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EOQ models with stockouts for disassembly systems
9th IFAC Conference on Manufacturing Modelling, Management and 9th IFAC 9th IFAC Conference Conference on on Manufacturing Manufacturing Modelling, Modelling, Management Management and and Control 9th IFAC Conference on Manufacturing Modelling, Management and Control online at www.sciencedirect.com Control Berlin, Germany, August 28-30, 2019 Available 9th IFAC Conference on Manufacturing Modelling, Management and Control Berlin, Germany, August 28-30, 2019 Berlin, Germany, August 28-30, 2019 Control Berlin, Germany, August 28-30, 2019 Berlin, Germany, August 28-30, 2019
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IFAC PapersOnLine 52-13 (2019) 1681–1686
EOQ models with stockouts for disassembly systems EOQ models with stockouts for disassembly systems EOQ models with stockouts for disassembly systems Matthieu Godichaud*. Lionel Amodeo.* systems EOQ models with stockouts for disassembly Matthieu Godichaud*. Lionel Amodeo.*
Matthieu Godichaud*. Lionel Amodeo.* Matthieu Godichaud*. deLionel * Université de Technologie Troyes,Amodeo.* Troyes, France Matthieu Godichaud*. Lionel ** Université Université de Technologie de Troyes,Amodeo.* Troyes, France France de Technologielionel.amodeo}@utt.fr. de Troyes, Troyes, {matthieu.godichaud, de Troyes, Troyes, France * Université de Technologie {matthieu.godichaud, lionel.amodeo}@utt.fr. {matthieu.godichaud, lionel.amodeo}@utt.fr. * Université de Technologielionel.amodeo}@utt.fr. de Troyes, Troyes, France {matthieu.godichaud, {matthieu.godichaud, lionel.amodeo}@utt.fr. Abstract: Although economic order quantity (EOQ) problem is a widespread problem with many variants, Abstract: Although economic order quantity quantity (EOQ)which problem is aanew widespread problem with many variants, variants, Abstract: Although economic order (EOQ) problem is widespread with many few of them are dedicated to disassembly systems brings issues inproblem inventory management due Abstract: Although economic order quantity (EOQ)which problem is anew widespread problem with many variants, few of them are dedicated to disassembly systems which brings new issues in inventory management due few of them are dedicated to disassembly systems brings issues in inventory management due to their specificities. EOQ problem for disassembly systems consists in determining reorder time interval Abstract: Although economic order quantity (EOQ) problem is a widespread problem with many variants, few of them are dedicated to disassembly systems which brings new issues in inventory management due to their specificities. EOQ problem for disassembly systems consists in determining reorder time interval to their specificities. EOQ problem for disassembly systems consists in determining reorder time interval and/or quantity of end-of-life products which which must be disassembled to obtain their component. Three few of order them are dedicated to disassembly systems brings new issues in inventory management due to their specificities. EOQ problem for disassembly systems consists in determining reorder time interval and/ormodels order quantity quantity of end-of-life end-of-life products which must bestockout. disassembled to obtain obtain their their component.as Three and/or order of products which must be disassembled to component. Three new are proposed for the problem variant with The stockouts are considered full to their specificities. EOQ problem for disassembly systems consists in determining reorder time interval and/or order quantity of end-of-life which must bestockout. disassembled to obtain their component.as Three new models are proposed for the problem variant with The stockouts are full new models arelost proposed forpartial the products problem variant with stockout. Thecost stockouts areis considered considered as full backorders, full sales and backorders. For each problem the function derived according and/or order quantity of end-of-life products which must be disassembled to obtain their component. Three new modelsfull arelost proposed forpartial the problem variant stockout. Thecost stockouts areis as full backorders, sales and backorders. For each problem the function derived according backorders, full sales and backorders. Forwith eachThese problem the function is considered derived according to themodels assumptions and dedicated are proposed. models show that itare can be economically new arelost proposed forpartial themethods problem variant stockout. Thecost stockouts as full backorders, full lost sales and partial backorders. Forwith eachThese problem the cost function is considered derived according to the assumptions and dedicated methods are proposed. models show that it can be economically to the assumptions and dedicated methods areinput proposed. models show that it can be economically interested tofull allow depending on the parameters. © 2019 IFAC backorders, loststockout sales and partial backorders. For eachThese problem the cost function is derived according to the assumptions and dedicated methods are proposed. These models show that it can be economically interested to depending on input parameters. © 2019 interested to allow allow stockout stockout depending on the the parameters. 2019 IFAC IFAC to the assumptions and dedicated methods areinput proposed. These © models show that it can be economically interested todisassembly, allow stockout depending onAutomatic the input parameters. © 2019 IFAC © 2019, IFAC (International Federation of Control) backordering, Hosting by Elsevier Ltd. All rights reserved. Keywords: EOQ, lot sizing, inventory control, end-of-life product. interested to allow stockout depending on the input parameters. © 2019 IFAC Keywords: disassembly, disassembly, EOQ, EOQ, lot lot sizing, sizing, inventory inventory control, control, backordering, backordering, end-of-life end-of-life product. product. Keywords: Keywords: disassembly, EOQ, lot sizing, inventory control, backordering, end-of-life product. Keywords: disassembly, EOQ, lot sizing, inventory control, backordering, end-of-life product. products and store less components. Backordering (partial or 1. INTRODUCTION products and store less Backordering (partial or products and been storeconsidered, less components. components. Backordering (partial or full) has not as far the authors know, in the 1. INTRODUCTION INTRODUCTION 1. products and been storeconsidered, less components. Backordering (partial or full) has not as far the authors know, in the full) has not been considered, as far the authors know, in the 1. INTRODUCTION DEOQ related models. products and been storeconsidered, less components. Backordering (partial or Disassembly is a central activity in reverse supply chains since full) has not as far the authors know, in the 1. INTRODUCTION related models. DEOQ models. Disassembly is activity in supply since Disassembly is aa central central activity in reverse reverse supply chains chains since DEOQ full) hasrelated not been considered, as far the authors know, in the it allows separating the components of end-of-life products DEOQ related models. Disassembly is a central in reverse supply chains since The paper fills the gap between disassembly lot sizing and it allows the components of products it allows separating separating the activity components ofareend-of-life end-of-life products DEOQ related models. according to the channels allocated to. Insince this The Disassembly is arecovery central activity in they reverse supply chains paper the between disassembly lot The paper fills the gap gapAs between disassembly lot sizing sizing and it allows separating the components of end-of-life products EOQ relatedfills literature. presented in this section, as farand as according to the recovery channels they are allocated to. In this according to the recovery channels they are allocated to. In this The paper fills the gapAs between disassembly lot sizing and context, disassembly planning problems consist in it allows separating the components of end-of-life products EOQ related literature. presented in this section, as far as according to the recovery channels they are allocated to. In this the authors know, there is no model proposed for determining The paper fills the gap between disassembly lot sizing and context, disassembly planning problems consist in context, disassembly planning problems consist in the EOQ relatedknow, literature. As presented in this section, as far as determining the recovery quantitieschannels and the they timings of the disassembly according are allocated to. In this authors there isdisassembly no model proposed for determining context, to disassembly planning problems consist in EOQ with know, stockout in As systems. The proposed related literature. presented in this section, as far as determining the quantities and the timings of the disassembly determining the quantities and the timings of the disassembly the authors there is no model proposed for determining of end-of-life products to serve the demands of several context, disassembly planning problems consist in EOQ withfocus stockout in disassembly systems. The proposed determining theproducts quantitiesto andserve the timings of the disassembly references on deterministic EOQ models to highlight the the authors know, there is no model proposed for determining of end-of-life the demands of several of end-of-life the demands of several references EOQ withfocus stockout in disassembly systems. proposed recovery channels. The to supply of the a disassembly determining theproducts quantities andserve the process timings of onofdeterministic EOQ models toThe highlight the of end-of-life products to serve the demands of several main specificities disassembly systems compared to others EOQ with stockout in disassembly systems. The proposed recovery channels. The supply process of a disassembly recovery The of supply of a disassembly references focus onofdeterministic EOQ models to highlight the system is channels. the collection end-of-life atofcustomer of end-of-life products to serveprocess the products demands several main specificities disassembly systems compared recovery channels. The of process of a disassembly systems. Based paper, future research can beto onothers nonreferences focuson onofthis deterministic EOQ models to highlight the system is the end-of-life products at customer system is As theincollection collection ofsupply end-of-life products atmethod customer main specificities disassembly systems compared toon others locations. assembly systems, a right planning can recovery channels. The supply process of a disassembly systems. Based on this paper, future research can be nonsystem is As thein collection of end-of-life products atmethod customer deterministic lot-sizing as presented in [13]. EOQ model for main specificities of disassembly systems compared to others locations. assembly systems, a right planning can locations. As in assembly systems, a right planning method can systems. Based on this paper, future research can be on nonsave logistic costs and, in it can also improve system is As thein collection of disassembly, end-of-life products atmethod customer deterministic lot-sizing asbackorders, presented inshows [13]. that EOQit model for locations. assembly systems, a right planning can single item with planned can make systems. Based on this paper, future research can be on nonsave logistic costs and, disassembly, it improve save logistic costs and, in in disassembly, it can can also also improve deterministic lot-sizing presented inshows [13]. that EOQit model for product recovery rates. One of athe of can the single locations. As in assembly systems, rightcharacteristic planning method itemsense with plannedasbackorders, can save logistic costs and, in disassembly, it can also improve economic assuming that customers arethat willing to make wait. deterministic lot-sizing as presented in [13]. EOQ model for product recovery rates. One of the characteristic of the product recovery rates. One of the characteristic of the single item with planned backorders, shows it can make problems is that of characteristic one unitalso of improve product save logistic costs the and,disassembly inOne disassembly, it can economic sense assuming that customers are willing toshows wait. product recovery rates. of the of the Basic model can be found in [6] where the author also single item with planned backorders, shows that it can make problems is that the disassembly of one unit of product problems is that the disassembly of characteristic one unitIt implies of product economic sense that[6]customers are willing toshows wait. generates one or more units of each component. product rates. One of the of that the Basic canassuming bewith found wherethe thesolution author also problemsrecovery is that the disassembly of one unitIt of product that formodel thesense model fullin lost sales, is either to economic assuming that customers are willing toshows wait. generates one or more units of each component. implies that generates one or more units of each component. It implies that Basic model can be found in [6] where the author also the quantity of product disassembled is not necessarily exactly problems is that the units disassembly of one unitIt implies of product for the model with full lost sales, thePartial solution is either to generates one or more of each component. that that have no lost sales or to lose all demands. backordering Basic model can be found in [6] where the author also shows the quantity of product disassembled is not necessarily exactly the quantity of product disassembled is not necessarily exactly that for with full lost sales, thePartial solution is either to the number ofor demand for each component andItunnecessary generates one more units of each component. implies that have no the lostmodel sales or to lose all demands. backordering the quantity of product is not necessarily exactly models situations where customer arethe willing to wait if the that with full lost sales, solution is either to the number of demand for each component and unnecessary the number ofgenerated demanddisassembled for each component and unnecessary havefor no the lostmodel sales or to lose all demands. Partial backordering stock can be at each disassembly operation. The the quantity of product disassembled is not necessarily exactly models situations where customer are willing to wait if the the number ofgenerated demand for each component and unnecessary waiting timesales is short otherwise the sale is lost. Different have no lost or to lose all demands. Partial backordering stock can be at each disassembly operation. The stock can be generated at each disassembly operation. The models situations where customer are willing to wait if the planning method have for hence tocomponent be adaptedand to unnecessary handle The this waiting time is short otherwise the sale is lost. Different the number demand stock can method beofgenerated at each eachto customer impatience function can be taken into account [7]. A models where customer are willing to wait if the planning have be to this planning method have hence hence to disassembly be adapted adapted operation. to handle handle The this customer waiting situations time is short otherwise the sale is lost. Different characteristic. stock can be generated at each disassembly operation. impatience function can be taken into account [7]. A planning method have hence to be adapted to handle this review of EOQ models with partial backordering can be found waiting time is short otherwise the sale is lost. Different characteristic. characteristic. customer impatience function can bebackordering taken into account [7]. A planning method have hence to be adapted to handle this review EOQ with partial can found characteristic. in [8]. of In themodels casefunction of constant percentage of be demand customer impatience can bebackordering taken into account [7]. A Disassembly planning depends on the context (type of costs, review of EOQ models with partial can be found characteristic. in [8]. In the case of constant percentage of demand Disassembly planning depends on the context (type of costs, backordered over the length of the stockout interval, Pentico Disassembly planning depends on the context (type of costs, review of EOQ models with partial backordering can be found demands, product recovery process and quality of the product). in [8]. In the case of constant percentage of demand over the length of thewhich stockout interval, Pentico Disassembly planning depends on and the quality context (type of costs, backordered demands, product process the and Drake propose model simplifies demands, product recovery process quality