1. INTRODUCTION
In the recent years, most of the people are involved with the fashion industry either as the customers in purchasing the products or as the producers or manufacturing for managing the product life cycle. The variety of fashion goods is becoming larger where the accessories, jewellery, perfumes, clothes, cars, home and many other products can be considered as fashionable goods (Figure 1).
It is difficult for the decision makers to identify the fashionable goods as a particular product is a fashion good from a consumer point of view, but it may be considered as ordinary product for others. For example, buying a car for the middle class of a community is an ordinary good, but buying a car for a wealthy and welloffminded people is a fashionable good. Because of such complexity, scheduling of selling fashion items or production plan of these items is very important. Fashion is one of the most visible media of change. The fashion industry forms larger social and cultural phenomenon known as the “fashion system,” a concept that embraces not only the business of fashion but also the art of fashion, and not only production but also consumption. The peculiar nature of fashion industry can be characterized in terms of its volatility, velocity, variety, complexity and dynamism.
During the last decades, scientists pay attention to the fact that product demands is not constant between different periods of time necessarily and may be varied form a period to another subject to numerous variables. For example, climate change in different seasons, inflation rates, and technological advances are some of the reasons why they can reasonably change market demand. Moreover, the inflation rate and income level of different classes of society, which plays an important role in the purchasing power of the people, and it can change the market for the sales of fashion goods. For example, when a country's sweat fluctuates unevenly due to war or internal changes in political structure, the purchasing power of the people of society is diminished, and the demand for fashionable goods naturally diminishes. Latin American countries and some countries like Syria are the examples of widespread inflation.
On the other hand, the cultural structure of a country can also affect the consumption of fashionable goods. Asian countries are one of the main destinations for the sale of fashion goods in the world. For example, Saudi Arabia is the largest cosmetics importer in the world.
Therefore, the presentation of a planning method for the sale of manufactured goods can lead to the ordering and sale of these goods internally and even abroad, which naturally reduces the import of these goods from other countries that consequently increases the domestic economic prosperity of a country in will follow.
This research tries to answer this question that if an appropriate product distribution plan can reduce the product lost sale in a multimarket/multiretailer supply chain while the product demands are not fixed and can be varied from period to another. The objectives of this research is increasing the profit of selling fashion goods in a dynamic closed loop supply chain. A brief overview on fashion goods scheduling models shows that the scheduling of fashion goods in a multimarket/multiretailer supply chain in the presence of market demand uncertainty has not been addressed before. Therefore, the purpose of this study is to exclusively discuss the effects of possible market demand on the product distribution of fashion goods in the markets to maximize the profit of a supply chain.
2. LITERATURE REVIEW
In industrial world, supply chain management (SCM) is referred to the systematic and strategic management of value added flows of goods and services from purchasing the raw materials until delivery and after sale services. Such strategies require a comprehensive view of the links in the chain that work together efficiently to create customer satisfaction at the end point of delivery to the consumer. Figure 2 shows a classic supply chain.
2.1 Supply Chain Management
A supply chain involves all stages of direct and indirect procurement of goods / services. These steps can include all stages of order registration, material supply, material processing, quality control, packaging, sale, logistic and after sales services. A classical supply chain involves supplying raw materials from suppliers, producing products, sending to warehouses, distributing to wholesalers and retailers, and ultimately to the customer. In other words, a typical supply chain consists of suppliers, depots of raw materials, production centres, distributors, retailers and final customers. Each supply chain is guided by multiple components. These six components are: inventory, transportation, facilities, information, source finding, pricing (Figure 3).
2.2 Types of Supply Chain Management
In terms of the flow of materials and activities carried out in each supply chain, three main groups of supply chains are divisible. They include a forward supply chain, downward and a closed loop.
2.2.1 Froward Supply Chain
This definition represents a kind of classical chain that involves transferring raw materials from supplier to producer, producing final product and distributing it to the consumer. The goal of this supply chain is to increase the level of service and reduce the total cost of the system.
2.2.2 Downward Supply Chain
In this type of supply chain, organizations will attempt to recycle, rebuild, repair, and resell products to reduce waste generated in the environment (Chung and Wee, 2011).
