A simulation-optimisation approach for reconfigurable ...

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Int. J. Business Performance and Supply Chain Modelling, Vol. 5, No. 1, 2013

A simulation-optimisation approach for reconfigurable inventory space planning in remanufacturing facilities Aysegul Topcu* Structured Decisions Corporation, 1105 Washington Street Suite 1, West Newton, MA 02465, USA and Carroll School of Management, Boston College, 140 Commonwealth Avenue, Chestnut Hill, MA 02467, USA E-mail: [email protected] *Corresponding author

James C. Benneyan and Thomas P. Cullinane Department of Mechanical and Industrial Engineering, Northeastern University, 360 Huntington Avenue, Boston, MA 02115, USA E-mail: [email protected] E-mail: [email protected] Abstract: Although remanufacturing facilities are becoming increasingly vital components in some supply chains, significant variability over time in returned product volumes, reusable part yields, and refurbished item demand can result in significant variability in storage requirements over time. In response, manufacturers can implement reconfigurable inventory systems to accommodate off-setting swings in storage needs between types of components and processing activities, including temporary external storage. A Monte Carlo (MC) simulation-optimisation approach has first been developed to emulate a generalised remanufacturing facility with random receiving patterns, component yields, and refurbished demand. Then, a multi-dimensional golden section search algorithm is implemented to identify optimal storage capacities and reconfiguration decisions in each time period that minimise long-term expected total cost. In pilot applications, improvements over non-reconfigurable systems range from 9% to 33% reductions in total storage space costs. Keywords: Monte Carlo simulation; heuristic optimisation; reconfiguration; capacity planning; remanufacturing; reverse logistics; facility layout; supply chain modelling. Reference to this paper should be made as follows: Topcu, A., Benneyan, J.C. and Cullinane, T.P. (2013) ‘A simulation-optimisation approach for reconfigurable inventory space planning in remanufacturing facilities’, Int. J. Business Performance and Supply Chain Modelling, Vol. 5, No. 1, pp.86–114. Copyright © 2013 Inderscience Enterprises Ltd.

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Biographical notes: Aysegul Topcu joined Structured Decisions Corporation after getting her PhD in Industrial Engineering from Northeastern University. As a Senior Operations Researcher, she uses operations research methods to develop mathematical models for complex logistical operations in the service industries. She also works as a part-time Faculty at Boston College where she teaches in the Carroll School of Management. She received her BS and MS in Industrial Engineering from the University of Wisconsin-Madison. While doing her PhD at Northeastern University, she spent the summer of 2006 as an Intern Specialist in the Operations Department at Polaroid Corporation’s headquarters in Waltham, MA. James C. Benneyan is Full Professor of Industrial Engineering and Operations Research at Northeastern University in Boston, MA, where he also is Founding Director of the Quality and Productivity Research Laboratory and the Healthcare Systems Engineering Institute. He is past Vice President of the Institute for Industrial Engineers, Senior Fellow for the Institute for Healthcare Improvement, Advisor to the Center for Medicare and Medicaid Innovation, and serves on numerous editorial and advisory boards. Prior to joining Northeastern, he was Senior Industrial Engineer at IBM, Digital Equipment Corporation, and Harvard Community Health Plan and consults broadly in applied operations research. Thomas P. Cullinane is a Professor at Northeastern University (USA) and is the Director of the Engineering Management Program. He has served as the Director of the Facility Planning and Design Division of the Institute of Industrial Engineers, the President of the College Industry Council on Material Handling Education and the Chairperson of the Industrial Engineering Division of the American Society for Engineering Education. He has performed research for a wide range of industrial sponsors, the National Science Foundation, the US Air Force and NIST. His publications include a wide range of technical papers and a book on Project Management.

