A Hybrid Genetic Algorithm Approach for Optimizing Dynamic Multi-Commodity Supply Chains

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1 1 A Hybrid Genetic Algorithm Approach for Optimizing Dynamic Multi-Commodity Supply Chains H. Abbas * M. Hussein M. Etman Mechanical Engineering Dept. Helwan University Faculty of Engineering Cairo Egypt Abstract In this paper a dynamic supply chain network (SCN) model is developed in which both dynamic facility locations and dynamic distributed quantities of materials and products are assumed. The problem is formulated mathematically using the mixed integer linear programming (MILP). The solution methodology adopted in this research is the hybrid genetic algorithm (hga) comprising genetic algorithm (GA) and pattern search (PS) optimization techniques. A study is conducted in this research and experimental work is performed in order to validate the proposed hga. In this study several comparisons are made between the solutions of the designed hga and the solutions of the branch-and-bound algorithm used by the LINGO 12 as a commercial package. Results show that the developed hga is an appropriate and a useful solution methodology for largesize SCN with a small number of manufactured products and used raw materials. Keywords: Dynamic supply chains; Forward logistics networks; MILP; Genetic algorithm; Pattern search; hga. 1. Introduction A forward logistics supply chain is a set of echelons specifically suppliers manufacturing plants distribution centers and customers intended to convert set of materials into final products for the sake of satisfying customer demands. With the increased concerns of applying supply chains models to the real world manufacturing organizations adding dynamic features to the designed SCN models has a growing attention in the recent years. Serving new customer zones and opening new markets in different locations are the main causes of making adaptations to the SCN design. The design of an efficient SCN includes deciding various key variables affecting the performance of the chain. These design specifications result from optimizing the strategic the tactical and the operational decisions of the supply chain. On the strategic level s decisions it s important to determine which customer areas to be served in each planning period selecting the best locations for core facilities and suppliers making continuous product development to introduce new models for the existing products that may be matured or declined and introducing new products to the market. The tactical decisions focus on making benefit to the manufacturing organization within the defined set of organization s constraints. Examples for tactical decisions of a supply chain are adopting a certain manufacturing philosophy using public warehousing instead of constructing the organization s own warehouses the type of transportation modes used for transporting the products and the particular products to be developed to the customers markets. Operational decisions are the daily several decisions taken within the framework of the strategic and tactical decisions to ensure that the effective movement of the products along the supply chain and accordingly realize the maximum cost benefits to the supply chain. Keeping certain items in stock to prevent production stops keeping end products in stock relying on local suppliers instead of waiting for the contracted suppliers response relying on local logistics companies instead of using the third party logistics (3 PLs) companies are all examples of the operational decisions to be taken. Many research work have been published in the field of optimizing the forward logistics SCN. These researches focus on developing models that can help in making the strategic tactical or operational decisions. A limited research work focus on combining all these types of decisions in the same model. Multi period planning and dynamic working conditions for the supply chain are rarely found in the published researches. The focus of this paper is to develop a design of a multi period multi-commodity forward logistics SCN model that can be used as a decision support tool for the strategic tactical and operational decisions of the supply chain. The rest of this paper is organized as follows: section 2 introduces the reviewed articles and the analysis of these articles. The proposed model is presented in section 3. Section 4 explains the adopted solution methodology (hga). Section 5 presents the experimental work and the study of the behavior of the hga solutions of supply chains under changed parameters. The results of the conducted study and comparisons are included in section 6. The research s conclusion is presented in section Literature review Researches published on forward logistics supply chains can be categorized as: Static supply chains Dynamic supply chains * Corresponding author. Tel.: addresses: hthmabbas@yahoo.com (H. Abbas) moh_hus_has@yahoo.com (M. Hussein) dretman@gmail.com (M. Etman). 1016