of of the product). product). in [8]. In [9] the case ofa constant percentage of previous demand The first works inrecovery thisdepends field consider demands Disassembly planning on and the deterministic context (type of costs, and backordered over the length of the stockout interval, Pentico demands, product recovery process and quality of the product). Drake [9] propose a model which simplifies previous The first works in this field consider deterministic demands models by considering reorder and the percentage of The first works this field consider deterministic demands backordered over the length of interval thewhich stockout interval, Pentico which vary over in several periods ofand a finite horizon planning models demands, product recovery process quality of the product). and Drake [9] propose a model simplifies previous The first works in this field consider deterministic demands by considering reorder interval anddecision the percentage of which vary over periods of aa finite horizon planning demand that will be filled from stock as variables. which vary over inseveral several periods of finite horizon demands planning and Drake [9] propose a model which simplifies previous and apply a Material Requirement Planning algorithm [1]. The The first works this field consider deterministic models by considering reorder anddecision the percentage of that will be frominterval stock as variables. which varyaa Material over several periods of a finitealgorithm horizon planning and apply Requirement Planning [1]. The Partial canfilled be combined withas other to cope and apply Material Requirement Planning algorithm [1]. The demand modelsbackorders by considering reorder and thesettings percentage of problem can be extended by considering various typesplanning of costs which vary over several periods of a finite horizon demand that will be filled frominterval stock decision variables. and apply a Material Requirement Planning algorithm [1]. The Partial backorders can be combined with other settings to cope problem can be extended by various types of with some Taleizadeh etwith al.asother [10] combine partial problem can be extended by considering considering various types of costs costs demand thatreal willcases. be filled from stock decision variables. in particular fixed disassembly operation costs and inventory and apply a Material Requirement Planning algorithm [1]. The Partial backorders can be combined settings to cope problem can be extended by considering various types of costs backorders with some with real cases. Taleizadeh et al.The [10]objective combinefunction partial in particular fixed disassembly operation costs inventory special order and price. in particular fixed disassembly operation costs and and inventory backorders can be combined with other settings topartial cope costs as can in lot sizing problem [2]. A various model which problem be extended by considering types ofallows costs Partial with some real cases. Taleizadeh et al. [10] combine in particular fixed disassembly operation costs and inventory backorders with special order and price. The objective function costs as in lot sizing problem [2]. A model which allows is based on the difference between the cases with and without costs as in lot sizing problem [2]. A model which allows with some real cases. Taleizadeh et al. [10] combine partial stockout to lot decrease inventory surplus is proposed in [3]. backorders with special order and price. The objective function in particular fixed disassembly costs and inventory costs as in sizing problem operation [2]. A model which allows basedorder. on with theThe difference between the cases with and without stockout to decrease inventory surplus is in [3]. special solution method is based on the concavity stockout to lot decrease inventory surplus is proposed proposed in [3]. backorders special order and price. The objective function Disassembly economic order quantity (DEOQ) model apply to is costs as in sizing problem [2]. A model which allows is basedorder. on theThe difference between the cases with and without stockout to decrease inventory surplus is proposed in [3]. solution method is based on the concavity special Disassembly economic order quantity (DEOQ) model apply to of the objective function. Other variants related to the Disassembly economic orderand quantity (DEOQ) model apply toa is based on theThe difference between the casesare with and without deterministic, continuous constant over stockout to decrease inventory surplus is demand proposed in [3]. special order. solution method is based on the concavity Disassembly economic order quantity (DEOQ) model apply to the objective function. Other variants are related to the deterministic, continuous and constant demand over aa of timing of the payment of purchasing cost. Taleizadeh [9] deterministic, continuous and constant demand over special order. The solution method is based on the concavity planning horizon with the objective to balance order and Disassembly economic orderand quantity (DEOQ) model apply toa of the objective function. Other variants are related to the deterministic, continuous constant demand overand timing of the payment of purchasing cost. Taleizadeh [9] planning horizon with the objective to balance order proposes a model and its solution method for partial planning horizon with the objective to disposal balance order and of the objective function. Other variants are related to the inventory costs [4]. The authors consider decision to deterministic, continuous and constant demand over a timing of athemodel payment ofitspurchasing cost. Taleizadeh [9] planning horizon with the objective to balance order and proposes and solution method for partial inventory costs [4]. The authors consider disposal decision to backordering and advance payments for an evaporating item. inventory costs [4].surplus The the authors consider disposal to timing of athemodel payment cost. Taleizadeh [9] handle inventory inherent to to disassembly systems. planning horizon with objective balance decision order and proposes andofpayments itspurchasing solution method for partial inventory costs [4]. The authors consider disposal decision to backordering and advance for an evaporating item. handle inventory surplus inherent to disassembly systems. Another strategy consists of partial delayed payment [11]. handle inventory surplus inherent to can disassembly systems. proposes a model and payments its solution method for partial Fixed order cost for disposal operations also be considered inventory costs [4]. The authors consider disposal decision to backordering and advance for an evaporating item. handle inventory surplus inherent to can disassembly systems. Taleizadeh Another strategy consists of partial adelayed payment [11]. Fixed order cost disposal operations also considered etandal.advance [12] payments analyse model with partial Fixed order cost for disposal operations can also be bemodels considered for an evaporating item. to minimize the for overall cost [5]. The also backordering handle inventory surplus inherent to proposed disassembly systems. Another strategy consists of partial delayed payment [11]. Fixed order cost for disposal operations can also be considered Taleizadeh et al. [12] analyse a model with partial to minimize the overall cost [5]. models backordering incremental discount. In this case,partial two to minimize the for overall costoperations [5]. The The proposed models also strategy partial payment [11]. allows lost sales for the cases where theproposed disposal arealso too Another Fixed order cost disposal can also becosts considered Taleizadeh etand al.consists [12] of analyse adelayed model with to minimize the for overall cost where [5]. The proposed models also backordering and incremental discount. In this case, two allows lost sales the cases the disposal costs are too decision variables are used: the time cycle and the fraction of allows lost sales for the cases where the disposal costs are too Taleizadeh et al. [12] analyse a model with partial high and itsales canfor bethe more to disassemble less to minimize the overall costeconomical [5]. The proposed models also backordering and are incremental discount. In this case, two allows lost cases where the disposal costs are too decision variables used: the time cycle and the fraction of high and it can be more economical to disassemble less high and it can be more economical to disassemble less backordering and incremental discount. In this case, two allowsand lostitsales caseseconomical where the disposal costs are less too decision variables are used: the time cycle and the fraction of high canfor bethe more to disassemble decisionbyvariables are All used: thereserved. time cycle and the fraction of high and©it2019, canIFAC be more economical to disassemble less1702Hosting 2405-8963 IFAC (International Federation of Automatic Control) Elsevier Ltd. rights Copyright 2019 Copyright ©under 2019 responsibility IFAC 1702Control. Peer review© of International Federation of Automatic Copyright 2019 IFAC 1702 Copyright © 2019 IFAC 1702 10.1016/j.ifacol.2019.11.442 Copyright © 2019 IFAC 1702
2019 IFAC MIM 1682 Berlin, Germany, August 28-30, 2019 Matthieu Godichaud et al. / IFAC PapersOnLine 52-13 (2019) 1681–1686
items fill from stock which also implicitly set the optimal discount. Even if the model is quite complex (it requires a numerical solver), a sensitivity analysis based on several examples shows that the solutions are robust according to the estimation of input parameters. We extend these works to disassembly systems by taking into account their specificities. Three new models for DEOQ problems with stockout are proposed in this paper. Each model is analyzed to develop a solution method which find the optimal reorder interval and the percentage of demands which are fill from stock or backordered. Real life applications of DEOQ models are discussed in [4] and [5]. The proposed model in this paper shows that it can be can make economic sense to allow stockout in disassembly systems. Based on this model industrial managers can evaluate the opportunity of the stockout profit based on the data on their system. In section 2, basic setting, assumptions and notations of the DEOQ problem are presented. In section 3, a new model for DEOQ with full backordering is developed. An alternate, compared to [5], model for full lost sales is proposed in section 4. DEOQ problem with partial backordering is addressed in section 5. An illustrative example is proposed in section 6 and, finally, conclusion and perspectives are presented in section 7.
Fig. 1 presents the shape of the inventory over one cycle of a stationary policy. For one component, denoted by 𝑖𝑖, the slope of the inventory curve is equal to the demand per unit time (denoted by 𝑑𝑑𝑖𝑖 ). The quantity of each component received at the beginning of a cycle is equal to the quantity of product disassembled (denoted by 𝑄𝑄) times the yield of the component. There are two parts in each cycle for each inventory. In the first part the demands are served directly from stock and in the second parts they are backordered or lost.
2. ASSUMPTIONS AND NOTATIONS The models apply to two-level bill of materials with the first level representing the product to disassemble and the second level representing the components. The product contains a given number of each components, named yield and denoted by 𝑎𝑎𝑖𝑖 thereafter, and the disassembly of one product generates simultaneously all the components. The demands are for the components and they can be backorder under some conditions presented for each model hereafter. We consider the following assumptions:
Fig. 1. Component inventory with stockouts. The purpose of the model is to determine the cycle time (or the order quantity of product) and the percentage of demands for each components that will be served from stock to minimize the sum of setup, disassembly, inventory, stockout and disposal costs. The following notations are used:
each demand is characterized by a constant rate in units per unit time (per year for example),
planning horizon is considered as infinite,
the replenishments of component inventories (by disassembly operation) are instantaneous,
there is a fixed cost for each disassembly operation incurred whenever an order is placed,
there is an inventory holding cost for each part proportional to the inventory on-hand.
One of the characteristic of the disassembly systems is that the disassembly of one unit of product generates one or more units of each of its component. It implies that, for each components, the quantity of product disassembled is not necessarily exactly the quantity demanded between two orders and unnecessary stock can be generated at each disassembly operation. By considering stationary demands, these inventory surplus accumulate with time. Disposal decisions have to be taken into account to ensure the stability of the disassembly system and inventory surplus is disposed of as soon as it is receive when no disposal fixed cost are considered.
𝑖𝑖 = 1, … , 𝑁𝑁 are the index for the components,
𝑎𝑎𝑖𝑖 is the yield of component 𝑖𝑖 (number of component 𝑖𝑖 in the product),
𝑑𝑑𝑖𝑖 is the constant demand per unit time for component 𝑖𝑖,
ℎ𝑖𝑖 is the inventory cost per unit and per unit time of component 𝑖𝑖,
𝑏𝑏𝑖𝑖 is the backorder cost per unit and per unit time of component 𝑖𝑖, 𝑝𝑝𝑖𝑖 is the lost sales cost per unit of component 𝑖𝑖,
𝑟𝑟𝑖𝑖 disposal cost per unit of component 𝑖𝑖,
𝛽𝛽𝑖𝑖 , the fraction of stockouts of the item 𝑖𝑖 that will be backordered,
𝑘𝑘 is the disassembly setup cost or fixed ordering cost for the product,
𝑐𝑐 is the disassembly cost of the product,
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𝑄𝑄 is the order quantity (the received quantity of each component after disassembly is 𝑎𝑎𝑖𝑖 𝑄𝑄,
𝐹𝐹𝑖𝑖 is the percentage of demand for a component 𝑖𝑖 that will be filled from stock (fill rate),
𝑇𝑇 is the cycle time (time between two orders).
The total cost function in (1) is denoted by 𝐶𝐶(𝑇𝑇, 𝐹𝐹) where 𝑇𝑇 is the time between two orders and 𝐹𝐹 is a vector containing the variables 𝐹𝐹𝑖𝑖 for each component 𝑖𝑖 = 1 … 𝑁𝑁. Based on Fig.2 for a component 𝑖𝑖, 𝑑𝑑𝑖𝑖 𝑇𝑇𝐹𝐹𝑖𝑖2 ⁄2 is the mean inventory per unit time and 𝑑𝑑𝑖𝑖 𝑇𝑇(1 − 𝐹𝐹𝑖𝑖 )2 ⁄2 is the mean backorder per unit time. 𝑘𝑘
On Fig. 1, 𝑅𝑅𝑖𝑖 is an unnecessary surplus (due to disassembly characteristic) to dispose of for one cycle and 𝑅𝑅𝑖𝑖 = 𝑎𝑎𝑖𝑖 𝑄𝑄 − 𝑇𝑇(𝑑𝑑𝑖𝑖 𝐹𝐹𝑖𝑖 + 𝑑𝑑𝑖𝑖 𝛽𝛽𝑖𝑖 (1 − 𝐹𝐹𝑖𝑖 )) . 𝑆𝑆𝑖𝑖 is the total quantity of component 𝑖𝑖 stockout at the end of a cycle and 𝑆𝑆𝑖𝑖 = 𝑑𝑑𝑖𝑖 𝑇𝑇(1 − 𝐹𝐹𝑖𝑖 ) (based on the Fig. 1 directly). One part, equal to 𝑑𝑑𝑖𝑖 𝛽𝛽𝑖𝑖 𝑇𝑇(1 − 𝐹𝐹𝑖𝑖 ) of 𝑆𝑆𝑖𝑖 is backlogged and the other is lost according to the value of 𝛽𝛽𝑖𝑖 .
𝐶𝐶(𝑇𝑇, 𝐹𝐹) = + ∑𝑁𝑁 𝑖𝑖=1 [
The three models proposed in the following section vary according the value of the 𝛽𝛽𝑖𝑖 which reflect, as a data, the customer acceptance to wait. If 𝛽𝛽𝑖𝑖 = 1, all the demands are backordered whereas if 𝛽𝛽𝑖𝑖 = 0 are lost. Partial backorders are for 0 < 𝛽𝛽𝑖𝑖 < 1.
𝐹𝐹𝑖𝑖∗ =
3. DEOQ WITH FULL BACKORDERS The problem is to determine is to determine the reorder interval 𝑇𝑇 and the fill rate 𝐹𝐹𝑖𝑖 to minimise a mean total cost per unit of time in the case where 𝛽𝛽𝑖𝑖 = 1 for all 𝑖𝑖 = 1 … 𝑁𝑁. It means that all the demands are backordered and 𝑎𝑎𝑖𝑖 𝑄𝑄 = 𝑑𝑑𝑖𝑖 𝑇𝑇 for all components (see Fig. 2). During a cycle of length 𝑇𝑇, the inventory level for a component 𝑖𝑖 will be positive during 𝐹𝐹𝑖𝑖 𝑇𝑇 and negative (backorders) during (1 − 𝐹𝐹𝑖𝑖 )𝑇𝑇. If we note 𝐺𝐺 the component such as 𝑑𝑑𝐺𝐺 ⁄𝑎𝑎𝐺𝐺 = max𝑖𝑖 {𝑑𝑑𝑖𝑖 ⁄𝑎𝑎𝑖𝑖 } then we must have 𝑇𝑇 = 𝑎𝑎𝐺𝐺 𝑄𝑄⁄𝑑𝑑𝐺𝐺 otherwise some demands would be lost. For other components, an unnecessary inventory, equal to 𝑎𝑎𝑖𝑖 𝑄𝑄 − 𝑑𝑑𝑖𝑖 𝑇𝑇, is received. It is disposed and not kept in stock (since there is no fixed cost for disposal). The disassembly and the disposal costs per unit time are:
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𝑇𝑇
ℎ𝑖𝑖 𝑑𝑑𝑖𝑖 𝑇𝑇𝐹𝐹𝑖𝑖2 2
+
𝑏𝑏𝑖𝑖 𝑑𝑑𝑖𝑖 𝑇𝑇(1−𝐹𝐹𝑖𝑖 )2 2
]
(1)
The first derivative of 𝐶𝐶(𝑇𝑇, 𝐹𝐹) with respect to 𝐹𝐹𝑖𝑖 is: 𝜕𝜕𝜕𝜕(𝑇𝑇, 𝐹𝐹)⁄𝜕𝜕𝐹𝐹𝑖𝑖 = 𝑑𝑑𝑖𝑖 𝑇𝑇(ℎ𝑖𝑖 𝐹𝐹𝑖𝑖 − 𝑏𝑏𝑖𝑖 (1 − 𝐹𝐹𝑖𝑖 )),
and 𝜕𝜕𝜕𝜕(𝑇𝑇, 𝐹𝐹)⁄𝜕𝜕𝐹𝐹𝑖𝑖 = 0 gives the optimal value of 𝐹𝐹𝑖𝑖 in (2). 𝑏𝑏𝑖𝑖
(2)
𝑏𝑏𝑖𝑖 +ℎ𝑖𝑖
𝐹𝐹𝑖𝑖∗ is independent of 𝑇𝑇 and 𝐹𝐹𝑖𝑖∗ replace 𝐹𝐹𝑖𝑖 in (1). The first derivative of 𝐶𝐶(𝑇𝑇, 𝐹𝐹) with respect to 𝑇𝑇 by setting 𝐹𝐹𝑖𝑖 = 𝐹𝐹𝑖𝑖∗ gives: ∗ 2 𝜕𝜕𝜕𝜕(𝑇𝑇, 𝐹𝐹)⁄𝜕𝜕𝜕𝜕 = −𝑘𝑘 ⁄𝑇𝑇 2 + ∑𝑁𝑁 𝑖𝑖=1[ℎ𝑖𝑖 𝑑𝑑𝑖𝑖 (𝐹𝐹𝑖𝑖 ) ⁄2 + ∗ 2 𝑏𝑏𝑖𝑖 𝑑𝑑𝑖𝑖 𝑇𝑇(1 − 𝐹𝐹𝑖𝑖 ) ⁄2],
and 𝐶𝐶(𝑇𝑇, 𝐹𝐹)⁄𝜕𝜕𝜕𝜕 = 0 gives the optimal value of 𝑇𝑇 in (3). 𝜕𝜕 2 𝐶𝐶(𝑇𝑇, 𝐹𝐹)⁄(𝜕𝜕𝜕𝜕)2 = 2𝑘𝑘 ⁄𝑇𝑇 3 > 0 the cost function is convex with respect to 𝑇𝑇 (> 0) when the 𝐹𝐹𝑖𝑖 are fixed. 𝑇𝑇 ∗ = √∑𝑁𝑁
2𝑘𝑘
𝑖𝑖=1𝑑𝑑𝑖𝑖 ℎ𝑖𝑖 𝑏𝑏𝑖𝑖 ⁄(ℎ𝑖𝑖 +𝑏𝑏𝑖𝑖 )
(3)
Equations (2) and (3) provide directly, without using a search method, optimal values for the decision variables of the DEOQ problem with full backordering.
