Fashion supply chain management is challenging due to deman d uncertainty, short product life cycle, ﬁerce global compet ition, increasing expectations for diversity in production, and ke eping pace with the current and emerging trends. In this pape r, a brief review of the published works in the ﬁeld of fashion and cl othing supply chain is presented and an integrated linear mo del is developed to minimize the costs of a fashion brand, includ ing production and logistics costs, enabling the ﬁrm to make better decisions with regard to its production and replenishment p olicies and to respond in a reasonable time to customers dema nds for various designed items. We show that our model can succes sfully account for the complex structure of the supply chain consisting of designers, manufacturers, distributors, an d stores. The model is validated using the General Algebraic M odeling System (GAMS) with the data collected from an Iranian fashion b rand. Key words: Fashion Supply Chain Management; Operations research; Optim ization; Mathematical Modelling. 1 Introduction The fashion industry as a whole can consist of design, procurement of raw materials, production, distribution, marketing and sales. Mass production of clothes started in mid-19th century when producers started producing cloths that didn’t need the tailor to take measures for every customer individually. However, fashion did not becomeanindustryuntilthe20thcenturywhenthenet- work of local tailors transformed into bigger production businesses.AfterworldwarIItextilefactoriesandcloth- ing businesses experienced higher development due to higherdemandsandculturalandsocietalchangeswhich eventually resulted in less regularization and more di- versity in the clothing and fashion industry. At the be- ginning of the 20th century, signiﬁcant changes in the fashion business such as higher attention to creativity in design, European fashion houses, technological devel- opment such as sewing machines, the emergence of cap- italism, systematic development of factories and retail- ers made the fashion industry bigger and more powerful [9,2]. Email addresses: sarashoarinejad@gmail.com (Sara Shoarinejad), amin.shoarinejad@gmail.com (Amin Shoari Nejad).Today fashion industry is bigger than ever and in or- der to remain in the business it is very important to ef- ﬁciently manage the fashion supply chain especially in the presence of diﬀerent players such as suppliers, dis- tributors and producers that need to collaborate in har- monyinordertomeetthemarketdemand.Fashionsup- plychainmanagementrequiresconsideringtheindustry as a complex interconnected system in which businesses need to try balancing supply and demand, minimizing costs, maximizing proﬁt, managing risk, and unifying theirprocessesandtheirassociatedinformationﬂowsfor optimizingtheirsupplychainandobtainingcompetitive advantage [1,11]. In this paper, we aim at developing a mathematical modelthattakesintoaccountdemandformultiplefash- ion items in diﬀerent stores and helps businesses to de- cide about their production quantities and distribution plans while minimizing their costs. The paper is organized as follows. In section 2, a sum- mary of previous related works is provided. In section 3, we describe the problem that we try to solve. In sec- tion 4, we explain our modelling approach. In section 5 we discuss the model validation and results. In section 6, we discuss the implications of our method and how our model can help decision makers. We summerize the paper and conclude in 7. Preprint submitted to engrXiv.org 5 January 2022 2 Literature review [3]studiedaﬂexiblepricingschemethatallowstheprice tovarydependingoncustomer.Theirgoalwastoﬁndan optimalsolutionusingauniﬁedmodelthatconsidersthe ﬂexibility between the start and the end time of produc- tion and acceptance/rejection of the orders. They used amixed-integerprogrammingandaheuristicmethodto solve the problem and showed the signiﬁcant eﬀect of uniﬁed production on proﬁt considering other decisions of the supply chain. [7] focused on fashion products under deterministic de- mand, economic order quantity (EOQ) and limited re- plenishment.Fashionretailersusuallyusedynamicpric- ing (e.g. oﬀering discount) to deplete their stocks. They found the optimal order quantity and order point in or- der to maximize proﬁt and showed the eﬀect of them on demand by considering the customer saving price (the minimum price that a customer is willing to pay). They utilizeddiscreteanddynamicpricingschemesandalter- nated between them multiple times in order to satisfy customers demand in diﬀerent periods and minimized the inventory cost. [10] introduced a system