1

Introduction

Remanufacturing is becoming increasingly important worldwide, for reasons ranging from end-of-life take-back regulations (Toffel, 2003), growing consumer preference for environmentally conscious products (Canon, 2012; Xerox, 2011; Eastman Kodak, 2010; Sharp, 2012), secondary and global markets for refurbished goods (Meyer, 1999), potential raw material and lead time savings, and faster technology cycles that create early obsolescence of items that still have useful salvageable components. Although a significant amount of research over the past half century has focused on traditional manufacturing facility design, little work has focused on remanufacturing facility design nor on inventory planning within these systems. Recently, Alinovi et al. (2012) provide a framework to formulate a stochastic inventory control model in mixed manufacturing/remanufacturing systems, and evaluate the opportunity of establishing a return policy in those systems. Storage space requirements can vary significantly over time in such systems – both between part types and for different needs within the disassembly-remanufacturing-reassembly process – largely due to the compounding of several sources of variability. Examples include variable yield rates (within and between part types), variable returned product volumes, random demand for remanufactured

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products, and variable processing times dependent on age, wear, and condition. Chanintrakul et al. (2009) in their article recognise the importance of these reverse logistics operations, and specifically look at the development of closed loop supply chain models, where the uncertainty of supply and the dynamic interactions between forward and reverse flows are addressed. Remanufacturing processes also typically include additional operations such as disassembly, inspection, functionality testing, and cleaning that each require some amount of work-in-process storage capacity while inventory flows through the system. Multiple components typically are salvaged from each returned item, each having different yields and storage requirements. For example, Gupta et al. (2004) described a cellular phone remanufacturing facility in which 21 components (battery, printer circuit board, covers, antennas, etc.) are recaptured and refurbished. These types of facilities are partially governed by several endogenous and exogenous variables, including yield rates, returned product volumes, refurbished demand, and storage capacities for each type of part and at each location in the process. Since storage demands often may be offsetting over time, between product types, and at various locations in the process, it is beneficial to consider how to optimise overall space allocation, including reconfiguration of inventory capacities and temporary use of external storage. The remainder of this paper describes a simulation-based optimisation approach to identify optimal storage capacities that minimise total cost over several time periods, integrating a multi-dimensional golden section search algorithm with a Monte Carlo generalised remanufacturing inventory model. This simulation-optimisation model identifies optimal space allocations and reconfiguration decisions over a user-specified number of time periods so as to minimise long-term total expected costs. Section 2 presents a description of the remanufacturing inventory system and the general flow of parts and products along with storage needs and capacities as they go through disassembly, cleaning, and testing processes. In Section 3, a general description of the simulation-optimisation approach is given. Section 4 talks about how the simulation model is implemented using VB.NET, and explains the Monte Carlo logic in detail including running the model under different scenarios. In Section 5, we discuss about how the multi-dimensional golden section search algorithm is developed to identify optimal storage capacities for each storage and part type. Later in Section 6, we present the conclusions and some future work.

2 a

Remanufacturing inventory systems Remanufacturing systems

Remanufacturing a product generally is quite different from manufacturing the same item the first time. Typically, remanufacturing requires several additional operations, such as disassembly, inspection, functionality testing, and cleaning, each with additional storage needs. For further discussion, Topcu and Cullinane (2005) describe general differences between first-time manufacturing and remanufacturing operations for durable goods and identify facilities design features necessary for remanufacturing operations to be effective. As a result of variability in routings, processing times, and demand, remanufacturing systems also tend to be more dynamic and complex than traditional manufacturing systems, presenting new opportunities to optimise such processes. As

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examples, Kekre et al. (2003) maximised remanufacturing throughput via a line balancing simulation-based model, Lim and Noble (2006) used multi-criteria methods to evaluate and choose between different remanufacturing layout alternatives, and Franke et al. (2006) developed a combinatorial optimisation and discrete-event simulation method for continuously adapting remanufacturing facilities under quickly changing product, process, and market constraints. While a few authors have considered reconfigurable manufacturing systems (Lee, 1997; Mehrabi et al., 2000; Mellor, 2002; Elmaraghy, 2008; Benjaafar et al., 2002; Meng et al., 2004), little work if any has examined inventory storage planning issues. Borrowing ideas from reconfigurable manufacturing systems (Elmaraghy, 2008, Benjaafar et al., 2002, Meng et al., 2004), the current paper also introduces the possibility of re-allocating storage space between different functional needs and part types over time, including the possibility of temporarily paying for external storage. Some of the recent work in this area is performed by Nenes and Nikolaidis (2012) who develop a MILP model to optimise procurement, remanufacturing, stocking and salvaging decisions of returned products. Piplani and Saraswat (2012) also propose a MILP model to minimise the total cost subject to flow balance and logical constraints in the service network of a company providing repair and refurbishment services for its products, where the optimal network configuration depends on several uncertain factors such as the number of products returned, percent of faulty products and warranty fraction of modules. Kusumastuti et al. (2009) on the other hand combine both optimisation and simulation models to design a reverse logistics network for discrete product recovery, considering multiple objective functions, multi-period planning horizon and uncertainty with applications in computer manufacturing. The most recent literature review is performed in the article by Dowlatshahi (2012). He uses an exploratory case study research approach where the analysis results in the development of warehousing sub-factors, propositions and insights in reverse logistics. Bienstock et al. (2011) also utilise a case study approach where they discuss the strategic issues surrounding the effective management of product return services, and the importance of effective reverse logistics operations to the design of successful and profitable reverse supply chains. Lastly, Subramanian et al. (2010) investigate the integration of forward and reverse logistics with the aim to aid decision making on issues such as material procurement, production, distribution, recycling and disposal of end-of-life items.

