Guest Editors: Tsan-Ming Choi, Xiaohang Yue, Chun-Hung Chiu, and Pui-Sze Chow

Mathematical Modeling Research in Fashion and Textiles Supply Chains and Operational Control Systems

Mathematical Problems in Engineering

Mathematical Modeling Research inFashion and Textiles Supply Chains andOperational Control Systems

Mathematical Problems in Engineering

Mathematical Modeling Research inFashion and Textiles Supply Chains andOperational Control Systems

Guest Editors: Tsan-Ming Choi, Xiaohang Yue,Chun-Hung Chiu, and Pui-Sze Chow

Copyright © 2013 Hindawi Publishing Corporation. All rights reserved.

This is a special issue published in “Mathematical Problems in Engineering.” All articles are open access articles distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided theoriginal work is properly cited.

Editorial Board

Mohamed Abd El Aziz, EgyptEihab M. Abdel-Rahman, CanadaRashid K. Abu Al-Rub, USASarp Adali, South AfricaSalvatore Alfonzetti, ItalyIgor Andrianov, GermanySebastian Anita, RomaniaW. Assawinchaichote, ThailandErwei Bai, USAEzzat G. Bakhoum, USAJosé Manoel Balthazar, BrazilR. K. Bera, IndiaChristophe Bérenguer, FranceJonathan N. Blakely, USAStefano Boccaletti, SpainStephane P. A. Bordas, USADaniela Boso, ItalyM. Boutayeb, FranceMichael J. Brennan, UKSalvatore Caddemi, ItalyPiermarco Cannarsa, ItalyJose E. Capilla, SpainCarlo Cattani, ItalyMarcelo Cavalcanti, BrazilDiego J. Celentano, ChileMohammed Chadli, FranceArindam Chakraborty, USAYong-Kui Chang, ChinaMichael J. Chappell, UKXinkai Chen, JapanKui Fu Chen, ChinaKue-Hong Chen, TaiwanJyh-Hong Chou, TaiwanSlim Choura, TunisiaCesar Cruz-Hernandez, MexicoSwagatam Das, IndiaFilippo de Monte, ItalyMaria de Pinho, PortugalAntonio Desimone, ItalyYannis Dimakopoulos, GreeceBaocang Ding, ChinaJoao B. R. Do Val, BrazilDaoyi Dong, AustraliaB. Dubey, IndiaHorst Ecker, Austria

M. Onder Efe, TurkeyElmetwally Elabbasy, EgyptAlex Elias-Zuniga, MexicoAnders Eriksson, SwedenVedat S. Erturk, TurkeyQi Fan, USAMoez Feki, TunisiaRicardo Femat, MexicoRobertt Fontes Valente, PortugalClaudio Fuerte-Esquivel, MexicoZoran Gajic, USAUgo Galvanetto, ItalyFurong Gao, Hong KongXin-Lin Gao, USABehrouz Gatmiri, IranOleg V. Gendelman, IsraelDidier Georges, FrancePaulo Batista Gonçalves, BrazilOded Gottlieb, IsraelFabrizio Greco, ItalyQuang Phuc Ha, AustraliaM. R. Hajj, USAThomas Hanne, SwitzerlandKatica Hedrih, SerbiaM. I. Herreros, SpainWei-Chiang Hong, TaiwanJaromir Horacek, Czech RepublicGordon Huang, CanadaChuangxia Huang, ChinaHuabing Huang, ChinaYi Feng Hung, TaiwanHai-Feng Huo, ChinaAsier Ibeas, SpainAnuar Ishak, MalaysiaReza Jazar, AustraliaZhijian Ji, ChinaJun Jiang, ChinaJ. J. Judice, PortugalTadeusz Kaczorek, PolandTamas Kalmar-Nagy, USATomasz Kapitaniak, PolandHamid Reza Karimi, NorwayMetin O. Kaya, TurkeyNikolaos Kazantzis, USAFarzad Khani, Iran

Kristian Krabbenhoft, AustraliaRen-Jieh Kuo, TaiwanJurgen Kurths, GermanyClaude Lamarque, FranceMarek Lefik, PolandStefano Lenci, ItalyRoman Lewandowski, PolandShanling Li, CanadaMing Li, ChinaJian Li, ChinaShihua Li, ChinaTeh-Lu Liao, TaiwanPanos Liatsis, UKJui-Sheng Lin, TaiwanYi-Kuei Lin, TaiwanShueei M. Lin, TaiwanWanquan Liu, AustraliaBin Liu, AustraliaYuji Liu, ChinaPaolo Lonetti, ItalyVassilios C. Loukopoulos, GreeceChien-Yu Lu, TaiwanJunguo Lu, ChinaAlexei Mailybaev, BrazilManoranjan K. Maiti, IndiaOluwole Daniel Makinde, South AfricaRafael Martinez-Guerra, MexicoDriss Mehdi, FranceRoderick Melnik, CanadaXinzhu Meng, ChinaYuri V. Mikhlin, UkraineGradimir Milovanovic, SerbiaEbrahim Momoniat, South AfricaTrung Nguyen Thoi, VietnamHung Nguyen-Xuan, VietnamBen T. Nohara, JapanSotiris Ntouyas, GreeceGerard Olivar, ColombiaClaudio Padra, ArgentinaBijaya Ketan Panigrahi, IndiaFrancesco Pellicano, ItalyMatjaz Perc, SloveniaVu Ngoc Phat, VietnamAlexander Pogromsky, The NetherlandsSeppo Pohjolainen, Finland

Stanislav Potapenko, CanadaSergio Preidikman, USACarsten Proppe, GermanyHector Puebla, MexicoJusto Puerto, SpainDane Quinn, USAKumbakonam Rajagopal, USAGianluca Ranzi, AustraliaSivaguru Ravindran, USAG. Rega, ItalyPedro Ribeiro, PortugalJ. Rodellar, SpainRosana Rodriguez-Lopez, SpainAlejandro J. Rodriguez-Luis, SpainIgnacio Romero, SpainHamid Ronagh, AustraliaCarla Roque, PortugalRubén Ruiz Garcı́a, SpainManouchehr Salehi, IranMiguel A. F. Sanjuán, SpainIlmar Ferreira Santos, DenmarkNickolas S. Sapidis, GreeceEvangelos J. Sapountzakis, GreeceBozidar Sarler, SloveniaAndrey V. Savkin, AustraliaMassimo Scalia, ItalyMohamed A. Seddeek, EgyptAlexander P. Seyranian, RussiaLeonid Shaikhet, UkraineCheng Shao, China

Bo Shen, GermanyDaichao Sheng, AustraliaTony Sheu, TaiwanJian-Jun Shu, SingaporeZhan Shu, UKDan Simon, USALuciano Simoni, ItalyGrigori M. Sisoev, UKChristos H. Skiadas, GreeceDavide Spinello, CanadaSri Sridharan, USARolf Stenberg, FinlandJitao Sun, ChinaXi-Ming Sun, ChinaChangyin Sun, ChinaAndrzej Swierniak, PolandYang Tang, GermanyAllen Tannenbaum, USACristian Toma, RomaniaIrina N. Trendafilova, UKAlberto Trevisani, ItalyJung-Fa Tsai, TaiwanKuppalapalle Vajravelu, USAVictoria Vampa, ArgentinaJosep Vehi, SpainStefano Vidoli, ItalyDan Wang, ChinaYouqing Wang, ChinaYongqi Wang, GermanyMoran Wang, China

Cheng C. Wang, TaiwanYijing Wang, ChinaXiaojun Wang, ChinaGerhard-Wilhelm Weber, TurkeyJeroen Witteveen, USAKwok-Wo Wong, Hong KongZheng-Guang Wu, ChinaLigang Wu, ChinaWang Xing-yuan, ChinaX. Frank Xu, USAXuping Xu, USAXing-Gang Yan, UKJun-Juh Yan, TaiwanSuh-Yuh Yang, TaiwanMahmoud T. Yassen, EgyptMohammad I. Younis, USAHuang Yuan, GermanyS. P. Yung, Hong KongIon Zaballa, SpainAshraf Zenkour, Saudi ArabiaJianming Zhan, ChinaYingwei Zhang, ChinaXu Zhang, ChinaLu Zhen, ChinaLiancun Zheng, ChinaJian Guo Zhou, UKZexuan Zhu, ChinaMustapha Zidi, France

Contents

Mathematical Modeling Research in Fashion and Textiles Supply Chains and Operational ControlSystems, Tsan-Ming Choi, Xiaohang Yue, Chun-Hung Chiu, and Pui-Sze ChowVolume 2013, Article ID 470567, 4 pages

Fashion Brand Purity and Firm Performance, Jin-hui Zheng, Zixia Cao, Xin Dai, and Chun-Hung ChiuVolume 2013, Article ID 363095, 18 pages

The Impact of the Strategic Advertising on Luxury Fashion Brands with Social Influences, Jin-Hui Zheng,Bin Shen, Pui-Sze Chow, and Chun-Hung ChiuVolume 2013, Article ID 534605, 16 pages

No Refund or Full Refund: When Should a Fashion Brand Offer Full Refund Consumer Return Servicefor Mass Customization Products?, Tsan-Ming Choi, Na Liu, Shuyun Ren, and Chi-Leung HuiVolume 2013, Article ID 561846, 14 pages

Collective Recycling Responsibility in Closed-Loop Fashion Supply Chains with a Third Party: FinancialSharing or Physical Sharing?, Jiajia Nie, Zongsheng Huang, Yingxue Zhao, and Yuan ShiVolume 2013, Article ID 176130, 11 pages

