Optimal ordering and clearance strategies under demand uncertainty

9th IFAC Conference on Manufacturing Modelling, Management and 9th 9th IFAC IFAC Conference Conference on on Manufacturing Manufacturing Modelling, Modelling, Management Management and and Control Control 9th IFAC Conference on Manufacturing Modelling, Management and Available online at www.sciencedirect.com Control Berlin, Germany, August 28-30, 2019 9th IFAC IFAC Conference Conference on on Manufacturing Manufacturing Modelling, Modelling, Management Management and 9th and Berlin, Control Berlin, Germany, Germany, August August 28-30, 28-30, 2019 2019 Control Control Berlin, Germany, August 28-30, 2019 Berlin, Germany, Germany, August August 28-30, 28-30, 2019 2019 Berlin,

ScienceDirect

IFAC PapersOnLine 52-13 (2019) 583–588

Optimal Optimal ordering ordering and and clearance clearance strategies strategies Optimal ordering and clearance strategies under demand uncertainty Optimal ordering and clearance strategies Optimalunder ordering and clearance strategies demand uncertainty under demand uncertainty under demand demand uncertainty under uncertainty Mechmech, R*. Harbi, S*. B.Hadj-Alouane, A*.

Mechmech, S*. B.Hadj-Alouane, Mechmech, R*. R*. Harbi, Harbi, S*.S**. B.Hadj-Alouane, A*. A*. Sboui, Mechmech, R*. Harbi, S*. B.Hadj-Alouane, A*. Sboui, S**. Sboui, S**. Mechmech, R*. Harbi, S*. B.Hadj-Alouane, Mechmech, R*. Harbi, S*.S**. B.Hadj-Alouane, A*. A*. Sboui, Sboui, S**. Sboui, S**. University of Tunis El Manar, 1002, Tunis *National Engineering school of Tunis, OASIS Laboratory, *National Engineering school school of of Tunis, Tunis, OASIS OASIS Laboratory, Laboratory, University University of of Tunis Tunis El El Manar, Manar, 1002, 1002, Tunis Tunis *National(e-mail: Engineering TUNISIA [email protected]; [email protected]; [email protected] ). *National Engineering school of Tunis, OASIS Laboratory, University of Tunis El Manar, 1002, Tunis TUNISIA (e-mail: [email protected]; [email protected]; [email protected] ). TUNISIA (e-mail: [email protected]; [email protected]; [email protected] ). *National Engineering school of Tunis, OASIS Laboratory, University of Tunis El Manar, 1002, Tunis ** SQLI Services, 2010, Manouba, TUNISIA (e-mail: [email protected]) *National Engineering school of Tunis, OASIS Laboratory, University of Tunis El Manar, 1002, Tunis TUNISIA (e-mail: [email protected]; [email protected]; [email protected] ). ** SQLI Services, 2010, Manouba, TUNISIA (e-mail: [email protected]) ** [email protected]; SQLI Services, 2010, Manouba, TUNISIA (e-mail: [email protected]) TUNISIA [email protected]; [email protected] TUNISIA (e-mail: (e-mail: [email protected]; [email protected]; [email protected] ). ). ** SQLI Services, 2010, Manouba, TUNISIA (e-mail: [email protected]) ** SQLI Services, 2010, Manouba, TUNISIA (e-mail: [email protected]) ** SQLI Services, 2010, Manouba, TUNISIA (e-mail: [email protected]) Abstract: The management of unsold inventory represents a significant issue for many manufacturing firms Abstract: The management of unsold represents aa significant issue for manufacturing firms Abstract: TheSome management ofstrategies unsold inventory inventory represents significant for many many manufacturing firms and retailers. clearance have been proposed in previousissue researches to drain this undesirable Abstract: The management of unsold inventory represents a significant issue for many manufacturing firms and retailers. Some clearance strategies have been proposed in previous researches to drain this undesirable and retailers. Some clearance strategies have been proposed in previous researches to drain this undesirable Abstract: The management of unsold inventory represents a significant issue for many manufacturing firms inventory. This paper addresses the problem ofrepresents the single-period ordering decisions and the selection of Abstract: The management ofstrategies unsold inventory a significant issue for many manufacturing firms and retailers. Some clearance have been proposed in previous researches to drain this undesirable inventory. This paper addresses the problem of the single-period ordering decisions and the selection of inventory. This paper addresses the problem of the single-period ordering decisions and the selection of and retailers. Some clearance strategies have been proposed in previous researches to drain this undesirable the appropriate clearance strategies under uncertain demand. We develop an MILP model to study the and retailers. Some clearance strategies have been proposed in previous researches to drain this selection undesirable inventory. This paper addresses the problem of the single-period ordering decisions and the of the clearance strategies under uncertain demand. We develop an MILP model to study the the appropriate appropriate clearance strategies under uncertain demand. Weordering develop antotal MILP model to study the inventory. This paper paper addresses theThe problem of the single-period single-period ordering decisions and the profit selection of retailer’s optimal order quantity. main objective is to maximize thedecisions expected while inventory. This addresses the problem of the and the selection of the appropriate clearance strategies under uncertain demand. We develop an MILP model to study the retailer’s optimal order quantity. The main objective is to maximize the total expected profit while retailer’s optimal order quantity. The main objective is to maximize the total expected profit while the appropriate clearance strategies under uncertain demand. We develop develop an MILP MILP model to to study study the considering the risk and the performance of the clearance strategies as constraints. We considered the case the appropriate clearance strategies under uncertain demand. We an model the retailer’s optimal quantity. The main is to maximize the totalWe expected profit considering the and performance of clearance strategies as considered thewhile case considering the risk riskorder and the the performance of the theobjective clearance as constraints. constraints. We considered case retailer’s order quantity. The main objective is to the expected profit of uniformoptimal demand and two clearance The piecewise linearization approach was used to while solve retailer’s optimal order quantity. Thestrategies. main objective isstrategies to maximize maximize the total total expected profitthe while considering the risk and the performance of the clearance strategies as constraints. We considered the case of uniform demand two clearance strategies. The piecewise linearization approach was used to solve of uniform demand and two clearance strategies. The piecewise linearization approach was used to solve considering the risk the performance of the clearance strategies as constraints. We considered the case the model. Copyright © the 2019 IFAC. strategies. considering the risk and and performance of the clearance strategies as constraints. We considered the case of uniform demand two clearance The piecewise linearization approach was used to solve the model. Copyright © 2019 IFAC. the model. Copyright © 2019 IFAC. of uniform demand and two clearance strategies. The piecewise linearization approach was used to solve of uniform demand and two clearance strategies. The piecewise linearization approach was used to solve Keywords: unsold inventory, newsvendor model, salvage price, clearance strategies. the model. Copyright © 2019 IFAC. © 2019, IFAC (International Federation Control) Hosting by Elsevier Ltd. All rights reserved. the model. Copyright © IFAC. Keywords: unsold newsvendor model, the model. Copyright © 2019 2019 IFAC. of Automatic Keywords: unsold inventory, inventory, newsvendor model, salvage salvage price, price, clearance clearance strategies. strategies. Keywords: unsold inventory, newsvendor model, salvage price, clearance strategies. Keywords: Keywords: unsold unsold inventory, inventory, newsvendor newsvendor model, model, salvage salvage price, price, clearance clearance strategies. strategies. on the leftover amount. Lately, Mitra (2018) proposes an NV on the leftover amount. Lately, Mitra Mitra (2018) (2018) proposes proposes an an NV 1. INTRODUCTION on the leftover amount. Lately, model while considering the salvage/clearance priceanasNV a 1. INTRODUCTION 1. INTRODUCTION on the leftover amount. Lately, Mitra (2018) proposes model while considering the salvage/clearance price asNV model while considering the salvage/clearance price as aa on the leftover amount. Lately, Mitra (2018) proposes an NV 1. INTRODUCTION decision variable. on the leftover amount. Lately, Mitra (2018) proposes an NV 1. model while considering the salvage/clearance price as a decision variable. 