2 2 A supply chain is said to be static if no changes occur in the specifications or planning decisions in the supply chain s logistics model. Fixed facility locations deterministic product demand and unchanged number of facilities over time are characteristics of this type of supply chain. In this field of research single commodity supply chains models were developed in many published researches. For example Syarif YoungSu and Yun (2002) developed a multi-stage logistics network model formulated in a 0-1 MILP and solved using a spanning tree-based GA. Altiparmark Gen and Lin (2005) also designed a mixed integer nonlinear model for a multi-objective supply chain network. The supply chain was designed for a single product of a plastic company. The model was solved with a developed solution procedure based on genetic algorithms. On the other hand multi-commodity supply chains problems were tackled in fewer researches. A supply chain with single assignment problems was modeled by Crdeau Laporte and Pasin (2008) using a mixed integer nonlinear programming. Iterated local search heuristic was used for solving the model. YoungSu Moon and Kim (2009) developed a four-echelon supply chain model with complicated routes. An hga was used for solving the model comprises hga and a local search heuristic. According to Hinojosa et al. (2008) dynamic supply chain is a supply chain in which one or more of the following characteristics could occur; dynamic relocation is required for the main facilities of the chain increasing or decreasing the number of facilities over time in any echelon changing the variables of the supply chain and/or the dynamic environment of the supply chain. Melo Nickel and Saldanha (2005) provided a branch-and-bound-based solution for the dynamic multi-commodity capacitated facility location problem. The model is formulated in a MIP. The problem of building new facilities and closing down existing facilities at two different distribution levels over a given time horizon was adopted by Hinojosa et al. (2008). The problem modeled is a two-echelon multi-commodity supply chain with considering holding inventory and outsourcing aspects. The problem is formulated in a MILP and solved through heuristic solution tailored for the model. A computational study is then attached to evaluate the performance of the heuristic. Eltawil Elwany and Megahed (2007) developed a MILP model for a capacitated three echelon multi-commodity dynamic production distribution model with inventory accumulation consideration (CTMDPI). Thanh Bostel and Peton (2008) developed a dynamic production-distribution supply chain with multi-commodity deterministic product demand chain. Dynamic facility location of the suppliers and manufacturing facilities were considered with the options of capacity expansion capacity contraction product subcontracting and the supplier price discount. The branch-and-bound heuristic solution methodology was used to solve the model. Also Lin Gen and Wang (2009) designed a MILP-based multi-stage logistics network. The model was solved by a hybrid evolutionary algorithm composed of Wagner- Whitin algorithm GA and fuzzy logic controller. The location-allocation problem with fuzzy demands was discussed by Zhou and Liu (2007). The objective of this study is to locate new facilities to supply fuzzy customers demands. A hybrid intelligent algorithm is used for solution comprising fuzzy simulations embedded in genetic algorithm together with the network simplex algorithm. Another two-echelon supply chain model that deals with fuzzy customers demands and fuzzy manufacturing costs is developed by Zhou Zhao and Tang (2008). A chance-constrained programming and two different game structures for pricing are used to enrich the developed model. A numerical example is given to compare pricing strategies under different game structures of supply chains. Wang and Shu (2007) adopted the uncertainties in the supply chains producing new products where uncertain demands are expressed in fuzzy sets in a probabilistic supply chain model. The model is formulated in a fuzzy programming model and solved with GA. A stochastic model of the multi-stage global supply chain network problem was developed by Goh Lim and Meng (2007). The model is formulated in an integer linear convex programming. A new solution methodology is provided based on the Moreau Yosida regularization. Cristobal Escodero and Monge (2009) adopted a stochastic dynamic programming approach and multi-stage scenario tree approach to represent the problem of uncertainty in the supply chain parameters. A 0-1 MILP model for the whole scenario tree of the supply chain is developed. El-Sayed Afia and El-Kharbotly (2010) built a mathematical model to solve a three-echelon multi-interval supply chain for a single product under stochastic demand. Experimental work was conducted in the study to develop direct relationships between supply chain input parameters and the total profit gained. Shukla et al. (2010) developed a simulation-based dynamic SCN and a Taguchi method for parameter design. The problem is formulated mathematically using regression analysis and then solved with a created psychoclonal algorithm and an artificial immune systems (AIS). Also Tiwari et al. (2010) developed a combined Taguchi-AIS technique to optimize a multi-stage supply chain in terms of defining the best locations of the facilities of the chain. Hsu and Li (2011) developed a reliability evaluation method to evaluate the performance of the supply chain s plants under demand fluctuations. They integrated two mathematical programming models to determine the optimal adjustment decisions regarding production reallocation among plants under different fluctuating demands. A probabilistic linear programming approach is adopted by Kabak and Ülengin (2011) for supply chain networking decisions using fuzzy set theory-based model to deal with those uncertainties. The resource-planning decisions are made using fuzzy demand forecasts and fuzzy yield rates as well as other inputs such as costs and capacities. Analysis of published papers in the field of forward logistics chains The reviewed forward logistics papers can be classified according to the degree of complexity and the level of dynamic features included in the problem modeling into different categories: single period planning or multi period planning single commodity or multi-commodity deterministic or stochastic product demand two three or four echelons logistics. Upon the 1017