On the other hand, with the advancement of productions manufacturing technology, the need for the use of raw materials will be reduced, and today, in many products, a percentage of recycled materials is usable. Therefore, the use of reverse supply chain from the point of view of profitability is also important. In the other definition, reverse logistics can be defined as the sum of products used or consumed by consumers, recycling or rebuilding, repairing parts. Today, legal and civilian institutions and organizations are working to establish regulations in preventing the demolition or pollution of the environment. For this reason, the use of reverse supply chain is very important (Yang et al., 2013).
2.2.3 Closed Loop Supply Chain
The closed loop chain is achieved by integrating the forward supply chain and the downward supply chain (Amin and Zhang, 2013).
From another point of view, supply chains are separable into forward and backward supply chains. In the forward supply chain, the movement of primary materials from the supplier to the customer is considered and in the backward supply chain, the system moves from the supplier's products (including items sold, used, returned, or auctioned) to primary materials (Vahdani et al., 2012). In recent years, more focus has been placed on closed loop supply chains.
2.3 Fashion Goods and Supply Chains
This area of science points to items that lose their value over time due to rapid technological changes or the production of a new product by a rival. Fashion goods may lose their value over time and may be outdated.
Today, the attention of many scholars and researchers has focused on the topic of fashion or obsolete goods. With the growth of fashion industry over the past few years, it has a significant share of the economy of the countries.
AbecassisMoedas (2006), factors such as changing the trend of the fashion industry, short life cycle of products associated with this industry, the intense competition for countries with lowcost labor, the unprecedented growth of emerging markets have brought about significant changes in the traditional models of this Industry.
2.3.1 Competition and Pricing in SCM
Nagurney and Yu (2012) presented a new model of exclusive competition for the supply chain of multiproduct fashion goods with consideration of environmental issues. Chen and Chang (2013) conducted a study in which an analytical decision framework was presented, in conditions that each sale market varies both in terms of product type and in terms of sales time. Mehrjoo and Pasek (2014) developed a modelling method for longterm and dynamic management in the fashion commodity industry, the focus of their model was on the interaction between physical processes, information flows, and the management of the clothing supply chain to create dynamic variables such as diverse products, inventories, costs, and profits. In the following, Zhou et al. (2015) presented two models of optimal pricing methods for fashion product companies by which the optimal strategy style was determined. Brito et al. (2015) conducted two data mining methods for customer segmentation.
2.3.2 Product Distribution and Storage
In supply chain models, product distribution is considered as a significant parameter in the success of the supply chain.
In many cases, products were provided to retailers by various transmission methods, as well as through various routes. Şen (2008) showed that by using a field study, the communication between manufacturers and retailers in the fashion industry would improve supply chain performance. Mula et al. (2010) conducted a comprehensive research on a variety of mathematical programming models in the supply chain focusing on production and transportation. Peidro et al. (2010) presented a linear programming model for intelligent planning of a supply chain system that was multiproduct multilevel and in a number of time periods with the goal of simultaneous decisionmaking. There also some researches that focused on quick response to customer demand (Castelli and Brun, 2010) and also the coordination of the fashion supply chain (Wang et al., 2012). Ni and Fan (2011) provided a model for shortterm and longterm dynamic sales forecasts for fashion products. He et al. (2010) made a solution to optimize the use of raw materials and replenishment of these materials on corrosive products. In multi market issues, due to different parts of sales in different markets, production planning and inventory control are very complicated. Qin et al. (2011) used a selective control system for the transmission of products. Jia and Bai (2011) introduced an approach in 2011 to develop a production strategy based on qualitative parameters. In the following, they used fuzzy theory to integrate the model in order to reduce the ambiguity in decision making. Liang et al. (2011) used mathematical programming to solve the problem of integrated production planning, which in their model variety of products and different production periods would work in a fuzzy environment with the goal of reducing overall system costs and taking into account available inventory levels, the level of available human resources, the capacity of machinery, stockpile, and budget. ElBaz (2011) presented the decisionmaking approach to measure the supply chain performance by combining fuzzy theory and AHP. Olugu and Wong (2012) used an expert system to evaluate closed loop supply chain management in terms of parameters such as efficiency, effectiveness, and economic strategies. Lo et al. (2012) presented a study on the impacts of environmental management systems on profits and the improvement of the performance of fashionproducing factories. Li et al. (2012) focused on the issue of returning fashion goods among members of the supply chain model for cost optimization and ordering policies. Dye and Hsieh (2012), provided an inventory model with variable rate of deterioration and a slight downgrade which considered the amount of capital involved in the product conservation technology that would measure the maximum profit. Basu and Nair (2014) presented a multiperiod inventory control formula in 2013 in a dynamic random programming