Clarifying Cutting and Sewing Processes with Due Windows Using an Effective Ant ColonyOptimization, Rong-Hwa Huang and Shun-Chi YuVolume 2013, Article ID 182598, 12 pages

Optimal Product Quality of Supply Chain Based on Information Traceability in Fashion and TextilesIndustry: An Adverse Logistics Perspective, Zhaolin Cheng, Jinghua Xiao, Kang Xie, and Xiaoli HuangVolume 2013, Article ID 629363, 13 pages

A Heuristic Approach to Proposal-Based Negotiation: With Applications in Fashion Supply ChainManagement, Stefania Costantini, Giovanni De Gasperis, Alessandro Provetti, and Panagiota TsintzaVolume 2013, Article ID 896312, 15 pages

Loss-Averse Inventory and Borrowing Decisions with Constraints on Working Capital in Fashion andTextiles Industry, Lijun Ma, Weili Xue, Yingxue Zhao, and Xudong LinVolume 2013, Article ID 657641, 9 pages

A Coordination of Risk Management for Supply Chains Organized as Virtual Enterprises, Min Huang,Xingwei Wang, Fu-Qiang Lu, and Hua-Ling BiVolume 2013, Article ID 931690, 11 pages

A Novel Ensemble Learning Approach for Corporate Financial Distress Forecasting in Fashion andTextiles Supply Chains, Gang Xie, Yingxue Zhao, Mao Jiang, and Ning ZhangVolume 2013, Article ID 493931, 9 pages

Vehicle Routing Problem for Fashion Supply Chains with Cross-Docking, Zhi-Hua Hu, Yingxue Zhao,and Tsan-Ming ChoiVolume 2013, Article ID 362980, 10 pages

The Impact of the Subsidy Policy on Total Factor Productivity: An Empirical Analysis of China’s CottonProduction, Yanwen Tan, Jianbo Guan, and Hamid Reza KarimiVolume 2013, Article ID 248537, 8 pages

Coordinating Contracts for Two-Stage Fashion Supply Chain with Risk-Averse Retailer andPrice-Dependent Demand, Minli Xu, Qiao Wang, and Linhan OuyangVolume 2013, Article ID 259164, 12 pages

Quantity Discount Supply Chain Models with Fashion Products and Uncertain Yields, Hongjun Peng andMeihua ZhouVolume 2013, Article ID 895784, 11 pages

Sales Rebate Contracts in Fashion Supply Chains, Chun-Hung Chiu, Tsan-Ming Choi, Ho-Ting Yeung,and Yingxue ZhaoVolume 2012, Article ID 908408, 19 pages

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013, Article ID 470567, 4 pageshttp://dx.doi.org/10.1155/2013/470567

EditorialMathematical Modeling Research in Fashion andTextiles Supply Chains and Operational Control Systems

Tsan-Ming Choi,1 Xiaohang Yue,2 Chun-Hung Chiu,3 and Pui-Sze Chow4

1 Institute of Textiles and Clothing, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong2 Lubar School of Business, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA3 Sun Yat-Sen Business School, Sun Yat-Sen University, Guangzhou 510275, China4 Institute of Textiles and Clothing, Faculty of Applied Science and Textiles, The Hong Kong Polytechnic University,Hung Hom, Hong Kong

Correspondence should be addressed to Tsan-Ming Choi; jason.choi@polyu.edu.hk

Received 11 March 2013; Accepted 11 March 2013

Copyright © 2013 Tsan-Ming Choi et al.This is an open access article distributed under theCreativeCommonsAttribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The fashion and textiles industry is one of themost importantindustries in the world and it has rapidly developed over thepast decade. With its fast development, the correspondingsupply chain systems become much more complex and a lotof challenging optimization-related issues arise. These issuesspan over topics such as product quality control, shippingand distribution arrangements, coordination mechanism,and inventory management. This special issue is devoted topublishing the latest and significant results on mathematicalmodeling research related to fashion and textiles supplychains and operational control systems. This special issueis on the interface between engineering and business andfeatures both analytical and empirical studies. This specialissue features 15 technical papers and we describe each oneof them concisely as follows.

In “A heuristic approach to proposal-based negotiation:with applications in fashion supply chain management,” S.Costantini et al. explore a heuristic approach to proposal-based negotiation with applications in fashion supply chainmanagement. In their proposed model, they present andassess experimentally an extended approach which allowstwo supply chain agents to potentially make offers that arean internal point of its negotiation space. They report thatthe extended approach functions well and identify the casesin which it outperforms the original approach as reported inthe literature.They discuss the usefulness of the approach forsupply chain management in the fashion industry, which is

a field of growing importance in economy and e-commerce.They further argue that the efficient computational modelssuch as the one that they proposed can greatly facilitate thework of humanmanagers in fashion (both sellers and buyers)who have to negotiate every day with multiple counterpartson multiple issues.

In “Sales rebate contracts in fashion supply chains,” C. H.Chiu et al. investigate the sales rebate contract, which is acommonly adopted supply chain contract in the fashionindustry, with the use of real industrial data from fivecompanies. In their analysis, they find that sales-rebatescontract can help to achieve coordination in the supply chaininwhich both the retailer and themanufacturer are benefited.One interesting finding is that the risk levels for attainingcoordination by sales rebates contract are also higher for boththe retailer and the manufacturer. They hence propose thatthe optimal parameters of the sales-rebate contracts shouldbe determined with a good balance between the benefit(expected profit) and the risk (variance of profit). They arguethat an inappropriate setting of sales rebate could eitherhurt the expected profit for the retailer or the manufacturer.Finally, they show that in the presence of sales effort, higherexpected profits and lower risks for the retailers and themanufacturers can be found by using sales rebates contract.

In “Quantity discount supply chain models with fashionproducts and uncertain yield,” H. Peng and M. Zhou explorethe analytical quantity discount coordination models in

2 Mathematical Problems in Engineering

the fashion supply chain with uncertain yields and ran-dom demand. They establish a stochastic gaming model foremploying quantity discount to achieve coordination andemploy it to analyze how to optimally determine the quantitydiscount scheme. They find that their proposed quantity dis-count coordination scheme can increase the manufacturer’sordering quantity for the raw material, which contributes tothe desirable result of achieving the best profitability levelin the fashion supply chain. They also reveal that a largeruncertainty of the yields and the demand in the supply chainwill significantly affect the coordination mechanism and alsothe resulting fashion supply chain profit.

In “A coordination of risk management for supply chainsorganized as virtual enterprises,” F. Lu et al. investigate therisk management for fashion supply chains organized asvirtual enterprises. The aim of their study is to find theproper decision mechanisms that can improve the overallperformance of risk management for the whole supplychain system and its agents as a virtual enterprise (VE). Inorder to deal with this challenge, they develop a centralizedmechanism as a base case and then derive a novel distributeddecision-making (DDM) method. Moreover, the particleswarm optimization (PSO) algorithm is designed to solvethe resulting optimization problem. They find that theirproposedmechanism can help to achieve a win-win situationin the fashion supply chain. They argue that the strategicpartnerships among the members in a fashion supply chainorganized as a VE are essential for improving the fashionsupply chain’s performance and responsiveness.

In “Loss-averse inventory and borrowing decisions withconstraints on working capital in fashion and textiles industry,”L. Ma et al. observe that decision makers in real worldhave different decision preferences. As a result, they usethe prospect theory to model the loss-averse behavior ofdecision makers in a fashion supply chain and study theinventory control problems with financial constraints. Byusing dynamic programming, they characterize the optimalinventory replenishment policy as a capital-dependent base-stock policy and study how the policy parameters changewith respect to the accumulated wealth and the loss-averseindicator.

In “The impact of the strategic advertising on luxury fash-ion brands with social influences,” J. H. Zheng et al. studythe optimal strategic advertising decisions on luxury fashionbrands with social influences. They observe that consumersfor fashion products are influenced significantly by theirsocial needs such as the need for uniqueness and the need ofconformity whenmaking a purchase decision. In their paper,they focus on investigating the impacts of two competingsocial needs which separate customers into two groups whoexhibit different buying behaviors. They start by consideringdifferent advertising allocation strategies with social needsand then develop the model under which a luxury fashionbrand will suffer a loss when the advertising efforts todifferent customer groups are not up to a certain level. Inorder to reveal the impact of the strategic advertising onluxury fashion brands with social influences, they derive themechanism for the fashion brands to identify the optimal

strategies and decisions in pricing and advertising budgetallocation.

In “Coordinating contracts for two stage supply chain withrisk-averse retailer and price dependent demand,” M. Xu et al.incorporate the fashion retailer’s risk aversion attitude alongwith pricing factor into a two-stage fashion supply chain viathe classic mean-variance approach. They consider a price-dependent demand in the supply chain and discuss bothsingle contracts and combined contracts for achieving supplychain coordination. When the risk averse fashion retailer’soptimal pricing-ordering decision is equal to the verticallyintegrated supply chain’s optimal decision, the fashion supplychain is coordinated and the whole supply chain gains themaximum profit. They find that under the single contractsscenario, the coordinating revenue sharing contract and two-part tariff contract for the supply chains with risk neutralagents could still coordinate the supply chain with a risksensitive retailer. However, a more complicated “sales rebateand penalty” contract fails to do so. When consideringcombined contracts, they find that the joint revenue sharingwith two-part tariff contract is able to achieve supply chaincoordination.