dynamics model to analyze the fashion supply chain. They evaluated the eﬀect of work duration and transportation delays on the perfor- mance of a fast fashion supply chain including three lev- els of producer, distributor and retailer. They showed how each level is aﬀected by the delays. [8] developed a causal loop diagram for fashion supply chain. They divided the supply process into two parts: pre sale season and the sale season. The former includes collectionplanning,demandforecasting,purchase,logis - ticsandreplenishmentandthelatterincludesbalancing supply and demand. [5] focused on the problem of determining initial order quantity and replenishment of a retail product to min- imize the opportunity cost, cost of returned products and expired inventory. They framed the problem into a two-step stochastic dynamical programming model and suggested a novel heuristic method to ﬁnd the optimal solution. They showed that their solution could double the proﬁt. [6] in collaboration with ZARA, designed and tested a support system model for replenishment and distribu- tion. Their model is data-driven and is designed to up- datedemandforecastingandallocateinventorythrough time using dynamical optimization methods. Fastfashionisacompetitiveareainthefashionindustry and new and diverse trends, highly uncertain demand, andhighinventoryaresomeofitsimportantcharacteris- tics.Therefore,inthisarea,decisionmakersshouldhave Fig. 1. Schematic of the fashion supply chain. high operations ﬂexibility to respond to demand in a timely manner. [4] developed an optimization model for a supply chain of a fashion retailer for a fast response to demand and having an optimized replenishment policy. Their model is cable of calculating the optimal quantity that needs to be transferred between stores in order to meet the demand in diﬀerent locations. They suggested thattheirmodelcanhelpinternationalretailerstomake better decisions for replenishing their stocks according to feedback from the market and sales data. 3 Problem deﬁnition Inthissectionweoutlinetheproblem.Thegoalistode- velop a model for a multilevel fashion supply chain en- abling fast response to customers demands. We suggest a model considering a direct supply chain and we aim at developing a decision support system for production and replenishment quantities. We use data provided by Queen Manto, a fashion brand based in Iran. Our sug- gested model could be an extension to the [4] by con- sidering four levels of fashion supply chain instead of one.Thesupplychainofourproblemincludesdesigning center, manufacturing centers, distribution centres and stores. The functionality of each level are as follows: (1)Designing center : Determining the kind and quantities of diﬀerent designs to be produced by the manufacturing centers. (2)Manufacturing centers : Producing the placed ordersandsendingthemtothedistributioncenters. (3)Distribution centers : Products are stored in these centers before being sent to stores. (4)Stores: Products received from the distribution centers are sold to customers in stores. 4 Modelling procedure In this section we explain the details of our modelling approach for the supply chain schematically shown in ﬁgure 1. We consider the total cost, including ﬁxed and 2 Table 1. Nomenclature used in the model. Indices Description i:i∈ {1,2,...,I} designs j:j∈ {1,2,...,J} manufacturers k:k∈ {1,2,...,K} distribution centers e:e∈ {1,2,...,E} stores Decision variables Description trans ijk number of item i to be sent from manufacturer j to distributio n center k trans ike number of item i to be sent from distribution center k to store e xij number of item i to be produced by manufacturer j Model parameters Description cij cost of producing item i by manufacturer j ctfjk ﬁxed cost of sending item i from manufacturer j to distributio n center k ctvjk variable cost of sending item i from manufacturer j to distri bution center k ctfke ﬁxed cost of sending item i from distribution center k to store e ctvke variable cost of sending item i from distribution center k to store e capi capacity of manufacturer j capk capacity of manufacturer k cape capacity of store e demie demand for item i in store e distjk distance from manufacturer j to distribution center k distke distance from distribution center k to store e varying logistics costs for transferring the products be- tween the entities of the supply chain and also produc- tion costs, as the objective function that needs to be minimised. Table 1 describes the nomenclature used in the model. We develop a linear programming model as follows: •The cost Zis deﬁned as: Minz =/summationdisplay i/summationdisplay jxijcij+ /summationdisplay j/parenleftBigg ctfj+ctvj/parenleftBigg/summationdisplay kdistjk/parenleftBigg/summationdisplay itransijk/parenrightBigg/parenrightBigg/parenrightBigg + /summationdisplay k/parenleftBigg ctfk+ctvk/parenleftBigg/summationdisplay edistke/parenleftBigg/summationdisplay itransike/parenrightBigg/parenrightBigg/parenrightBigg (1) In this function the ﬁrst sum accounts for the cost of producing the designed cloths by the producers and the 2nd and the 3rd sums are accounting for the cost of transportation of the goods. •Thequantityoftheproducedgoodsmustbelessthantheir capacity: /summationdisplay ixij≤capj∀j (2) •Theamountoftransferredgoodsfromtheproducerto the distribution center must beless than the available quantity of that product in the producer’s warehouse: xij−/summationdisplay ktransijk≥0∀i∀j (3) •The amount of transferred goods to the distribution centers must be less than their capacity: /summationdisplay i/summationdisplay jtransijk≤capk∀k (4) •The amount of goods transferred to the stores must be less than the amount of goods transferred from the 3 manufacturer to the distribution center. /summationdisplay jtransijk−/summationdisplay etransike≥0∀i∀k (5) •Theamountofproductstransferredtothestoresmust be less than their capacity: /summationdisplay i/summationdisplay ktransike≤cape∀e (6) •Theamountofproductstransferredtothestoresmust must satisfy the demand: /summationdisplay ktransike≥demie∀i∀e (7) •Necessary boundaries: transike,transijk,xij≥0 (8) 5 Model validation To validate the suggested model, historical data associ- atedwithcostsanddemandsoftheQueenMantobrand are being used. The direct supply chain of our problem consists of ten diﬀerent designed products, three manu- facturer, four distribution centers and three stores. We used GAMS to solve the problem. The solver suc- cessfullyconvergedundertwosecondsandfoundtheop- timal value of the objective function to be Z= 935209. The decision variables values for the solution are given in ﬁgures 2-4. 6 Discussion Supply chain managers in the fashion industry have to make reasonable and fast decisions due to high demand anditscomplexdynamics.Ournovelmodelcanhelpthe managers to make better decisions about their produc- tion,replenishment,andlogisticspoliciesinareasonabl e time. As opposed to previous works such as [4] that was limited to only manufacturers and stores, our suggested model considers the decisions of designers, manufactur- ers, distribution centers and stores in a more compre- hensive way. We believe managers in the fashion industry equipped withourmodelandwithananalyticalpointofview,can have better control over their supply chain components, Fig. 2. The optimal quantity of item ito be produced by manufacturer j. Fig. 3. The optimal quantity of item iproduced by manufac- turerj(i-jiscodedonthey-axis)tobesenttodistributor k. such as logistics between diﬀerent centers, and make de- cisions that result in more eﬃcient use of resources. For instance,ifthemanagerscometorealizethattheamount of goods transferred between two centers are frequently high in diﬀerent periods, they might want to use more suitable and eﬃcient transportation vehicles for those routes to further minimize their costs. Insummary,ourmodelconsistingofmultiplelevelsfrom the designer to the customers can give valuable insights about the diﬀerent parts of the supply chain and enable the decision makers to optimize their operations encom- passing production quantities, logistics, inventory lev- els, and sales. However, our model has some limitations which we believe can be addressed by adding some ex- tensions to it and making it more comprehensive and more ﬂexible. Accordingly, future works could possibly include these extensions: considering dynamic pricing, 4 Fig. 4. The optimal quantity of item ithat distributor k(i-k is coded on the y-axis) has to send to store e. including promotions eﬀects, and stochastic demand. 7 Conclusion We developed a model to minimize production and lo- gistics costs for fast response to customer demands and eﬃcient replenishment in fashion supply chain with four major levelsincludingdesigner,manufacturers, distribu - tion centers, and stores. We validated our model using data from a fashion brand and solved for the optimal solution using GAMS. We reported the results of our model including the optimal quantity for production of the designers’ designed items by each manufacturer, the optimal quantity of goods that need to be transferred from each manufacturer to each distribution center, and the optimal quantity of goods that need to be sent from the distribution centers to the stores in order to satisfy demand and minimise the costs.