model. MacCarthy and Jayarathne (2013), from the point of view of retailers, it has been shown that there are differences between the components of the distribution chain in apparel retailers and significant consistency between the type of retailer and the distribution chain is visible. The focus of the strategy of geographical diversity in the fashion industry has been identified by many authors. Caniato et al. (2014) also expanded the scope of studies by designing a comprehensive framework for examining the integration of the new product development plan and international retailing in the fashion industry with a probable approach. Iannone et al. (2015) focused on the relationships between the decision making variables of a fashion supply chain to integrate the retailer network. The model provided by them was able to assist in the process of prepurchase decision, delivery, and replenishment steps. Macchion et al. (2015) by carrying out statistical analysis on the information of 132 Italian factories producing fashion goods, identified three different branches of factories in which different ways were found to organize production and distribution network with specific competitive preferences. Delgoshaei et al. (2016a) focused on machineload variation as a major shortcoming in manufacturing systems. For this purpose, a new method is proposed for scheduling dynamic manufacturing systems in the presence of bottleneck and parallel machines. They showed that the condition of dynamic costs affects the routing of materials in process and may induce machineload variation. Delgoshaei et al. (2017) presented a new method for shortterm period scheduling of dynamic manufacturing systems in a dual resource constrained environment. The aim of this method is to find best production strategy of inhouse manufacturing using worker assignment (both temporary and skilled workers) and outsourcing, while part demands are uncertain and can be varied periodically.
2.3.3 Product Design and Market Forecasting
From the point of view of varying the demand for various goods over time, mathematical models are divided into two main parts. The first part of the model is the demand level of the predefined various products, and the second set of models is that the demand for products which can be predicted with regard to past data or market existing analyses.
Ouyang et al. (2006) by presenting an inventory model for corrupt items, focusing on determining repository replenishment optimal policies and proposing practical solutions to reduce annual inventory costs. Another form of definite demand is the pricebased demand. Qin et al. (2006) in their study, concentrated on those corruptive items whose demand was pricedependent. Pishvaee et al. (2009) presented an integer programming (IP) model for designing a direct and reverse logistics integrated network. Hsu et al. (2010) provided a model to determine the optimal level of inventory control. In this study, the researchers focused on investing in new technologies to improve the maintenance of corrosive items. Zhao et al. (2010) focused on the uncertainty and dynamic changes of supply chain models. They found that issues such as machine failures, special sales orders, and some similar items could affect the increase in uncertainty and emphasized that the presented models require specific coordination to establish mechanisms for allocating available resources in order to achieve production goals. Tong et al. (2010) used an Adaptive Fuzzy Method in 2009 to investigate the uncertainty estimation in supply chain logistics systems. Regulwar and Gurav (2011) introduced a fuzzy multiobjective programming model to examine the uncertainty in supply chain systems. Widyadana and Wee (2011) considered a constant demand inventory control model for corruptive items in a production system with constant production rates and the possibility of machine failure. Musa and Sani (2012) presented the mathematical model for the inventory system of corruptive items with the ability of products credit purchase. Mishra et al. (2013) focused on the effect of time on demand, maintenance costs, and rate of corruption. So, they provided a model for backlog deficiency with the goal of minimizing costs. Sicilia et al. (2014) presented an inventory model for corrosive goods. Montagna (2015) and Delgado and Albuquerque (2015) looked at cultural issues and its relationship with the fashion industry. Yang and Ng (2014) have presented a strategy model with variable capacity and a multiperiod market for uncertain demand and investment constraints. Delgoshaei et al. (2016b) compared different material transferring models that are developed by scientists in the scheduling manufacturing systems. In modern production systems, planning and scheduling are integrated to achieve the fastest response to customer demand.
A review over the opted researches shows that the idea distributing products in the multimarket/multiretailer supply chain while the product demands are to fixed and may be varied in the different periods has been less developed. Table 1 compares a number of opted researches with similar targets.
The contribution of this research is proposing a new scheduling model for the distributing of fashion goods in order to increase profit while product demands are varied from a period to another.
3. RESEARCH METHODOLOGY
3.1 Selling Models
As mentioned in the previous section, one of the important members of the supply chain management is sales. This is why sales models are very important in supply chain management. So far, many sales models have been introduced in supply chain models. Each has followed the issue of sales from a particular point of view.
3.1.1 Objective Functions in Supply Chain Models
So far, various objective functions have been considered by scientists in modelling sales models. Some of these target functions are:

Maximizing sales profits

Maximizing the amount of delivered goods

Maximizing customer satisfaction

Minimizing delays in delivery

Minimizing transportation costs (explaining that transportation models include a wide range of sales models.)
The results of the study of sales models show that in most scientific papers, sales models are considered as singlegoal models, and the use of mathematical models of several objective functions is very limited.
3.1.2 Constraints in Supply Chain Models:
Constraints on sales models can be categorized as follows:

Production capacity constraints

Restricted demand in the market

Transportation capacity constraints

Material limitations
The results of this section show that the use of constraints on the probable demand for commodity market has been highly respected in recent years, and in many papers over the past two decades, probable functions for estimating demand for products in different sales periods has been used. However, the use of these limitations increases the complexity of the model, but it helps to make the results of the study more realistic. In this section, the research method will be displayed as the flowchart (Figure 4). The proposing model, with the input information like the amount of market demand, retailer capacity, sales profits, as well as transport costs from the producer to the retailer, from the retailer to the market, Sellers and returns to the manufacturer. Further the mathematical model can find best (nearest) retailer and can achieved through retailers to the best market (a market that has more demand and less transportation cost). If the number of products received by each retailer is sold during the market period, there will be no need to transfer it to the upcoming period. Otherwise, the surplus of products will be auctioned at a later time between retailers and, if not sold during this period (regarding to fashion goods situation which explain before), this additional amount of the products will be fined, deducted from the total price of the product and return to producer. We want to find the best distribution of productions to retailers and then to distribute these goods in the target markets in such a way that the highest profitability of the commodity will be earn by the producer.
3.2 Developing a Model for Planning
We consider demand for products in different planning periods is not constant and is estimated from the triangular distribution function. The reason for considering the triangular distribution function is the ability to consider three optimistic, pessimistic, and most probable values for each demand that these market conditions (bestseller, moderate, and lowselling) are applicable. In summary, the features of the model presented in this study can be summarized as follows:

Scheduling fashion goods

MultiProduct/MultiMarket/MultiScheduling Period

Considering the uncertain market demands for sales planning

Auction of extra products

Considering the return of the remained products
3.3 Assumptions
To find best product transferring solutions in the model, the following assumptions are taken into consideration:

The demand for various time periods is not constant and follows the triangular distribution function.

Each market has a specific rate of demand.

If the goods are not sold at a later period, they will be auctioned at a lower price if they are not sold in this period. They are returned to the producer with a fine.

Transportation of products between the manufacturer's origin and retailers’ costs.

Transportation of products from retailers to different markets has shipping costs.

Transportation of goods between different retailers’ costs.

Returning products from retailers to the origin of production costs.

The initial inventory level is zero and the inventory of the end of the last period is considered zero too.
3.4 Indexes
The variables and parameters are defined as bellow:
3.5 Parameters

D_{ist} : Market demand for the ith product over the period $t\to {D}_{ist}~tri({\delta}_{ist},\hspace{0.17em}{\gamma}_{ist})$

δ_{ist} : Upper point limit for triangular probability function

γ_{ist} : Lower point limit for triangular probability function

CT_{ijt} : Cost of transportation a unit of product i from the manufacturer to the retailer j at the time t

BN_{ijst} : Sale price of one unit of ith product by the jth retailer in the market s and in the tth period

α_{ijt} : Discount rate on the sale of each iproduct by jth retailer in auction in the time period t

β_{ijt} : Penalty cost that set for the retailer j while returning each unit of the product i to the manufacturer within the time period t

ρ_{i}: Allowed amount of auction of product i at the end of each period

ω_{i}: Allowed amount of on returning product i to the manufacturer at the end of each period