In “The impact of subsidy policy on total factor produc-tivity: an empirical analysis for China’s cotton production,”Y. W. Tan et al. investigate the impact of subsidy policy ontotal factor productivity for China’s cotton production. Theirstudy is motivated by an important observation that since2007, China has carried out one subsidy policy on cottonseed in high quality (simplified as “seed subsidy”) in orderto motivate the farmers to produce more cotton. However,Tan et al. find that the seed subsidy policy has not achievedthe devising aim to increase the productivity and the outputof cotton. Motivated by this important observation, Tan etal. investigate, using a theoretical mathematic model, therelationship between the subsidy policy related to arableland and the productivity. They also explore the impact ofthe subsidy policy on the total factors productivity (TFP)of China’s cotton production using Malmquist Index. Theyargue that their proposed mathematical model can interpretwhat would happen about the TFP after the subsidy policyis implemented. Their findings also indicate that there existsa negative relationship between the subsidy policy andTFP if the subsidy is related to planting area. They arguethat promoting investment in research and development ofagriculture, and enhancing technical progress in agriculturemay be a better way to increase the agricultural TFP than thesubsidy policy.

In “Vehicle routing problem for fashion supply chains withcross-docking,” Z. H. Hu et al. address the issue of how tooptimize the overall traveling time, distance, and waitingtime at a cross-docking center in a fashion supply chain.In their study, they formulate the optimal route selectionproblems from suppliers to cross-docking center and fromcross-docking center to customers as the respective vehiclerouting problems (VRPs). By an integration of the twoindependent VRP models, they identify the optimal solu-tions. They propose that the fashion supply chains, especiallythe textiles supply chains, can use the model developed inthe paper to form strategies that can significantly improve

Mathematical Problems in Engineering 3

the overall operations performance of their cross-dockingcenters. Furthermore, even though the VRP is NP-hard, withtheir proposed mixed 0/1 integer programming model, theissue under study can be solved within a reasonable time,which significantly facilitates the practical applicability of thecross-docking model.

In “A novel ensemble learning approach for corporate fina-ncial distress forecasting in fashion and textiles supply chains,”G. Xie et al. explore the prediction problem of corporatefinancial distress related to fashion enterprises. To achievegood forecasting performance, they propose a novel fore-casting model in which a logistics regression (LR) model isintegrated with artificial intelligence tools, that is, supportvector machine (SVM) and back-propagation neural net-works (BPNN). To be specific, in their proposed forecastingframework, the forecasting results by LR are introduced intothe SVM and BPNN which can help recognizing the fore-casting errors in fitness by LR. Their computational resultssuggest that artificial intelligence tools are better than LR andthe proposed novel ensemble learning approach can achievebetter forecasting performance than that of the individualmodels. By using the proposed approach,managers in fashionand textiles companies can predict the financial state of theirsuppliers, manufacturers, and retailers in advance which cangreatly enhance their operations.

In “Fashion brand purity and firm performance,” J.Zheng et al. develop a dynamic brand dilution model basedon the economics demand curves and two fashion consumergroups: targeted group (leading group) and nontargetedgroup (following group).They consider the situation inwhichfashion companies can adjust product prices to change salesvolumes, according to the demand curve, and further changethe composition of LG and FG that reflects the degree ofbrand dilution. One significant aspect of their study is that“brand purity” is used to measure brand dilution level andthis concept can quantitatively illustrate the relationshipbetween brand dilution and firm performance includingrevenue and profit. They derive propositions and conductnumerical studies to explain the dilution situations thatmanyfashion brands experienced. They argue that (1) with thestrong motivation to expand customer base, fashion firmstend to dilute the brands unintentionally. (2) Brand dilutioncan harm firm performance significantly and can be easilyaggravated. (3) To increase brand purity, firms need toexclude the FG consumers and focus on its LG consumers.

In “Clarifying cutting and sewing processwith duewindowsusing an effective ant colony optimization,” R. H. Huang and S.C. Yu explore the cutting and sewing process in textiles andclothing with due windows. They formulate the problem asa two-stage flexible flow shop scheduling problem with theobjective of minimizing earliness, tardiness, and makespan.They develop a novel “effective ant colony optimization(EACO)” method to solve the problem. Their computationalresults indicate that for either small or large problems, EACOis more effective than the particle swarm optimization algo-rithm and the traditional ant colony optimization algorithm.They also argue that EACO can be used to solve complex flowshop scheduling problem efficiently.

In “Optimal product guality of supply chain based oninformation traceability in fashion and textiles industry: anadverse logistics perspective,” Z. Cheng et al. investigate andcompare two quality control methods, namely, the inspectioncontrol and the traceability control, to optimize supply chainquality in the fashion and textiles industry. They considerquality as a controllable variable indicating the level ofopportunistic behavior. They model “return” as a functionof quality in which a higher quality implies a lower return.Based on theories of resource-based view (RBV) anddynamiccapability view (DCV), they propose that the product type,the manufacturer’s technology capability, and the standard-ization level all affect the manufacturer’s choice of qualitycontrol methods (inspection versus traceability). They fur-ther reveal that the manufacturer’s optimal strategy of qualitycontrol is to impose appropriate punishment measures giventhe high cost of replacing suppliers. Taking into account bothquality and inventory quantity, they propose a newsvendormodel-based adverse logistics model to analyze and comparethe two different methods for supply chain quality control.Furthermore, they analyze and discuss the differences inthe applications and scopes of the two control methods.Through examining the model and the numerical example,they discuss the four propositions and their implications tooptimal supply chain quality management in the fashion andtextiles industry. With the improved industrial standardiza-tion and technology, they believe that there is much room forthe manufacturer to use information technology to improveproducts quality in fashion and textiles industry in the future.

In “No refund or full refund: when should a fashion brandoffer consumer return with full refund for mass customizationproducts?” T. M. Choi et al. explore the mass customizationproblem in fashion supply chains with the considerationof consumer demand uncertainty and risk aversion of thecompany. Under the mean-variance framework, they studyhow the degree of risk aversion affects the company’s decisionon optimal retail pricing and modularity level. They alsoexamine the interesting research question on when the MCcompanies should offer consumer return with full refund.They find that for the scenario with full refund and return,the optimal retail price and the optimal modularity levelwould vary monotonically with respect to the return servicecharge and the salvage value when some analytical conditionsare satisfied. For the case when return is not allowed, theyfind that the optimal retail price and the optimal modularitylevel are decreasing in the MC company’s level of riskaversion and the demand uncertainty. They also find thatthe optimal retail pricing decision is linearly proportional tothe optimal modularity decision for both cases under study.They finally reveal that whether it is beneficial for the MCfashion company to offer consumer return with full refund(compared to no return) depends on the demand-returncorrelation coefficient.

In “Collective recycling responsibility in closed-loop fash-ion supply chains with a third party: Financial sharing orphysical sharing?” J. Nie et al. study the collective recyclingresponsibility in closed-loop fashion supply chain systemswith a third party. They examine three models where themanufacturer may either offer a financial or physical support

4 Mathematical Problems in Engineering

(called collective recycling responsibility) for a third partyto collect the used goods for remanufacturing. They revealthat if the manufacturer offers either kind of support tothe third-party firm, the retail price will decrease and thedemand will increase. In addition, all supply chain agentsin this closed-loop supply chain will prefer the presence ofthe financial or the physical support. The situation underwhich the financial support outperforms the physical supportdepends on a critical factor which is “the transfer price of theused product.”

We believe that this special issue presents many interest-ing research on fashion and textile supply chain managementproblems. The published papers also provide a good foun-dation for future research related to mathematical modelsfor fashion and textile supply chains and operational controlproblems.

Tsan-Ming ChoiXiaohang Yue

Chun-Hung ChiuPui-Sze Chow

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013, Article ID 363095, 18 pageshttp://dx.doi.org/10.1155/2013/363095

Research ArticleFashion Brand Purity and Firm Performance

Jin-hui Zheng,1 Zixia Cao,2 Xin Dai,3 and Chun-Hung Chiu3

1 Institute of Textiles and Clothing, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong2Department of Management, Marketing and General Business, West Texas A&M University, Canyon, TX 79016, USA3 Sun Yat-Sen Business School, Sun Yat-Sen University, No. 135, Xingang West Road, Guangzhou 510275, China

Correspondence should be addressed to Xin Dai; daixin@mail.sysu.edu.cn

Received 7 December 2012; Accepted 12 February 2013

Academic Editor: Tsan-Ming Choi

Copyright © 2013 Jin-hui Zheng et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A large number of prior empirical research and case studies used qualitative methodology to discuss the fashion brand dilutionresulting from consumer base extension from the target group(s) to the nontarget groups and its impacts. From a differentperspective, this paper establishes a dynamic brand dilution and performance model, demonstrating how dynamic changes of salesvolumes involving the two consumer groups affect the degree of brand dilution and the performance of the brand. We incorporatethe factor “brand purity” to the model as a quantitative measure of brand dilution level that affects firm annual revenue and profitchange comprehensively in iteration. Our model suggests that fashion brands, especially luxury brands, can be easily diluted underthe pressure of firm growth, and the brands suffer the significant negative impact on their revenues and profit. While increasingsales volume can aggravate the negative consequences, brand purity can be increased through limiting the consumer base to thetarget group only.

1. Introduction

The challenge faced by many fashion brands is that whetherto stay within the existing target market to avoid branddilution or to extend the brand to a larger market with therisks of diluting the brand [1]. Increasing consumer baseseems to facilitate firm growth, but prior research found thatconsumers tend to value the product (or the brand) less whena larger group of consumers owns it [2]. Because consumers’decision to buy a “conspicuous” branded product dependson not only the product’s functionality, but also the brand’ssocial benefits such as its prestige [2–6], the fashion brandneeds to maintain its exclusivity. A brand is no longer worthits vertiginous price if it is owned by too many consumers [7].