1. INTRODUCTION INTRODUCTION decision variable. model while considering the salvage/clearance price as Consequently, to tackle thethe problem of unsold inventory model while considering salvage/clearance price asweaa decision variable. Consequently, to tackle the problem of unsold inventory we Nowadays, having an important unsold inventory represents decision Consequently, to tackle problem of unsold inventory we variable. may underline three basicthe approaches in the literature: the NV decision variable. Nowadays, having having an an important unsold inventory Nowadays, important unsoldencountered inventory represents represents to tackle the problem of unsold inventory we may underline three basic approaches in the the literature: the NV NV one of the most common challenges by many Consequently, may underline three basic approaches in literature: the Consequently, to tackle the problem of unsold inventory we Nowadays, having an important unsold inventory represents model, the markdown policies andof the price-dependent Consequently, to tackle the problem unsold inventory we one the common challenges encountered by one of ofHowever, the most most common challenges encountered by many many Nowadays, having an important important unsold inventory represents may underline three basic approaches in the literature: the NV model, the markdown policies and the price-dependent firms. it may constitute an economic asset and bring Nowadays, having an unsold inventory represents model, the markdown policies and the price-dependent may underline three basic approaches in the literature: the NV one of the most common challenges encountered by many demand. Using the NV model, scientists aim to maximize the may underline three basic approaches in the literature: the NV firms. However, it may constitute an economic asset and bring firms. However, it may constitute an economic asset and bring one of of the the financial most common common challenges encountered by 2006). many demand. model, the markdown policies and the price-dependent Using the scientists aim maximize the significant gain if challenges it is well managed (Sboui, one most encountered by many demand. Using the NV NV model, model, scientists aim to to maximize the model, the markdown policies and the price-dependent firms. However, it may constitute an economic asset and bring profit optimizing the ordered considering model, while the markdown policies and quantity the price-dependent significant financial gain if well managed (Sboui, 2006). significant financial gain if it it is is well managed (Sboui, 2006). firms. However, it may constitute an economic asset and bring demand. Using the NV model, scientists aim to maximize the profit while optimizing the ordered quantity considering In the academic literature, several research papers were firms. However, it may constitute an economic asset and bring profit while the ordered quantity considering demand. Usingoptimizing the NV NV model, scientists aim to tosituations. maximize the significant financial gain if it is well managed (Sboui, 2006). simultaneously the out of stock and overstock The demand. Using model, scientists aim maximize the In the academic literature, several research papers were In the academic literature, several research papers were significant financial gain if it is well managed (Sboui, 2006). profit while optimizing the ordered quantity considering simultaneously the out out of and overstock situations. The interested infinancial the management ofwell unsold inventories. Most of simultaneously significant gain if it is managed (Sboui, 2006). the of stock stock and overstock situations. The profit while while optimizing the ordered ordered quantity considering In the academic literature, several research papers were markdown policies are especially represented by the clearance profit optimizing the quantity considering interested in the management of unsold inventories. Most of interested inonthe management of unsold inventories. Most of simultaneously In the academic literature, several research papers were the are out of stock and overstock The markdown policies policies are especially represented bysituations. the clearance clearance them using removal strategies such as the donations In thefocus academic literature, several research papers were markdown represented by the simultaneously out of stock and overstock situations. The interested management of unsold inventories. Most of sales. Theypolicies arethe adopted enhance sales and liquidate the simultaneously the out especially of to stock and overstock situations. The them focusin onthe using removal strategies strategies such as the the donations donations them focus on using removal such as interested in the management of unsold inventories. Most of markdown are especially represented by the clearance sales. They are adopted to enhance sales and liquidate the and the recycling (Islam and Vate, 2013). More recently, interested in the management of unsold inventories. Most of sales. They are adopted to enhance sales and liquidate the markdown policies are especially represented by the clearance them focus on using removal strategies such as the donations greatest part from the remaining products. The pricemarkdown policies are especially represented by the clearance and the recycling (Islam and Vate, 2013). More recently, and the recycling (Islam and Vate, 2013). More recently, them focus on using removal strategies such as the donations sales. They are adopted to enhance sales and liquidate the greatest part from the remaining products. The priceMechmech et al. (2017) have presented eight strategies used them focus on using removal strategies such as the donations greatest part from the remaining products. The pricesales. They are adopted to enhance sales and liquidate the and the recycling (Islam and Vate, 2013). recently, dependent method is the based on regulating and sales. They arefrom adopted toremaining enhance sales and adjusting liquidate the Mechmech et al. al. (2017) (2017) have presented eight More strategies used Mechmech et have presented eight strategies used and the recycling (Islam and Vate, 2013). More recently, greatest part products. The pricedependent method is based on regulating and adjusting the to liquidate unsold items designated “the clearance strategies”. and the recycling (Islamhave andpresented Vate, 2013). More recently, dependent method is based on regulating and adjusting the greatest part from the remaining products. The priceMechmech et al. (2017) eight strategies used sales’ price so that companies may boost the consumers’ greatest part from the remaining products. The priceto liquidate unsold items designated “the clearance strategies”. to liquidate unsold items designated “the clearance strategies”. Mechmech etare al.interested (2017) have presented eight strategies used dependent method is based on regulating and adjusting the sales’ price so that companies may boost the consumers’ Other papers in the use of the markdown policies Mechmech et al. (2017) have presented eight strategies used sales’ price so that companies may boost the consumers’ dependent method is based on regulating and adjusting the to liquidate unsold items designated “the clearance strategies”. purchases. We conclude that to remaining stocks and dependent method is companies based onliquidate regulating and adjusting the Other papers are in use of markdown policies Other paperssales are interested interested in the the use“the of the the markdown policies to liquidate liquidate unsold itemsetdesignated designated “the clearance strategies”. price so that may boost the consumers’ purchases. We conclude that remaining stocks and increase (Shen al., 2013; Huang et al., 2014). This sales’ to unsold items clearance strategies”. purchases. that to to liquidate liquidate remaining stocks sales’ price so that may boost the consumers’ Other papers are interested in the use of the markdown policies theWe demand, scientists were focusing on and the sales’ price soconclude that companies companies mayglobally boost the consumers’ to increase sales (Shen et al., al., 2013; Huang et al., al., 2014). 2014). This increase to increase sales (Shen et 2013; Huang et This Other papers are interested in the use of the markdown policies purchases. We conclude that to liquidate remaining stocks and increase demand, scientists were focusing on the alternative could represent an efficient the2014). standpoint Other papers are interested in the use Huang oftool the from markdown policies increase the theWe demand, scientists were globally globally focusing on and the purchases. We conclude that to todifferent liquidate remaining stocks and to increase sales (Shen et al., 2013; et al., This discounted-prices through types of stocks reduction purchases. conclude that liquidate remaining alternative could represent an efficient efficient tool from from the2014). standpoint alternative could represent an tool the standpoint to increase sales (Shen et al., 2013; Huang et al., This increase the demand, scientists were globally focusing on discounted-prices through different types of reduction of Mantrala and Rao (2001). They underline that high to increase sales (Shen et al., 2013; Huang et al., 2014). This discounted-prices through different types of reduction increase the the sales, demand, scientists were globally globally focusing on the the alternative could represent an efficient tool from the standpoint (clearances multiple discounts, markdowns policies, increase demand, scientists were focusing on the of Mantrala and Rao (2001). They underline that high of Mantrala andrepresent Rao (2001). They underline that high discounted-prices alternative could an tool from the through different types of reduction (clearances sales, discounts, markdowns policies, markdowns percentages do not They necessarily to lesser alternative could represent an efficient efficient toolunderline fromlead the standpoint standpoint (clearances sales,themultiple multiple discounts, markdowns policies, discounted-prices through different types of reduction of Mantrala and Rao (2001). that high etc.). Moreover, use of the removal ways was completely discounted-prices through different types of reduction markdowns percentages do not necessarily lead to lesser markdowns percentages do Hausman not They necessarily lead that to (2010) lesser of Mantrala and Rao underline high (clearances sales,the multiple discounts, markdowns policies, Moreover, use the removal ways was completely completely supplier profits. and Thorbeck of Mantrala andMoreover, Rao (2001). (2001). They underline that high etc.). etc.). Moreover, useofof ofstrategies the removal ways was (clearances multiple discounts, markdowns policies, markdowns percentages do not necessarily lead to lesser distinct fromsales, thethe use providing cash. We also (clearances sales, multiple discounts, markdowns policies, supplier profits. Moreover, Hausman and Thorbeck (2010) supplier profits. Moreover, Hausman and Thorbeck (2010) markdowns percentages do not necessarily lead to lesser etc.). Moreover, the use of the removal ways was completely distinct from the use of strategies providing cash. We also highlight that the application of the markdown policies in the markdowns percentages do not necessarily lead to lesser distinct from the use of strategies providing cash. We also etc.). Moreover, the use of the removal ways was completely supplier profits. Moreover, Hausman and Thorbeck (2010) highlight that the main objective of the previous works is to etc.). Moreover, the use of the removal ways was completely highlight that the application of the markdown policies in the highlight that the application of the markdown