b

General material flow

The general flow of parts through a remanufacturing facility is dependent on the necessary disassembly, cleaning, and testing processes; age and amount of wear; and demand for remanufactured items. Figure 1 illustrates a generalised flow of recaptured material through a remanufacturing system and the various storage types. When a returned item arrives at a remanufacturing facility, the condition of its components usually is not fully known until the product is disassembled. After disassembly, parts typically are tested or inspected to determine their functionality and then put into storage. Uncertainty about each part’s usability creates uncertainty as to how much storage space should be allocated for each part type, which in turn impacts the layout of the remanufacturing facility. Other factors that influence the layout are physical characteristics of the recovered and remanufactured units (e.g., dimensions/size, volume,

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weight, fragility), work to be performed on the returned products, and storage requirements (e.g., temperature or humidity range). A fundamental characteristic of a remanufacturing environment is the inbound flow of used products, where most of the components or subassemblies have probabilistic yields with a random number of parts not suitable for re-use. This stochastic nature of returned products affects predictability, safety stock, production targets, rework (such as additional cleaning, testing, inspection, and reassembly), and waste. Required materials or equipment, such as cleaning solvents or particular tools, depend on the condition and nature of a part, resulting in product flow and equipment needs that are uncertain until used products arrive at a facility. Reclaimed, returned, and remanufactured items all must be stored in inventory until they are needed to satisfy demand. All these sources of uncertainty in space needs make the design of a remanufacturing facility more complex than in traditional settings. Generalised reconfigurable remanufacturing process and W.I.P. (Work-in-process) inventory

Di sp os ab

le

pa r

ts

Figure 1

c

Storage needs and capacities

As product flows through a remanufacturing process, items or components are stored in several locations. Generally, upon receipt, returned products initially are routed to returned product storage, possibly with excess product stored externally (at a cost). After disassembly, the constituent parts are sorted into usable or disposable items, with the former stored in usable part storage until they are inspected and tested; parts exceeding this storage capacity are sent to external storage. Disposable parts exceeding the disposal shipping capacity are held elsewhere, in our example in returned product storage by

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reallocating some of this space away from returned products use. After inspection, parts requiring repair are held in repairable part storage, accepted parts are routed to finished part storage, and non-repairable parts are sent to disposable part storage. If there is not enough space to store repairable parts, space is reallocated from elsewhere for this purpose – here from usable part storage, if capacity exists – with any overflow stored externally. Once the necessary work on repairable parts is performed, these components are held in finished part storage. In each time period, demand for remanufactured products either is completely met by some or all of the available finished parts or additional new parts are purchased from an OEM (original equipment manufacturer). If demand is less than the available finished parts, excess parts remain in finished part inventory until the next time period. However, if the converse is true, new parts are ordered in batches of one or more (user-specified). Extra new parts are stored either by reconfiguring space from finished parts inventory or externally. Finally, after re-assembly is complete, remanufactured products are stored in remanufactured product inventory before being shipped to customers. If the capacity of remanufactured product inventory is less than the number of remanufactured products, then the excess items are stored externally and shipped directly to customers from there. Shipments to customers are made every period. Throughout this process, storage space for a particular part type can be reallocated either for another use for that part type or for the same use for a different part type. Each time period begins with the arrival of returned products in batches and ends with all demand fully satisfied.