We have witnessed the failures of excessivemarket expan-sions where some fashion brands were diluted. Lacoste, awell-established French high-end apparel brand, was popularamong many young people in the poorly developed outersuburban area of African. The expanded market segmentof Lacoste conflicted with its brand positioning [7]. Thecompany’s advertising manager, Didier Calon, viewed thephenomenon during an interview in 2005: “It obviously had

negative effects on the brand image. Certain of our consumerswere upset that we could pursue this target, when in factwe have no control over it. Our sales decreased for threeor four years. . .” [8]. The classic high-end British brandBurberry became popular with the British football casualcult during the 1970s, leading it to be associated with chavs,hooligans, and members of football companies by the 1990s.Even Burberry admitted that “Burberry is now synonymouswith chavs and thugs” at that time [9]. To revitalize thebrand, Burberry spent a huge amount of resources on theadvertising campaign [10]. Pierre Cardin is another exampleof the fashion brand that overextended the market at thecost of degrading its brand value and losing shares in itsprimary target market. Pierre Cardin’s licensing operationsproliferated so much that the name had been lent to morethan 800 products by the 1980s, including toilet-seat covers.In the end, despite his talents as a couturier, his name becametoo common for many high-fashion consumers [11]. A recentrelevant observation is that when Louis Vuitton hired his newCEO who was formerly executive vice president of Frenchfood giant’s division in the end of 2012, lots of connoisseursworried that the management change provides a hint of

2 Mathematical Problems in Engineering

its plan of entering the mass market that could dilute LVbrand. These phenomena are known as “brand dilution,”referring to the weakening of a brand due to its overuse.Brand dilution frequently happens as a result of unsuccessfulbrand extension [12–14].

Price cutting might increase sales volume but might alsodamage brand equity [11]. Brand extension, which extends theoriginal brand to the newmarkets, could exert dilution effectson the parent brand when the attributes of extended brandsare inconsistent with the original brand positioning [1, 15–17].Many related studies in fashion industry explained why somefactors are critical for brand extension success [18–20]. Thispaper focuses on expanding the customer base to nontargetconsumers by cutting price.This brand dilution factor resultsfrom social influences, such as conspicuous consumption [2,3, 11, 20].

Han et al. [21] demonstrate with field experiments andmarket data that a market segment’s preference for con-spicuously (or inconspicuously) branded luxury product issignificantly related with consumers’ desire to be associatedor dissociated with the members of that segment. Priorstudies have identified two competing social needs amongconsumers: a need for uniqueness and a countervailing needfor conformity [22, 23]. Research in the area of referencegroup suggests the differences between consumer groups.For example, consumers from the elite group would liketo distinguish themselves from the masses in consumption,but the masses seek to emulate the choices of the elites [24,25]. Therefore, we build the model based on two groups ofconsumers, namely, the leader-group (LG) consumers andthe follower-group (FG) consumers. LG consumers seek forthe uniqueness and devalue a brand owned by too many FG,whereas FG consumers would like to follow LG and makethe purchasing decisions when prices are acceptable [7, 26].Assuming that the fashion firm can change sales volume byadjusting product prices, we expect that the demand from FGwill be increased when price is lowered. A higher demand ofLG induces a higher demand of FG, while a higher demandof FG leads to a lower demand of LG.

The impact of brand dilution on firm performance iswidely observed and discussed [12, 27, 28], especially infashion industry [29]. Some empirical studies measured theindexes of brand dilution and its moderating effect [13, 14, 30,31]. Here we use the concept of brand purity to denote thedegree of brand dilution that is defined in the mathematicmodel.

While many cases suggest the essential tradeoff thatfashion brands need to make between increasing marketshare and degrading brand value, there is little research thathas analyzed the dynamic mechanism of brand dilution.Therefore, we establish a dynamic brand dilution model andinvestigate analytically and numerically the situations wherebrand dilution occurs, the effects of brand dilution on brandfirm performance, and the strategies to cope with branddilution. The findings provide managerial implications forfashion firms regarding how to achieve the balance betweenmarket extension and brand dilution to optimize the firmperformance.

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𝑃

𝑞

Figure 1: Demand curve and sales revenue.

For the numerical analysis, we use the System Dynamics(SD) that is a widely used methodology and mathematicalmodeling technique that clearly visualizes the model. Byusing SD model, the dynamic brand dilution model is vividand is easier to understand. In addition, SD model is suitableto conduct stimulation for a particular case and to produceiteration results. It also acts as a delicate what-if tool formanagers to conveniently modify specific variables of themodel based on their needs and observe stimulation resultsunder different conditions.

This paper contributes to the literature by (1) firstintroducing the idea of the brand purity and proposing aquantitative measure of brand dilution level; (2) establishingthe dynamic brand dilution model; and (3) demonstratinganalytically and numerically how dynamic changes of salesvolumes affect the brand purity and the firm performance.

We develop an economic model for the mechanism ofbrand dilution in Section 2 and analyze the process of howbrand dilution and firm performance develops in Section 3.The related system dynamics model and the numerical anal-ysis appear in Section 4. Simulations on luxury fashion brandbackground further validate the process of brand dilution inSection 5. Conclusions are summarized in Section 6.

2. Model

2.1. Price-Sales Volume Relation. To simplify the configu-ration, we use demand curves to describe the relationshipbetween the selling price and the quantity (sales volume)of a fashion brand in Figure 1. Given the existing productportfolio of the fashion brand, let 𝑝 be the highest price ofthe branded products within the product portfolio and 𝑞the corresponding sales volume of these products. Within acertain period, price will be arbitrarily modified to adapt tothe market, thus gradually falling below the starting price 𝑝.Let 𝑝 = 𝑝(𝑞), where 𝑝 > 0, 𝑞 > 0, 𝑑𝑝/𝑑𝑞 < 0, 𝑑2𝑝/𝑑𝑞2 ≥ 0, bethe price function. The price function indicates that the finalsale volume is 𝑞 if the final retail price is set at 𝑝(𝑞). The firmcan reduce price to boost sales.

The revenue of the fashion brand is calculated by accu-mulating the product of the price and the quantity on thedemand curve; for example, if the final price is 𝑝(𝑞), thenthe revenue of all the goods with 100% brand purity (thedefinition of brand purity and its effects on revenue arediscussed later in this section) is given by the sum of areasI, II, and III in Figure 1.

Mathematical Problems in Engineering 3

Price can effectively distinguish LG and FG consumers ofthe fashion brand [32]. For simplicity, we assume 𝑝∗ is thecutoff price for distinguishing these two types of consumers;that is, LG consumers only purchase the product of the brandwhen 𝑝 > 𝑝∗, and FG consumers only purchase the productof the brandwhen𝑝 ≤ 𝑝∗, where𝑝∗ = 𝑝(𝑞∗).The cutoff price𝑝

∗ between LG and FG is rational simplification to reality, inwhichLGandFGcould have their own affordable price spans,and the two price spans can be overlapped or isolated. Notethat 𝑞∗ is called the “critical point” of the sales volume thatdifferentiates LG consumers and FG consumers.

Although brands can set price to control the sales volume,it is still difficult for them to prevent brand dilution, becausethey might not actually discern what is the exact 𝑞∗ and areeasy tomake actual sales volume bigger than the unknown 𝑞∗(the lowest price is set lower than 𝑝∗), where FG consumersare involved and brand dilution occurs.

2.2. Brand Purity. The brand purity represents the mix ofconsumer amongst consumer groups. Let 𝑞

𝐿be the sales

volume of LG consumers and 𝑞𝐹the sales volume of FG

consumers. The total sales volume is given by 𝑞 = 𝑞𝐿+ 𝑞

𝐹.

Let 𝑠 be the brand purity. We consider that the brand is purewith 𝑠 = 1 when only LG consumers purchase the productsof the brand, and the brand is diluted with 𝑠 < 1 whenFG consumers also purchase the products of the brand too.The smaller the 𝑠 is, the more the FG consumers purchasethe products and the more the brand is diluted. Noting thatthe measure of the brand purity 𝑠 can be defined in manyways, 𝑠 = 𝑞

𝐿/(𝑞

𝐿+ 𝑞

𝐹) is a simple example of the measure.

However, it does not reflect the dynamic feature of the brandpurity, so we do not adopt this measure. A dynamic brandpuritymeasure is introduced in Section 2.3 later in this paper.The brand purity affects the purchasing intension of the LGconsumers (but does not affect the purchasing intension ofthe FG consumers). For simplicity, we consider that, for 0 ≤𝑠 ≤ 1, only 𝑠 portion of the demand of LG consumers will betransformed into the sales volume of the brand products.

For any given 0 ≤ 𝑠 ≤ 1 and the final price 𝑝(𝑞), therevenue of the brand for one selling season is given by

𝑅 = 𝑝 (𝑞) 𝑞𝑠 + ∫

𝑞

𝑞

𝑝 (𝑞) 𝑠𝑑𝑞, for 𝑞 ≤ 𝑞∗, (1a)

𝑅 = 𝑝 (𝑞) 𝑞𝑠 + ∫

𝑞∗

𝑞

𝑝 (𝑞) 𝑠𝑑𝑞 + ∫

𝑞

𝑞∗

𝑝 (𝑞) 𝑑𝑞, for 𝑞 > 𝑞∗,

(1b)

where 𝑝(𝑞)𝑞 is area I in Figure 1, ∫𝑞𝑞𝑝(𝑞)𝑑𝑞 is the front part of

area II (for 𝑞 ≤ 𝑞∗), ∫𝑞∗

𝑞𝑝(𝑞) is area II in Figure 1 (for 𝑞 > 𝑞∗),

and ∫𝑞𝑞∗𝑝(𝑞)𝑑𝑞 is III in Figure 1 (for 𝑞 > 𝑞∗). Moreover, the

profit of the brand for one selling season is given by

𝑂 = (𝑝 (𝑞) − 𝑐) 𝑞𝑠 + ∫

𝑞

𝑞

(𝑝 (𝑞) − 𝑐) 𝑠𝑑𝑞 − 𝐶, for 𝑞 ≤ 𝑞∗,

(2a)

𝑂 = (𝑝 (𝑞) − 𝑐) 𝑞𝑠 + ∫

𝑞∗

𝑞

(𝑝 (𝑞) − 𝑐) 𝑠𝑑𝑞 − 𝐶

+ ∫

𝑞

𝑞∗

(𝑝 (𝑞) − 𝑐) 𝑑𝑞 − 𝐶, for 𝑞 > 𝑞∗,(2b)

where 𝑐 is the variable cost and𝐶 is the fixed cost of the brand.