CD_{ijst} : Cost of transferring each ith unit from the jth retailer to the market s during the time t

CB_{ijt} : Cost of transferring each ith product unit from the jth in the auction in the time t

CR_{ijt} : Return cost of each product unit i by retailer j in the time interval t

CP_{j} : Retail capacity j

ϕ_{it} : Rate of reduction in the price each ith product unit in the time t
3.6 Decision Variables
This model contains 4 series of integer variables as follows:

X_{ijt} : Number of ith products delivered to the jth retailer during the time period t. (Integer)

Y_{ijst} : Number of ith products sold during the time period t by the jth retailer in the sth market. (Integer)

Z_{ijt} : Number of ith products that are sold by jth retailer in auction in the time period t. (Integer)

P_{ijt} : Number of type i product that is returned to the manufacturer in the time interval t. (Integer)
3.7 Model formulation
The mathematical model presented in this study is an integer programming model. This leads to attention to the use of metaheuristic algorithms. On the other hand, the results of the literature review show that about 80% of the researches in this field have used metaheuristic algorithms for similar models.
S.T:
The objective function of this model is to calculate the amount of profits from the sale of products by retailers. The first sentence is the amount of sales profits after deduction of the cost of transportation to the destination. The second sentence is the cost of transporting the origin of the produce to the grower’s place. The third sentence shows the cost of the auction in the retailer, and the last sentence indicates the amount of redistribution and repatriation from the distributor to the original origin.
The first constraint of the model shows that the total number of imported products of type i from all retailers to a market in each period should not exceed the demand for that market. The second constraint ensures that for each retailer the sum of the quantities sent from one type of product to all markets, plus the quantities sent to other retailers at the next auction for the auction, and the amounts returned in the two subsequent periods with fines must be equal or less than the values receive from the manufacturer. The third constraint shows that the amount of product receipt for each retailer in each period is less than the capacity of the retailer, this capacity can be expressed in terms of storage space or available capital amount. The fourth constraint is for those who indicate that less than ρ_{i} (for example 20%) of the products in each retailer in the next period can be auctioned and 80% of them should be sold on the market. This restriction, usually imposed by manufacturers, is aimed at encouraging retailers to sell more. The fifth constraint is similar to the previous constraint to prevent returning more than ω_{i} (for example 30%) of the unsold distributed goods by retailers. Such strategy is sometimes used by manufacturer as a penalty to encourage retailers to increase their efforts to sell the products as much as possible. Since the retailers know that they are also responsible for financial loss. Noted that the ρ_{i} and ω_{i} are parameters that will be set by manager to control the distribution goods in the market and maybe varied from a product to another. Therefore, If the ρ_{i} and ω_{i} are considered 1, there is no force in selling auction or returning unsold goods to the manufacturer. The constraints sixth and seventh show the initial values of the variables Z and P which means there would be no auction in the first period and also no returned goods in the first and the second periods. The last four constraints represent the scope of the variables of the model.
3.8 Contribution of the Model
Table 2 shows the contribution and novelties of this research comparing to some of the similar researches in literature:
In the constraints number 2, 4 and 6 (which are shown by equations 3, 5, and 6 respectively), one variable is divided into another one which increase the complexity of the model and consequently the chance of using heuristics and metaheuristics.
4. A HYBRID GENETIC ALGORITHM AND SIMULATED ANNEALING
Each of the metaheuristic algorithms has unique characteristics and features that speed up the search for the feasible solution space. In this research we will use a hybrid Genetic and Simulate Annealing algorithms (GASA). Figure 5 shows the performance of a classical genetic algorithm. Table 3 shows the pseudo code of the proposed GASA. The reasons for using the genetic algorithm are:

This algorithm has been used frequently used in similar researches.

This algorithm utilizes a mutation operator to escape from local optimum points.

This algorithm can provide convergence by using the crossover operator.
The contributions of the proposed A Hybrid Genetic and Simulated Annealing Algorithms can be summarized as:

Using Product demand operator for estimating product in the various markets.

Using the crossover operator for generating new solution strings.

Using Mutation operator to search more spaces in the solution space.