policies in the supplier profits. Moreover, Hausman and Thorbeck (2010) distinct from the use of strategies providing cash. We also highlight that main objective of the previous works is to fashion products can even bring significant profit for all supply supplier profits. Moreover, Hausman and Thorbeck (2010) highlight that main objective of the previous works is to distinct from the use of strategies providing cash. We also highlight that the application of the markdown policies in the maximize the total profit. To the best of our knowledge, no distinct from the use of strategies providing cash. We also fashion products can even bring significant profit for all supply fashion products even bring profit for allmention supply highlight thatHowever, thecan application ofsignificant theand markdown policies in the the highlight that the main objective of the previous works is to maximize the total profit. To the best of our knowledge, no chain actors. Avittathur Biswas (2017) highlight that the application of the markdown policies in maximize the total profit. To the best of our knowledge, no highlight that the main objective of the previous works is to fashion products can even bring significant profit for all supply paper has that dealt with the objective importance oftheoptimal ordering highlight the main previous workswhile is no to chain actors. However, Avittathur and Biswas (2017) mention chain Avittathur and Biswas (2017) fashion products can even even bring significant profit for all allmention supply maximize the total To the of best of our knowledge, paper has dealt dealt withprofit. the importance importance of optimal optimal ordering while while that inactors. someHowever, situations the retailer must dispose of the paper fashion products can bring significant profit for supply has with the of ordering maximize the total profit. To of knowledge, no chain Avittathur and Biswas (2017) mention considering selection of the thebest most suitable clearance maximize thethe total profit. To the best of our our knowledge, no that in inactors. someHowever, situations the retailer retailer must dispose dispose of the that some situations the must of the chain actors. However, Avittathur and Biswas (2017) mention paper has dealt with the importance of optimal ordering while considering thewith selection of the the of most suitable clearance clearance inventory for free retailer in and order to dispose avoid chain actors. However, Avittathur Biswas (2017) negative mention considering the selection of most suitable clearance paper has has to dealt thetoimportance importance optimal ordering while that in some situations of the strategies sell and/or dispose of the excess inventory, their paper dealt with the of optimal ordering while clearance inventory for the free retailer in order to avoid negative clearance inventory for free in ordermust to dispose avoid negative that in some situations the must of the considering the selection of the most suitable clearance strategies to sell and/or to dispose of the excess inventory, their revenues. The adoption of a dynamic-price strategy also that in some situations the retailer must dispose of the strategies to sell and/or to dispose of the excess inventory, their consideringand thetheir selection ofThus, the most most suitable clearance clearance inventory for free in order to avoid negative efficiency impact. in this paper we aim to considering the selection of the suitable clearance revenues. The adoption of a dynamic-price strategy also revenues. The adoption of a in dynamic-price strategy also strategies clearance inventory for free order to avoid negative to sell and/or to dispose of the excess inventory, their efficiency and their impact. Thus, in this paper we aim to represents an efficient way to attract consumers and to improve clearance inventory for free in order to avoid negative efficiency and their impact. Thus, in this paper we aim to strategies to sell and/or to dispose of the excess inventory, their revenues. The adoption of a dynamic-price strategy also develop an NV model to order the optimal quantity while strategies to sell and/or to dispose of the excess inventory, their represents an efficient way to attract consumers and to improve represents an efficient way of to attract consumers and to improve revenues. The adoption aa dynamic-price strategy also efficiency and their impact. Thus, in this paper we aim to develop an NV model to order the optimal quantity while sales (Huang et al., 2014). According to Ma et al. (2017), the revenues. The adoption of dynamic-price strategy also develop an NV model to order the optimal quantity while efficiency and their impact. Thus, in this paper we aim to represents an efficient way to attract consumers to improve considering thetheir appropriate clearance strategies. Weaim differ and impact. Thus, inoptimal this paper we to sales (Huang et al., 2014). According to Ma Ma et et and al. (2017), (2017), the efficiency sales (Huang et al., 2014). According to al. the represents an efficient way to attract consumers and to improve develop an NV model to order the quantity while considering the appropriate clearance strategies. We differ selling-prices analysis, also known as the price-dependent represents an efficient way to attract consumers and to improve considering the appropriate clearance strategies. We differ develop an NV model to order the optimal quantity while sales (Huang et al., 2014). According to Ma et al. (2017), the from existing works by: develop an NV model to order the optimal quantity while selling-prices analysis, also known as the price-dependent selling-prices also known as theworks. sales (Huang et al., 2014). to et al. the considering appropriate clearance strategies. We differ existing works by: demand, has been studied byAccording many scientific Moreover, sales (Huang etanalysis, al., 2014). According to Ma Ma etprice-dependent al. (2017), (2017), the from from existingthe works by: considering the appropriate clearance strategies. We selling-prices analysis, also known as the price-dependent the appropriate clearance strategies. based We differ differ demand, has been studied by many scientific works. Moreover, •from Considering simultaneously the strategies on demand, has been studied by schemes many scientific works. Moreover, considering selling-prices analysis, also known assell theexcess price-dependent existing works by: • Considering simultaneously the strategies based on the use of multiple discounts to inventories selling-prices analysis, also known as the price-dependent from existing works by: •from Considering simultaneously theremoval strategies based on demand, has been studied by many scientific works. Moreover, existing works by: the use of multiple discounts schemes to sell excess inventories discounted prices as well as the ones. We also the use of multiple discounts schemes to sell excess inventories demand, has hasa been been studied by byarea many scientific works. Moreover, Moreover, •• Considering simultaneously the strategies based on discounted prices as well as the removal ones. We also represents well-studied in the newsvendor problem demand, studied many scientific works. discounted prices as well as the removal ones. We also Considering simultaneously the strategies based on the use of multiple discounts schemes to sell excess inventories represents well-studied area in the the newsvendor problem • Considering examine theprices use of as strategies using external waysbased based simultaneously theremoval strategies on represents aaconstitutes well-studied area in newsvendor problem the use multiple discounts schemes to excess discounted well as the ones. We also examine the use strategies using external ways based on (NVP) and an efficient solution to getinventories rid of the the use of of multiple discounts schemes to sell sell excess inventories examine the use of of as strategies using external ways based on discounted prices well as the removal ones. We also represents aaconstitutes well-studied area in the newsvendor problem the list of clearance strategies presented by Mechmech et (NVP) and an efficient solution to get rid of the discounted prices as well as the removal ones. We also (NVP) and constitutes an efficient solution to get rid of the represents well-studied area in the newsvendor problem examine the use of strategies using external ways based the list of clearance strategies presented by Mechmech et undesirable amount from an early stage. For a limited represents a well-studied area in the newsvendor problem the (2017). list ofthe clearance strategiesusing presented byways Mechmech et examine the use of of strategies strategies using external ways based on on (NVP) and constitutes an efficient solution to get rid of the al. undesirable amount from an early stage. For a limited examine use external based on undesirable amount Biswas from an early stage. For arid limited (NVP) and constitutes an efficient solution to get of the the list of clearance strategies presented by Mechmech al. (2017). clearance inventory, and Avittathur (2018) propose an (NVP) and constitutes an efficient solution to get rid of the al. (2017). the list list of of clearance clearance strategies strategies presented presented by by Mechmech Mechmech et et undesirable amount from an early stage. For a limited clearance inventory, Biswas and Avittathur (2018) propose an the et clearance and Avittathur (2018) propose an undesirable amount from an For aa depends limited al. (2017). NV modelinventory, considering a variable salvagestage. revenue undesirable amount Biswas from an early early stage. Forthat limited al. (2017). clearance inventory, Biswas and Avittathur (2018) propose an NV model considering a variable salvage revenue that depends al. (2017). NV modelinventory, considering a variable salvage revenue that depends clearance Biswas and (2018) propose an clearance inventory, Biswas and Avittathur Avittathur (2018)that propose an NV model considering aa variable salvage revenue depends NV model considering variable salvage revenue that depends NV model© considering a variable salvage revenue that depends 2405-8963 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