2.3. Dynamic Model. In this paper, we consider a multipleselling season situation (for simplicity, we consider 1 yearfor 1 selling season). In each year, the brand will pushnew products to the market. Denote by 𝑠

𝑖, 𝑅𝑖, and 𝑂

𝑖(we

use subscript to represent the year for all other notations),respectively, the brand purity, the revenue, and the profit ofthe brand, in the 𝑖th year which are included, where 𝑖 is anonnegative integer. We consider the dynamics model of thebrand purity as follows:

𝑠

𝑖= 𝛼𝑠

𝑖−1+ (1 − 𝛼)

𝑞

𝐿,𝑖

𝑞

𝑖

, 𝑠

0= 1, (3)

where 0 < 𝛼 < 1, 𝑞𝐿,𝑖

is the sales volume of LG consumersin 𝑖th year, and 𝑞

𝑖is the total sales volume of both LG

and FG consumers. By noting the dynamic brand purity (3)follows the format of exponential weighted moving average(EWMA), that is, 𝑠

𝑖is the weighted sum on the last brand

purity status 𝑠𝑖−1

, and the current mix of the sales volume ofthe two groups of consumers 𝑞

𝐿,𝑖/𝑞

𝑖.We set 𝑠

0= 1 and assume

that there is no brand dilution at the beginning. Starting fromthe 1st year, the brand dilution begins when FG consumersstart to purchase this brand. Let 𝑞

𝐹,𝑖be the sales volume of

FG consumers in the 𝑖th year. By following (3), the degree ofbrand purity is accumulated from year 0 up to year 𝑖 − 1.

By following (1b), the revenue of the brand in year 𝑖 is

𝑅

𝑖= 𝑝 (𝑞) 𝑞𝑠

𝑖−1+ ∫

𝑞∗

𝑞

𝑝 (𝑞) 𝑠

𝑖−1𝑑𝑞

+ ∫

𝑞𝑖+(1−𝑠

𝑖−1)𝑞∗

𝑞∗

𝑝 (𝑞) 𝑑𝑞, for 𝑝𝑖< 𝑝 (𝑞

∗) .

(4)

In (4), the base of LG consumers of year 𝑖 shrinks by anamount proportional to the brand purity 𝑠

𝑖−1. As 𝑞𝐿,𝑖= 𝑠

𝑖−1𝑞

∗

and 𝑞𝐹,𝑖= 𝑞

𝑖−𝑠

𝑖−1𝑞

∗ when 𝑝 < 𝑝∗, the final price will declineto 𝑝(𝑞

𝑖+(1−𝑠

𝑖−1)𝑞

∗) = 𝑝(𝑞

∗+𝑞

𝐹,𝑖) to obtain 𝑞

𝐹,𝑖sales volume

from FG. Moreover, the annual profit of the brand is given by

𝑂

𝑖= (𝑝 (𝑞) − 𝑐) 𝑞𝑠

𝑖−1+ ∫

𝑞∗

𝑞

(𝑝 (𝑞) − 𝑐) 𝑠

𝑖−1𝑑𝑞

+ ∫

𝑞𝑖+(1−𝑠

𝑖−1)𝑞∗

𝑞∗

(𝑝 (𝑞) − 𝑐) 𝑑𝑞 − 𝐶, for 𝑝𝑖< 𝑝 (𝑞

∗) .

(5)

When the fashion brand wants to achieve more annualrevenue and profit through lowering the price to enlargethe sales volume, brand dilution, which will undermine thefirm performance (revenue and profit), occurs once the salesvolume exceeds the critical point 𝑞∗ and FG consumers areattracted to the brand.

4 Mathematical Problems in Engineering

3. Analysis

The increase of sales volume of the fashion brand is oftenconsidered as an indicator of brand growth. Under thepressure from the stakeholders, the brand is highly motivatedto continue increasing its sales volume overtime. Although, toincrease the sales volume, the brand has to decrease marginalprice, it still has plenty of space to cut prices, as the priceof fashion product is usually significantly higher than theproduct cost. The growth mechanisms of brands with a purebrand (𝑠 = 1) and with diluted brand (𝑠 < 1) are different aswe discuss below.

3.1. The Brand Is Pure. When the brand is pure, increasingsales volume is a good strategy for a firm to enjoy continuedincreasing revenue and profit.

Proposition 1. Suppose that 𝑠𝑖−1

= 1; 𝑠𝑖< 1 only if sales

volume 𝑞𝑖> 𝑞

∗ (or 𝑝𝑖≤ 𝑝(𝑞

∗)). (All proofs are presented in

Appendix A.)

Discussion of Proposition 1. As shown in Proposition 1, dilu-tion does not occur when only LG consumers purchase theproducts, or dilution occurs until FG consumers start topurchase the brand.

Proposition 2. Suppose that 𝑠𝑖−1

= 1; if 𝑠𝑖= 1, 𝑅

𝑖and 𝑂

𝑖are

both increasing functions of 𝑞.

Discussion of Proposition 2. As the maximum 𝑞 that keepsthe brand pure is 𝑞∗ (by Proposition 1), according toProposition 2, the revenue and the profit of the brand aremaximized at 𝑞 = 𝑞∗ when the brand is pure, and cuttingprice to increase sales (less than 𝑞∗) volume leads to positiveconsequences (higher revenue and profit) over years.

3.2. The Brand Is Diluted. Once the firm gets used to theexperience of benefiting from increasing sales volume, thefirm may have the tendency to oversell the products to FGconsumers and suffer the negative impacts of brand dilution.To focus on brand dilution effects, our analysis starts from thecritical point of brand dilution. Moreover, for simplicity, weassume that the fashion brand firmwill increase sales volumeat an annual rate of 𝜃

𝑖> 0. Hence, the sales volume in year 𝑖

is given by

𝑞

𝑖= 𝑞

∗

𝑖

∏

𝑗=1

(1 + 𝜃

𝑗) , 𝑞

0= 𝑞

∗. (6)

Following the changes in sales volume, the brandpurity is alsoaltered:

𝑠

𝑖= 𝛼𝑠

𝑖−1+ (1 − 𝛼)

𝑞

𝐿,𝑖

𝑞

𝑖

= 𝛼𝑠

𝑖−1+ (1 − 𝛼)

𝑞

∗𝑠

𝑖−1

𝑞

∗∏

𝑖

𝑗=1(1 + 𝜃

𝑗)

= (𝛼 +

1 − 𝛼

∏

𝑖

𝑗=1(1 + 𝜃

𝑗)

) 𝑠

𝑖−1,

(7)

or

𝑠

𝑖=

𝑖

∏

𝑘=1

(𝛼 +

1 − 𝛼

∏

𝑘

𝑗=1(1 + 𝜃

𝑗)

) . (8)

According to (8), 𝑠𝑖will decrease at an accelerating rate of

(𝛼+ (1−𝛼)/(1 + 𝜃)

𝑖); namely, the brand purity will be eroded

severely as the sales volume continues to rise. The brand willgradually lose its superiority to LG consumerswho contributethe majority of its revenue, inevitably resulting in a shrinkingof LG consumer base.

Proposition 3. If 𝑠0= 1 and 𝑠

𝑖is given by (8), with 𝜃

𝑖> 0,

𝑖 = 1, 2, . . ., then 𝑠𝑖+2/𝑠

𝑖+1< 𝑠

𝑖+1/𝑠

𝑖< 1.

Discussion of Proposition 3. Once the increasing sale volumeexceeds the critical point 𝑞∗ and the brand purity declines,the process of brand dilution will accelerate, as sales volumekeeps increasing. So things will deteriorate swiftly whenbrand dilution happens. Is there anything we can do for it?For example, to keep the annual sales volume unchanged afterbrand dilution occurs, let𝐻 > 1 be an integer.

Proposition 4. Suppose that 𝜃𝑘> 0 for 𝑘 = 1, . . . , 𝐻 − 1 and

𝜃

𝑘= 0 for 𝑘 ≥ 𝐻. If 𝑠

𝑖is given by (8), then 𝑠

𝑖+2/𝑠

𝑖+1= 𝑠

𝑖+1/𝑠

𝑖<

1 for 𝑖 > 𝐻.

Discussion of Proposition 4. Once the brand purity starts todecline, stopping increasing the sales volume cannot preventthe deterioration, and the brand purity will still decrease at aconstant rate. This case is a little bit better than decreasing atan accelerating rate but is still bad because it cannot stop thebrand to ruin. So how about cutting the sales volume after thebrand is diluted?

Proposition 5. 𝑠𝑖≥ 𝑠

𝑖−1, 𝑖 ∈ 𝑁, if and only if 𝑞

𝑖≤ 𝑞

∗.