Using the local escaping rate of the simulated annealing which helps the algorithm escape from local optimum tales.
The steps in this algorithm will be listed as follows:
Step 1: Import Data of Model
At this stage, inputs of the mathematical model will be entered. Which include:

Number of products

Number of retailers

Number of planning courses

The number of markets

The demand for each period (triangular function)

Transportation costs from the manufacturer to each retailer

The cost of shipment from retailers to each market

The cost of returning the goods

The cost of carriage between retailers (at auction)

Rate of reduction of products
Step 2: Input GASA Parameters
At this point, the inputs of the algorithm will be entered. The main inputs of the genetic algorithm used in this research are:
Step 3: Forcasting Market Demands
In this step the amount of market demands will be forcasted using triangular probability function.
Step 4: Crossover Operator
The operator selects the products in priority (sales profits) and selects them for transfer to retailers. In the future, these products are transmitted from retailers to the applicant's markets. This operator selects the product, retailer, and market of the applicant in such a way that the most benefit from the producer.
Step 5: Fitness Function
After assigning the algorithm, the amount of the target function is checked to determine if this allocation has improved the model. In other words, does the algorithm have a tendency to accept a new answer?
Step 6: Mutation Operator
If, after assigning different products to retailers and then target markets, the response from the target function was not improved. The genetic algorithm with the probability of this allocation is allocated in the hope of better improvements in the process of the process. Although the probability of this allocation is very low, however, it is necessary to use it to cross the optimal local points. The amount of LER will be determined by Taguchi Method.
Step 7: Termination Criteria
In this step, the genetic algorithm examines the exit conditions of the search process, and if these conditions are met, the algorithm stops, and the best answer is shown. Each of the above steps is fully explained, the mathematical form is shown, and the appropriate rates are chosen.
4.1 Number of Generations
The number of generations in the genetic algorithm actually represents the number of repetitive algorithms for improving the chromosomes of community members in different periods. Obviously, the more generations to be considered, the higher possibility of getting better answers, but it should be noted that, the higher volume of generations of an algorithm increases the problemsolving time and may lead to inefficiency of the algorithm. Therefore, proper selection of generations is necessary. For this purpose, the problems studied in this research are divided into three groups of small issues (having less than 25 variables), middle class issues having more than 26 variables and less than 100 variables, and large group issues with dimensions of more than 101 variables. In Design of Experiment section (DOE), the best values for each of the parameters are estimated.
4.2 Population Size
Community members in every generation is an important factor in the nswer improvement. The greater the number of members of the community in each generation, the wider the range of searches and subsequently the better chance for better answers. But similar to the number of generations, taking large numbers for this factor will also increase the solving time and possibly the inefficiency of the model.
4.3 Market Demands Operator
One of the features of the mathematical model presented in this research is to consider the stochastic demand of products. This feature helps the realization of the model, because in reality, the demand for fashion goods market is usually not predictable and is subject to many different factors that are beyond the scope of this research. For this reason, triangular distribution function has been used to estimate the demand for products.
It is noted that delta and gamma values are estimated by the market study and can be different for each product. But alpha and beta values that indicate the increase or decrease in demand estimation can be considered based on the decision maker opinion.
4.4 Calculate Fitness Function
After generating a result, to determine how efficient it is, it should be compared to. In this research, as in most similar researches, the objective function presented in the model will be used as the fit function. This operator function is such that the amount obtained from objective function is being compared to the best result from the objective function in the searching space so far. If the amount obtained from the current level is better than the previous, it will be used as a solution in the next generation.
4.5 Crossover Operator
The purpose of this operator is selection of genes in the chromosomes. For such purpose, if a market is selected at a single stage in each generation and increases the fitness function, then the chance for being chosen for future generations will increase. In addition, in each period, 2 responses are selected as inputs, and their valuation would be done product by product and the one with better result is located at the location of the relative gene in the chromosome (Figure 6).

1) Choose two parents. Find their chromosome string.

2) For each gene in the chromosome string, calculate the improvement of the fitness function.

3) To choose a gene, pick the one in parent that results a greater improvement rate.
4.6 Mutation Operator

Consider a number as the mutation rate (Figure 7).

Calculate the fitting function for a new generated member and compare it to the best value obtained from objective function.

If the amount of fitting function calculated for a newly generated member is less than the objective function, then accept the answer with the probability of a mutation rate.