3

Modelling framework

To determine optimal storage capacities in each time period that minimise total expected cost, a simulation-optimisation approach shown in Figure 2 was developed. A Monte Carlo program emulates the general logic and stochastic events as material flows through the remanufacturing facility, including random return volumes, component yields, and refurbished demand. The model runs for a user-specified number of time periods and replications, tracking the overall costs of total, internal, reconfigured, external, and unused space for any given set of inputs (storage capacities, costs). This simulation program can be used either to conduct what-if analysis or within an optimisation framework. For the later, a multi-dimensional golden section search algorithm was developed that iteratively calls the Monte Carlo program, passing it a vector of storage capacities for evaluation, and repeating this logic to search for the combination of each decision variable (storage capacity) that minimises long-term expected total cost. Hybrid simulation-optimisation has been used in similar cases where the stochastic and combinatorial aspects of the problem make other approaches somewhat intractable (Ferrari et al., 2003). Prior work discussed in Benneyan et al. (2010) and Topcu et al. (2008) developed stochastic recourse models to determine optimal storage capacities and reconfiguration schedules that proved somewhat intractable for realistically sized problems, with model size (number of constraints and variables) and computation time increasing dramatically as a function of the number of periods. Other prior work illustrated that accounting for variability in returned product volumes and yield rates clearly is important, with even a simple example based on three components causes capacity requirements to fluctuate significantly (Topcu et al., 2007). All these sources of

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randomness affect inventory accumulation, resulting in random capacity needs that can substantially impact the layout of a remanufacturing facility. There are controllable and uncontrollable variables inherent in these types of systems, where the subscripts p, t, and r indicate the part type (1,…, P), time period (1,…, T), and simulation replication (1,…, R), respectively, and with uncontrollable variables including yield percentages, number of returned products per time period, and demand for remanufactured items. Overall yield quantities for each type of recaptured component are functions of four specific yield rates: usable parts, repairable parts, repaired parts, and acceptable parts. Usable parts are defined as those that appear good upon disassembly, whereas acceptable parts are those that then are confirmed as good upon inspection and testing. Process performance measures include space (internal, reconfigured, external, unused) and associated storage cost (total, internal, reconfigured, external) measures, where total cost includes all storage costs over all periods plus the opportunity cost of unused space, TC p ,t = IC p ,t + RC p ,t + EC p ,t + US p , t Figure 2

(1)

Overall optimisation framework

ε

In model output variables, please note that p is dropped and becomes returned or remanufactured when referring to the products. The simulation model is fairly flexible and can be used for a product composed of any number of part types. The user can specify whether parts stored externally will be used in the current or next period, whether ordering new parts is an option to satisfy demand, the batch size of new parts, the type of probability distributions for the arrival of returned products to the remanufacturing facility and remanufactured product demand, and the unit costs of storing each part type internally, externally, or in reconfigured space.

A simulation-optimisation approach for reconfigurable inventory space

4 a

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Simulation model User interface

The simulation model is implemented in Microsoft VB.NET, an object-oriented computer language running Visual Basic on Microsoft’s .NET framework, includes a front-end GUI user interface. The user enters the inputs in two separate screens, one for macro-level information and the other for detailed cost data, as illustrated in Figures 3 and 4. On the first input screen, the user specifies the mean and standard deviation of the number of returned products, demand for remanufactured products, number of components that can be recaptured from each product, internal and external storage costs for returned and remanufactured items, and simulation run conditions (number of time periods and replications). The number of returned products and demand for remanufactured products both follow a normal distribution with a mean and standard deviation specified by the user on the first screen. On the second screen, the user enters the number of recaptured parts per item, batch size of externally sourced replacement parts, and yield rates and associated unit storage costs for each of the four processes through which parts flow (disassembly, inspection, repair, reassembly). Alternately, all information can be imported from a Microsoft Excel file. After the simulation executes, results are exported automatically to an Excel file, including the mean, standard deviation, minimum, and maximum of total, internal, reconfiguration, external, unused space, and new part demand costs. Figure 3

Simulation model macro-level input screen (see online version for colours)

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Figure 4

b

Simulation model detailed cost and part input screen (see online version for colours)

