH H 0 L H M VL VH 50 H H VH VH H 80 L VH H H VH VL M H VH H 5 VL VH H VH VH 8 L H M M H M H M H H 5 L H H VH H 8 L H H VH VH 3 L L H VH H 53 VL M H VH H 83 VH H M H H 4 VL H H VH VH 54 M H M M H 84 M H M VH H 5 L VH M VH VH 55 M M H VH H 85 VH H L H VH 6 M H M VH H 56 L H H H H 86 M VH M H VH 7 L VH M H VH 57 M H M H VH 87 L H M H H 8 H VH H VH H 58 L VH M L VH 88 M VH M VH VH 9 M H M H VH 59 M H M H VH 89 L H M VH H 30 M VH H VH VH 60 M M M VH VH 90 M VH M VH H

6 F. ORFI E AL. 69 able 5. Parameter estimates and SSR for model. No. Apprximate- distance (Y ) Nominal demand Real demand ( Ŷ ) ( Y ) SSR based on Apprximate- distance d Y, Y d Y, Y ˆ H L L L L M L L L L L M L L L L L H L L L L L L L M L L M H L L M VL L L VH M L L L L L L H H L L VH H L L L L VL L L L VL M L M VL L M M H M

7 70 F. ORFI E AL L L L H VL L L L L L L L VL L L L L M L L L L VL L L H L L L L VL H L L M H M VH L L L L L L M H L L L M M L H M L L L M L L L L M H L L L L L H L L 0 0

8 F. ORFI E AL L L L L L M L L M H L L M L L L VH M L L L L L L H H L L VH H L L L L VL L L L VL M SUM Figure. SSR between actual demand and regression model, with real demand and nominal demand.

9 7 F. ORFI E AL. model based on the algorithm. herefore the presented regression model calculated the demands of each node. Step 4. Simultaneousl with the measurements of the real 90 nodes, the nominal demands were calculated from the compan s documents and data. Step 5. Calculate SSR based on Apprximate-distance and Real demand and SSR based on Nominal demand and Real demand b Equation () and the result of step 3 and step 4 are shown in ables 5. he results in able 5 show that the SSR s based on the distance d Y, Y and d, ˆ Y Y. Based on this able, the last step found the preference for the compan s documents and Approximate-distance approach as follows:,, d Y Y d Y Y ˆ 5. Computational Results he demands derived from the Approximate-distance method, are similar to the results obtained from real demand. he results of approximate-distance fuzz least-squares also substantiated the results of real environment, whereas the SSR s in the Approximate-distance d Y, Y less than SSR s in the Nominal de- method mand ˆ, d Y Y. Results suggest that the application of the fuzz sstematic evaluation in the estimation problems can reduce the risk in decision-making processes. Comparison regression model and the real demand, with real demand and nominal demand, are shown in Figure. 6. Conclusions he paper aimed at a critical analsis of estimation method to address the demand management in transport routing sstem. For this purpose the methodolog involved proposing a developed version of the intervalistance, fuzz least square as a realistic approach in addressing and solving the problem. Results indicated that the use of the fuzz logic proved a much closer solution to the real-world situation than its comparative methods. Results further showed that the method which was applied here ielded better solution to the problem than was possible from the data on demand obtained from the compan documents. It was further observed that under certain cases where the three crucial components of inputs, outputs and parameters are somewhat vague and stochastic, the fuzz linear regression is a more powerful analtical tool and as such, would be more preferable than others. he conclusion being that it would be more viable to appl trapezium fuzz numbers as an analtical framework for industrial case studies. 7. References [] R. J. Petch and S. Salhi, A Multi-Phase Constructive Heuristic for the Vehicle Routing Problem with Multiple rips, Discrete Applied Mathematics, Vol. 33, No. -3, 004, pp doi:0.06/s066-8x(03) [] D. Mester and O. Bras, Active Guided Evolution Strategies for Large-Scale Vehicle Routing Problems with ime Windows, Computers and Operations Research, Vol. 3, No. 6, 005, pp doi:0.06/j.cor [3] A. S. Maria, F. Elena and L. Gilbert, Heuristic and Lower Bound for a Stochastic Location-Routing Problem, European Journal of Operational Research, Vol. 79, No. 3, 007, pp doi:0.06/j.ejor [4] F. orfi, R. Z. Farahani and S. Rezapour, Fuzz AHP to Determine the Relative Weights of Evaluation Criteria and Fuzz OPSIS to Rank the Alternatives Applied Soft Computing, Vol. 0, No., 00, pp doi: 0.06/j.asoc [5] F. orfi, R. Z. Farahani and N. Hedaat, A New Model to Determine the Weights of Multiple Objectives in Combinational Optimization Problems, Proceedings of ICCESSE International Conference on Computer, Electrical, Sstems Science and Engineering, 07, Paris, 00, pp [6] H. anaka, S. Vegima and K. Asai, Linear Regression Analsis with Fuzz Model, IEEE ransactions on Sstems, Man, and Cbernet, Vol., No. 6, 98, pp [7] H. anaka and H. Ishibuchi, Identification of Positivistic Linear Sstems b Quadratic Membership Functions of Fuzz Parameters, Fuzz Sets and Sstems, Vol. 4, 99, pp doi:0.06/065-04(9)908-f [8] H. anaka, H. Ishibuchi and S.Yoshikawa, Exponential Possibilit Regression Analsis, Fuzz Sets and Sstems, Vol. 69, No. 3, 995, pp doi:0.06/065-04(94)0079-b [9] M. Sakawa and H. Yano, Multiobjective Fuzz Linear Regression Analsis for Fuzz Input-Output Data, Fuzz Sets and Sstems, Vol. 47, 99, pp doi: 0.06/065-04(9) [0] M. Albrecht, Approximation of Functional Relationships to Fuzz Observations, Fuzz Sets and Sstems, Vol. 49, No. 3, 99, pp doi:0.06/065-04(9) [] M. S. Yang and C.H. Ko, On Cluster-Wise Fuzz Regression Analsis, IEEE ransactions on Sstems, Man, and Cbernet, Vol. 7, No., 997, pp. -3. doi: 0.09/ [] P. Diamond, Fuzz Least Squares, Information Science, Vol. 46, 988, pp doi:0.06/ (88)

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