3 3 previous classifications and after overlapping these categories the reviewed articles are allocated in these categorizes as shown in Table 1. After the general classification of the forward logistics papers the papers that include SCN models with dynamic production-distribution with multi period multi-commodity multi-echelon networks will be shown in more details in Table 2 and Table 3. These tables show the dynamic features considered in the models developed in these papers. The sign ( ) means that this feature is included in the model. The abbreviation DC means distribution centers. After reviewing the developed research work in this field the following limitations can be concluded: All the papers ignore the dynamic change in the customer locations which is a very important feature in any supply chain is to open new markets. The quantities of raw materials delivered from suppliers to manufacturing plants are ignored or considered to be unchanged over time. No more than two-echelon simultaneous change in facility location is conducted in the previous studies especially in the manufacturing plants and the distribution centers. Changing supply chain parameters are included in only few papers. The problem of supplier selection is conducted in only one paper of the logistics network design problem. Rarely included in the reviewed articles is experimental work to study the effect of different parameters on the behavior of the supply chain. The total number of echelons in most of the developed SCN models is not more than three echelons. Table 1 Forward logistics papers classification Single-Period Single-Commodity Multi-Commodity Number Deterministic Stochastic Deterministic Stochastic of echelons Single echelon Zhou (2007) Two echelons Goh (2007) Zhou (2008) Duramz (2009) Syarif (2002) Three echelons youngsu (2009) Altiparmak (2005) Wang (2007) Melachrinoudis (2007) Cordeau (2008) Azaron (2008) Tiwari (2010) Multi-Period Single-Commodity Multi-Commodity Deterministic Stochastic Deterministic Stochastic Cristobal (2009) Goetschalckx (2002) Melo (2005) Lin Lin (2009) Hinojosa (2008) El-Sayed (2010) Eltawil (2007) Thanh (2008) 1018

4 4 Table 2 Dynamic features concerned with facilities locations Research Goetschalckx (2002) Number of Echelons Two Network Organization Plant-Finishing Facility Supplier Producer Dynamic Facility Location Warehouse or DC. Melo (2005) Three Plant-DC-Customer Retailer Customer Lin Lin (2009) Hinojosa (2008) Eltawil (2007) Thanh (2008) El-Sayed (2010) Four Three Three Four Four Plant-DC-Retailer- Customer Customer Plant-Warehouse- Plant-Warehousecustomer Supplier-Plant- Warehouse-Customer Supplier-Plant-DC- Customer Table 3 Dynamic features concerned with delivered quantities Research Stochastic or Deterministic Raw materials Dynamic Quantities Produced or subcontracted Warehouse DC Customer Inventory Accumulation Goetschalckx (2002) Deterministic Melo (2005) Deterministic Lin Lin (2009) Deterministic Hinojosa (2008) Deterministic Eltawil (2007) Deterministic Thanh (2008) Deterministic El-Sayed (2010) Stochastic 3. The proposed model The supply chain network proposed in this work comprises four echelons representing the raw materials (or components) suppliers the manufacturing plants the supply chains warehouses and the customers. The proposed model is a multi period multi-echelon capacitated model with dynamic facility locations and dynamic number of nodes in the echelons of the network. The possibility of adding or closing plants and warehouses is handled by the model in order to adapt the production plan to the customer demand. The model enables the period-by-period adaptation (increase or decrease) of the supplier s list and using multiple raw materials which is a unique feature of the developed model. Also the uniqueness of this model is mainly shown in the simultaneous changes of the four echelons. Inventory accumulation is considered in this model where stored quantities of raw materials and end products can be held in stock to be used in the next planning intervals. 3.1 Model assumptions The model assumptions can be stated as follows: 1. The planning horizon is known and consists of multi time periods with equal lengths. 2. The locations of the potential network s facilities are known in each period but not fixed. They can be changed in the positions and the number of potential facilities of the same type. 3. The customers demand from the end products is known in each period. 4. Opening or closing plants and warehouses is performed at the beginning of the planning period. 5. Adaptation of the number and locations of suppliers and the number and locations of customers is done at the beginning of each period. 1019