Discussion of Proposition 5. Once the brand purity declines,the firm can enhance brand purity by reducing sales lessthan 𝑞∗. Noting that sales volume is less than 𝑞∗ does notimply all the sales volume is made up of LG consumers,especially fashion brand which had ever diluted. To enhancebrand image, sales volume of current year should be confinedwithin the range. In other words, only cutting sales volumea little cannot promise improving the brand purity when italready exceeds the critical point 𝑞∗. Therefore, brand puritywill continue to decline unless sales volume decreases tobelow 𝑞∗. However, in practice, recovering a diluted brand isalways difficult because firms are often reluctant to adopt suchradical method. Moreover, Proposition 5 once again showsthe importance of the critical point 𝑞∗. No matter how manyexceeding sales volume accumulated by years, to return to thecritical point is the key for keeping brand purity.

To make the model more comprehensive, the constrainton the annual sales volume is relaxed as 𝜃

𝑖> −1. So the

annual sales volume can be increased (𝜃𝑖> 0) or be decreased

(−1 < 𝜃𝑖< 0). Note that, for this case, 𝑞

𝑖= 𝑞

∗∏

𝑖

𝑗=1(1 + 𝜃

𝑗) <

𝑞

∗𝑠

𝑖−1could occur. As 𝑞∗𝑠

𝑖−1is the maximum sales volume of

LG in year 𝑖, if 𝑞𝑖< 𝑞

∗𝑠

𝑖−1, then 𝑞

𝐿,𝑖= 𝑞

𝑖; that is, only LG

Mathematical Problems in Engineering 5

consumers purchase the brand in year 𝑖. Together with thefact that demand of LG consumers is always be fulfilled first,we have 𝑞

𝐿,𝑖= min{𝑞∗𝑠

𝑖−1, 𝑞

𝑖}. Moreover, 𝑞

𝑖≤ 𝑞

∗𝑠

𝑖−1means

that not all the demand of LG consumers of year 𝑖 will befulfilled. In other words, the brand controls the annual salesvolume to meet part of LG’s demand only. The brand puritybecomes

𝑠

𝑖=

{

{

{

𝛼𝑠

𝑖−1+ (1 − 𝛼)

𝑞

𝐿,𝑖

𝑞

𝑖

, for 𝑞𝑖> 𝑞

𝐿,𝑖,

𝛼𝑠

𝑖−1+ (1 − 𝛼) , for 𝑞

𝑖= 𝑞

𝐿,𝑖.

(9)

If 𝑞𝑖> 𝑞

𝐿,𝑖, for all 𝑖 ∈ 𝑁 (𝑁 denotes natural number),

that is, 𝑞∗∏𝑖𝑗=1(1 + 𝜃

𝑗) > 𝑞

∗𝑠

𝑖−1, the equation can be deduced

as same as (8), even through 𝜃𝑖can be less than 0. If 𝑞

𝑖=

𝑞

𝐿,𝑖, the 𝑖th increment of the sales volume in year 𝑖 is 𝜃

𝑖<

(𝑠

𝑖−1/∏

𝑖−1

𝑗=1(1 + 𝜃

𝑗)) − 1, which means the firm will cut the

sales volume dramatically that very few brands would adoptit.

Next, we explore the impacts of the annual sales volumeon the revenue and profits.

Proposition 6. Suppose that 𝜃𝑘> 0 for 𝑘 = 1, . . . , 𝐻 − 1 and

𝜃

𝑘= 0 for 𝑘 ≥ 𝐻. If 𝑠

1< 1, then 𝑅

𝑖< 𝑅

𝑖−1, 𝑂𝑖< 𝑂

𝑖−1𝑂

𝑖−

𝑂

𝑖−1< 0 for all 𝑖 > 𝐻.

Discussion of Proposition 6. Once the brand is diluted, evenif the sales volume remains unchanged thereafter, the annualrevenue and profit will continue to decline.

Then what if the sales volume continues to increaseafter the brand is diluted? What is the speed of decline ofthe annual revenue and profit? And what are the factorsthat account the fluctuation? Proposition 3 indicates that theprocess of dilution will accelerate, once sales volume exceedsthe critical point and keeps increasing every year.The revenueand profit could decline because newly added FG consumerswill make part of LG consumers leave. Thus, the profit drops.However, it is possible that the negative effects of reducedrevenue and profits caused by shrinking LG consumer basecan be compensated by the benefit of an enlarged consumersize due to the expansion to FG consumers. It is the neteffect of these variations that determines the impact of brandpurity. Meanwhile, the direction and magnitude of changesof revenue and profit are uncertain due to the influence offactors such as specific shape of demand (or sales) curve,growth rate, and other variables.

To answer these questions, as a further step, we conductnumerical stimulations, which enable us to have a clearerpicture of the interactions of different factors and how theyaffect the revenue and profit. Specifically, we use the SD thatis a widely used methodology and mathematical modelingtechnique that clearly visualizes the model. Moreover, byusing SD model, the demonstration of propositions raisedabove will be vivid and easier to understand (the results areavailable from the authors upon request).

s

aqL

O

k b cc C

R

qi

q∗

q

Figure 2: Causal loop diagram of the model.

Table 1: Correspondence between mathematics model and SDmodel.

Variables in mathematics model Variables in SD model𝑞

𝑖qi

𝑞

∗ q∗𝑞

𝐿,𝑖qL

𝛼 a𝑞 q𝐶 C𝑐 cc𝑠

𝑖s

𝑅

𝑖R

𝑂

𝑖O

4. System Dynamics Model(Numerical Analysis)

The mathematical model in Sections 2 and 3 can also beenpresented by an SD model. SD method is an effective toolfor numerical analysis and simulation.The modeling processmainly consisted of two parts: causal loop diagram(s) andmathematical expressions accordingly. The model can helpidentify the essential factors that positively or negatively affectfirm performance by increasing sales volume.

4.1. Causal Loop Diagram. Equation (4) helps build thecausal loop diagram accordingly. The variable 𝑞

𝑖is the sales

volume of year 𝑖, which is set as a cumulative quantity(stocks). Its dynamic change (flow) is proportional to its sizedenoted by 𝜃

𝑗in (6). Variable 𝑠

𝑖, the brand purity in (3), is

affected by the sales volume of the 𝑖th year 𝑞𝑖, the sales volume

of 𝑖th year from LG 𝑞𝐿,𝑖, and the cumulative weight of brand

purity 𝛼. Similarly, the annual revenue (𝑅) and the annualprofit 𝑂 could be linked together as in Figure 2, according to(4) and (5), respectively.We use Vensim PLEV5.11 as SD tool,which helps present the causal loop diagram as Figure 2.

In Vensim, the names of the variables do not includethe superscripts, subscripts, and Greek letters. Therefore thenames of variables in the mathematics model are differentfrom the names in the SDmodel, while bothmodels represent

6 Mathematical Problems in Engineering

the same thing. Table 1 shows the corresponding variablenames between the two models.

In the SD model, the 𝑘 and 𝑏 are parameters for thedemand curve function 𝑝 = 𝑝(𝑞). In the linear function, 𝑘 isslope, and 𝑏 is intercept on ordinate (showed in Section 4.3).

4.2. Mathematical Expression in SD. An SD model forsimulation should use a concrete function with numericalparameters.We present an SDmodel example by formulatingit with the linear demand curve 𝑝(𝑞) = 𝑏 − 𝑘𝑞 with 𝑏 = 1400and 𝑘 = 0.02. For example, the derived firm’s annual revenue𝑅

𝑖from (4) is

𝑅

𝑖= (𝑏 − 𝑘𝑞) 𝑞𝑠

𝑖−1+ (2𝑏 − 𝑘 (𝑞

∗+ 𝑞)) (𝑞

∗− 𝑞) 𝑠

𝑖−1

+ (2𝑏 − 𝑘 (𝑞

𝑖+ (1 − 𝑠

𝑖−1) 𝑞

∗+ 𝑞

∗))

× ((𝑞

𝑖+ (1 − 𝑠

𝑖−1) 𝑞

∗+ 𝑞

∗)) .

(10)

The corresponding formula in Vensim PLE, the SD tool,isR = (b − k ∗ q) ∗ q ∗ DELAY1 (s, 1)

+ (

1

2

) ∗ (2 ∗ b − k ∗ ("q ∗ " + q))

∗ ("q ∗ " − q) ∗ DELAY1 (s, 1) + (12

)

∗ (2 ∗ b−k ∗ (qi+(1− DELAY1 (s, 1)) ∗ "q ∗ "+"q ∗ "))

∗ (qi + (1 − DELAY1 (s, 1)) ∗ "q ∗ " − "q ∗ ") .(11)

Because the “∗” in variable q∗ could be recognized asproduct sign, variable q∗ is represented as “q∗” in the tool.To be consistent with this style, all formulas in Section 2 arefilled into the causal link (in Figure 2).Thewholemathematicformulae in SD model are shown in Appendices B and C.Models engaged with other demand curves as below areproduced likewise by set of corresponding demand curveparameters, respectively.

4.3. Results. To investigate how annual revenue and profitwill change when the sales volume expands beyond the LGconsumer base and leads to brand dilution, in the followingpart, we simulate 3 different types of general demand curve:(1) straight line; (2) fold line; (3) convex curve.