𝑐𝑐(𝑑𝑑𝐺𝐺 ⁄𝑎𝑎𝐺𝐺 ) and ∑𝑁𝑁 𝑖𝑖=1[𝑟𝑟𝑖𝑖 (𝑎𝑎𝑖𝑖 (𝑑𝑑𝐺𝐺 ⁄𝑎𝑎𝐺𝐺 ) − 𝑑𝑑𝑖𝑖 )],
which are independent of 𝑇𝑇 and 𝐹𝐹𝑖𝑖 .
Fig. 3. Component inventory with full lost sales.
Fig. 2. Component inventory with full backorders.
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4. DEOQ WITH FULL LOST SALES The problem with full lost sales is for 𝛽𝛽𝑖𝑖 = 0 for all 𝑖𝑖 = 1 … 𝑁𝑁. Fig. 3 illustrates the shape of the inventory of one component during one cycle. It means that the customer do not want to wait and all the demands are lost during the stockout phase of the cycle. Compared to the model in [5], the addition of the variables 𝐹𝐹𝑖𝑖 allows having both lost sales and disposal for some component inventories during one cycle. It means it could be more economical to dispose of some units at the beginning of a cycle (and lose some demands on the end of the cycle) instead of keeping them in stock to serve demands. However, by analysing the related cost function in this section, we retrieve the results in [5] where, for a given component inventory, there are either lost sales or disposals. Compared to the previous model in section 3, both 𝑇𝑇 and 𝑄𝑄 have to be considered as decision variables since we can have 𝑎𝑎𝑖𝑖 𝑄𝑄 < 𝑑𝑑𝑖𝑖 𝑇𝑇 for the component inventories with lost sales. We use 𝑇𝑇 and 𝑋𝑋 = 𝑄𝑄 ⁄𝑇𝑇 as decision variables to ease the analysis. The mean total cost per unit time in (4) is then a function of 𝑇𝑇, 𝑋𝑋 and F, the vector containing the 𝐹𝐹𝑖𝑖 s. 𝑘𝑘
𝐶𝐶(𝑇𝑇, 𝑋𝑋, 𝐹𝐹) = + 𝑐𝑐𝑐𝑐 + ∑𝑁𝑁 𝑖𝑖=1 [ 𝑇𝑇
𝑎𝑎
ℎ𝑖𝑖 𝑑𝑑𝑖𝑖 𝑇𝑇𝐹𝐹𝑖𝑖2 2
+ 𝑝𝑝𝑖𝑖 𝑑𝑑𝑖𝑖 (1 − 𝐹𝐹𝑖𝑖 ) +
𝑟𝑟𝑖𝑖 𝑑𝑑𝑖𝑖 (( 𝑖𝑖 ) 𝑋𝑋 − 𝐹𝐹𝑖𝑖 )]
(4)
𝑑𝑑𝑖𝑖
We retrieve in (4) the mean order cost, 𝑘𝑘⁄𝑇𝑇 , and the mean inventory cost for each component, ℎ𝑖𝑖 𝑑𝑑𝑖𝑖 𝑇𝑇𝐹𝐹𝑖𝑖2 ⁄2. Compared to (1), the disassembly cost, 𝑐𝑐𝑐𝑐, and the disposal costs, 𝑟𝑟𝑖𝑖 𝑑𝑑𝑖𝑖 ((𝑎𝑎𝑖𝑖 ⁄𝑑𝑑𝑖𝑖 )𝑋𝑋 − 𝐹𝐹𝑖𝑖 ), are not constant. The mean lost sales cost per unit time, 𝑝𝑝𝑖𝑖 𝑑𝑑𝑖𝑖 (1 − 𝐹𝐹𝑖𝑖 ), correspond to the stockout phase of one cycle (see Fig. 3). The variables 𝑋𝑋 and 𝐹𝐹𝑖𝑖 are here subject to several constraints. For a given item 𝑖𝑖:
if 𝑋𝑋 ≥ 𝑑𝑑𝑖𝑖 ⁄𝑎𝑎𝑖𝑖 , the item may not have lost sales and 𝐹𝐹𝑖𝑖 ≤ 1,
if 𝑋𝑋 < 𝑑𝑑𝑖𝑖 ⁄𝑎𝑎𝑖𝑖 , the item has at least some lost sales and 𝐹𝐹𝑖𝑖 ≤ 𝑎𝑎𝑖𝑖 𝑋𝑋.
𝑘𝑘
𝐶𝐶(𝑇𝑇, 𝑋𝑋) = + 𝑐𝑐𝑐𝑐 + ∑𝑁𝑁 𝑖𝑖=1 [
𝑟𝑟𝑖𝑖 𝑑𝑑𝑖𝑖 (𝑝𝑝𝑖𝑖 +𝑟𝑟𝑖𝑖 ) 𝑇𝑇ℎ𝑖𝑖
𝑇𝑇
𝑑𝑑𝑖𝑖 (𝑝𝑝𝑖𝑖 +𝑟𝑟𝑖𝑖 )2 2𝑇𝑇ℎ𝑖𝑖
] + ∑𝑁𝑁 𝑖𝑖=1[𝑝𝑝𝑖𝑖 𝑑𝑑𝑖𝑖 + 𝑟𝑟𝑖𝑖 𝑎𝑎𝑖𝑖 𝑋𝑋]
By denoting:
−
𝑝𝑝𝑖𝑖 𝑑𝑑𝑖𝑖 (𝑝𝑝𝑖𝑖 +𝑟𝑟𝑖𝑖 ) 𝑇𝑇ℎ𝑖𝑖
− (6)
2
𝐴𝐴𝑖𝑖 = 𝑑𝑑𝑖𝑖 (𝑝𝑝𝑖𝑖 + 𝑟𝑟𝑖𝑖 ) ⁄(2𝑇𝑇ℎ𝑖𝑖 ) − 𝑝𝑝𝑖𝑖 𝑑𝑑𝑖𝑖 (𝑝𝑝𝑖𝑖 + 𝑟𝑟𝑖𝑖 )⁄(𝑇𝑇ℎ𝑖𝑖 ) −
𝑟𝑟𝑖𝑖 𝑑𝑑𝑖𝑖 (𝑝𝑝𝑖𝑖 + 𝑟𝑟𝑖𝑖 )⁄(𝑇𝑇ℎ𝑖𝑖 ),
2
and by developing it, it reduces to 𝐴𝐴𝑖𝑖 = − 𝑑𝑑𝑖𝑖 (𝑝𝑝𝑖𝑖 ) ⁄(2𝑇𝑇ℎ𝑖𝑖 ) − 𝑑𝑑𝑖𝑖 (𝑟𝑟𝑖𝑖 )2 ⁄(2𝑇𝑇ℎ𝑖𝑖 ) − 𝑑𝑑𝑖𝑖 𝑝𝑝𝑖𝑖 𝑟𝑟𝑖𝑖 ⁄(2𝑇𝑇ℎ𝑖𝑖 ) which is negative. The cost function 𝐶𝐶(𝑇𝑇, 𝑋𝑋) is of the form: 𝐶𝐶(𝑇𝑇, 𝑋𝑋) = 𝐴𝐴⁄𝑇𝑇 + 𝐵𝐵𝐵𝐵 + 𝐶𝐶 𝑁𝑁 𝑁𝑁 with 𝐴𝐴 = 𝑘𝑘 + ∑𝑁𝑁 𝑖𝑖=1 𝐴𝐴𝑖𝑖 , 𝐵𝐵 = 𝑐𝑐 + ∑𝑖𝑖=1 𝑟𝑟𝑖𝑖 𝑎𝑎𝑖𝑖 and 𝐶𝐶 = ∑𝑖𝑖=1 𝑝𝑝𝑖𝑖 𝑑𝑑𝑖𝑖 . The analysis of 𝐶𝐶(𝑇𝑇, 𝑋𝑋) is this form leads to the same result as in [5]. 𝑋𝑋 has to be as small as possible but it is subject to the constraints 𝑋𝑋 ≥ (𝑑𝑑𝑖𝑖 ⁄𝑎𝑎𝑖𝑖 )𝐹𝐹𝑖𝑖 for all 𝑖𝑖 = 1 … 𝑁𝑁. The best value of 𝑇𝑇 depends on the sign of 𝐴𝐴. If 𝐴𝐴 ≥ 0, 𝑇𝑇 has to be as large as possible and it means that all the demands are lost. If 𝐴𝐴 < 0, the value of 𝑇𝑇 has to be as small as possible which is reached for 𝐹𝐹𝑖𝑖 = 𝑚𝑚𝑚𝑚𝑚𝑚{1, (𝑎𝑎𝑖𝑖 ⁄𝑑𝑑𝑖𝑖 )𝑋𝑋}. It is the same to assume that an inventory cannot have both disposal and lost sales as in [5] and the remaining of the analysis is the same. The solution method consists of breaking pieces the cost function and searching for optimal values of 𝑋𝑋 and 𝑇𝑇 for each pieces. In each piece, the set of components is divided into two subsets: one with components which have lost sales and the other with components which have inventory surplus. 4. DEOQ WITH PARTIAL BACKORDERS Partial backordering occurring when 𝛽𝛽𝑖𝑖 , the fraction of stockouts of the item 𝑖𝑖 that will be backordered, is 0 < 𝛽𝛽𝑖𝑖 < 1. Based on Fig.4, 𝑇𝑇[𝑑𝑑𝑖𝑖 (𝐹𝐹𝑖𝑖 + 𝛽𝛽𝑖𝑖 (1 − 𝐹𝐹𝑖𝑖 ))] is the quantity requested to serve the backorders and the demands of component 𝑖𝑖 for one cycle of length 𝑇𝑇. The values of variables 𝐹𝐹𝑖𝑖 are feasible if 𝑄𝑄 ≥ 𝑇𝑇[(𝑑𝑑𝑖𝑖 ⁄𝑎𝑎𝑖𝑖 )(𝐹𝐹𝑖𝑖 + 𝛽𝛽𝑖𝑖 (1 − 𝐹𝐹𝑖𝑖 ))] for all 𝑖𝑖 = 1 … 𝑁𝑁. To set a replenishment policy, we use the variables 𝑇𝑇, 𝑋𝑋 (= 𝑄𝑄 ⁄𝑇𝑇) and 𝐹𝐹𝑖𝑖 .
𝑑𝑑𝑖𝑖
The problem can be then stated as follow: Min 𝐶𝐶(𝑇𝑇, 𝑋𝑋, 𝐹𝐹) subject to 0 ≤ 𝐹𝐹𝑖𝑖 ≤ 𝑚𝑚𝑚𝑚𝑚𝑚 {1, 𝑋𝑋 ≥ max { 𝑖𝑖=1…𝑁𝑁
𝑑𝑑𝑖𝑖 𝐹𝐹𝑖𝑖 𝑎𝑎𝑖𝑖
}
𝑎𝑎𝑖𝑖
𝑑𝑑𝑖𝑖
𝑋𝑋} ∀𝑖𝑖 = 1 … 𝑁𝑁 and (5)
The analysis can be made for 𝐶𝐶(𝑇𝑇, 𝑋𝑋, 𝐹𝐹) starting from 𝐹𝐹𝑖𝑖 with 𝑇𝑇 and 𝑋𝑋 considered as fixed. It gives 𝐹𝐹∗𝑖𝑖 (𝑇𝑇) the optimal value of 𝐹𝐹𝑖𝑖 when 𝑋𝑋 and 𝑇𝑇 are fixed (which is just a function of 𝑇𝑇):
𝐹𝐹∗𝑖𝑖 (𝑇𝑇) =
𝑝𝑝𝑖𝑖 +𝑟𝑟𝑖𝑖 𝑇𝑇ℎ𝑖𝑖
.
By replacing 𝐹𝐹𝑖𝑖 by 𝐹𝐹∗𝑖𝑖 (𝑇𝑇) in 𝐶𝐶(𝑇𝑇, 𝑋𝑋, 𝐹𝐹), we obtain 𝐶𝐶(𝑇𝑇, 𝑋𝑋) in (6).
Fig. 4. Component inventory with partial backorders.
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with 𝐶𝐶𝑖𝑖 = 𝑝𝑝𝑖𝑖 + 𝑟𝑟𝑖𝑖 − (𝑐𝑐 + ∑𝑁𝑁 𝑘𝑘=1 𝑟𝑟𝑘𝑘 𝑎𝑎𝑘𝑘 )⁄𝑎𝑎𝑖𝑖 if 𝑖𝑖 = 𝑞𝑞 and 𝐶𝐶𝑖𝑖 = 𝑝𝑝𝑖𝑖 + 𝑟𝑟𝑖𝑖 otherwise.