5 5 6. Storing raw materials is allowed in the plants and product storage is allowed in the warehouses. 7. Shortage of end products is allowed in the model. 8. Manufacturing capacities of the plants can be extended or reduced in any period and also the assigned capacities for end products are changed. 9. The products are delivered according to the service level of each customer from each product. 10. All cost parameters (fixed manufacturing transportation holding shortage and non-utilized capacity costs) are known for each time period. 3.2 Mathematical model formulation The proposed model has been formed mathematically in a MILP technique. The sets parameters and decision variables used in the mathematical model are all included in Table 4. Table 4 Indices parameters and decision variables used in the mathematical model Indices and superscripts materials supplier index ( ). distance between supplier and plant. manufacturer index (Comprising old manufacturer and new manufacturer ). distance between plant and warehouse. distance between warehouse and customer. utilization of material per unit of product. service level of customer from product. warehouse index ( ). customer index ( ). material type index ( ). assigned capacity of supplier for material type. product or resource type index ( ). assigned manufacturing capacity of manufacturer time period index ( ). for product type. Sets storage capacity of material type in the possible suppliers for material. manufacturing facility. possible manufacturers for product. storage capacity of product type in warehouse. potential number of warehouses. potential customer locations. Integer Decision Variables set of used materials and components. set of finished products. quantity of material type purchased and transported set of time periods. from supplier to manufacturer. Parameters and coefficients purchasing cost per unit of material m from supplier i. transportation cost per unit of material m from supplier to manufacturing facility. manufacturing cost per unit of product in plant. cost per unit of transporting product type from manufacturer to warehouse location. cost per unit of transporting stored product type from warehouse to customer location. cost per unit of holding in stock material type in facility location. cost per unit of holding in stock product type in the warehouse location. shortage cost per unit of product in the warehouse. fixed cost of opening a new manufacturing facility ( ) at time period. quantity of product type produced at manufacturer. quantity of product type transported from manufacturer to warehouse. quantity of product type transported from warehouse to customer. quantity held in stock of material type. quantity held in stock of product type. Binary decision variables = 1 if supplier is selected at time period to deliver material type ; 0 otherwise. = 1 if manufacturing facility is active and selected to manufacture product ; 0 otherwise ( is used for old manufacturers and is used for new manufacturers). fixed cost of closing an existing manufacturing facility ( at time period 1. fixed cost of keeping an existing manufacturing facility open ( ). = 1 if warehouse is active and selected at time period to store product ; 0 otherwise. = 1 if customer is selected to as a destination node to transport product at time period ; 0 otherwise. 3.3 Objective function The objective of this model is to minimize the total cost of the proposed supply chain over the whole planning horizon. Cost Items The total cost function of any of the following cost items is calculated as the sum of this item s cost over the planning horizon T. The cost of any item (purchasing manufacturing transportation holding shortage) is the direct multiplication of the unit s cost and the quantity transferred from the selected origin node to the selected destination node at time period t. the detailed cost items are shown as follows: 1020

6 6 (1) Materials purchasing costs (2) Materials Transportation Costs (1) (2) (3) Products manufacturing costs (3) (4) Costs of products transportation to warehouses (5) Costs of products transportation to customers (4) (5) (6) Materials holding cost (6) is given by the following equation: (7) is calculated by the following formula: (8) (7) Products holding cost (9) (8) Products shortage cost (10) (11) The shortage quantity can be given by the following equation: max 0 (12) (9) Non-Utilized capacity cost (13) 1021

7 7 is given by the following equation: max0 } (14) (10) Fixed costs The fixed costs of the logistics chain manufacturing facility can be divided into several types: a) Fixed costs of keeping the existing plants open. b) Fixed costs of opening new plants. c) Fixed costs of closing non-useful existing facilities. d) Fixed costs of keeping existing warehouses open. e) Fixed costs of opening new warehouses. f) Fixed costs of closing non-useful existing warehouses The following equation describes the total value of the fixed cost. (15) 3.4 Constraints The proposed SCN model has the following constraints (16) (17) (18) (19) (20) (21) (22) (23) (materials purchasing cost + materials transportation cost + products manufacturing cost + products transportation to warehouses cost + products transportation to customers cost + materials holding cost + products holding cost + shortage cost + non utilized capacity cost + fixed costs) t (24) 1022