Program logic

The Monte Carlo model operates according to the following assumptions and general logic, where the circles in Figures 5 through 10 denote seven general capacitated storage locations: C.1 denotes returned products, C.2 usable parts, C.3 disposable parts, C.4 repairable parts, C.5 new parts, C.6 finished parts, and C.7 remanufactured products. As shown in Figure 5, a quantity of XR,t returned products arrives to the remanufacturing facility in each time period t. The internal storage cost for returned products is TR,t ⋅ cr,internal, where TR,t = min (XR,t, ϑR,t) i.e., the number of returned products that are stored in the returned product storage area, with any excess items ER,t = max (0, XR,t – TR,t) stored externally at a cost of ER,t ⋅ cR,external. After disassembly of a returned product, component types eligible for recapture are sorted into either usable parts (stored in usable part inventory) or disposable parts (stored in disposable part inventory), as shown in Figures 6a and 6b. If the number of usable parts is less than the usable part storage capacity (for that part type), the internal usable parts storage cost is TU p ,t ⋅ cU p ,internal ; otherwise, additional parts ( EU p ,t ) first are stored in any available usable part storage for other component types, at a reconfiguration and storage cost of RU p ,t ⋅ cU p ,recon , with any remaining usable parts (if not enough local ‘overflow’ space is available) stored externally at a higher cost of ( EU p ,t − RU p ,t ) cU p ,external . In addition to the above flow, components arrive to disposable part inventory from two other activities, both discussed below: 1

inspection and testing

2

repair.

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The cost of storing these parts (in disposable part inventory) before they are disposed (possibly shipped to an external material processing or hazardous waste facility) is TD p ,t ⋅ cD p ,internal . If the disposable part storage capacity is exceeded, additional parts ( ED p ,t ) first are stored in any available disposable part storage for other component types at a cost of

RD p ,t ⋅ cD p ,recon

and then externally at a cost of space

( ED ,t − RD ,t ) cD ,external . p

Figure 5

p

p

Arrival logic for returned products

Accept the returned products X R, t that arrive at the remanufacturing facility

The total number of returned products is the returned products from the current period plus the extra products from previous period:

TR , t = X R , t + E R , (t −1) C.1

TR, t < ϑR, t

NO

S R, t = 0

YES

The number of extra returned products:

E R, t = TR, t − ϑ R, t

The unused space in returned product inventory:

S R, t = ϑ R, t − TR, t

Store extra returned products externally Send returned products to disassembly External storage

Disassemble products

Sort parts

Sorted part type?

A. Topcu et al.

96 Figure 6

(a) Disassembly and sorting process logic – disposable parts, (b) disassembly and sorting process logic – usable parts Disposable parts

Disposable parts from inspection and repair processes

Sorted part type?

The number of disposable part from the current period:

X Dp , t = TR, t . λ Dp , t The total number of disposable parts is the disposable parts from the current period, plus the extras from previous period, plus the disposable parts from inspection and repair processes:

TDp , t = X Dp , t + E Dp , (t −1) + X D1p , t + X D 2 p , t C.3

NO

SDp , t = 0

XDp,t 0

X D2 p , t = TLp , t . λD2 p , t

TW p , t = TL p , t . λW p , t

TFP , t = TAp , t + TWp , t + E Fp , (t −1) + E F1p , (t −1)

(b)

Is E L p , t still > 0

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Usable parts are sent to cleaning, inspection, and testing operations where they are deemed acceptable, repairable, or disposable. Accepted parts are stored in finished part inventory, disposable parts are directed to disposable part inventory, and repairable parts are sent to repairing area, as shown in Figures 7(a) and 7(b). The internal storage cost for repairable parts is TLp ,t ⋅ cLp ,internal . Parts beyond the repairable part storage capacity

( EL ,t ) p

are first stored in any available repairable part storage for other component types

at a cost of RL p ,t ⋅ cLp ,recon and then in usable part storage space, if available, at a reconfiguring and storage cost of is RLU p ,t ⋅ c p ,urecon . Finally, any remaining parts are stored externally at a cost of ( EL p ,t − RL p ,t − RLU p ,t ) ⋅ cLp ,external . Once repaired, these parts are stored in finished part inventory, along with accepted parts, at a cost of TFp ,t ⋅ cFp ,internal ; parts that cannot be repaired are disposed following the earlier logic. Note that both acceptable and repaired parts are sent to finished part inventory storage. Figure 8