The resulting mean total cost per unit of time is given in (7). 𝑘𝑘
𝐶𝐶(𝑇𝑇, 𝑋𝑋, 𝐹𝐹) = + 𝑐𝑐𝑐𝑐 + ∑𝑁𝑁 𝑖𝑖=1 [ 𝑇𝑇
𝑝𝑝𝑖𝑖 𝑑𝑑𝑖𝑖 (1 − 𝛽𝛽𝑖𝑖 )(1 − 𝐹𝐹𝑖𝑖 )] + 𝐹𝐹𝑖𝑖 )}]
ℎ𝑖𝑖 𝑑𝑑𝑖𝑖 𝑇𝑇𝐹𝐹𝑖𝑖2 2
+
𝑏𝑏𝑖𝑖 𝑑𝑑𝑖𝑖 𝑇𝑇𝛽𝛽𝑖𝑖 (1−𝐹𝐹𝑖𝑖 )2 2
∑𝑁𝑁 𝑖𝑖=1[𝑟𝑟𝑖𝑖 𝑑𝑑𝑖𝑖 {(𝑎𝑎𝑖𝑖 ⁄𝑑𝑑𝑖𝑖 )𝑋𝑋
+
The values obtained from (11) and (12) are optimal for the subproblem of minimising 𝐶𝐶𝑞𝑞 (𝑇𝑇, 𝐹𝐹) if the constraints (9) and
− 𝐹𝐹𝑖𝑖 + 𝛽𝛽𝑖𝑖 (1 −
(7)
We have in (7) the mean order cost, the disassembly cost, the mean inventory, backorder and lost sales costs for each component from the previous models, in sections 3 and 4. The disposal quantity at each cycle is 𝑎𝑎𝑖𝑖 𝑋𝑋 − 𝑑𝑑𝑖𝑖 (𝐹𝐹𝑖𝑖 + 𝛽𝛽𝑖𝑖 (1 − 𝐹𝐹𝑖𝑖 )) which gives the last parts of (7) corresponding to the mean diposal costs. One notes, by developing this last part and eliminating the constant term, minimising (7) is equivalent to minimise (8). 𝑘𝑘
𝐶𝐶(𝑇𝑇, 𝑋𝑋, 𝐹𝐹) = + (𝑐𝑐 + 𝑏𝑏𝑖𝑖 𝑑𝑑𝑖𝑖 𝑇𝑇𝛽𝛽𝑖𝑖 (1−𝐹𝐹𝑖𝑖 )2 2
𝑇𝑇
∑𝑁𝑁 𝑖𝑖=1 𝑟𝑟𝑖𝑖 𝑎𝑎𝑖𝑖 )𝑋𝑋
+
ℎ𝑖𝑖 𝑑𝑑𝑖𝑖 𝑇𝑇𝐹𝐹𝑖𝑖2 ∑𝑁𝑁 𝑖𝑖=1 [ 2
+ (𝑝𝑝𝑖𝑖 + 𝑟𝑟𝑖𝑖 )𝑑𝑑𝑖𝑖 (1 − 𝛽𝛽𝑖𝑖 )(1 − 𝐹𝐹𝑖𝑖 )]
+ (8)
Minimising (8) is subject to the constraints (9) and (10). Constraints (9) set the fill rates bounds (1 for no stockout and 0 for all stockout). Constraints (10) force to disassembled the quantity of product to serve the all demands required with respect to the fill rates. 0 ≤ 𝐹𝐹𝑖𝑖 ≤ 1, ∀𝑖𝑖 = 1 … 𝑁𝑁
(9)
X ≥ (𝑑𝑑𝑖𝑖 ⁄𝑎𝑎𝑖𝑖 )(𝐹𝐹𝑖𝑖 + 𝛽𝛽𝑖𝑖 (1 − 𝐹𝐹𝑖𝑖 )), ∀𝑖𝑖 = 1 … 𝑁𝑁
(10)
As in the model with full lost sales, we have 𝜕𝜕𝐶𝐶(𝑇𝑇, 𝑋𝑋, 𝐹𝐹)⁄𝜕𝜕𝜕𝜕 = 𝑐𝑐 + ∑𝑁𝑁 𝑖𝑖=1 𝑟𝑟𝑖𝑖 𝑎𝑎𝑖𝑖 > 0 and then 𝑋𝑋 must be then chosen as small as possible subject to the constraints (10) and then we have 𝑋𝑋 = max{ (𝑑𝑑𝑖𝑖 ⁄𝑎𝑎𝑖𝑖 )(𝐹𝐹𝑖𝑖 + 𝛽𝛽𝑖𝑖 (1 − 𝐹𝐹𝑖𝑖 ))}. 𝑖𝑖
When the component 𝑞𝑞 = argmax{(𝑑𝑑𝑘𝑘 ⁄𝑎𝑎𝑘𝑘 )(𝐹𝐹𝑘𝑘 + 𝛽𝛽𝑘𝑘 (1 − 𝑘𝑘
𝐹𝐹𝑘𝑘 ))} is known, 𝑋𝑋 is replaced by (𝑑𝑑𝑞𝑞 ⁄𝑎𝑎𝑞𝑞 ) (𝐹𝐹𝑞𝑞 + 𝛽𝛽𝑞𝑞 (1 − 𝐹𝐹𝑞𝑞 )) in 𝐶𝐶(𝑇𝑇, 𝑋𝑋, 𝐹𝐹) which becomes a function of 𝑇𝑇 and 𝐹𝐹 only, denoted by 𝐶𝐶𝑞𝑞 (𝑇𝑇, 𝐹𝐹). The solution method presented in this paper consists in solving 𝑁𝑁 sub-problem 𝐶𝐶𝑞𝑞 (𝑇𝑇, 𝐹𝐹) for 𝑞𝑞 = 1 … 𝑁𝑁 and select the one with the minimal cost. The stationary points of 𝐶𝐶𝑞𝑞 (𝑇𝑇, 𝐹𝐹) are given by taking the first partial derivatives with respect to 𝑇𝑇 and 𝐹𝐹𝑖𝑖 for 𝑖𝑖 = 1 … 𝑁𝑁. Taking the partial derivative of 𝐶𝐶𝑞𝑞 (𝑇𝑇, 𝐹𝐹) with respect to 𝑇𝑇 and setting it equal to 0 gives 𝑇𝑇(𝐹𝐹), the optimal value of 𝑇𝑇 when 𝐹𝐹 is fixed, in (11). 𝑇𝑇(𝐹𝐹) = √∑𝑁𝑁
2𝑘𝑘
2 2 𝑖𝑖=1 𝑑𝑑𝑖𝑖 [ℎ𝑖𝑖 𝐹𝐹𝑖𝑖 −𝑏𝑏𝑖𝑖 𝛽𝛽𝑖𝑖 (1−𝐹𝐹𝑖𝑖 ) ]
(11)
Taking the partial derivative of 𝐶𝐶𝑞𝑞 (𝑇𝑇, 𝐹𝐹) with respect to with respect to 𝐹𝐹𝑖𝑖 and setting it equal to 0 gives 𝐹𝐹𝑖𝑖 (𝑇𝑇) in (12). We note that 𝐹𝐹𝑖𝑖 (𝑇𝑇), the optimal value of 𝐹𝐹𝑖𝑖 when 𝑇𝑇 is fixed, depend only on 𝑇𝑇 but not on the other 𝐹𝐹𝑗𝑗 , 𝑗𝑗 = 1 … 𝑁𝑁 and 𝑗𝑗 ≠ 𝑖𝑖. 𝐹𝐹𝑖𝑖 =
𝐶𝐶𝑖𝑖 (1−𝛽𝛽𝑖𝑖 )+𝑏𝑏𝑖𝑖 𝛽𝛽𝑖𝑖 𝑇𝑇 𝑇𝑇(ℎ𝑖𝑖 +𝑏𝑏𝑖𝑖 𝛽𝛽𝑖𝑖 )
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(12)
(𝑑𝑑𝑞𝑞 ⁄𝑎𝑎𝑞𝑞 ) (𝐹𝐹𝑞𝑞 + 𝛽𝛽𝑞𝑞 (1 − 𝐹𝐹𝑞𝑞 )) ≥ (𝑑𝑑𝑖𝑖 ⁄𝑎𝑎𝑖𝑖 )(𝐹𝐹𝑖𝑖 + 𝛽𝛽𝑖𝑖 (1 − 𝐹𝐹𝑖𝑖 )),
∀𝑖𝑖 = 1 … 𝑁𝑁, 𝑖𝑖 ≠ 𝑞𝑞
(13)
are satisfied. Given these constraint, we propose the following iterative method to find the best value of the decision variables. Step 1 is the same for all sub-problem while Step 2 and 3 are repeated for each sub-problem for 𝑞𝑞 = 1 … 𝑁𝑁. (0)
Step 1 (Initialisation). Set 𝐹𝐹𝑖𝑖 = 1 for 𝑖𝑖 = 1 … 𝑁𝑁, 𝑋𝑋 (0) = max{ (𝑑𝑑𝑖𝑖 ⁄𝑎𝑎𝑖𝑖 )(𝐹𝐹𝑖𝑖 + 𝛽𝛽𝑖𝑖 (1 − 𝐹𝐹𝑖𝑖 ))}, 𝑇𝑇 (0) = √2𝑘𝑘 ⁄∑𝑁𝑁 𝑖𝑖=1 𝑑𝑑𝑖𝑖 ℎ𝑖𝑖 and 𝑖𝑖
(0) the total cost is 𝐶𝐶 (0) = √2𝑘𝑘 ∑𝑁𝑁 + 𝑖𝑖=1 𝑑𝑑𝑖𝑖 ℎ𝑖𝑖 + 𝑐𝑐𝑋𝑋 (0) ∑𝑁𝑁 − 𝑑𝑑𝑖𝑖 )] (it is the problem without stockout, see 𝑖𝑖=1[𝑟𝑟𝑖𝑖 (𝑎𝑎𝑖𝑖 𝑋𝑋 [4] for details). 𝑢𝑢 ← 0. (𝑢𝑢)
Step 2. 𝑢𝑢 ← 𝑢𝑢 + 1. Set, for 𝑖𝑖 = 1 … 𝑁𝑁, 𝐹𝐹𝑖𝑖
=
𝑚𝑚𝑚𝑚𝑚𝑚 {1, (𝐶𝐶𝑖𝑖 (1 − 𝛽𝛽𝑖𝑖 ) + 𝑏𝑏𝑖𝑖 𝛽𝛽𝑖𝑖 𝑇𝑇 ) ⁄( + 𝑏𝑏𝑖𝑖 𝛽𝛽𝑖𝑖 ))}. If the constraints (13) are satisfied go to Step 3 otherwise go to another sub-problem by changing 𝑞𝑞. (𝑢𝑢−1
Step 3. Set 𝑇𝑇 (𝑢𝑢) =
2
𝑇𝑇 (𝑢𝑢−1) (ℎ𝑖𝑖
2
(𝑢𝑢) (𝑢𝑢) √2𝑘𝑘 ⁄∑𝑁𝑁 ) − 𝑏𝑏𝑖𝑖 𝛽𝛽𝑖𝑖 (1 − 𝐹𝐹𝑖𝑖 ) ] 𝑖𝑖=1 𝑑𝑑𝑖𝑖 [ℎ𝑖𝑖 (𝐹𝐹𝑖𝑖
(𝑢𝑢)
and 𝐶𝐶 (𝑢𝑢) = 𝐶𝐶𝑞𝑞 (𝑇𝑇 (𝑢𝑢) , 𝐹𝐹 (𝑢𝑢) ), (𝐹𝐹 (𝑢𝑢) is the set of variable 𝐹𝐹𝑖𝑖 at iteration 𝑢𝑢. If 𝐶𝐶 (𝑢𝑢) − 𝐶𝐶 (𝑢𝑢−1) > 𝜀𝜀 then go to Step 2, else the best values for 𝑇𝑇 and 𝐹𝐹𝑖𝑖 , 𝑖𝑖 = 1 … 𝑁𝑁, has been found. We note there is always a solution at Step 1 (the solution without stockout is a solution of the problem). At Step 2, if an inequality of (13) is not satisfied, the decision variable must be changed to obtain an equality which is the same as changing the sub-problem (i.e. changing 𝑞𝑞). In Step 3, 𝜀𝜀 must be chosen according to the desired precision and the computing time. 5. ILLUSTRATIVE EXAMPLE An illustrative example for a product with five components, 𝑁𝑁 = 5, is proposed in this section to facilitate the understanding of the model and method developed for the DEOQ with partial backorder (section 4). The setup cost is 𝑘𝑘 = 5 and the disassembly cost is 𝑐𝑐 = 1. The other data for each component 𝑖𝑖 = 1 … 𝑁𝑁 are presented in Table 1.
At the Step 1, the solution without stockout (for 𝐹𝐹𝑖𝑖 = 1, 𝑖𝑖 = 1 … 5) has a total cost 𝐶𝐶 (0) = 731.14 with 𝑋𝑋 (0) = 20.67 and 𝑇𝑇 (0) = 1.58. At Step 2 for 𝑞𝑞 = 1, applying (12) with 𝑇𝑇 = 𝑇𝑇 (0) gives 𝐹𝐹1 = 1.12, 𝐹𝐹2 = 0.68, 𝐹𝐹3 = 1.84, 𝐹𝐹4 = 0.82, 𝐹𝐹5 = 1.28. The values superior to 1 are corrected to 1. This leads to the violation of constraint (13) for 𝑖𝑖 = 3,4,5 and the values of 𝐹𝐹𝑖𝑖 must be
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change until the constraints are satisfied. It leads to obtain equalities for constraint (13) which is equivalent to change 𝑞𝑞 and solve another sub-problem.
Table 1. Component data for the example 𝑖𝑖
𝑎𝑎𝑖𝑖
𝑑𝑑𝑖𝑖
ℎ𝑖𝑖
𝑝𝑝𝑖𝑖
𝑟𝑟𝑖𝑖
𝑏𝑏𝑖𝑖
𝛽𝛽𝑖𝑖
1 2 3
4 5 3
26 33 62
2.47 2.32 2.08
13.11 7.24 13.68
0.99 0.19 0.99
2 2.48 1.91
0.6 0.8 0.5
4 5
3 3
53 43
1.67 0.98
7.78 12.52
0.22 0.33
2.47 1.65
0.8 0.8
Fig. 5. Evolution of the cost with respect to the iterations.
[4] M. Godichaud and L. Amodeo, Economic order quantity for multistage disassembly systems, International Journal of Production Economics, vol. 199, pp. 16-25, 2018. [5] M. Godichaud and L. Amodeo, EOQ inventory models for disassembly systems with disposal and lost sales, International Journal of Production Research, 2018. [6] P. Zipkin, Foundations of inventory management. New York: McGraw-Hill, 2000. [7] P.L. Abad, Optimal Pricing and Lot-Sizing under Conditions of Perishability and Partial Backordering, Management Science, vol. 42, no. 8, pp. 1093-110, 1996. [8] D.W. Pentico and M.J. Drake, A survey of deterministic models for the EOQ and EPQ with partial backordering, European Journal of Operational Research, vol. 214, pp. 179-198, 2011. [9] A.A. Taleizadeh, An EOQ Model with Partial Backordering and Advance Payments for an Evaporating Item, International Journal of Production Economics, vol. 155, pp. 185–193, 2014. [10] A.A. Taleizadeh, D. W. Pentico, M. Aryanezhad and S. M. Ghoreyshi, An Economic Order Quantity Model with Partial Backordering and a Special Sale Price, European Journal of Operational Research vol. 221, no.3, pp. 571– 583, 2012. [11] A.A. Taleizadeh, D.W. Pentico, M. S. Jabalameli, and M. Aryanezhad,An EOQ Model with Partial Delayed Payment and Partial Backordering, Omega, vol. 41, no. 2, pp. 354–368, 2013. [12] A.A. Taleizadeh, I. Stojkovska, and D. W. Pentico, “An Economic Order Quantity Model with Partial Backordering and Incremental Discount.” Computers & Industrial Engineering vol. 82, pp. 21–32, 2015. [13] M.A. Aloulou, A. Dolgui and M.Y. Kovalyov A bibliography of non-deterministic lot-sizing models Mohamed Ali Aloulou, Alexandre Dolgui Mikhail Y. Kovalyov, vol. 52, no. 8, pp. 2293-2310, 2014.
6. CONCLUSIONS Disassembly systems necessitates new models for their inventory management due to their specificities. Three problems with stockouts (full backorders, full lost sales and partial backorders) have been addressed in this paper. The models are non-linear optimization model with constraints. Their analysis shows that closed-form or fast methods can be derived for each of them. Their application allows to investigating the economic advantages of stockouts for disassembly management. One extension of this work is to develop a sensitive analysis to get additional managerial implications. Future research can also the integration of environmental constraints or objective, with specifically the impact of stockout on them, which is an important motivation to perform disassembly activities. REFERENCES [1] S.M. Gupta and K.N. Taleb, Scheduling disassembly, International Journal of Production Research, vol. 32, no. 8, pp.1857-1866, 1994. [2] H.-J. Kim, D.-H. Lee and P. Xirouchakis, Disassembly scheduling: literature review and future research directions, International Journal of Production Research, vol. 45, no. 18-19, pp.4465-4484, 2007. [3] M. Hrouga, M. Godichaud and L. Amodeo, Heuristics for multi-product capacitated disassembly lot sizing with lost sales, IFAC-PapersOnLine, vol. 49, no. 12, 2016, pp. 628633.