8 (25) 01 (26) Constrains (16) and (17) are the demand satisfaction constraints. Constraint (16) guarantees that each manufacturing facility receives enough material quantity. Demand satisfaction is either from the materials supplier or from the stored material in the manufacturing facility itself. Constraint (17) ensures that each customer will be satisfied from the ordered products within the service levels of each customer from the manufactured products of the supply chain. Constraints from (18) to (21) are the capacity constraints. Constraint (18) is intended to ensure that the total quantities of any type materials transported from any supplier do not exceed the capacity of this supplier at any time period. Constraint (19) is used to limit the total quantities of any type of products transported from any plant to the assigned products manufacturing capacities of this plant at any time period. Constraint (20) ensures that the quantity held in stock of any material type doesn t exceed the assigned material s storage capacity for this material at the manufacturing plants at any period. Constraint (21) limits the quantity held in stock of any product type to the assigned product s storage capacity for this product at warehouses at any time period. Constraint (22) and constraint (23) are intended to put lower limits for the number of manufacturing plants and the number of warehouses that must be open at any time period. Constraint (24) is the budgeting constraint to work within a defined budget. Finally constraints (25) and (26) are the variables domains constraints. 4. Proposed solution methodology This section is intended to provide the proposed solution methodology for the developed mathematical model that represents the problem of dynamic supply chain. The proposed solution methodology adopted in this research is the hga methodology which is composed of GA and PS optimization techniques. The stages are sequential as shown in Figure 1; where the GA gives a reasonable initial feasible solution point that can be used for further search by the PS to develop the final solution point. Figure 1 Stages of the proposed solution methodology 4.1 GA stage Genetic algorithms are stochastic search algorithms that mimic the mechanism of natural selection and natural genetics. According to YoungSu Moon and Kim (2009) many key parameters must be controlled in the design of the GA which are: Gene representation Chromosome selection Crossover operator Mutation operator Migration operation 4.2 PS stage PS is an optimization algorithm that searches a set of points around the current solution point looking for one where the value of the objective function is better than the value at the current point. This set of points called a mesh is formed by adding the current point to a scalar multiple of a set of vectors called a pattern. There are three main steps must be managed in constructing the pattern search algorithm which are: Initialization step Mesh generation step Poll step 1023

9 9 The Matlab R2010a package was used for the optimization process by using the Global Optimization and the Mathematical Programming toolboxes. Table 5 shows the developed GA initialization process the chromosome selection process and the GA operators. The proposed PS procedure is shown in Table 6. Table 5 Performance of various solution methodologies Procedure: GA initial steps Initialization. Model representation stage Set f (objective function) as Set C (constraints) as Gene representation Use real-valued representation in each chromosome Use 160 individuals as initial population Chromosome selection Step 1. Fitness assignment Use the linear ranking fitness assignment method Set the maximum survival rate as 2. Procedure: GA operators Crossover operator. Use the intermediate crossover algorithm Set the ratio d=1 Mutation operator. Use the uniform mutation method Set the ratio R=0.25(25 % of the variables will be mutated) Migration model. Use the migration method in both directions (forward and backward) Set the migration fraction as 0.2(ratio of interchanged individuals) Set the migration interval as 20 generations Step 2. Individuals selection Use the tournament selection method Set the tournament size as 4 Set the number of elitist individuals as N1 Set the number for crossover operations as N2 Set the number for mutation operations as N3 Table 6 The proposed PS procedure PS Initialization. PS Algorithm. Set of Inputs f (objective function) f Є Rn A b (the constraints coefficients and the constraints right hand side) A b Є Rn x0 (start solution point) where x0 Є Rn Step 1. Mesh Generation Let {vi}be the pattern vectors For i=1: N (no. of constraints) vi Generate the direction vectors {di} by multiplying {vi} by a scalar (mesh size=1) Calculate Pk:={ xk ± di : i Є N} End Step 2. Poll Step For k = 0 1 If f(t) < f(xk) for some t Є Pk Set xk+1= xt Set k+1=2* k (Expansion) Otherwise Set xk+1= xk Set k+1= k/2 (Contraction) If stopping conditions are not met return to step 1. End 5. Analysis of the developed hga The developed solution methodology is a heuristic solution technique and was adopted in this research after numerous experiments. In these experiments a simple dynamic supply chain was designed and solved using the proposed hga and other heuristic solution optimization techniques. The results show that the hga out performs other techniques in terms of the value of the fitness function. In order to validate the developed hga multiple comparisons were made between the hga solutions and the LINGO12 solutions as a commercial package which uses mainly the branch-and-bound algorithm. In these comparisons a comprehensive study was made to examine the behavior and performance of the proposed hga (in terms of the accuracy of the obtained solution) under different values of the parameters affecting the supply chain. 5.1 Design of Experiments Initial experiments for a designed dynamic supply chain show that the most significant factors affecting the accuracy of the solution gained by the hga are all concerned with the size of the SCN and the number and quantities of transported items through the chain. The main factors included in the study are: 1024