Finished parts logic The total number of finished parts:

TFP , t = TAp , t + TWp , t + EFp , (t −1) + EF1p , (t −1) New parts

NO

S Fp , t = 0

C.6

TFp , t < ϑ Fp , t YES

The number of extra finished parts due to capacity:

E Fp , t = T Fp , t − ϑ Fp , t The unused space in finished part inventory:

SFp , t = ϑFp , t − EFp , t

Store extra finished parts externally

External storage

TFp , t > D p, t

NO

YES The number of extra finished parts after demand is met:

E F1p , t = TFp , t − D p, t

The new part demand is:

TN p , t = Dp, t −TFp , t

As shown in Figure 8, if the number of finished parts is larger than the finished part storage capacity (for that part type), the additional parts ( EFp ,t ) first are stored in any available finished part storage for other component types, at a reconfiguration and storage cost of RFp ,t ⋅ cFp ,recon , with any remaining finished parts stored externally at a higher cost of ( EFp ,t − RFp ,t ) ⋅ cFp ,external .

100 Figure 9

A. Topcu et al. Demand fulfilment logic

TN p , t = Dp, t − TFp , t

TNp , t > ENp , (t−1)

TOp , t = 0

TOp , t = TN p , t − E N p , (t −1)

TOp , t + ENp , (t −1) < ϑNp , t

SNp ,t = 0

E N p , t = TOp , t + E N p , (t −1) − ϑ N p , t SNp , t =ϑNp , t −TOp ,t − ENp , (t−1)

Is E N p , t still > 0

ENp , t = TOp , t + ENp , (t −1) −TNp , t

S Fp , t > 0

Is E N p , t still > 0

If demand exceeds the supply of finished refurbished parts, the availability and number of any extra parts in new parts inventory from previous periods are checked as shown in Figure 9. If demand is still higher, new parts are ordered (in user-specified batch sizes) to meet demand and stored at a temporary storage cost of TN p ,t ⋅ cN p ,internal . If the number of new parts ordered is higher than the new part storage capacity (for that part type), the additional parts ( E N p ,t ) first are stored in any available new part storage for other component types, at a reconfiguration and storage cost of RN p ,t ⋅ cN p ,recon , with any

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remaining new parts stored in finished part storage space, if available, at a reconfiguring and storage cost of is RNFp ,t ⋅ c p , frecon , or externally at a higher cost of

( EN ,t − RN ,t − RNF ,t ) ⋅ cN ,external . New parts are assumed to arrive within the same time p

p

p

p

period ordered, such that the remanufacturing process is not delayed by delivery lags. Once finished refurbished and new (if any needed) parts are assembled into remanufactured products, they are stored in the remanufactured product inventory at a cost of TM,t ⋅ cM,internal before shipping to the customer, with extras stored externally at a cost of EM,t ⋅ cM,external as shown in Figure 10. Demand for remanufactured products every time period always is satisfied, and new part supply is assumed to be infinite (so no backordering occurs of either remanufactured items or replacement parts). Figure 10

Assembly process logic Assemble parts into products

Send remanufactured products into remanufactured product inventory

C.7

The number of extra remanufactured products:

EM , t = TM , t −ϑM , t

NO

SM,t = 0

EM , t < ϑM , t YES

Store extra remanufactured products externally

External storage

The unused space in remanufactured product inventory:

S M , t = ϑM , t − E M , t

Ship remanufactured products to customers

The total cost for the entire remanufacturing process is the sum of the internal, reconfigured, and external storage space costs plus the opportunity cost of unused space. TC p ,t = cunused ( S R ,t + SU p ,t + S D p ,t + S Lp ,t + S Fp ,t + S N p ,t + S M ,t ) + IC p ,t + RC p ,t + EC p ,t

where

(2)

102

A. Topcu et al. IC p ,t = ( cR ,internal ⋅ TR ,t ) + ( cU p ,internal ⋅ TU p ,t ) + ( cD p ,internal ⋅ TD p ,t ) + ( cL p ,internal ⋅ TLp ,t ) + ( cFp ,internal ⋅ TFp ,t ) + ( cN p ,internal ⋅ TN p ,t ) + ( cM ,internal ⋅ TM ,t ) ,