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IFAC PapersOnLine 52-13 (2019) 1681–1686
EOQ models with stockouts for disassembly systems EOQ models with stockouts for disassembly systems EOQ models with stockouts for disassembly systems Matthieu Godichaud*. Lionel Amodeo.* systems EOQ models with stockouts for disassembly Matthieu Godichaud*. Lionel Amodeo.*
Matthieu Godichaud*. Lionel Amodeo.* Matthieu Godichaud*. deLionel * Université de Technologie Troyes,Amodeo.* Troyes, France Matthieu Godichaud*. Lionel ** Université Université de Technologie de Troyes,Amodeo.* Troyes, France France de Technologielionel.amodeo}@utt.fr. de Troyes, Troyes, {matthieu.godichaud, de Troyes, Troyes, France * Université de Technologie {matthieu.godichaud, lionel.amodeo}@utt.fr. {matthieu.godichaud, lionel.amodeo}@utt.fr. * Université de Technologielionel.amodeo}@utt.fr. de Troyes, Troyes, France {matthieu.godichaud, {matthieu.godichaud, lionel.amodeo}@utt.fr. Abstract: Although economic order quantity (EOQ) problem is a widespread problem with many variants, Abstract: Although economic order quantity quantity (EOQ)which problem is aanew widespread problem with many variants, variants, Abstract: Although economic order (EOQ) problem is widespread with many few of them are dedicated to disassembly systems brings issues inproblem inventory management due Abstract: Although economic order quantity (EOQ)which problem is anew widespread problem with many variants, few of them are dedicated to disassembly systems which brings new issues in inventory management due few of them are dedicated to disassembly systems brings issues in inventory management due to their specificities. EOQ problem for disassembly systems consists in determining reorder time interval Abstract: Although economic order quantity (EOQ) problem is a widespread problem with many variants, few of them are dedicated to disassembly systems which brings new issues in inventory management due to their specificities. EOQ problem for disassembly systems consists in determining reorder time interval to their specificities. EOQ problem for disassembly systems consists in determining reorder time interval and/or quantity of end-of-life products which which must be disassembled to obtain their component. Three few of order them are dedicated to disassembly systems brings new issues in inventory management due to their specificities. EOQ problem for disassembly systems consists in determining reorder time interval and/ormodels order quantity quantity of end-of-life end-of-life products which must bestockout. disassembled to obtain obtain their their component.as Three and/or order of products which must be disassembled to component. Three new are proposed for the problem variant with The stockouts are considered full to their specificities. EOQ problem for disassembly systems consists in determining reorder time interval and/or order quantity of end-of-life which must bestockout. disassembled to obtain their component.as Three new models are proposed for the problem variant with The stockouts are full new models arelost proposed forpartial the products problem variant with stockout. Thecost stockouts areis considered considered as full backorders, full sales and backorders. For each problem the function derived according and/or order quantity of end-of-life products which must be disassembled to obtain their component. Three new modelsfull arelost proposed forpartial the problem variant stockout. Thecost stockouts areis as full backorders, sales and backorders. For each problem the function derived according backorders, full sales and backorders. Forwith eachThese problem the function is considered derived according to themodels assumptions and dedicated are proposed. models show that itare can be economically new arelost proposed forpartial themethods problem variant stockout. Thecost stockouts as full backorders, full lost sales and partial backorders. Forwith eachThese problem the cost function is considered derived according to the assumptions and dedicated methods are proposed. models show that it can be economically to the assumptions and dedicated methods areinput proposed. models show that it can be economically interested tofull allow depending on the parameters. © 2019 IFAC backorders, loststockout sales and partial backorders. For eachThese problem the cost function is derived according to the assumptions and dedicated methods are proposed. These models show that it can be economically interested to depending on input parameters. © 2019 interested to allow allow stockout stockout depending on the the parameters. 2019 IFAC IFAC to the assumptions and dedicated methods areinput proposed. These © models show that it can be economically interested todisassembly, allow stockout depending onAutomatic the input parameters. © 2019 IFAC © 2019, IFAC (International Federation of Control) backordering, Hosting by Elsevier Ltd. All rights reserved. Keywords: EOQ, lot sizing, inventory control, end-of-life product. interested to allow stockout depending on the input parameters. © 2019 IFAC Keywords: disassembly, disassembly, EOQ, EOQ, lot lot sizing, sizing, inventory inventory control, control, backordering, backordering, end-of-life end-of-life product. product. Keywords: Keywords: disassembly, EOQ, lot sizing, inventory control, backordering, end-of-life product. Keywords: disassembly, EOQ, lot sizing, inventory control, backordering, end-of-life product. products and store less components. Backordering (partial or 1. INTRODUCTION products and store less Backordering (partial or products and been storeconsidered, less components. components. Backordering (partial or full) has not as far the authors know, in the 1. INTRODUCTION INTRODUCTION 1. products and been storeconsidered, less components. Backordering (partial or full) has not as far the authors know, in the full) has not been considered, as far the authors know, in the 1. INTRODUCTION DEOQ related models. products and been storeconsidered, less components. Backordering (partial or Disassembly is a central activity in reverse supply chains since full) has not as far the authors know, in the 1. INTRODUCTION related models. DEOQ models. Disassembly is activity in supply since Disassembly is aa central central activity in reverse reverse supply chains chains since DEOQ full) hasrelated not been considered, as far the authors know, in the it allows separating the components of end-of-life products DEOQ related models. Disassembly is a central in reverse supply chains since The paper fills the gap between disassembly lot sizing and it allows the components of products it allows separating separating the activity components ofareend-of-life end-of-life products DEOQ related models. according to the channels allocated to. Insince this The Disassembly is arecovery central activity in they reverse supply chains paper the between disassembly lot The paper fills the gap gapAs between disassembly lot sizing sizing and it allows separating the components of end-of-life products EOQ relatedfills literature. presented in this section, as farand as according to the recovery channels they are allocated to. In this according to the recovery channels they are allocated to. In this The paper fills the gapAs between disassembly lot sizing and context, disassembly planning problems consist in it allows separating the components of end-of-life products EOQ related literature. presented in this section, as far as according to the recovery channels they are allocated to. In this the authors know, there is no model proposed for determining The paper fills the gap between disassembly lot sizing and context, disassembly planning problems consist in context, disassembly planning problems consist in the EOQ relatedknow, literature. As presented in this section, as far as determining the recovery quantitieschannels and the they timings of the disassembly according are allocated to. In this authors there isdisassembly no model proposed for determining context, to disassembly planning problems consist in EOQ with know, stockout in As systems. The proposed related literature. presented in this section, as far as determining the quantities and the timings of the disassembly determining the quantities and the timings of the disassembly the authors there is no model proposed for determining of end-of-life products to serve the demands of several context, disassembly planning problems consist in EOQ withfocus stockout in disassembly systems. The proposed determining theproducts quantitiesto andserve the timings of the disassembly references on deterministic EOQ models to highlight the the authors know, there is no model proposed for determining of end-of-life the demands of several of end-of-life the demands of several references EOQ withfocus stockout in disassembly systems. proposed recovery channels. The to supply of the a disassembly determining theproducts quantities andserve the process timings of onofdeterministic EOQ models toThe highlight the of end-of-life products to serve the demands of several main specificities disassembly systems compared to others EOQ with stockout in disassembly systems. The proposed recovery channels. The supply process of a disassembly recovery The of supply of a disassembly references focus onofdeterministic EOQ models to highlight the system is channels. the collection end-of-life atofcustomer of end-of-life products to serveprocess the products demands several main specificities disassembly systems compared recovery channels. The of process of a disassembly systems. Based paper, future research can beto onothers nonreferences focuson onofthis deterministic EOQ models to highlight the system is the end-of-life products at customer system is As theincollection collection ofsupply end-of-life products atmethod customer main specificities disassembly systems compared toon others locations. assembly systems, a right planning can recovery channels. The supply process of a disassembly systems. Based on this paper, future research can be nonsystem is As thein collection of end-of-life products atmethod customer deterministic lot-sizing as presented in [13]. EOQ model for main specificities of disassembly systems compared to others locations. assembly systems, a right planning can locations. As in assembly systems, a right planning method can systems. Based on this paper, future research can be on nonsave logistic costs and, in it can also improve system is As thein collection of disassembly, end-of-life products atmethod customer deterministic lot-sizing asbackorders, presented inshows [13]. that EOQit model for locations. assembly systems, a right planning can single item with planned can make systems. Based on this paper, future research can be on nonsave logistic costs and, disassembly, it improve save logistic costs and, in in disassembly, it can can also also improve deterministic lot-sizing presented inshows [13]. that EOQit model for product recovery rates. One of athe of can the single locations. As in assembly systems, rightcharacteristic planning method itemsense with plannedasbackorders, can save logistic costs and, in disassembly, it can also improve economic assuming that customers arethat willing to make wait. deterministic lot-sizing as presented in [13]. EOQ model for product recovery rates. One of the characteristic of the product recovery rates. One of the characteristic of the single item with planned backorders, shows it can make problems is that of characteristic one unitalso of improve product save logistic costs the and,disassembly inOne disassembly, it can economic sense assuming that customers are willing toshows wait. product recovery rates. of the of the Basic model can be found in [6] where the author also single item with planned backorders, shows that it can make problems is that the disassembly of one unit of product problems is that the disassembly of characteristic one unitIt implies of product economic sense that[6]customers are willing toshows wait. generates one or more units of each component. product rates. One of the of that the Basic canassuming bewith found wherethe thesolution author also problemsrecovery is that the disassembly of one unitIt of product that formodel thesense model fullin lost sales, is either to economic assuming that customers are willing toshows wait. generates one or more units of each component. implies that generates one or more units of each component. It implies that Basic model can be found in [6] where the author also the quantity of product disassembled is not necessarily exactly problems is that the units disassembly of one unitIt implies of product for the model with full lost sales, thePartial solution is either to generates one or more of each component. that that have no lost sales or to lose all demands. backordering Basic model can be found in [6] where the author also shows the quantity of product disassembled is not necessarily exactly the quantity of product disassembled is not necessarily exactly that for with full lost sales, thePartial solution is either to the number ofor demand for each component andItunnecessary generates one more units of each component. implies that have no the lostmodel sales or to lose all demands. backordering the quantity of product is not necessarily exactly models situations where customer arethe willing to wait if the that with full lost sales, solution is either to the number of demand for each component and unnecessary the number ofgenerated demanddisassembled for each component and unnecessary havefor no the lostmodel sales or to lose all demands. Partial backordering stock can be at each disassembly operation. The the quantity of product disassembled is not necessarily exactly models situations where customer are willing to wait if the the number ofgenerated demand for each component and unnecessary waiting timesales is short otherwise the sale is lost. Different have no lost or to lose all demands. Partial backordering stock can be at each disassembly operation. The stock can be generated at each disassembly operation. The models situations where customer are willing to wait if the planning method have for hence tocomponent be adaptedand to unnecessary handle The this waiting time is short otherwise the sale is lost. Different the number demand stock can method beofgenerated at each eachto customer impatience function can be taken into account [7]. A models where customer are willing to wait if the planning have be to this planning method have hence hence to disassembly be adapted adapted operation. to handle handle The this customer waiting situations time is short otherwise the sale is lost. Different characteristic. stock can be generated at each disassembly operation. impatience function can be taken into account [7]. A planning method have hence to be adapted to handle this review of EOQ models with partial backordering can be found waiting time is short otherwise the sale is lost. Different characteristic. characteristic. customer impatience function can bebackordering taken into account [7]. A planning method have hence to be adapted to handle this review EOQ with partial can found characteristic. in [8]. of In themodels casefunction of constant percentage of be demand customer impatience can bebackordering taken into account [7]. A Disassembly planning depends on the context (type of costs, review of EOQ models with partial can be found characteristic. in [8]. In the case of constant percentage of demand Disassembly planning depends on the context (type of costs, backordered over the length of the stockout interval, Pentico Disassembly planning depends on the context (type of costs, review of EOQ models with partial backordering can be found demands, product recovery process and quality of the product). in [8]. In the case of constant percentage of demand over the length of thewhich stockout interval, Pentico Disassembly planning depends on and the quality context (type of costs, backordered demands, product process the and Drake propose model simplifies demands, product recovery process quality of of the product). product). in [8]. In [9] the case ofa constant percentage of previous demand The first works inrecovery thisdepends field consider demands Disassembly planning on and the deterministic context (type of costs, and backordered over the length of the stockout interval, Pentico demands, product recovery process and quality