Copyright © 2019 IFAC 590 Copyright 2019 IFAC 590 Peer review© of International Federation of Automatic Copyright ©under 2019 responsibility IFAC 590Control. Copyright © 2019 IFAC 590 10.1016/j.ifacol.2019.11.221 Copyright © 2019 IFAC 590 Copyright © 2019 IFAC 590

2019 IFAC MIM 584 Berlin, Germany, August 28-30, 2019

R. Mechmech et al. / IFAC PapersOnLine 52-13 (2019) 583–588

• Selecting the adequate clearance strategies before the beginning of the season period (without knowing the amount of remaining product). • Aiming to optimize the total profit as well as satisfying two evaluation criteria: the clearance performance and risk. • Developing an NV model while considering simultaneously several salvage prices belonging each one to a specific clearance strategy under uncertain demand. This paper is organized as follows: Section 2 describes the research problem; Section 3 presents the relevant literature on the NVP and the clearance concept. In Section 4, we present our proposed approach. A numerical illustration is depicted in Section 5. And finally, Section 6 draws the main concluding remarks and future insights.

introduce four main disposal options: return or reuse; liquidation; disposal; and donation. The main objective for a company having unsold inventory, even it is a retailer, a manufacturer or a wholesaler, is to drain the total amount and realize maximum benefit. Brown and Narayanan (2010) present different kinds of retails helping companies to liquidate the unsold inventories through the secondary market. They call this channel as an after-retail excess inventory channel. According to Rogers et al. (2012), companies should focus on liquidating the maximum quantity of their unsold items within the shortest possible period since the products’ value is forfeited over time. This represents the primary goal of the clearance chain which has been introduced by Mechmech et al. (2017). They also present eight clearance strategies shown in Table 1. These strategies may be grouped into disposal (removal) and cash recovery categories. Each one has its advantageous and drawbacks (for example, according to Chu et al. (2018) donations enhance the tax deduction and the firm’s social image).

2. PROBLEM STATEMENT AND ASSUMPTIONS

Considering the case where a company has to choose the appropriate clearance strategies for an item before the beginning of the selling period. Thus, the decision maker has to deal with an uncertain demand as well as an uncertain unsold quantity. Moreover, it must carefully fix the item’s quantity to order which maximizes the expected profit. Based on the NVP approach, we aim in this paper to optimize the ordered quantity while maximizing the expected profit under the use of one or more clearance strategies. In this situation, we consider that the firm may consider two costs; the shortage cost and the salvage one. The shortage cost is incurred if the company does not satisfy its customer demand. It may reflect the cost of outsourcing operations used to provide the unsatisfied demand, the market lost sales, etc. Otherwise, if the ordered quantity exceeds the demand, the unsold products must be sold through an efficient way at a salvage price that is generally lower than the selling-period one. In that case, the application of the clearance strategy may represent a costeffective solution. However, their use may lead to significant costs affecting the firm’s business. Consequently, they should be properly selected and carefully applied. This selection is considered as a strategic decision. Thus, our main objective is to help managers choose the most adequate ordered quantity that optimizes the expected firm’s profit while considering the use of the clearance strategies to dispose of the excess inventory.

Table 1. Clearance strategies Strategies 𝑆𝑆1 : Clearance sales 𝑆𝑆2 : Factory outlet 𝑆𝑆3 : Online clearance 𝑆𝑆4 : Discounters 𝑆𝑆5 : Clearance wholesalers 𝑆𝑆6 : Recycling 𝑆𝑆7 : Destruction 𝑆𝑆8 : Donation The three first strategies constitute the in-house ones. Their application does not require the use of external resources. The discounters and clearance wholesalers are external strategies. Finally, the recycling, destruction, and donation represent the removal strategies. In most cases, they are applied after the use of the in-house ones. An evaluation of those strategies was conducted by Mechmech et al. (2017). It is based on a riskperformance approach using the AHP method. Based on the company’s vision and goals, the decision maker can assess the performance (𝑃𝑃𝑖𝑖 ) and the risk (𝑅𝑅𝑖𝑖 ) of each strategy. The performance, 𝑃𝑃𝑖𝑖 , was assessed according to the clearance price, the flow speed and the drained volume. Similarly, the risk, 𝑅𝑅𝑖𝑖 , was evaluated based on the cannibalization, the deterioration of the brand image and the customer buying behavior changes risks. In this paper, we will use the indicators 𝑃𝑃𝑖𝑖 and 𝑅𝑅𝑖𝑖 to incorporate the clearance strategies’ evaluation in our proposed model.

3. LITERATURE REVIEW

In this section, we present a literature review on the clearance strategies within the supply chain and the single-period ordering under uncertainty, mainly the newsvendor problem (NVP).