Otherwise, reject it.
4.7 Local Optimum Escaping Operator
Falling into local optimum traps is a big concern in optimizing problems. One dominant feature of the proposed GASA is using the ability of Simulated Annealing algorithm to escape from local optimum traps (Figure 8). After calculating the fitness function for the developed solution in iterations (say ${\text{X}}_{\text{k}}^{\text{itr}}$), the GASA algorithm checks them with the best observed value achieved so far (F^{best;itr−1}). If the fitness function value (${\text{F}}_{\text{k}}^{\text{itr}}$) is less than the F^{best;itr−1} GASA replaces the ${\text{X}}_{\text{k}}^{\text{itr}}$ with X^{best;itr−1}. But at the same time if the value of ${\text{F}}_{\text{k}}^{\text{itr}}$ is more than F^{best;itr} it will be withdrawn immediately.
As shown by Figure 8, in proposed method, even after achieving the worse fitness function, the algorithm provides a base to keep them with a small probability. Such strategy lets the algorithm to keep searching to find better solutions as shown by Figure 8. Such local escaping operator is added to the algorithm by using a function which described:
where R is a normal random number between (0, 1) and LER is a local escaping rate which is defined by decision maker. Note that the exact amount of LER cannot be determined and may be different from case to case but it can be approximately estimated using design of experiments.
4.8 Stopping Criteria
Genetic algorithm would stop searching in the solution space when one of the following scenarios occurs:

The number of generations will be fully spent.