of the product). Drake [9] propose a model which simplifies previous The first works in this field consider deterministic demands models by considering reorder and the percentage of The first works this field consider deterministic demands backordered over the length of interval thewhich stockout interval, Pentico which vary over in several periods ofand a finite horizon planning models demands, product recovery process quality of the product). and Drake [9] propose a model simplifies previous The first works in this field consider deterministic demands by considering reorder interval anddecision the percentage of which vary over periods of aa finite horizon planning demand that will be filled from stock as variables. which vary over inseveral several periods of finite horizon demands planning and Drake [9] propose a model which simplifies previous and apply a Material Requirement Planning algorithm [1]. The The first works this field consider deterministic models by considering reorder anddecision the percentage of that will be frominterval stock as variables. which varyaa Material over several periods of a finitealgorithm horizon planning and apply Requirement Planning [1]. The Partial canfilled be combined withas other to cope and apply Material Requirement Planning algorithm [1]. The demand modelsbackorders by considering reorder and thesettings percentage of problem can be extended by considering various typesplanning of costs which vary over several periods of a finite horizon demand that will be filled frominterval stock decision variables. and apply a Material Requirement Planning algorithm [1]. The Partial backorders can be combined with other settings to cope problem can be extended by various types of with some Taleizadeh etwith al.asother [10] combine partial problem can be extended by considering considering various types of costs costs demand thatreal willcases. be filled from stock decision variables. in particular fixed disassembly operation costs and inventory and apply a Material Requirement Planning algorithm [1]. The Partial backorders can be combined settings to cope problem can be extended by considering various types of costs backorders with some with real cases. Taleizadeh et al.The [10]objective combinefunction partial in particular fixed disassembly operation costs inventory special order and price. in particular fixed disassembly operation costs and and inventory backorders can be combined with other settings topartial cope costs as can in lot sizing problem [2]. A various model which problem be extended by considering types ofallows costs Partial with some real cases. Taleizadeh et al. [10] combine in particular fixed disassembly operation costs and inventory backorders with special order and price. The objective function costs as in lot sizing problem [2]. A model which allows is based on the difference between the cases with and without costs as in lot sizing problem [2]. A model which allows with some real cases. Taleizadeh et al. [10] combine partial stockout to lot decrease inventory surplus is proposed in [3]. backorders with special order and price. The objective function in particular fixed disassembly costs and inventory costs as in sizing problem operation [2]. A model which allows basedorder. on with theThe difference between the cases with and without stockout to decrease inventory surplus is in [3]. special solution method is based on the concavity stockout to lot decrease inventory surplus is proposed proposed in [3]. backorders special order and price. The objective function Disassembly economic order quantity (DEOQ) model apply to is costs as in sizing problem [2]. A model which allows is basedorder. on theThe difference between the cases with and without stockout to decrease inventory surplus is proposed in [3]. solution method is based on the concavity special Disassembly economic order quantity (DEOQ) model apply to of the objective function. Other variants related to the Disassembly economic orderand quantity (DEOQ) model apply toa is based on theThe difference between the casesare with and without deterministic, continuous constant over stockout to decrease inventory surplus is demand proposed in [3]. special order. solution method is based on the concavity Disassembly economic order quantity (DEOQ) model apply to the objective function. Other variants are related to the deterministic, continuous and constant demand over aa of timing of the payment of purchasing cost. Taleizadeh [9] deterministic, continuous and constant demand over special order. The solution method is based on the concavity planning horizon with the objective to balance order and Disassembly economic orderand quantity (DEOQ) model apply toa of the objective function. Other variants are related to the deterministic, continuous constant demand overand timing of the payment of purchasing cost. Taleizadeh [9] planning horizon with the objective to balance order proposes a model and its solution method for partial planning horizon with the objective to disposal balance order and of the objective function. Other variants are related to the inventory costs [4]. The authors consider decision to deterministic, continuous and constant demand over a timing of athemodel payment ofitspurchasing cost. Taleizadeh [9] planning horizon with the objective to balance order and proposes and solution method for partial inventory costs [4]. The authors consider disposal decision to backordering and advance payments for an evaporating item. inventory costs [4].surplus The the authors consider disposal to timing of athemodel payment cost. Taleizadeh [9] handle inventory inherent to to disassembly systems. planning horizon with objective balance decision order and proposes andofpayments itspurchasing solution method for partial inventory costs [4]. The authors consider disposal decision to backordering and advance for an evaporating item. handle inventory surplus inherent to disassembly systems. Another strategy consists of partial delayed payment [11]. handle inventory surplus inherent to can disassembly systems. proposes a model and payments its solution method for partial Fixed order cost for disposal operations also be considered inventory costs [4]. The authors consider disposal decision to backordering and advance for an evaporating item. handle inventory surplus inherent to can disassembly systems. Taleizadeh Another strategy consists of partial adelayed payment [11]. Fixed order cost disposal operations also considered etandal.advance [12] payments analyse model with partial Fixed order cost for disposal operations can also be bemodels considered for an evaporating item. to minimize the for overall cost [5]. The also backordering handle inventory surplus inherent to proposed disassembly systems. Another strategy consists of partial delayed payment [11]. Fixed order cost for disposal operations can also be considered Taleizadeh et al. [12] analyse a model with partial to minimize the overall cost [5]. models backordering incremental discount. In this case,partial two to minimize the for overall costoperations [5]. The The proposed models also strategy partial payment [11]. allows lost sales for the cases where theproposed disposal arealso too Another Fixed order cost disposal can also becosts considered Taleizadeh etand al.consists [12] of analyse adelayed model with to minimize the for overall cost where [5]. The proposed models also backordering and incremental discount. In this case, two allows lost sales the cases the disposal costs are too decision variables are used: the time cycle and the fraction of allows lost sales for the cases where the disposal costs are too Taleizadeh et al. [12] analyse a model with partial high and itsales canfor bethe more to disassemble less to minimize the overall costeconomical [5]. The proposed models also backordering and are incremental discount. In this case, two allows lost cases where the disposal costs are too decision variables used: the time cycle and the fraction of high and it can be more economical to disassemble less high and it can be more economical to disassemble less backordering and incremental discount. In this case, two allowsand lostitsales caseseconomical where the disposal costs are less too decision variables are used: the time cycle and the fraction of high canfor bethe more to disassemble decisionbyvariables are All used: thereserved. time cycle and the fraction of high and©it2019, canIFAC be more economical to disassemble less1702Hosting 2405-8963 IFAC (International Federation of Automatic Control) Elsevier Ltd. rights Copyright 2019 Copyright ©under 2019 responsibility IFAC 1702Control. Peer review© of International Federation of Automatic Copyright 2019 IFAC 1702 Copyright © 2019 IFAC 1702 10.1016/j.ifacol.2019.11.442 Copyright © 2019 IFAC 1702
2019 IFAC MIM 1682 Berlin, Germany, August 28-30, 2019 Matthieu Godichaud et al. / IFAC PapersOnLine 52-13 (2019) 1681–1686
items fill from stock which also implicitly set the optimal discount. Even if the model is quite complex (it requires a numerical solver), a sensitivity analysis based on several examples shows that the solutions are robust according to the estimation of input parameters. We extend these works to disassembly systems by taking into account their specificities. Three new models for DEOQ problems with stockout are proposed in this paper. Each model is analyzed to develop a solution method which find the optimal reorder interval and the percentage of demands which are fill from stock or backordered. Real life applications of DEOQ models are discussed in [4] and [5]. The proposed model in this paper shows that it can be can make economic sense to allow stockout in disassembly systems. Based on this model industrial managers can evaluate the opportunity of the stockout profit based on the data on their system. In section 2, basic setting, assumptions and notations of the DEOQ problem are presented. In section 3, a new model for DEOQ with full backordering is developed. An alternate, compared to [5], model for full lost sales is proposed in section 4. DEOQ problem with partial backordering is addressed in section 5. An illustrative example is proposed in section 6 and, finally, conclusion and perspectives are presented in section 7.
Fig. 1 presents the shape of the inventory over one cycle of a stationary policy. For one component, denoted by 𝑖𝑖, the slope of the inventory curve is equal to the demand per unit time (denoted by 𝑑𝑑𝑖𝑖 ). The quantity of each component received at the beginning of a cycle is equal to the quantity of product disassembled (denoted by 𝑄𝑄) times the yield of the component. There are two parts in each cycle for each inventory. In the first part the demands are served directly from stock and in the second parts they are backordered or lost.
2. ASSUMPTIONS AND NOTATIONS The models apply to two-level bill of materials with the first level representing the product to disassemble and the second level representing the components. The product contains a given number of each components, named yield and denoted by 𝑎𝑎𝑖𝑖 thereafter, and the disassembly of one product generates simultaneously all the components. The demands are for the components and they can be backorder under some conditions presented for each model hereafter. We consider the following assumptions:
Fig. 1. Component inventory with stockouts. The purpose of the model is to determine the cycle time (or the order quantity of product) and the percentage of demands for each components that will be served from stock to minimize the sum of setup, disassembly, inventory, stockout and disposal costs. The following notations are used:
each demand is characterized by a constant rate in units per unit time (per year for example),
planning horizon is considered as infinite,
the replenishments of component inventories (by disassembly operation) are instantaneous,
there is a fixed cost for each disassembly operation incurred whenever an order is placed,
there is an inventory holding cost for each part proportional to the inventory on-hand.
One of the characteristic of the disassembly systems is that the disassembly of one unit of product generates one or more units of each of its component. It implies that, for each components, the quantity of product disassembled is not necessarily exactly the quantity demanded between two orders and unnecessary stock can be generated at each disassembly operation. By considering stationary demands, these inventory surplus accumulate with time. Disposal decisions have to be taken into account to ensure the stability of the disassembly system and inventory surplus is disposed of as soon as it is receive when no disposal fixed cost are considered.
𝑖𝑖 = 1, … , 𝑁𝑁 are the index for the components,
𝑎𝑎𝑖𝑖 is the yield of component 𝑖𝑖 (number of component 𝑖𝑖 in the product),
𝑑𝑑𝑖𝑖 is the constant demand per unit time for component 𝑖𝑖,
ℎ𝑖𝑖 is the inventory cost per unit and per unit time of component 𝑖𝑖,
𝑏𝑏𝑖𝑖 is the backorder cost per unit and per unit time of component 𝑖𝑖, 𝑝𝑝𝑖𝑖 is the lost sales cost per unit of component 𝑖𝑖,
𝑟𝑟𝑖𝑖 disposal cost per unit of component 𝑖𝑖,
𝛽𝛽𝑖𝑖 , the fraction of stockouts of the item 𝑖𝑖 that will be backordered,
𝑘𝑘 is the disassembly setup cost or fixed ordering cost for the product,
𝑐𝑐 is the disassembly cost of the product,
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𝑄𝑄 is the order quantity (the received quantity of each component after disassembly is 𝑎𝑎𝑖𝑖 𝑄𝑄,
𝐹𝐹𝑖𝑖 is the percentage of demand for a component 𝑖𝑖 that will be filled from stock (fill rate),
𝑇𝑇 is the cycle time (time between two orders).
The total cost function in (1) is denoted by 𝐶𝐶(𝑇𝑇, 𝐹𝐹) where 𝑇𝑇 is the time between two orders and 𝐹𝐹 is a vector containing the variables 𝐹𝐹𝑖𝑖 for each component 𝑖𝑖 = 1 … 𝑁𝑁. Based on Fig.2 for a component 𝑖𝑖, 𝑑𝑑𝑖𝑖 𝑇𝑇𝐹𝐹𝑖𝑖2 ⁄2 is the mean inventory per unit time and 𝑑𝑑𝑖𝑖 𝑇𝑇(1 − 𝐹𝐹𝑖𝑖 )2 ⁄2 is the mean backorder per unit time. 𝑘𝑘
On Fig. 1, 𝑅𝑅𝑖𝑖 is an unnecessary surplus (due to disassembly characteristic) to dispose of for one cycle and 𝑅𝑅𝑖𝑖 = 𝑎𝑎𝑖𝑖 𝑄𝑄 − 𝑇𝑇(𝑑𝑑𝑖𝑖 𝐹𝐹𝑖𝑖 + 𝑑𝑑𝑖𝑖 𝛽𝛽𝑖𝑖 (1 − 𝐹𝐹𝑖𝑖 )) . 𝑆𝑆𝑖𝑖 is the total quantity of component 𝑖𝑖 stockout at the end of a cycle and 𝑆𝑆𝑖𝑖 = 𝑑𝑑𝑖𝑖 𝑇𝑇(1 − 𝐹𝐹𝑖𝑖 ) (based on the Fig. 1 directly). One part, equal to 𝑑𝑑𝑖𝑖 𝛽𝛽𝑖𝑖 𝑇𝑇(1 − 𝐹𝐹𝑖𝑖 ) of 𝑆𝑆𝑖𝑖 is backlogged and the other is lost according to the value of 𝛽𝛽𝑖𝑖 .
𝐶𝐶(𝑇𝑇, 𝐹𝐹) = + ∑𝑁𝑁 𝑖𝑖=1 [
The three models proposed in the following section vary according the value of the 𝛽𝛽𝑖𝑖 which reflect, as a data, the customer acceptance to wait. If 𝛽𝛽𝑖𝑖 = 1, all the demands are backordered whereas if 𝛽𝛽𝑖𝑖 = 0 are lost. Partial backorders are for 0 < 𝛽𝛽𝑖𝑖 < 1.
𝐹𝐹𝑖𝑖∗ =
3. DEOQ WITH FULL BACKORDERS The problem is to determine is to determine the reorder interval 𝑇𝑇 and the fill rate 𝐹𝐹𝑖𝑖 to minimise a mean total cost per unit of time in the case where 𝛽𝛽𝑖𝑖 = 1 for all 𝑖𝑖 = 1 … 𝑁𝑁. It means that all the demands are backordered and 𝑎𝑎𝑖𝑖 𝑄𝑄 = 𝑑𝑑𝑖𝑖 𝑇𝑇 for all components (see Fig. 2). During a cycle of length 𝑇𝑇, the inventory level for a component 𝑖𝑖 will be positive during 𝐹𝐹𝑖𝑖 𝑇𝑇 and negative (backorders) during (1 − 𝐹𝐹𝑖𝑖 )𝑇𝑇. If we note 𝐺𝐺 the component such as 𝑑𝑑𝐺𝐺 ⁄𝑎𝑎𝐺𝐺 = max𝑖𝑖 {𝑑𝑑𝑖𝑖 ⁄𝑎𝑎𝑖𝑖 } then we must have 𝑇𝑇 = 𝑎𝑎𝐺𝐺 𝑄𝑄⁄𝑑𝑑𝐺𝐺 otherwise some demands would be lost. For other components, an unnecessary inventory, equal to 𝑎𝑎𝑖𝑖 𝑄𝑄 − 𝑑𝑑𝑖𝑖 𝑇𝑇, is received. It is disposed and not kept in stock (since there is no fixed cost for disposal). The disassembly and the disposal costs per unit time are:
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𝑇𝑇
ℎ𝑖𝑖 𝑑𝑑𝑖𝑖 𝑇𝑇𝐹𝐹𝑖𝑖2 2
+
𝑏𝑏𝑖𝑖 𝑑𝑑𝑖𝑖 𝑇𝑇(1−𝐹𝐹𝑖𝑖 )2 2
]
(1)
The first derivative of 𝐶𝐶(𝑇𝑇, 𝐹𝐹) with respect to 𝐹𝐹𝑖𝑖 is: 𝜕𝜕𝜕𝜕(𝑇𝑇, 𝐹𝐹)⁄𝜕𝜕𝐹𝐹𝑖𝑖 = 𝑑𝑑𝑖𝑖 𝑇𝑇(ℎ𝑖𝑖 𝐹𝐹𝑖𝑖 − 𝑏𝑏𝑖𝑖 (1 − 𝐹𝐹𝑖𝑖 )),
and 𝜕𝜕𝜕𝜕(𝑇𝑇, 𝐹𝐹)⁄𝜕𝜕𝐹𝐹𝑖𝑖 = 0 gives the optimal value of 𝐹𝐹𝑖𝑖 in (2). 𝑏𝑏𝑖𝑖
(2)
𝑏𝑏𝑖𝑖 +ℎ𝑖𝑖
𝐹𝐹𝑖𝑖∗ is independent of 𝑇𝑇 and 𝐹𝐹𝑖𝑖∗ replace 𝐹𝐹𝑖𝑖 in (1). The first derivative of 𝐶𝐶(𝑇𝑇, 𝐹𝐹) with respect to 𝑇𝑇 by setting 𝐹𝐹𝑖𝑖 = 𝐹𝐹𝑖𝑖∗ gives: ∗ 2 𝜕𝜕𝜕𝜕(𝑇𝑇, 𝐹𝐹)⁄𝜕𝜕𝜕𝜕 = −𝑘𝑘 ⁄𝑇𝑇 2 + ∑𝑁𝑁 𝑖𝑖=1[ℎ𝑖𝑖 𝑑𝑑𝑖𝑖 (𝐹𝐹𝑖𝑖 ) ⁄2 + ∗ 2 𝑏𝑏𝑖𝑖 𝑑𝑑𝑖𝑖 𝑇𝑇(1 − 𝐹𝐹𝑖𝑖 ) ⁄2],
and 𝐶𝐶(𝑇𝑇, 𝐹𝐹)⁄𝜕𝜕𝜕𝜕 = 0 gives the optimal value of 𝑇𝑇 in (3). 𝜕𝜕 2 𝐶𝐶(𝑇𝑇, 𝐹𝐹)⁄(𝜕𝜕𝜕𝜕)2 = 2𝑘𝑘 ⁄𝑇𝑇 3 > 0 the cost function is convex with respect to 𝑇𝑇 (> 0) when the 𝐹𝐹𝑖𝑖 are fixed. 𝑇𝑇 ∗ = √∑𝑁𝑁
2𝑘𝑘
𝑖𝑖=1𝑑𝑑𝑖𝑖 ℎ𝑖𝑖 𝑏𝑏𝑖𝑖 ⁄(ℎ𝑖𝑖 +𝑏𝑏𝑖𝑖 )
(3)
Equations (2) and (3) provide directly, without using a search method, optimal values for the decision variables of the DEOQ problem with full backordering.