Several assumptions are set: a fashion brand increases 10%of sale volume per year; at the 10th year the sales volumereaches the critical point 𝑞∗, which means during the 0–10th years the sales volume covered only the LG consumers,and the brand purity is kept to be 1; from the 11th year, themarginal increased sales volume comes from FG consumers,brand begins diluting, and some LG consumers abandon thebrand. In Sections 2 and 3, we discuss propositions from yearof initial brand dilution; so 𝑖 = 0 in Sections 2 and 3 is equalto 𝑖 = 10 in this section. Similarly, 𝑖 = 1 above is equal to𝑖 = 11 here, and so on. We insert 10 years period of prelude,in order to show and contrast what happens before and afterbrand dilution.

Table 2: Straight line with various slopes.

Line 𝑏 𝑘 (𝑝∗, 𝑞∗)1 5000 0.20 (1000, 20000)2 4000 0.15 (1000, 20000)3 3000 0.10 (1000, 20000)4 2600 0.08 (1000, 20000)5 2000 0.05 (1000, 20000)6 1400 0.02 (1000, 20000)7 1200 0.01 (1000, 20000)

4.3.1. Straight Line. Choose different slopes and intercepts in𝑝(𝑞) = 𝑏 − 𝑘𝑞 as below. In order to compare conveniently,assume the critical point (𝑝∗, 𝑞∗) betweenLGandFG is (1000,20000), and all straight lines pass through this point.

Figure 3 shows the result of the simulation test when 𝛼 =0.8. As the slope gets gentler (𝑘 gradually decreases), the turn-down year of annual revenue and profit happens later andlater (from line 1 to line 7). In Figure 3 annual profits of lines1 and 2 turn down at 12th; those of lines 3 and 4 do at 13th;those of lines 5, 6, and 7 do at 14th, 18th, and 24th. Annualrevenues of those lines have similar features. In other words,with sales growth of 10% annually, when the brand dilutionstarts (after the 10th year), the slope of lines mainly decideshow fast the bad effect on fashion firm performance (drop ofannual revenue and profit) appears because of brand dilution.What is more is brand dilution is often tricky and does notshow its true faces immediately. Firms frequently did not findthe problem until it had become heavy. As lines 5, 6, and 7,brand dilution happened at the 11th year but would show itstrue faces after growing several years.

If all the other parameters remain the same and changethe accumulated weighted average index into 𝛼 = 0.5(Figure 4), which means that the brand dilution’s impactin the current year plays a more important role than theaccumulation by past years, the dilution process will go faster.In general, the decrease of annual revenue and profit startsearlier, and other characteristics are in line with Figure 3.Thus, the influence of𝛼 is not as strong as the slope.Therefore,the following simulations will use 𝛼 = 0.8 as examples todemonstrate the situations.

4.3.2. Fold Line. Generally, the scale of LG is always some-what smaller than that of the FG. So the demand curve ofthe LG is more slanting than that of FG. To better reflect thedemand elasticity of the two groups, fold lines are adoptedin Figure 5(a). As above, the former part of the fold linesrepresents sales to LG, and the latter represents the sales toFG. As the straight lines, (1000, 2000) points remain as thecritical point which is set as the fold point of each fold line.

Figure 5(b) numerates fold lines with 8 combinations.Table 3 lists these combinations and the number of linecorresponding to Table 2.

In Figure 6, the simulation test reveals that from the 10thyear since the brand dilution started, as the sales volumekeeps increasing, whether the annual revenue and profit fallor notmostly depends on the slope of the fold line pair. If both

Mathematical Problems in Engineering 7

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Figure 3: Annual revenue and profit under various straight lines (𝛼 = 0.8).

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(M)

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Figure 4: Annual revenue and profit under various straight lines (𝛼 = 0.5).

parts slant a lot (e.g., the combination of fold line 1 in Table 3),significant decrease will appear earlier; on the contrary (e.g.,the combination of fold line 8 in Table 3), decreasewill appearlater or even not appear.This is consistent with the simulationresults of the straight lines. From fold lines 1–4 in the firstgroup and fold lines 5-6 in the second group (Figure 6), wecan see that when the former part of the fold line remains thesame, if the latter part is less slanting, performance decreasewill appear more delayed. Comparing fold lines 1 and 5 (or 2and 4, 3 & 7, 4 and 8), we can conclude that while the slope ofthe latter part remains constant, the less slanting the former

part is, the latter the performance decrease will occur. It couldbe dangerous for fashion brand firms, if brand dilution doesnot signal immediately, because firms could not be aware ofthe constant losing of LG with the increasing sales volume,and actions to remedy the harmful brand dilution will delay.On the other side, if the slope of the latter part of fold line isgently (much more elastic), which means that demand fromFG ismuch greater than LG, it is feasible that the fashion firmwill not interferewith the brand dilution even if the brandwasdegraded. However, to compare these two strategies requiresanother paper to discuss.

8 Mathematical Problems in Engineering

𝑞∗

𝑃

𝑞

(a)

𝑞∗

𝑃

𝑞

(b)

Figure 5: (a) 𝑝(𝑞) of fold line. (b) 𝑝(𝑞) of fold lines pairing.

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(M)

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Figure 6: Annual revenue and profit under various fold line pairs.

Proposition 7. Once brand purity declines, whether the an-nual revenue and profit will rise or drop is mainly decided bythe elasticity of demand curve of both LG and FG: the lesselastic (bigger of slope) the LG demand curve is, the higherthe probability for the annual revenue and profit to drop is;the more elastic (smaller of slope) the FG demand curve is, thehigher the probability for the annual revenue and profit to riseis.

Discussion of Proposition 7. The demonstration in straightline above has showed the property, as well as the fold linesituation here. The bigger the slope of the front part of foldline is (demand curve of LG), the greater the loss in revenueand profit generated from those high-end consumers is. Asthe nadirs of the former part of fold lines are same, thegreater the slope is, the bigger the area in the part II inFigure 1 is. Bigger area with multiple same proportion lossof consumer produces a bigger drop of revenue and profit.

On the other side, the smaller the slope of the hind part is(demand curve of FG), the greater the increment in revenueand profit generated from those FG consumers is.

In the following, some typical convex curve examples willshow if Proposition 7 works.

4.3.3. Convex Curve. Here we use common convex curvefunction,𝑝(𝑞) = 𝑘/𝑞𝑏. Table 4 presents 5 different curveswithvarious 𝑘 and 𝑏.

For easier comparison, the parametric setting of thesecurves allows them to pass through or be extremely closeto the critical point (𝑝∗, 𝑞∗). The graph is presented inFigure 7.

The results of the curve got from the simulating test areconsistent with those of the fold lines. Because the slope nearthe critical point in lines 1 and 2 is greater, the decreaseof annual revenue and profit occurs earlier (Figure 8); since

Mathematical Problems in Engineering 9

Table 3: Fold lines pairing composition in Figure 5(b).

Fold Line Line 𝑏 𝑘 (𝑝∗, 𝑞∗) Line 𝑏 𝑘1 1 5000 0.20 (1000, 20000) 4 2600 0.082 1 5000 0.20 (1000, 20000) 5 2000 0.053 1 5000 0.20 (1000, 20000) 6 1400 0.024 1 5000 0.20 (1000, 20000) 7 1200 0.015 3 3000 0.10 (1000, 20000) 4 2600 0.086 3 3000 0.10 (1000, 20000) 5 2000 0.057 3 3000 0.10 (1000, 20000) 6 1400 0.028 3 3000 0.10 (1000, 20000) 7 1200 0.01

Table 4: Convex curve 𝑝(𝑞) with various parameters.

Curve 𝑏 𝑘 (𝑝∗, 𝑞∗)1 3/2 2828430000 (1000, 20000)2 4/3 542884000 (1000, 20000)3 1 20000000 (1000, 20000)4 2/3 736806 (1000, 20000)5 1/2 141421 (1000, 20000)

4000

3000

2000

1000

00 5000 10000 15000 20000 25000 30000 35000 40000

𝑝 = 2828430000/𝑞(3/2)

𝑝 = 542884000/𝑞(4/3)

𝑝 = 20000000/𝑞

𝑝 = 736806/𝑞(2/3)

𝑝 = 141421/𝑞(1/2)

Figure 7: Convex curves with various parameters.

the slope near the critical point in lines 4 & 5 is gentler,obvious decrease does not occur at observed period.

From the above numerical analysis, we can see that, afterbrand purity decline happens, when sales increase, the changeof sales revenue and profit will be determined by the mutualeffect of the following two parts: (A) the loss of revenuesand profit resulting from the sales decrease due to the lossof the LG consumers, which is negative; (B) the increase ofrevenues and profit resulting from the sales increase due tothe marginal increasing FG consumers, which is positive.Obviously, this relationship will most strongly affect the slopeof the demand curve before and after the 𝑞∗. The steeper thepart before 𝑞∗ is, the bigger A will be; the flatter the partafter 𝑞∗ is, the bigger B will be. Proposition 7 works in convexcurve situations (We have test simulation under 20%/yearincrement on sales volume and get similar results as 10%

increment. We would like to provide the simulation result toanyone interested.).

Along with the propositions in Sections 2 and 3 and thesimulation in Section 4, thismodel well describes the impactsthat brand purity has exerted on annual revenue and profit offashion firms. This model severs as a good reference whenanalyzing the influence of brand purity on firm performanceunder different circumstances.