10 10 1. The number of operating facilities (the number of nodes in each echelon of the chain). a. Number of suppliers in the suppliers list. b. Number of working plants. c. Number of active warehouses. d. Number of served customers. 2. Number of manufactured products with different manufacturing costs. 3. Number of used raw materials components and subassemblies with different purchasing prices. 4. Demanded quantities of final products. To accomplish this study a 4-period 4-echelon SCN was designed and modeled using selected nominal values for the studied parameters. These selected values are shown in Table 7. Table 7 Nominal values of the studied parameters of SCN Parameter Value Parameter Value Number of available suppliers in any time period (minimum maximum) (46) Materials suppliers capacity (units) 5000 Number of working plants in any time period Production capacities of each product (units) 5000 (25) (minimum maximum) Materials storage capacity for each material 2000 Number of active warehouses in any time period (minimum maximum) Number of customers in any time period (minimum maximum) Number of raw materials or components in any time period (minimum maximum) Number of manufactured products in any time period (minimum maximum) Products demand quantities for each product ~ N (μ σ) (13) (47) (36) (13) N (100020) Change in product demand mean (units) Results and discussions Products storage capacities for each product 2000 In order to conduct this study number of scenarios for experiments must be built in order to involve all the mentioned studied parameters. The number of possible scenarios is which needs much time to build and solve. An effective way to minimize this number is to consider some factors constant after a certain number of experiments. In this way the number of experiments has decreased to 75. The SCN problems were built and modeled through the programming with the Matlab R2010a package. The following sections present the behavior of the developed hga in terms of the accuracy of the obtained solutions with the change of the parameters in concern. The accuracy of the hga solutions is expressed by the value of the error percentage calculated by the following equation: 100 % (27) The following sections will discuss the effect of the factors values on the value of the error percentage. 6.1 Effect of the number of operating facilities The relation between the number of suppliers for different demand mean values and the error percentage is shown in Figure 2. There is an inverse relation between the number of suppliers echelon nodes and the error. The error decreases as the number of suppliers increases. The demand mean factor serves as a supporter for lowering the value of the error. The results plotted in the figure show also that there are high differences in error values as the number of suppliers increases. 1025

11 11 Error pecentage (%) suppliers 5 suppliers 6 suppliers Demand mean (units) Figure 2 Effect of the number of suppliers Figure 3 shows that the number of plants as a factor affects inversely the error percentage. Increasing the number of operating plants decreases the error percentage. The plotted curves show that there are observable differences in the error values from the case of operating 2 plants and the case of operating 3 plants. The error values differences are small among the last three levels (3plants 4 plants and 5 plants). Error percentage (%) plants 3 plants 4 plants 5 palnts Demand mean (units) Figure 3 Effect of the number of plants The behavior of the error percentage change as shown Figure 4 with the increase of the number of active warehouses is similar to that obtained with the two previous factors (the inverse relationship). The demand mean factor has no effect when combined with the number of warehouses factor. The differences in error values are very small for the different demand means. Figure 5 shows that as the number of customers increases the solution accuracy is improved (i.e. error decreases). The best level of this factor is to serve 6 customers or 7 customers. The difference between error values of these two levels is too little especially at a demand mean of at least 2500 units. Error percentage (%) warehouses 3 warehouses 4 warehouses Demand mean (units) Figure 4 Effect of the number warehouses 1026