3.2 Ordering decision under demand uncertainty: The NVP Our paper is concerned with ordering decisions under demand uncertainty. The NVP figures among the most classical problem that have been studied through the academic literature in the inventory management field. The main characteristics of the classical NVP are to consider a single-period selling season where the ordering decision is made at once under uncertain

3.1 The clearance strategies Many manufacturing firms and retailers are faced with an important amount of excess inventory due to the supply chain risks and uncertainties, seasonal products, short life cycle products, forecasting errors, etc. Islam and Vate (2013) 591

2019 IFAC MIM Berlin, Germany, August 28-30, 2019

R. Mechmech et al. / IFAC PapersOnLine 52-13 (2019) 583–588

585

relative to the investment, staff, space rental, etc. It does not depend on the liquidated quantity. The variable cost represents the costs generated proportionally to the quantity of unsold products and usually expressed by the “unit cost”. It reflects expenses such as the storage cost (the management and handling operations’ costs), the shipment cost, etc. The clearance revenue can be expressed in terms of clearance (salvage) price (for S1, S2, S3, S4) or tax deduction (for S6, S8) or null (for S7). Parameters: ̃ : random demand 𝐷𝐷 𝐶𝐶𝑣𝑣 : selling price in the market (seasonal) period. 𝐶𝐶𝐶𝐶𝑖𝑖 : clearance price in the selling-off period by 𝑆𝑆𝑖𝑖 (salvage price) 𝑃𝑃𝑃𝑃: purchase cost per unit, 𝑆𝑆𝑆𝑆: shortage cost, 𝑃𝑃𝑖𝑖 : performance of 𝑆𝑆𝑖𝑖 , 𝑃𝑃𝑚𝑚𝑚𝑚𝑚𝑚 : minimum value of the performance indicator fixed by the managers, 𝑅𝑅𝑖𝑖 : risk value of 𝑆𝑆𝑖𝑖 , 𝑅𝑅𝑚𝑚𝑚𝑚𝑚𝑚 : maximum accepted strategy-risk value, 𝐶𝐶𝑖𝑖 : capacity of 𝑆𝑆𝑖𝑖 , 𝐶𝐶𝐶𝐶𝑖𝑖 : fixed strategy implementation cost. 𝐶𝐶𝐶𝐶𝐶𝐶𝑖𝑖 : variable strategy implementation cost.

customer demand. Then, companies find themselves with an excess inventory (that should be sold at a salvage price) or in an out-of-stock situation (and have to bear a shortage cost). Consequently, the classical NV mathematical model aims to optimize the expected profit while considering the shortage cost and the salvage revenue. A number of extensions of the NV classical model were studied in the literature. Khouja (1999) presents 11 main extensions categories. We find those related to the supplier pricing policies, the newsvendor pricing policies and discounting structures, the multi-product, etc. Another classification was recently proposed by Ma et al. (2017). It is based on four categories: the supplier (quantity discounts, multiple suppliers…), the newsvendor (its objective, initial inventory…), the product (product substitution, returns, etc.) and the consumer (price-dependent demand, etc.). Research works belonging to the consumer extension category show a great interest in the management of excess inventory. Their main objective is to control the consumer demand to avoid ending up with excess inventories. Indeed, relying on the fact that the customer demand increases with a price decrease, companies choose to carefully fix the selling price in order to control sales and liquidate the total inventory (Sana, 2012; Lau and Lau, 1988). This issue is known as the NV pricedependent demand. Qin et al. (2011) comprehensively review the NVP in the context of modeling consumer demand. Other researches are focusing on proposing a number of successive discounts schemes to sell their remaining products (Khouja, 2000; Khouja, 1995). More recently, Ma et al. (2017) propose to consider simultaneously a price-dependent NVP and multiple discounts to sell excess inventory, optimize the ordered quantity and maximize the expected profit.

Since the expected value of the sold items is represented by the ̃, minimum between the ordered quantity 𝑍𝑍 and the demand 𝐷𝐷 ̃ )]. it can be formulated as 𝐸𝐸[𝑀𝑀𝑀𝑀𝑀𝑀 (𝑍𝑍, 𝐷𝐷 ̃ )] 𝑄𝑄̃ = 𝑍𝑍 − 𝐸𝐸[𝑀𝑀𝑀𝑀𝑀𝑀 (𝑍𝑍, 𝐷𝐷 (1)

According to Baghalian et al. (2013), the second term of (1) can be written as: ̃ )] = 𝑍𝑍 − 𝐸𝐸[𝑀𝑀𝑀𝑀𝑀𝑀(0, 𝑍𝑍 − 𝐷𝐷 ̃ )] 𝐸𝐸[𝑀𝑀𝑀𝑀𝑀𝑀(𝑍𝑍, 𝐷𝐷 (2) 𝑍𝑍 ̃ (3) 𝐸𝐸[𝑀𝑀𝑀𝑀𝑀𝑀(0, 𝑍𝑍 − 𝐷𝐷 )] = ∫ 𝐹𝐹𝐷𝐷̃ (𝑡𝑡)𝑑𝑑𝑑𝑑 0

4. MODEL DEVELOPMENT

Where: 𝐹𝐹𝐷𝐷̃ : represents the cumulative distribution function of the ̃ . Consequently, the quantity of uncertain demand variable 𝐷𝐷 unsold items is formulated as: 𝑍𝑍 (4) 𝑄𝑄̃ = ∫0 𝐹𝐹𝐷𝐷̃ (𝑡𝑡)𝑑𝑑𝑑𝑑

In this section, we propose to incorporate the selection of the clearance strategies in the NV mathematical model. We present the model and the solution approach in Sections 4.1 and 4.2 respectively.

Decision variables: • 𝑍𝑍: the quantity to purchase. • 𝑥𝑥̃𝑖𝑖 : quantity to liquidate by 𝑆𝑆𝑖𝑖 . • 𝑦𝑦𝑖𝑖 : binary variable equals to 1 if 𝑥𝑥̃𝑖𝑖 > 0.

4.1 Model formulation The main objective of our model is to optimize the expected profit of a company while identifying the optimal quantity to order and to liquidate under uncertain demand. Additionally, the selection of the appropriate clearance strategies represents another crucial goal. As we have mentioned previously, the excess items are sold at a salvage price or disposed of in order to optimize the expected profit. The main idea here is how to estimate the clearance strategies’ cost and revenues. Furthermore, we are interested in how to integrate the impact of the clearance strategy on the firm’s business. To evaluate the risk and performance of each clearance strategy (𝑆𝑆𝑖𝑖 ) we will rely on the performance-risk based AHP method developed by Mechmech et al. (2017). Moreover, for each 𝑆𝑆𝑖𝑖 , an implementation cost 𝐶𝐶𝐶𝐶𝑖𝑖 has also been assigned. It is composed of two elements, a fixed cost 𝐶𝐶𝐶𝐶𝑖𝑖 and a variable one 𝐶𝐶𝐶𝐶𝐶𝐶𝑖𝑖 . The fixed cost generally represents the expenses

Objective function: The objective function can be written as: 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 − 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝ℎ𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 − 𝑠𝑠ℎ𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝑐𝑐𝑐𝑐𝑠𝑠𝑡𝑡 + 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 } 𝜋𝜋 = 𝑀𝑀𝑀𝑀𝑀𝑀 { − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 Where: ̃ )] ∗ 𝐶𝐶𝑣𝑣 . • Total revenue = 𝐸𝐸[min(𝑄𝑄, 𝐷𝐷 • Purchase cost = 𝑍𝑍 ∗ 𝑃𝑃𝑃𝑃. ̃ − 𝑍𝑍)] ∗ 𝑆𝑆𝑆𝑆. • Shortage cost = 𝐸𝐸[max(0, 𝐷𝐷 ̃ )] ∗ 𝐶𝐶𝐶𝐶𝑖𝑖 • Clearance revenue = 𝐸𝐸[max(0, 𝑍𝑍 − 𝐷𝐷