5. Application Simulation forDifferent Strategies of Sales

In order to better demonstrate the application of the model,this section discusses, in the context of fashion luxury brands,howbrand purity and the firmperformance differwhen usingdifferent strategies of sales. Fashion luxury brand is chosenas the context because the luxury brand, compared to otherfactors, has greater influence on the price of the product. Forthe simulation, we choose a set of parameters that reflectthe luxury brand context well. Due to the difference in thepurchasing power of the LG and the FG, the slopes of demandcurve for LG and FG are different in general. The differencesare exhibited by the convex curve. An intuitive observationsuggests investigating a model in the form of the SD model.Take line 3, displayed in Table 4, as the original demandcurve: 𝑝(𝑞) = 𝑘/𝑞. In order to be closer to the real scenario,parameters are set as follows: 𝑘 = 20000000, 𝑞 = 1000,𝑞

∗= 20000, 𝑐 = 300, and 𝐶 = 20000000. In reality, luxury

brands such as LV, the annual sales volumes of each categoryare generally 2millions, 100 times of 𝑞∗ = 20000. 𝑘/𝑞∗ = 1000dollar (pecuniary unit is dollar if not state) and 𝑘/𝑞 = 20000are, respectively, theminimumand themaximumprices of anLV product in principle. Fashion firms often adopt the pricestrategy of high-priced new arrival and off-season discountsor provide different types of products among the 1000–20000price range. The variable cost 𝑐 = 300 is one of the mainfactors of unit cost that are related to the quantity change.The variable cost includes marketing costs, additional laborcosts, and costs of variable inputs. 𝐶 = 20000000 is annualfixed cost, which generally represents the corporate cost.To observe the trend of firm profits, factor 𝐶, served as aconstant, is almost negligible.

We conduct the following simulations to illustrate twocommonproblems of fashion brands. (1) Firms are vulnerableto suffering band dilution when they are blinded by their

10 Mathematical Problems in Engineering

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Figure 8: Simulation results of the 5 curves in Table 4.

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Figure 9: Contrast of two cases: without (line 1) and with brand purity decline (line 2).

strong motivation of growing business. (2) Different salesstrategies can change brand purity and affect firm perfor-mance.

5.1. Two Cases of Brand Dilution. Maximizing profitability isthe objective ofmany businesses, and increasing sales volumeis the common tactics often used by not only fashion firms butalsomany other firms to enhance profits. However, firmsmaybe stuck in the following two brand dilution problems, if theydo not manage the increase of sales volume cautiously.

5.1.1. Case 1: Consumer Group Has Not Yet Been Segmented.If the target fashion brand consumers can be, but are not,divided into potential LG and FG, the firm could encounterunexpected performance deterioration due to indulging inenlarging sales volume to enhance profit. The example willshow the difference between expectation (line 1 in Figure 9)and unexpected result (line 2 in Figure 9).

Example 8. Set the corresponding parameters as the basic set,and 𝛼 = 0.8, 𝜃 = 10%. Contrast the two situations with andwithout brand purity decline.

The firm may expect the sales volume to increase in thisway: the demand increases as themarginal price decreases. Aslong as the declined product price is greater than the productvariable cost, the firm is able to earn profit. Therefore, withcontinuing growth of sales volume, annual profit will increaseuntil the 23rd year. However, with the unknown influence ofbrand dilution, LG consumers keep abandoning this brand,which causes profit decreases 2 years later after sales volumeexceeds the critical point at 10th year.That is beyond the firm’sexpectation.

5.1.2. Case 2: The Boundary of FG and LG Is Unknown. Themodel simulation shows that expanding market share in LGthrough continuously raising sales volume benefits firms’

Mathematical Problems in Engineering 11

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Figure 10: Contrast of profits resulting from different sales volumes in the LG target group.

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Figure 11: Contrast of profits when sales volumes stop growing at different years under brand dilution.

profit and revenue. In Figure 10, line 1 holds the critical pointof LG consumer base. If the sales volume does not exceed it,the closer it is to this point, the higher profit the firm willgain. If the firm stops increasing sales volume earlier (as inFigure 10 line 3 shows), a big amount of potential profit is leftunearned. Actually Proposition 2 has predicted the result.

Moreover, the firm will try to approach to the idealboundary by reducing the price and bringing up the salesvolume. However, as it is difficult to estimate the criticalpoint close to line 1, during this process, the sales volumemight exceed the boundary and produce brand dilution, andconsequently influence the LG consumer base, revenue, andprofit.

Line 1 in Figure 11 shows the ideal situation. Once thebrand is diluted, stopping sales volume growth at different

years (lines 2-3) is little better than keeping sales volumegrowing (line 4) but still cannot prevent annual profit fromcontinuous declining. In short, even if firms are aware ofbrand dilution, they could be in trouble when the criticalpoint is hard to identify.

Obviously, the effects of brand purity are significant inthe development journey of fashion firms. Once the brandis diluted, the firm’s profits are continuously harmed, whichunderscores the necessity of identifying the critical point ofbrand dilution.

5.2. Model Simulation of Sales Changing Trend

5.2.1. Strategy 1: Increase Sales Volume Each Year. As the firmhas inertia to increase sales volume, for example, by a certain

12 Mathematical Problems in Engineering

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Figure 12: Simulation results under 10% of sales increasing rate.

proportion (0 < 𝜃 < 1), if the discussed initial sales volumeis 𝑞∗, sales volume of year 𝑖 will be 𝑞

𝑖= 𝑞

∗(1 + 𝜃)

𝑖. Whatperformance will the firm operate?

Example 9. Set the same parameters as Example 8. Afternearly 10 years of growth, sales volume from LG is close tosaturate the target market (critical point). If the firm stillrequests a 10% rising rate, then the majority of the newconsumers will be FG. We conduct a simulation of 30-yeardevelopment that consisted of profit, revenue, sales volume,and brand purity.

According to the results displayed in Figure 12, with therising sales volume, the profit (𝑂) of the first 10 years is onthe raise, and the brand purity keeps still. But as the salesto the LG tend to be saturated, FG join the consumer baseand brings brand purity down quickly (Proposition 3 has alsopromised these properties), resulting in the continuous lossof high quality consumers of LG. Annual profit and the firmrevenue decrease.

Market observations suggest that when firm grows withincreasing sales volume and expandingmarket share, the firmperformance is continuously strengthened. But after reachinga certain point, the situation begins to deteriorate, and

revenue and profit go down as sales volume increases. Manyfashion luxury brand firms have somewhat gone throughsimilar stories as we mentioned at the beginning.

5.2.2. Strategy 2: Stop Increasing Sales When Profit Con-tinuously Declines. The increase of sales volume does notnecessarily suggest profit increase. It is not hard for luxurybrand firms to control sales volume, when the brand is stillperceived by consumers as luxury. In the following part, wediscuss the situation where sales volume is kept fixed whenthe firm finds profit decreasing.

Example 10. The setting is the same as the setting inExample 8. Assume that the firm found profit declines withsales increase in the 12th year, and then it began keeping thesales volume fixed since the 16th year.

Line 1 in Figure 13 is the same as that in Figure 12. And line2 is the simulation result of the new strategy. According to thesimulation result, although the purpose of the new strategyis to mitigate the brand dilution caused by the increase ofFG, the brand dilution (Figure 13(d)) does not slow downsubstantially. Proposition 4 has also predicted the results.Theprofit (Figure 13(a)) and the revenue (Figure 13(b)) continue

Mathematical Problems in Engineering 13

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Figure 13: Contrast of trend when stopping or continuing sales increase.

to decline at the same time, and the revenue performanceis even worse than the profit. Keeping the sales volume thesame (Figure 13(c)) is obviously not a good strategy either,because once brand is diluted, some LG consumers willleave continuously and only low-value FG consumers willcome. While it is not viable to mitigate the brand dilution bystopping the sales volume growth, we test whether increasingthe sales and lowering the product cost are helpful as anotherstrategy.

5.2.3. Strategy 3: Lower the Cost When ProfitContinuously Declines

Example 11. The setting is still the same with the setting asExample 8. Assume that the firm finds profit declines withsales increase in the 12th year and begins to cut down the costin the 16th year. The variable cost is 10% lower each year, andthe comparative cost is reduced to 60 (line 1), 30 (line 2), and15 (line 3), respectively.We iterate the operation for fifty years.

According to the simulation result (Figure 14(a)), fromthe 16th year to the 30th year, cutting cost cannot prevent

profit from rapid decline. After this, there is a rising trend ofthe profit during the 30th-to-50th-year period due to the costcutting of line 2 and line 3, but the brand purity is almost zero(Figure 14(b)). As variable cost goes down from 300 to 60, oreven 15, the perceived quality of commodities and serviceswould be greatly discounted, which can no longer (may alsoseem unnecessary to) meet the LG’s needs. The brand wouldbe no longer for the LG group and degraded into a brand forthe FG group since then.

5.2.4. Strategy 4: Reduce the Sales Volume Gradually WhenProfit Continuously Declines. Tofind a good reduction policy,we reduce the sales volume to different extents when profitcontinuously declines.

Example 12. The setting is still the same as the setting inExample 8. The firm starts to reduce the sales volume in the16th year after being aware of profit decline. Contrasting theresults between that sales volume will stay less than and veryclose to the critical point 𝑞∗ (keep shrinking at 10% per year,5 years in total, line 1 in Figure 15) and that sales volume will

14 Mathematical Problems in Engineering

80

60

40

20

O

00

5 10 15 20 25 30 35 40 45 50

Time (year)

1

1

1 11

1

1 1 1 1 1 11

1

1

1

2

22

2 2

2

22 2 2

2 2 2 2

2

3

33

3 3

33

3 3 3

33 3

33

(M)

O: NOAd N full50 01 08 300 1000 15d60O: NOAd N full50 01 08 300 1000 15d30O: NOAd N full50 01 08 300 1000 15d15

(a) Annual profit

s: NOAd N full50 01 08 300 1000 15d60s: NOAd N full50 01 08 300 1000 15d30s: NOAd N full50 01 08 300 1000 15d15

1

0.75

0.5

0.25

s

00