𝑐𝑐(𝑑𝑑𝐺𝐺 ⁄𝑎𝑎𝐺𝐺 ) and ∑𝑁𝑁 𝑖𝑖=1[𝑟𝑟𝑖𝑖 (𝑎𝑎𝑖𝑖 (𝑑𝑑𝐺𝐺 ⁄𝑎𝑎𝐺𝐺 ) − 𝑑𝑑𝑖𝑖 )],
which are independent of 𝑇𝑇 and 𝐹𝐹𝑖𝑖 .
Fig. 3. Component inventory with full lost sales.
Fig. 2. Component inventory with full backorders.
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4. DEOQ WITH FULL LOST SALES The problem with full lost sales is for 𝛽𝛽𝑖𝑖 = 0 for all 𝑖𝑖 = 1 … 𝑁𝑁. Fig. 3 illustrates the shape of the inventory of one component during one cycle. It means that the customer do not want to wait and all the demands are lost during the stockout phase of the cycle. Compared to the model in [5], the addition of the variables 𝐹𝐹𝑖𝑖 allows having both lost sales and disposal for some component inventories during one cycle. It means it could be more economical to dispose of some units at the beginning of a cycle (and lose some demands on the end of the cycle) instead of keeping them in stock to serve demands. However, by analysing the related cost function in this section, we retrieve the results in [5] where, for a given component inventory, there are either lost sales or disposals. Compared to the previous model in section 3, both 𝑇𝑇 and 𝑄𝑄 have to be considered as decision variables since we can have 𝑎𝑎𝑖𝑖 𝑄𝑄 < 𝑑𝑑𝑖𝑖 𝑇𝑇 for the component inventories with lost sales. We use 𝑇𝑇 and 𝑋𝑋 = 𝑄𝑄 ⁄𝑇𝑇 as decision variables to ease the analysis. The mean total cost per unit time in (4) is then a function of 𝑇𝑇, 𝑋𝑋 and F, the vector containing the 𝐹𝐹𝑖𝑖 s. 𝑘𝑘
𝐶𝐶(𝑇𝑇, 𝑋𝑋, 𝐹𝐹) = + 𝑐𝑐𝑐𝑐 + ∑𝑁𝑁 𝑖𝑖=1 [ 𝑇𝑇
𝑎𝑎
ℎ𝑖𝑖 𝑑𝑑𝑖𝑖 𝑇𝑇𝐹𝐹𝑖𝑖2 2
+ 𝑝𝑝𝑖𝑖 𝑑𝑑𝑖𝑖 (1 − 𝐹𝐹𝑖𝑖 ) +
𝑟𝑟𝑖𝑖 𝑑𝑑𝑖𝑖 (( 𝑖𝑖 ) 𝑋𝑋 − 𝐹𝐹𝑖𝑖 )]
(4)
𝑑𝑑𝑖𝑖
We retrieve in (4) the mean order cost, 𝑘𝑘⁄𝑇𝑇 , and the mean inventory cost for each component, ℎ𝑖𝑖 𝑑𝑑𝑖𝑖 𝑇𝑇𝐹𝐹𝑖𝑖2 ⁄2. Compared to (1), the disassembly cost, 𝑐𝑐𝑐𝑐, and the disposal costs, 𝑟𝑟𝑖𝑖 𝑑𝑑𝑖𝑖 ((𝑎𝑎𝑖𝑖 ⁄𝑑𝑑𝑖𝑖 )𝑋𝑋 − 𝐹𝐹𝑖𝑖 ), are not constant. The mean lost sales cost per unit time, 𝑝𝑝𝑖𝑖 𝑑𝑑𝑖𝑖 (1 − 𝐹𝐹𝑖𝑖 ), correspond to the stockout phase of one cycle (see Fig. 3). The variables 𝑋𝑋 and 𝐹𝐹𝑖𝑖 are here subject to several constraints. For a given item 𝑖𝑖:
if 𝑋𝑋 ≥ 𝑑𝑑𝑖𝑖 ⁄𝑎𝑎𝑖𝑖 , the item may not have lost sales and 𝐹𝐹𝑖𝑖 ≤ 1,
if 𝑋𝑋 < 𝑑𝑑𝑖𝑖 ⁄𝑎𝑎𝑖𝑖 , the item has at least some lost sales and 𝐹𝐹𝑖𝑖 ≤ 𝑎𝑎𝑖𝑖 𝑋𝑋.
𝑘𝑘
𝐶𝐶(𝑇𝑇, 𝑋𝑋) = + 𝑐𝑐𝑐𝑐 + ∑𝑁𝑁 𝑖𝑖=1 [
𝑟𝑟𝑖𝑖 𝑑𝑑𝑖𝑖 (𝑝𝑝𝑖𝑖 +𝑟𝑟𝑖𝑖 ) 𝑇𝑇ℎ𝑖𝑖
𝑇𝑇
𝑑𝑑𝑖𝑖 (𝑝𝑝𝑖𝑖 +𝑟𝑟𝑖𝑖 )2 2𝑇𝑇ℎ𝑖𝑖
] + ∑𝑁𝑁 𝑖𝑖=1[𝑝𝑝𝑖𝑖 𝑑𝑑𝑖𝑖 + 𝑟𝑟𝑖𝑖 𝑎𝑎𝑖𝑖 𝑋𝑋]
By denoting:
−
𝑝𝑝𝑖𝑖 𝑑𝑑𝑖𝑖 (𝑝𝑝𝑖𝑖 +𝑟𝑟𝑖𝑖 ) 𝑇𝑇ℎ𝑖𝑖
− (6)
2
𝐴𝐴𝑖𝑖 = 𝑑𝑑𝑖𝑖 (𝑝𝑝𝑖𝑖 + 𝑟𝑟𝑖𝑖 ) ⁄(2𝑇𝑇ℎ𝑖𝑖 ) − 𝑝𝑝𝑖𝑖 𝑑𝑑𝑖𝑖 (𝑝𝑝𝑖𝑖 + 𝑟𝑟𝑖𝑖 )⁄(𝑇𝑇ℎ𝑖𝑖 ) −
𝑟𝑟𝑖𝑖 𝑑𝑑𝑖𝑖 (𝑝𝑝𝑖𝑖 + 𝑟𝑟𝑖𝑖 )⁄(𝑇𝑇ℎ𝑖𝑖 ),
2
and by developing it, it reduces to 𝐴𝐴𝑖𝑖 = − 𝑑𝑑𝑖𝑖 (𝑝𝑝𝑖𝑖 ) ⁄(2𝑇𝑇ℎ𝑖𝑖 ) − 𝑑𝑑𝑖𝑖 (𝑟𝑟𝑖𝑖 )2 ⁄(2𝑇𝑇ℎ𝑖𝑖 ) − 𝑑𝑑𝑖𝑖 𝑝𝑝𝑖𝑖 𝑟𝑟𝑖𝑖 ⁄(2𝑇𝑇ℎ𝑖𝑖 ) which is negative. The cost function 𝐶𝐶(𝑇𝑇, 𝑋𝑋) is of the form: 𝐶𝐶(𝑇𝑇, 𝑋𝑋) = 𝐴𝐴⁄𝑇𝑇 + 𝐵𝐵𝐵𝐵 + 𝐶𝐶 𝑁𝑁 𝑁𝑁 with 𝐴𝐴 = 𝑘𝑘 + ∑𝑁𝑁 𝑖𝑖=1 𝐴𝐴𝑖𝑖 , 𝐵𝐵 = 𝑐𝑐 + ∑𝑖𝑖=1 𝑟𝑟𝑖𝑖 𝑎𝑎𝑖𝑖 and 𝐶𝐶 = ∑𝑖𝑖=1 𝑝𝑝𝑖𝑖 𝑑𝑑𝑖𝑖 . The analysis of 𝐶𝐶(𝑇𝑇, 𝑋𝑋) is this form leads to the same result as in [5]. 𝑋𝑋 has to be as small as possible but it is subject to the constraints 𝑋𝑋 ≥ (𝑑𝑑𝑖𝑖 ⁄𝑎𝑎𝑖𝑖 )𝐹𝐹𝑖𝑖 for all 𝑖𝑖 = 1 … 𝑁𝑁. The best value of 𝑇𝑇 depends on the sign of 𝐴𝐴. If 𝐴𝐴 ≥ 0, 𝑇𝑇 has to be as large as possible and it means that all the demands are lost. If 𝐴𝐴 < 0, the value of 𝑇𝑇 has to be as small as possible which is reached for 𝐹𝐹𝑖𝑖 = 𝑚𝑚𝑚𝑚𝑚𝑚{1, (𝑎𝑎𝑖𝑖 ⁄𝑑𝑑𝑖𝑖 )𝑋𝑋}. It is the same to assume that an inventory cannot have both disposal and lost sales as in [5] and the remaining of the analysis is the same. The solution method consists of breaking pieces the cost function and searching for optimal values of 𝑋𝑋 and 𝑇𝑇 for each pieces. In each piece, the set of components is divided into two subsets: one with components which have lost sales and the other with components which have inventory surplus. 4. DEOQ WITH PARTIAL BACKORDERS Partial backordering occurring when 𝛽𝛽𝑖𝑖 , the fraction of stockouts of the item 𝑖𝑖 that will be backordered, is 0 < 𝛽𝛽𝑖𝑖 < 1. Based on Fig.4, 𝑇𝑇[𝑑𝑑𝑖𝑖 (𝐹𝐹𝑖𝑖 + 𝛽𝛽𝑖𝑖 (1 − 𝐹𝐹𝑖𝑖 ))] is the quantity requested to serve the backorders and the demands of component 𝑖𝑖 for one cycle of length 𝑇𝑇. The values of variables 𝐹𝐹𝑖𝑖 are feasible if 𝑄𝑄 ≥ 𝑇𝑇[(𝑑𝑑𝑖𝑖 ⁄𝑎𝑎𝑖𝑖 )(𝐹𝐹𝑖𝑖 + 𝛽𝛽𝑖𝑖 (1 − 𝐹𝐹𝑖𝑖 ))] for all 𝑖𝑖 = 1 … 𝑁𝑁. To set a replenishment policy, we use the variables 𝑇𝑇, 𝑋𝑋 (= 𝑄𝑄 ⁄𝑇𝑇) and 𝐹𝐹𝑖𝑖 .
𝑑𝑑𝑖𝑖
The problem can be then stated as follow: Min 𝐶𝐶(𝑇𝑇, 𝑋𝑋, 𝐹𝐹) subject to 0 ≤ 𝐹𝐹𝑖𝑖 ≤ 𝑚𝑚𝑚𝑚𝑚𝑚 {1, 𝑋𝑋 ≥ max { 𝑖𝑖=1…𝑁𝑁
𝑑𝑑𝑖𝑖 𝐹𝐹𝑖𝑖 𝑎𝑎𝑖𝑖
}
𝑎𝑎𝑖𝑖
𝑑𝑑𝑖𝑖
𝑋𝑋} ∀𝑖𝑖 = 1 … 𝑁𝑁 and (5)
The analysis can be made for 𝐶𝐶(𝑇𝑇, 𝑋𝑋, 𝐹𝐹) starting from 𝐹𝐹𝑖𝑖 with 𝑇𝑇 and 𝑋𝑋 considered as fixed. It gives 𝐹𝐹∗𝑖𝑖 (𝑇𝑇) the optimal value of 𝐹𝐹𝑖𝑖 when 𝑋𝑋 and 𝑇𝑇 are fixed (which is just a function of 𝑇𝑇):
𝐹𝐹∗𝑖𝑖 (𝑇𝑇) =
𝑝𝑝𝑖𝑖 +𝑟𝑟𝑖𝑖 𝑇𝑇ℎ𝑖𝑖
.
By replacing 𝐹𝐹𝑖𝑖 by 𝐹𝐹∗𝑖𝑖 (𝑇𝑇) in 𝐶𝐶(𝑇𝑇, 𝑋𝑋, 𝐹𝐹), we obtain 𝐶𝐶(𝑇𝑇, 𝑋𝑋) in (6).
Fig. 4. Component inventory with partial backorders.
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with 𝐶𝐶𝑖𝑖 = 𝑝𝑝𝑖𝑖 + 𝑟𝑟𝑖𝑖 − (𝑐𝑐 + ∑𝑁𝑁 𝑘𝑘=1 𝑟𝑟𝑘𝑘 𝑎𝑎𝑘𝑘 )⁄𝑎𝑎𝑖𝑖 if 𝑖𝑖 = 𝑞𝑞 and 𝐶𝐶𝑖𝑖 = 𝑝𝑝𝑖𝑖 + 𝑟𝑟𝑖𝑖 otherwise.