592

2019 IFAC MIM 586 Berlin, Germany, August 28-30, 2019

Consequently: 𝜋𝜋 = 𝑀𝑀𝑀𝑀𝑀𝑀

R. Mechmech et al. / IFAC PapersOnLine 52-13 (2019) 583–588

̃ )]∗ 𝐶𝐶𝑣𝑣 ) ( Z ∗ 𝐶𝐶𝑣𝑣 − 𝐸𝐸[max(0, 𝑍𝑍 − 𝐷𝐷 ̃ ] ∗ 𝑆𝑆𝑆𝑆 − (𝑍𝑍 ∗ 𝑃𝑃𝑃𝑃) − 𝐸𝐸[𝐷𝐷

̃ ) ∗ 𝑆𝑆𝑆𝑆] +𝑍𝑍 ∗ 𝑆𝑆𝑆𝑆 − 𝐸𝐸[max(0, 𝑍𝑍 − 𝐷𝐷 8 {+(∑i=1 𝑥𝑥̃𝑖𝑖 ∗ (𝐶𝐶𝐶𝐶𝑖𝑖 − 𝐶𝐶𝐶𝐶𝐶𝐶𝑖𝑖 )) − ∑8𝑖𝑖=1 𝑦𝑦𝑖𝑖 ∗ 𝐶𝐶𝐶𝐶𝑖𝑖 }

clearance wholesalers strategy may significantly reduce this cost. 𝑡𝑡𝑠𝑠𝑠𝑠

(5) Seasonal sales period

Z ̃ ] ∗ 𝑆𝑆𝑆𝑆 Z ∗ 𝐶𝐶1 − [∫0 FD̃ (t)dt ∗ 𝐶𝐶2 ] − 𝐸𝐸[𝐷𝐷

𝐶𝐶1 = (𝐶𝐶𝑣𝑣 + 𝑆𝑆𝑆𝑆 − 𝑃𝑃𝑃𝑃) 𝐶𝐶2 = (𝐶𝐶𝑣𝑣 + 𝑆𝑆𝑆𝑆)

𝑠𝑠. 𝑡𝑡 ∑8𝑖𝑖=1 𝑥𝑥̃𝑖𝑖 ∗ 𝑃𝑃𝑖𝑖 ≥ ∑8𝑖𝑖=1 𝑥𝑥̃𝑖𝑖 ∗ 𝑃𝑃𝑚𝑚𝑚𝑚𝑚𝑚 ∑2𝑖𝑖=1 𝑅𝑅𝑖𝑖 ∗ 𝑦𝑦𝑖𝑖 ≤ 𝑅𝑅𝑚𝑚𝑚𝑚𝑚𝑚 𝑥𝑥̃ ≤ 𝑀𝑀 ∗ 𝑦𝑦𝑖𝑖 { 𝑖𝑖 𝑦𝑦𝑖𝑖 ≤ 𝑀𝑀 ∗ 𝑥𝑥̃𝑖𝑖 Z ∑8i=1 𝑥𝑥̃𝑖𝑖 = ∫0 FD̃ (t)dt 0 ≤ 𝑥𝑥̃𝑖𝑖 ≤ 𝐶𝐶𝑖𝑖 𝑦𝑦𝑖𝑖 ∈ {0,1} 𝑥𝑥̃𝑖𝑖 , 𝑍𝑍 ≥ 0

…

𝑡𝑡𝑐𝑐𝑐𝑐

𝑡𝑡𝑐𝑐𝑐𝑐

Clearance sales period

Fig. 1. Time-lag between the seasonal and clearance periods

𝜋𝜋 = 𝑀𝑀𝑀𝑀𝑀𝑀 { } (6) +(∑8i=1 𝑥𝑥̃𝑖𝑖 ∗ (𝐶𝐶𝐶𝐶𝑖𝑖 − 𝐶𝐶𝐶𝐶𝐶𝐶𝑖𝑖 )) − ∑8𝑖𝑖=1(𝑦𝑦𝑖𝑖 ∗ 𝐶𝐶𝐶𝐶𝑖𝑖 ) Where:

𝑡𝑡𝑠𝑠𝑠𝑠

We underline that the strategies’ selection depends on the risk and performance levels expected by managers, their evaluation and their capacities. Based on researches of Baghalian et al. (2013) and Chelly et al. (2017), an uncertain customer demand may be approximated by the uniform distribution. Thus, we suppose that it is uniformly distributed in the interval [𝑎𝑎, 𝑏𝑏]. The main changes affect only the objective function (6) and constraint (12) where the cumulative function 𝐹𝐹𝐷𝐷̃ is replaced with 1 𝑍𝑍 2 𝑎𝑎2 ̃ ] with {𝑏𝑏−𝑎𝑎}. { ∗ ( − 𝑎𝑎𝑎𝑎 + )} and 𝐸𝐸[𝐷𝐷

(7) (8) (9) ( 10 )

𝑏𝑏−𝑎𝑎

( 11 )

2

2

2

As the cumulative function, 𝐹𝐹𝐷𝐷̃ , represents a nonlinear term, we propose the use of the piecewise linearization approach which has been widely used in various applications (Lin, 2013). Bazaraa (1993) and Vielma (2010) present different piecewise linearization methods. In parallel, Croxton et al. (2003) indicate that the linear functions are equivalent to each other. Their main objective is to transform the non-linear functions into a piecewise linear function. The main idea is to divide the interval [𝑎𝑎, 𝑏𝑏] into a well-defined number of (M) subintervals and to approximate the relative functions with a sequence of (M) linear segments. This operation brings extra binary variables and constraints. On each interval m of those M intervals, we will approximate FD̃ using the linear function {𝛼𝛼𝑚𝑚 ∗ 𝑧𝑧𝑚𝑚 + 𝛽𝛽𝑚𝑚 , } where 𝛼𝛼𝑚𝑚 represents the slope and 𝛽𝛽𝑚𝑚 represents the intercept of the linear segment. As for 𝑧𝑧𝑚𝑚 , it constitutes the decision variable reflecting the quantity to produce in the interval 𝑚𝑚 (𝑚𝑚 𝜖𝜖 [1. . 𝑀𝑀]). Thus, we will add the following parameters and consider the following decision variables. Parameters: • 𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑚𝑚 : upper bound of the interval 𝑚𝑚. • 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝑚𝑚 : lower bound of the interval 𝑚𝑚.

( 12 ) ( 13 ) ( 14 ) ( 15 )

Constraints (9) and (10) present the levels of clearance performance and risk required by the firm. Indeed, 𝑅𝑅𝑚𝑚𝑚𝑚𝑚𝑚 and 𝑃𝑃𝑚𝑚𝑚𝑚𝑚𝑚 are generally fixed by managers after a well-studied benchmark reflecting the firm’s strategy and vision. We underline that the clearance strategy performance value strongly depends on the liquidated quantity. However, the clearance risk is considered only if 𝑆𝑆𝑖𝑖 is used. Constraint (11) shows the relation between the decision variables 𝑥𝑥̃𝑖𝑖 and 𝑦𝑦𝑖𝑖 . Constraint (12) enforces the liquidation of the total unsold amount. Constraint (13) ensures that the liquidated quantity by a strategy does not exceed its capacity. Finally, constraints (14) and (15) represent the restriction and the non-negativity constraints. 4.2 Solution approach In order to develop our model, we have made a number of assumptions which are: considering a single-period inventory management; ordering the quantity of product before the beginning of the selling period and having a seasonal product. Thus, the start of the seasonal period is denoted by 𝑡𝑡𝑠𝑠𝑠𝑠 and its end by 𝑡𝑡𝑠𝑠𝑠𝑠 (Fig. 1). Generally, companies in practice are not required to use the eight strategies simultaneously. Therefore, to solve our model we will consider only two strategies: the clearance wholesalers and the clearance sales (we denote by 𝑡𝑡𝑐𝑐𝑐𝑐 and 𝑡𝑡𝑐𝑐𝑐𝑐 for the beginning and end of the clearance sales period, respectively). This choice is made to highlight the impact of the fixed strategy implementation cost. Indeed, as it is shown in Fig. 1, an extended time may exist between the seasonal and the clearance sales periods. Thus, if the company chooses to apply the clearance sales strategy, it has to consider the storage cost arising from the waiting time. However, the selection of the