If more than 30% of generations do not succeed progressively. For example, if the number of generations is 100, then the search stops when the algorithm does not find any progress in 30 successive generations.
5. DESIGN OF EXPERIMENTS
The experimental design helps estimate the significance of each setting parameter and also determine the interactions between these parameters. In this research, Taguchi method designs are used using Minitab® 17.
For DOE experiments, the levels of each parameter must be estimated. In order to estimate levels of an input parameter a range of data for a parameter can be used and then appropriate value(s) can be estimated considering other condition of the problem like, the scale of the problem and complexity of the problem. The best estimated values for each parameter of metaheuristics are shown by the Table 4:
The levels of factors, which are shown by Table 4, are then used for DOE in order to find the best estimating value for each parameter, significant parameters and interactions between them. Upon implementing the experiments designed for the Taguchi method, the obtained results show that the algorithm is sensitive to the number of generations, population size, Mutation and local escaping rate (9).
The high slope of the curves related to the local escaping rate shows that while using GASA for small, medium and large scale problems, L2 provides better results. With the same logic, number of generations should not have more than 2 members for small scale problems. For medium and large scale problems, L2 must be considered for number of the generations. The numbers of generations and population size have moderate impact on minimizing the objective function. Figure 9 also shows that the local escaping rate has the highest impact on maximizing the algorithm power (maximizing the fitness function).
Therefore, the appropriate ranges for number of generations and number of populations can be set as what shown by Table 5:
6. RESULTS AND DISCUSSION
6.1 Solving Numerical Examples and Analysing Results
In this section, numerical examples will be solved to demonstrate the validity and performance of the proposed algorithm. These numerical examples are designed to accommodate the different. In this situation, we shall consider the problem in small, medium and largescale categories. Our aim is to illustrate if this algorithm is appropriate to solve problems in large scale mode or not. We assume that example ranges start from 5 products, 5 retailers and 5 target markets to 100 products, 30 retailers and 30 target markets (Table 6).
As the main reason for using metaheuristic models is their speed in searching the solution space and reaching to acceptable result, the computer specifications being used should be specified, as the CPU, RAM and processor type will certainly be very effective on computer speed. In this study, a personal laptop with an Intel ® Core ™ i5 processor with 2.50 GH CPU and 4 GB RAM being used. Table 7
By applying the model on various numerical examples, different results have been obtained. As seen, the developed GASA algorithm is able to solve all numerical examples in different dimensions and with different inputs (Figure 10). Therefore, using this algorithm seems to be appropriate. The results show that the proposed genetic algorithm can solve small, medium and large dimensions in reasonable CPU time (Figure 11).
The results show that the results obtained in the first generation up to the last generation in the algorithm have been improved by the range of 3% to 28% in different examples (Figure 12). Therefore, it can be concluded that the developed genetic algorithm has a good ability to search to the optimal path.
Comparing the results of genetic algorithm with forwarding serial planning method shows that the developed genetic algorithm has given better result and is able to make much suitable improvement. This means crossover, and mutation operators have good functionality.
The results of the studying the conditions of constant demand with the probable demand indicate that when market demand is probable, the amount of profit is highly affected by this difference and can be up to 40% (in the examples solved in this research) depending on the probable rate of change market slowly. This issue is very important and should be considered in different markets.
6.2 Verifying the Model Using a Practical Example
In this section, to verify the operation of the model in the real world, information from a company manufacturing cloth manufacturer in Malaysia has been collected and used. The company sends 5 types of its products to the four retailers in different states. These retailers sell their products in each of these three provinces. It should be noted that the sample numbers have been slightly modified according to the manufacturer’s request.
The rest of the example information is shown as follows (Table 8):
Table 9 represents the estimated value of the product demands after using the triangular function. Demand in various markets for these products were estimated using MATLAB using Triangular operator:
The capacity of retailer is shows in Table 10:
The remaining parameters of the genetic algorithm are considered as follows:
After solving the model, the overall result of the model was obtained as follows:
The complete solution of the model by genetic algorithm is shown below:
Part I Response:
The received amount of each type of goods produced by the retailers. This type of categorization can be considered different from the decision maker based on circumstances of the retailers (such as geographic location, population distribution, etc.) (Table 11).
In this section, the distributed the goods will be sold by the retailers based on the market needs in a way that firstly, the minimum shipping cost is obtained, and secondly, this amount wouldn’t increase the amount of goods that the retailers have received (Table 12).
As shown by Table 12 the delivered goods from any retailer does not exceed the capacity that has been considered for them. But from the comparison of Table 12 and the previous table, it can be concluded that some retailers still have a high proportion of the products produced, with surplus amounts being auctioned in the next period. As with mathematical symmetry development, none of products must being sold in the sale more than 20% of the amounts of goods received by the retailer. For example, for the first retailer (Table 13):
Table 13 shows, 10 of the third type goods, 7 of the fourth type goods and 3 of the second type goods, these values should be sold in the auction market. In the case of the first retailer, all surplus products can be sold at auction. But with regard to the second, third and fourth retailers, due to the surplus of more than the products in the warehouse, all these quantities cannot be auctioned on the market, so the residual products with a dropin value are returned to the producer. Table 14 represents the allowed amounts of unsold goods to be returned to the manufacturer at the end of each production period. The return rate is considered 0.5.
Table 15 shows the Information of sold goods, goods to sale, goods to be returned by each of the retailers. The outcomes indicated that the algorithm can successfully distributed goods between the retailers and customers.
The remained products at retailer may get thrown away because of nonreturn acceptance by the producer. It should be noted that if we consider 1 for the rate of goods return, that means all the products that hasn’t been sold or auctioned, and for the rate of 0 none of the goods at such state would get returned to the producer and so will be thrown away.
7. CONCLUSIONS
This paper presents an applicable method for scheduling dynamic supply chain systems of fashion goods in the presence of the market demand uncertainty. The main aim of the research is evaluating the impact of market uncertainty on product distributing by manufacturer, wholesalers and retailers in dynamic SCM.
In this regard an integer programming model is developed for sailing fashion goods in multimarket multiproduct problem where the objective was calculating the amount of profits from the sale of products by retailers. Since the developed model was Nphard, a hybrid Genetic and Simulated Annealing algorithms (GASA) was proposed to solve the problem. The outcomes indicted that uncertainty in market demands for fashion goods has significantly affects on sailing such products. The results is also showed that using different rates auctions and product return to manufacturer can play a key role in maximizing the profit of the supply chain. As a subsidiary conclusion, the results also showed that using improper vehicles for transferring the product can decrease the productivity of the supply chain. In the other words, if the vehicles are not scheduled appropriately, more number of travels is needed that will increase the system costs.
The proposed algorithm is then applied for a clothing company in Malaysia. The outcomes show that the proposed algorithm can successfully being used in real industries and there is no extra encoding and decoding is needed. Future expansions of the proposed method by considering rival’s strategies on sailing of a dynamic supply chain is suggested. It is also recommended to consider the impact of market demand uncertainty as an objective for pricing of fashion goods.