12 12 Error percentage (%) Customers 5 Customers 6 Customers 7 Customers Demand Mean (units) Figure 5 Effect of the number customers The behavior of the developed SCN model and the developed hga under the discussed four previous factors can be explained in the following points: The hga starts its solution search mechanism by assigning initial values for the quantities of raw materials and final products to all branches of the SCN in order to cover a wide area of the search. Then the optimization steps create new solution individuals based on the principle of the survival of the fittest until the best solution is finally selected. In the case of small demand quantities of the product that can be fulfilled by a single plant or few number of plants and accordingly the raw materials quantities can be supplied by small number of suppliers the resulting error percentage of the hga is high. This happens because the optimal solution of the SCN model is existed in small number of branches of the network whereas the hga concentrates the search in a wide search area and this increases the tendency to develop a solution that has a considerable deviation from the optimal solution. When the demand mean is large require more than one plant for fulfillment and must be served by a larger number of suppliers and warehouses the optimal quantities of materials and products are distributed among a large number of network branches. This makes the hga solution approaches the optimal solution and the resulting error is greatly decreased. 6.2 Effects of the number of products and the number of materials As shown in Figure 6 the differences in error values are very small among the changed number of manufactured products. This is very clear at the plotted points where the points are approximately identical. It can be considered that this factor has minor effects on the solution s accuracy. Using different demand means combined with changing the number of products helps considerably in decreasing the values of the error percentage. On the other hand increasing the number of used raw materials has negative effects on the accuracy of the developed solutions. The error values increases with the increase of that factor. The results plotted in Figure 7 show that this factor has a significant effect as the differences between the errors values have relatively large magnitudes. As usual the factor of product demand mean affects positively affects the solution s accuracy (helps in decreasing the error). Error percentage (%) product 2 products 3 products Demand mean (units) Figure 6 Effect of the number products 1027

13 13 Error percentage (%) materials 4 materials 5 materials 6 materials Demand mean (units) Figure 7 Effect of the number raw materials The effect of these two studied factors on the accuracy of the hga can be explained as follows: Increasing the number of transported items through the SCN results in a great increase in the number of combinatorial solutions for the model (the problem is NP hard optimization problem). The designed hga needs longer time periods in order to allow the solution search mechanism covering all the solution areas of the model. Also the hga finds difficulties in detecting these areas precisely and this increases the possibility of stacking at local minimum areas and finding local minimum solutions rather than the global minimum one. This explains why the error percentages increase with the increase of the levels of the two factors. 7. Conclusion This paper presents a multi period multi-commodity multi-echelon dynamic SCN model formulated mathematically in a MILP. Research contribution of the developed model is obvious in the included dynamic features that were not tackled in the reviewed articles. From these features are: Considering multi dynamic raw material planning. Managing the dynamic supplier s list in each time period and allowing the increase and decrease of the number of suppliers in each time period. Allowing opening or closing operating facilities in any time period. Serving dynamic customers locations. Simultaneous change in four echelons in the same time period. A proposed hga is developed for solution comprising the GA and the PS optimization techniques. Experimental work is conducted in this research to study the effect of changing different dynamic parameters on the accuracy of the developed hga. Comparisons are made between the hga solutions and the solutions of a branch-and-bound based commercial package which develop exact solutions. Results show that the accuracy of the proposed hga is improved in the following cases: A large-size supply chain (large number of echelons nodes). A small number of manufactured products and small number of required raw materials. Large quantities of product demand. References Altiparmak F. Genan M. and Lin L. (2006) A genetic algorithm approach for multi-objective optimization of supply chain networks Computers and Industrial Engineering Vol. 51 No. 1 pp Azaron A. Brown K. Tarim S. and modarres M. (2008) A multi-objective stochastic programming approach for supply chain design considering risk International Journal of Production Economics Vol. 116 No. 1 pp Cordeau J. Laporte G. and Pasin F. (2008) An integrated local search heuristic for the logistics network design problem with single assignment International Journal of Production Economics Vol. 113 No. 2 pp Cristobal M. Escudero L. and Monge J. (2009) On stochastic dynamic programming for solving large-scale planning problems under uncertainty Computers and Operations Research Vol. 36 No. 8 pp Durmaz E. Aras N. and Altinel K. (2009) Discrete approximation heuristics for the capacitated continuous location allocation problem with probabilistic customer locations Computers and Operations Research Vol. 36 No. 7 pp Eltawil A. Elwany H. and Megahed A. (2007) Multi-commodity multi-period supply chain network design Proceedings of the 37 th International Conference on Computers and Industrial Engineering Vol. 182 pp

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