The resulting mean total cost per unit of time is given in (7). 𝑘𝑘
𝐶𝐶(𝑇𝑇, 𝑋𝑋, 𝐹𝐹) = + 𝑐𝑐𝑐𝑐 + ∑𝑁𝑁 𝑖𝑖=1 [ 𝑇𝑇
𝑝𝑝𝑖𝑖 𝑑𝑑𝑖𝑖 (1 − 𝛽𝛽𝑖𝑖 )(1 − 𝐹𝐹𝑖𝑖 )] + 𝐹𝐹𝑖𝑖 )}]
ℎ𝑖𝑖 𝑑𝑑𝑖𝑖 𝑇𝑇𝐹𝐹𝑖𝑖2 2
+
𝑏𝑏𝑖𝑖 𝑑𝑑𝑖𝑖 𝑇𝑇𝛽𝛽𝑖𝑖 (1−𝐹𝐹𝑖𝑖 )2 2
∑𝑁𝑁 𝑖𝑖=1[𝑟𝑟𝑖𝑖 𝑑𝑑𝑖𝑖 {(𝑎𝑎𝑖𝑖 ⁄𝑑𝑑𝑖𝑖 )𝑋𝑋
+
The values obtained from (11) and (12) are optimal for the subproblem of minimising 𝐶𝐶𝑞𝑞 (𝑇𝑇, 𝐹𝐹) if the constraints (9) and
− 𝐹𝐹𝑖𝑖 + 𝛽𝛽𝑖𝑖 (1 −
(7)
We have in (7) the mean order cost, the disassembly cost, the mean inventory, backorder and lost sales costs for each component from the previous models, in sections 3 and 4. The disposal quantity at each cycle is 𝑎𝑎𝑖𝑖 𝑋𝑋 − 𝑑𝑑𝑖𝑖 (𝐹𝐹𝑖𝑖 + 𝛽𝛽𝑖𝑖 (1 − 𝐹𝐹𝑖𝑖 )) which gives the last parts of (7) corresponding to the mean diposal costs. One notes, by developing this last part and eliminating the constant term, minimising (7) is equivalent to minimise (8). 𝑘𝑘
𝐶𝐶(𝑇𝑇, 𝑋𝑋, 𝐹𝐹) = + (𝑐𝑐 + 𝑏𝑏𝑖𝑖 𝑑𝑑𝑖𝑖 𝑇𝑇𝛽𝛽𝑖𝑖 (1−𝐹𝐹𝑖𝑖 )2 2
𝑇𝑇
∑𝑁𝑁 𝑖𝑖=1 𝑟𝑟𝑖𝑖 𝑎𝑎𝑖𝑖 )𝑋𝑋
+
ℎ𝑖𝑖 𝑑𝑑𝑖𝑖 𝑇𝑇𝐹𝐹𝑖𝑖2 ∑𝑁𝑁 𝑖𝑖=1 [ 2
+ (𝑝𝑝𝑖𝑖 + 𝑟𝑟𝑖𝑖 )𝑑𝑑𝑖𝑖 (1 − 𝛽𝛽𝑖𝑖 )(1 − 𝐹𝐹𝑖𝑖 )]
+ (8)
Minimising (8) is subject to the constraints (9) and (10). Constraints (9) set the fill rates bounds (1 for no stockout and 0 for all stockout). Constraints (10) force to disassembled the quantity of product to serve the all demands required with respect to the fill rates. 0 ≤ 𝐹𝐹𝑖𝑖 ≤ 1, ∀𝑖𝑖 = 1 … 𝑁𝑁
(9)
X ≥ (𝑑𝑑𝑖𝑖 ⁄𝑎𝑎𝑖𝑖 )(𝐹𝐹𝑖𝑖 + 𝛽𝛽𝑖𝑖 (1 − 𝐹𝐹𝑖𝑖 )), ∀𝑖𝑖 = 1 … 𝑁𝑁
(10)
As in the model with full lost sales, we have 𝜕𝜕𝐶𝐶(𝑇𝑇, 𝑋𝑋, 𝐹𝐹)⁄𝜕𝜕𝜕𝜕 = 𝑐𝑐 + ∑𝑁𝑁 𝑖𝑖=1 𝑟𝑟𝑖𝑖 𝑎𝑎𝑖𝑖 > 0 and then 𝑋𝑋 must be then chosen as small as possible subject to the constraints (10) and then we have 𝑋𝑋 = max{ (𝑑𝑑𝑖𝑖 ⁄𝑎𝑎𝑖𝑖 )(𝐹𝐹𝑖𝑖 + 𝛽𝛽𝑖𝑖 (1 − 𝐹𝐹𝑖𝑖 ))}. 𝑖𝑖
When the component 𝑞𝑞 = argmax{(𝑑𝑑𝑘𝑘 ⁄𝑎𝑎𝑘𝑘 )(𝐹𝐹𝑘𝑘 + 𝛽𝛽𝑘𝑘 (1 − 𝑘𝑘
𝐹𝐹𝑘𝑘 ))} is known, 𝑋𝑋 is replaced by (𝑑𝑑𝑞𝑞 ⁄𝑎𝑎𝑞𝑞 ) (𝐹𝐹𝑞𝑞 + 𝛽𝛽𝑞𝑞 (1 − 𝐹𝐹𝑞𝑞 )) in 𝐶𝐶(𝑇𝑇, 𝑋𝑋, 𝐹𝐹) which becomes a function of 𝑇𝑇 and 𝐹𝐹 only, denoted by 𝐶𝐶𝑞𝑞 (𝑇𝑇, 𝐹𝐹). The solution method presented in this paper consists in solving 𝑁𝑁 sub-problem 𝐶𝐶𝑞𝑞 (𝑇𝑇, 𝐹𝐹) for 𝑞𝑞 = 1 … 𝑁𝑁 and select the one with the minimal cost. The stationary points of 𝐶𝐶𝑞𝑞 (𝑇𝑇, 𝐹𝐹) are given by taking the first partial derivatives with respect to 𝑇𝑇 and 𝐹𝐹𝑖𝑖 for 𝑖𝑖 = 1 … 𝑁𝑁. Taking the partial derivative of 𝐶𝐶𝑞𝑞 (𝑇𝑇, 𝐹𝐹) with respect to 𝑇𝑇 and setting it equal to 0 gives 𝑇𝑇(𝐹𝐹), the optimal value of 𝑇𝑇 when 𝐹𝐹 is fixed, in (11). 𝑇𝑇(𝐹𝐹) = √∑𝑁𝑁
2𝑘𝑘
2 2 𝑖𝑖=1 𝑑𝑑𝑖𝑖 [ℎ𝑖𝑖 𝐹𝐹𝑖𝑖 −𝑏𝑏𝑖𝑖 𝛽𝛽𝑖𝑖 (1−𝐹𝐹𝑖𝑖 ) ]
(11)
Taking the partial derivative of 𝐶𝐶𝑞𝑞 (𝑇𝑇, 𝐹𝐹) with respect to with respect to 𝐹𝐹𝑖𝑖 and setting it equal to 0 gives 𝐹𝐹𝑖𝑖 (𝑇𝑇) in (12). We note that 𝐹𝐹𝑖𝑖 (𝑇𝑇), the optimal value of 𝐹𝐹𝑖𝑖 when 𝑇𝑇 is fixed, depend only on 𝑇𝑇 but not on the other 𝐹𝐹𝑗𝑗 , 𝑗𝑗 = 1 … 𝑁𝑁 and 𝑗𝑗 ≠ 𝑖𝑖. 𝐹𝐹𝑖𝑖 =
𝐶𝐶𝑖𝑖 (1−𝛽𝛽𝑖𝑖 )+𝑏𝑏𝑖𝑖 𝛽𝛽𝑖𝑖 𝑇𝑇 𝑇𝑇(ℎ𝑖𝑖 +𝑏𝑏𝑖𝑖 𝛽𝛽𝑖𝑖 )
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(12)
(𝑑𝑑𝑞𝑞 ⁄𝑎𝑎𝑞𝑞 ) (𝐹𝐹𝑞𝑞 + 𝛽𝛽𝑞𝑞 (1 − 𝐹𝐹𝑞𝑞 )) ≥ (𝑑𝑑𝑖𝑖 ⁄𝑎𝑎𝑖𝑖 )(𝐹𝐹𝑖𝑖 + 𝛽𝛽𝑖𝑖 (1 − 𝐹𝐹𝑖𝑖 )),
∀𝑖𝑖 = 1 … 𝑁𝑁, 𝑖𝑖 ≠ 𝑞𝑞
(13)
are satisfied. Given these constraint, we propose the following iterative method to find the best value of the decision variables. Step 1 is the same for all sub-problem while Step 2 and 3 are repeated for each sub-problem for 𝑞𝑞 = 1 … 𝑁𝑁. (0)
Step 1 (Initialisation). Set 𝐹𝐹𝑖𝑖 = 1 for 𝑖𝑖 = 1 … 𝑁𝑁, 𝑋𝑋 (0) = max{ (𝑑𝑑𝑖𝑖 ⁄𝑎𝑎𝑖𝑖 )(𝐹𝐹𝑖𝑖 + 𝛽𝛽𝑖𝑖 (1 − 𝐹𝐹𝑖𝑖 ))}, 𝑇𝑇 (0) = √2𝑘𝑘 ⁄∑𝑁𝑁 𝑖𝑖=1 𝑑𝑑𝑖𝑖 ℎ𝑖𝑖 and 𝑖𝑖
(0) the total cost is 𝐶𝐶 (0) = √2𝑘𝑘 ∑𝑁𝑁 + 𝑖𝑖=1 𝑑𝑑𝑖𝑖 ℎ𝑖𝑖 + 𝑐𝑐𝑋𝑋 (0) ∑𝑁𝑁 − 𝑑𝑑𝑖𝑖 )] (it is the problem without stockout, see 𝑖𝑖=1[𝑟𝑟𝑖𝑖 (𝑎𝑎𝑖𝑖 𝑋𝑋 [4] for details). 𝑢𝑢 ← 0. (𝑢𝑢)
Step 2. 𝑢𝑢 ← 𝑢𝑢 + 1. Set, for 𝑖𝑖 = 1 … 𝑁𝑁, 𝐹𝐹𝑖𝑖
=
𝑚𝑚𝑚𝑚𝑚𝑚 {1, (𝐶𝐶𝑖𝑖 (1 − 𝛽𝛽𝑖𝑖 ) + 𝑏𝑏𝑖𝑖 𝛽𝛽𝑖𝑖 𝑇𝑇 ) ⁄( + 𝑏𝑏𝑖𝑖 𝛽𝛽𝑖𝑖 ))}. If the constraints (13) are satisfied go to Step 3 otherwise go to another sub-problem by changing 𝑞𝑞. (𝑢𝑢−1
Step 3. Set 𝑇𝑇 (𝑢𝑢) =
2
𝑇𝑇 (𝑢𝑢−1) (ℎ𝑖𝑖
2
(𝑢𝑢) (𝑢𝑢) √2𝑘𝑘 ⁄∑𝑁𝑁 ) − 𝑏𝑏𝑖𝑖 𝛽𝛽𝑖𝑖 (1 − 𝐹𝐹𝑖𝑖 ) ] 𝑖𝑖=1 𝑑𝑑𝑖𝑖 [ℎ𝑖𝑖 (𝐹𝐹𝑖𝑖
(𝑢𝑢)
and 𝐶𝐶 (𝑢𝑢) = 𝐶𝐶𝑞𝑞 (𝑇𝑇 (𝑢𝑢) , 𝐹𝐹 (𝑢𝑢) ), (𝐹𝐹 (𝑢𝑢) is the set of variable 𝐹𝐹𝑖𝑖 at iteration 𝑢𝑢. If 𝐶𝐶 (𝑢𝑢) − 𝐶𝐶 (𝑢𝑢−1) > 𝜀𝜀 then go to Step 2, else the best values for 𝑇𝑇 and 𝐹𝐹𝑖𝑖 , 𝑖𝑖 = 1 … 𝑁𝑁, has been found. We note there is always a solution at Step 1 (the solution without stockout is a solution of the problem). At Step 2, if an inequality of (13) is not satisfied, the decision variable must be changed to obtain an equality which is the same as changing the sub-problem (i.e. changing 𝑞𝑞). In Step 3, 𝜀𝜀 must be chosen according to the desired precision and the computing time. 5. ILLUSTRATIVE EXAMPLE An illustrative example for a product with five components, 𝑁𝑁 = 5, is proposed in this section to facilitate the understanding of the model and method developed for the DEOQ with partial backorder (section 4). The setup cost is 𝑘𝑘 = 5 and the disassembly cost is 𝑐𝑐 = 1. The other data for each component 𝑖𝑖 = 1 … 𝑁𝑁 are presented in Table 1.
At the Step 1, the solution without stockout (for 𝐹𝐹𝑖𝑖 = 1, 𝑖𝑖 = 1 … 5) has a total cost 𝐶𝐶 (0) = 731.14 with 𝑋𝑋 (0) = 20.67 and 𝑇𝑇 (0) = 1.58. At Step 2 for 𝑞𝑞 = 1, applying (12) with 𝑇𝑇 = 𝑇𝑇 (0) gives 𝐹𝐹1 = 1.12, 𝐹𝐹2 = 0.68, 𝐹𝐹3 = 1.84, 𝐹𝐹4 = 0.82, 𝐹𝐹5 = 1.28. The values superior to 1 are corrected to 1. This leads to the violation of constraint (13) for 𝑖𝑖 = 3,4,5 and the values of 𝐹𝐹𝑖𝑖 must be
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change until the constraints are satisfied. It leads to obtain equalities for constraint (13) which is equivalent to change 𝑞𝑞 and solve another sub-problem.
Table 1. Component data for the example 𝑖𝑖
𝑎𝑎𝑖𝑖
𝑑𝑑𝑖𝑖
ℎ𝑖𝑖
𝑝𝑝𝑖𝑖
𝑟𝑟𝑖𝑖
𝑏𝑏𝑖𝑖
𝛽𝛽𝑖𝑖
1 2 3
4 5 3
26 33 62
2.47 2.32 2.08
13.11 7.24 13.68
0.99 0.19 0.99
2 2.48 1.91
0.6 0.8 0.5
4 5
3 3
53 43
1.67 0.98
7.78 12.52
0.22 0.33
2.47 1.65
0.8 0.8
Fig. 5. Evolution of the cost with respect to the iterations.
[4] M. Godichaud and L. Amodeo, Economic order quantity for multistage disassembly systems, International Journal of Production Economics, vol. 199, pp. 16-25, 2018. [5] M. Godichaud and L. Amodeo, EOQ inventory models for disassembly systems with disposal and lost sales, International Journal of Production Research, 2018. [6] P. Zipkin, Foundations of inventory management. New York: McGraw-Hill, 2000. [7] P.L. Abad, Optimal Pricing and Lot-Sizing under Conditions of Perishability and Partial Backordering, Management Science, vol. 42, no. 8, pp. 1093-110, 1996. [8] D.W. Pentico and M.J. Drake, A survey of deterministic models for the EOQ and EPQ with partial backordering, European Journal of Operational Research, vol. 214, pp. 179-198, 2011. [9] A.A. Taleizadeh, An EOQ Model with Partial Backordering and Advance Payments for an Evaporating Item, International Journal of Production Economics, vol. 155, pp. 185–193, 2014. [10] A.A. Taleizadeh, D. W. Pentico, M. Aryanezhad and S. M. Ghoreyshi, An Economic Order Quantity Model with Partial Backordering and a Special Sale Price, European Journal of Operational Research vol. 221, no.3, pp. 571– 583, 2012. [11] A.A. Taleizadeh, D.W. Pentico, M. S. Jabalameli, and M. Aryanezhad,An EOQ Model with Partial Delayed Payment and Partial Backordering, Omega, vol. 41, no. 2, pp. 354–368, 2013. [12] A.A. Taleizadeh, I. Stojkovska, and D. W. Pentico, “An Economic Order Quantity Model with Partial Backordering and Incremental Discount.” Computers & Industrial Engineering vol. 82, pp. 21–32, 2015. [13] M.A. Aloulou, A. Dolgui and M.Y. Kovalyov A bibliography of non-deterministic lot-sizing models Mohamed Ali Aloulou, Alexandre Dolgui Mikhail Y. Kovalyov, vol. 52, no. 8, pp. 2293-2310, 2014.
6. CONCLUSIONS Disassembly systems necessitates new models for their inventory management due to their specificities. Three problems with stockouts (full backorders, full lost sales and partial backorders) have been addressed in this paper. The models are non-linear optimization model with constraints. Their analysis shows that closed-form or fast methods can be derived for each of them. Their application allows to investigating the economic advantages of stockouts for disassembly management. One extension of this work is to develop a sensitive analysis to get additional managerial implications. Future research can also the integration of environmental constraints or objective, with specifically the impact of stockout on them, which is an important motivation to perform disassembly activities. REFERENCES [1] S.M. Gupta and K.N. Taleb, Scheduling disassembly, International Journal of Production Research, vol. 32, no. 8, pp.1857-1866, 1994. [2] H.-J. Kim, D.-H. Lee and P. Xirouchakis, Disassembly scheduling: literature review and future research directions, International Journal of Production Research, vol. 45, no. 18-19, pp.4465-4484, 2007. [3] M. Hrouga, M. Godichaud and L. Amodeo, Heuristics for multi-product capacitated disassembly lot sizing with lost sales, IFAC-PapersOnLine, vol. 49, no. 12, 2016, pp. 628633.
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