Decision variables: • 𝑧𝑧𝑚𝑚 : the quantity to purchase in the interval m. • 𝑡𝑡𝑚𝑚 : binary variable equals to 1 if the interval 𝑚𝑚 is selected. • 𝑥𝑥̃𝑖𝑖𝑖𝑖 : liquidated quantity by 𝑆𝑆𝑖𝑖 in the interval 𝑚𝑚. • 𝑦𝑦𝑖𝑖 : binary variable equals to 1 if 𝑥𝑥̃𝑖𝑖𝑖𝑖 > 0. ∑𝑀𝑀 𝑚𝑚=1 𝑧𝑧𝑚𝑚 ∗ 𝐶𝐶1 ∗ 𝛼𝛼𝑚𝑚 + 𝛽𝛽𝑚𝑚 ) ∗ 𝑡𝑡𝑚𝑚 ) ∗ 𝐶𝐶2 ] ′ 𝑏𝑏 −( − 𝑎𝑎⁄2) ∗ 𝑆𝑆𝑆𝑆 𝜋𝜋 = max +(∑2𝑖𝑖=1 ∑M ̃𝑖𝑖𝑖𝑖 ∗ (𝐶𝐶𝐶𝐶𝑖𝑖 − 𝐶𝐶𝐶𝐶𝐶𝐶𝑖𝑖 )) m=1 𝑥𝑥 2 − ∑𝑖𝑖=1 𝑦𝑦𝑖𝑖 ∗ 𝐶𝐶𝐶𝐶𝑖𝑖 { } Objective function:

−[(∑𝑀𝑀 𝑚𝑚=1(𝑧𝑧𝑚𝑚

593

( 16 )

2019 IFAC MIM Berlin, Germany, August 28-30, 2019

R. Mechmech et al. / IFAC PapersOnLine 52-13 (2019) 583–588

𝑠𝑠. 𝑡𝑡 ∑2𝑖𝑖=1(∑M ̃𝑖𝑖𝑖𝑖 ∗ 𝑃𝑃𝑖𝑖 ) ≥ ∑2𝑖𝑖=1 ∑M ̃𝑖𝑖𝑖𝑖 ∗ 𝑃𝑃𝑚𝑚𝑚𝑚𝑚𝑚 m=1 𝑥𝑥 m=1 𝑥𝑥 2 ∑𝑖𝑖=1 𝑅𝑅𝑖𝑖 ∗ 𝑦𝑦𝑖𝑖 ≤ 𝑅𝑅𝑚𝑚𝑚𝑚𝑚𝑚 𝑥𝑥̃ ≤ 𝑀𝑀 ∗ 𝑦𝑦𝑖𝑖 ∀ 𝑚𝑚 { 𝑖𝑖𝑖𝑖 𝑦𝑦𝑖𝑖 ≤ 𝑀𝑀 ∗ 𝑥𝑥̃𝑖𝑖𝑖𝑖 2 ∑𝑀𝑀 ̃𝑖𝑖𝑖𝑖 = ∑𝑀𝑀 𝑚𝑚=1(𝑧𝑧𝑚𝑚 ∗ 𝛼𝛼𝑚𝑚 + 𝛽𝛽𝑚𝑚 ) ∗ 𝑡𝑡𝑚𝑚 𝑚𝑚=1 ∑i=1 𝑥𝑥 M 0 ≤ ∑m=1 𝑥𝑥̃𝑖𝑖𝑖𝑖 ≤ 𝐶𝐶𝑖𝑖 ∀ 𝑖𝑖 𝑧𝑧𝑚𝑚 ≤ 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑚𝑚 ∗ 𝑡𝑡𝑚𝑚 𝑧𝑧𝑚𝑚 ≥ 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑚𝑚 ∗ 𝑡𝑡𝑚𝑚 𝑦𝑦𝑚𝑚 , 𝑡𝑡𝑚𝑚 ∈ {0,1} 𝑧𝑧𝑚𝑚 ; 𝑥𝑥̃𝑖𝑖𝑖𝑖 ≥ 0

encounters a loss (or must pay a penalty) estimated to 2€. Otherwise, an unsold item may be resold using the clearance strategies: at 7€ by the clearance sales and at 2.5€ if the firm chooses to adopt the clearance wholesalers. The application of a clearance strategy generates fixed and variable costs. Using these data, the optimal solution that guarantees the maximization of the total expected profit under uncertain demand is: • 𝑡𝑡1 = 0, 𝑡𝑡2 = 0, 𝑡𝑡3 = 1. • 𝑦𝑦1 = 1, 𝑦𝑦2 = 0. • 𝑧𝑧1 = 0, 𝑧𝑧2 = 0, 𝑧𝑧3 = 150000. • 𝑥𝑥̃11 = 0, 𝑥𝑥̃12 = 0, 𝑥𝑥̃13 = 22000. • 𝑥𝑥̃21 = 0, 𝑥𝑥̃22 = 0, 𝑥𝑥̃23 = 0.

( 17 ) ( 18 ) ( 19 ) ( 20 ) ( 21 ) ( 22 ) ( 23 ) ( 24 ) ( 25 )

Constraints (22) and (23) define the values of the intervals’ limits.

The outcomes of our model show that the total purchased quantity is equal to 150000 units in which 22000 units will be sold through the clearance sales strategy. The total expected profit is estimated to 728000€. For further analysis, we aim to highlight the impact of the clearance strategies prices (𝐶𝐶𝐶𝐶𝑖𝑖 ) on the chosen strategy, ordered quantity and the total expected profit (Table 4). The same data presented in Tables 2 and 3 are used.

5. ILLUSTRATION

In order to show the effectiveness and the applicability of the proposed method, we present an illustrative example using the data shown in Tables 2 and 3.

Table 4. Impact of 𝑪𝑪𝑪𝑪𝒊𝒊 on the global solution

𝐶𝐶𝐶𝐶1

Table 2. The used data for the demand linearization Parameters ̃ (× 103 ) D 𝑀𝑀 Lower𝑚𝑚 (× 103 ) Upper𝑚𝑚 (× 103 ) β𝑚𝑚 (× 103 ) α𝑚𝑚

6 5 4 3 7 6 5

Data 𝒰𝒰[100, 150] 3 [100, 120, 135] [120, 135, 150] [−29.8, −50, −75.5] [0.28, 0.45, 0.65]

Table 3. Data related to the strategies and cost parameters Parameters 𝑖𝑖 𝑅𝑅𝑖𝑖 𝑃𝑃𝑖𝑖 𝐶𝐶𝑖𝑖 (× 103 ) 𝐶𝐶𝐶𝐶𝑖𝑖 𝐶𝐶𝐶𝐶𝑖𝑖 𝐶𝐶𝐶𝐶𝐶𝐶𝑖𝑖 𝑃𝑃𝑚𝑚𝑚𝑚𝑚𝑚 𝑅𝑅𝑚𝑚𝑚𝑚𝑚𝑚 𝐶𝐶𝑣𝑣 𝑃𝑃𝑃𝑃 𝑆𝑆𝑆𝑆

0.3 0.7 10 6 2

𝐶𝐶𝐶𝐶2 6 5 4 3 4 4 4

Used 𝑆𝑆𝑖𝑖 𝑆𝑆2 𝑆𝑆2 𝑆𝑆2 𝑆𝑆2 𝑆𝑆1 𝑆𝑆1 𝑆𝑆1

𝑥𝑥̃𝑖𝑖𝑖𝑖

22000 22000 10750 10750 22000 22000 22000

𝑍𝑍

150000 150000 135000 135000 150000 150000 150000

Expected profit 715300 693300 672425 661675 728000 706000 684000

In our case, the realized gains from the clearance process application represent the major decision factor in an over-stock situation. Consequently, the decision in choosing a clearance strategy will depend enormously on its made profit in comparison with the purchase item cost (𝑃𝑃𝑃𝑃). For comparable values of 𝐶𝐶𝐶𝐶1 and 𝐶𝐶𝐶𝐶2 , the profit margin is identical for both strategies. Thus, the selection of a strategy will heavily depend on its fixed cost. Since the 𝐶𝐶𝐶𝐶1 > 𝐶𝐶𝐶𝐶2, the chosen solution is 𝑆𝑆2 . As for the three latest cases, given that 𝐶𝐶𝐶𝐶1 is better than 𝐶𝐶𝐶𝐶2 , the earned income provided by 𝑆𝑆1 exceeds the one provided from the application of 𝑆𝑆2 even when 𝐶𝐶𝐶𝐶1 > 𝐶𝐶𝐶𝐶2 . Consequently, 𝑆𝑆1 was the selected strategy. The case where 𝐶𝐶𝐶𝐶2 > 𝐶𝐶𝐶𝐶1 has not been analyzed because it does not reflect a real-world situation.

Data 1 0.5 0.46 150 7 1000 0.5

587

2 0.14 0.32 60 2.5 500 0.1

6. CONCLUSION

The considered customer demand is uniformly distributed between 100000 and 150000 units. The linearization approach has been applied while considering three main intervals. The expected performance level of the total chain is estimated by managers to be 0.3. The maximum accepted risk value is 0.7. Every unit of the product is bought at 6€ and sold in the market period at 10€. For an out-of-stock situation, the firm

In this study, we highlight the importance of considering the use of more than one clearance strategy into the NVP to maximize the total expected profit. Having a uniformly distributed demand, we have proposed an MILP model that was solved using the piecewise linearization approach. Our model reflects the case where a company needs to fix the 594

2019 IFAC MIM 588 Berlin, Germany, August 28-30, 2019

R. Mechmech et al. / IFAC PapersOnLine 52-13 (2019) 583–588

quantity to order at once, that is before the selling period. The total ordered quantity is sold based on two key phases: the first one represents the market period and the second phase is represented by the use of the clearance strategies. Two main clearance strategies are considered in this model. Nevertheless, companies in practice may have to combine more than two strategies based on their size, vision and their available resources. Therefore, we propose for future research to extend the developed model while considering more strategies from those presented on Table 1.

Khouja, M. (1999). The single-period (news-vendor) problem: literature review and suggestions for future research. Omega, 27(5), 537-553. Khouja, M. J. (2000). Optimal ordering, discounting, and pricing in the single-period problem. International Journal of Production Economics, 65(2), 201-216. Lau, A. H. L., & Lau, H. S. (1988). The newsboy problem with price-dependent demand distribution. IIE transactions, 20 (2), 168-175. Lin, M. H., Carlsson, J. G., Ge, D., Shi, J., & Tsai, J. F. (2013). A review of piecewise linearization methods. Mathematical problems in Engineering. Mantrala, M. K., & Rao, S. (2001). A decision-support system that helps retailers decide order quantities and markdowns for fashion goods. Interfaces, 31, 146-S165. Mitra, S. (2018). Newsvendor problem with clearance pricing. European Journal of Operational Research, 268(1), 193-202. Ma, S., Jemai, Z., Sahin, E., & Dallery, Y. (2017). Analysis of the Newsboy Problem subject to price dependent demand and multiple discounts. Journal of Industrial & Management Optimization, 14(3), 931-951. Mechmech, R., Harbi, S., B. Hadj-Alouane, A., Sboui, S., (2017), Identifying and ranking of clearance chain management (CCM) strategies. 7th International Conference on Industrial Engineering and Systems Management (IESM), Saarbrücken, Germany. Qin, Y., Wang, R., Vakharia, A. J., Chen, Y., & Seref, M. M. (2011). The newsvendor problem: Review and directions for future research. European Journal of Operational Research, 213(2), 361-374. Rogers, D. S., Melamed, B., & Lembke, R. S. (2012). Modeling and analysis of reverse logistics. Journal of Business Logistics, 33(2), 107-117. Sana, S. S. (2012). Price sensitive demand with random sales price–a newsboy problem. International Journal of Systems Science, 43(3), 491-498. Sboui, S. (2006, October). Unsold and Excess Inventory: between optimization and management: A New challenge for the Supply Chain Management. In Service Systems and Service Management, 2006 International Conference on (Vol. 1, pp. 283-289). IEEE. Systems Science, 43(3), 491-498 Shen, B., Choi, T. M., Wang, Y., & Lo, C. K. (2013). The coordination of fashion supply chains with a risk-averse supplier under the markdown money policy. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 43(2), 266-276. Vielma, J. P., Ahmed, S., & Nemhauser, G. (2010). A note on “A superior representation method for piecewise linear functions”. INFORMS Journal on Computing, 22(3), 493497.

ACKNOWLEDGEMENTS

This project is carried out under the MOBIDOC scheme, funded by the EU through the EMORI program and managed by the ANPR.

REFERENCES Avittathur, B., & Biswas, I. (2017). A note on limited clearance sale inventory model. International Journal of Production Economics, 193, 647-653. Baghalian, A., Rezapour, S., & Farahani, R. Z. (2013). Robust supply chain network design with service level against disruptions and demand uncertainties: A real-life case. European Journal of Operational Research, 227(1), 199215. Bazaraa, M. S., Sherali, H. D., & Shetty, C. M. (1993). Nonlinear programming: theory and algorithms. John Wiley & Sons, York, NY, USA, 2nd edition. Biswas, I., & Avittathur, B. (2018). The price-setting limited clearance sale inventory model. Annals of Operations Research, 1-17. Chelly, A., Hadj-Alouane, A., Noiura, I., & Frein, Y. (2017). A stochastic model for analyzing the combined effect of demand uncertainty and carbon tax within the context of Green Supply Chain Design. 7th International Conference on Industrial Engineering and Systems Management (IESM), Saarbrücken, Germany. Croxton, K. L., Gendron, B., & Magnanti, T. L. (2003). A comparison of mixed-integer programming models for nonconvex piecewise linear cost minimization problems. Management Science, 49(9), 1268-1273. Chu, L. Y., Li, G., & Rusmevichientong, P. (2018). Optimal pricing and inventory planning with charitable donations. Manufacturing & Service Operations Management, 20(4), 687-703. Huang, Y. S., Hsu, C. S., & Ho, J. W. (2014). Dynamic pricing for fashion goods with partial backlogging. International Journal of Production Research, 52(14), 4299-4314. Hausman, W. H., & Thorbeck, J. S. (2010). Fast fashion: Quantifying the benefits. In Innovative quick response programs in logistics and supply chain management (pp. 315329). Springer, Berlin, Heidelberg. Khouja, M. (1995). The newsboy problem under progressive multiple discounts. European Journal of Operational Research, 84(2), 458-466. 595