RC p ,t = ( cU p ,recon ⋅ RU p ,t ) + ( cD p ,recon ⋅ RD p ,t ) + ( cL p ,recon ⋅ RLp ,t ) + ( cFp ,recon ⋅ RFp ,t ) + ( cN p ,recon ⋅ RN p ,t ) + ( c p ,urecon ⋅ RLU p ,t ) + ( c p , frecon ⋅ RNFp ,t ) ,

(3)

(4)

and

(

EC p ,t = ( cR ,external ⋅ ER ,t ) + c p ,external cU p ,external ( EU p ,t − RU p ,t ) + cD p ,external

( ED ,t − RD ,t ) + cL ,external ( EL ,t − RL ,t − RLU ,t ) + cF ,external ( EF ,t − RF ,t ) + cN ,external ( EN ,t − RN ,t − RNF ,t ) ) + ( cM ,external ⋅ EM ,t ) p

p

p

p

p

p

p

p

p

p

p

p

p

(5)

In the below examples, the opportunity cost of unused space is cunused = $6 i.e., the cost of not using space allocated for a part or product in the specified period.

c

Examples

Example 1: Cell phones with two recaptured parts Gupta el al. (2004) described a cellular phone remanufacturing facility in which 21 components are recaptured and refurbished, with each phone requiring 4 processes: 1

disassembly and sorting

2

inspection and testing

3

repair and testing

4

assembly.

To simplify this example, we assumed just two parts are recaptured: 1

batteries

2

printed circuit boards (PCBs).

The yield percentages for each part type at each process, the number of returned cellular phones, and the demand for remanufactured phones all are random. Returned phones arrive to the facility with mean of 150 and a standard deviation of 10 per time period. Demand for remanufactured phones has a mean of 100 and a standard deviation of 5 per time period. Yield rates in each step of the process are summarised in Table 1. In the first scenario, returned product capacity is 150; usable, disposable, repairable, new, and finished part inventory capacities for both batteries and PCBs are 100, 30, 10, 5, and 80, respectively; and remanufactured product capacity is 100. In the second scenario, the finished part inventory for both parts is increased to 120. In the third scenario, the usable part inventory for both parts is increased to 130. An increase from 10 to 40 for the repairable part inventory is done in the fourth scenario. Internal and external storage costs per returned and remanufactured phones are $3 and $6, respectively. Internal, reconfiguration, and external storage costs per battery are $3, $3, and $6; whereas these costs for PCBs are $1, $1, and $2.

A simulation-optimisation approach for reconfigurable inventory space Table 1

103

Uncontrollable variables

Uncontrollable variables

Battery

PCB

Mean for number of returned products

Cover

Antenna

150

Standard deviation for number of returned products

10

Usable part yield percentage

90%

70%

81%

Acceptable part yield percentage

80%

65%

72%

92%

Repairable part yield percentage

75%

95%

89%

71%

Repaired part yield percentage

80%

90%

95%

78%

Mean for demand

100

Standard deviation for demand Table 2

73%

5

Example 1 inputs and outputs (Cell phones, two recaptured parts)

Run scenario

Results Total cost ($)

Internal cost ($)

Reconfigured cost ($)

External cost ($)

Unused space (units)

1

5,294 (225)

3,423 (24)

27 (4)

1,704 (71)

23 (4)

2

4,820 (123)

3,462 (32)

31 (5)

1,267 (47)

10 (2)

3

4,567 (156)

3,382 (46)

68 (6)

912 (18)

18 (4)

4

4,402 (168)

3,301 (12)

94 (2)

906 (22)

17 (5)

As seen by comparing the first two scenarios in Table 2, an increase of 50% in the finished part (C.6) inventory capacity (from 80 to 120 storage spaces) causes the total cost to decrease by 9% from $5,294 to $4,820. Increasing usable part (C.2) capacity from 100 to 130 in the third scenario decreases total cost by 7.3% from $4,820 to $4,567. In the fourth scenario, an increase from 10 to 40 in the repairable part (C.4) inventory capacity decreases total cost to $4,402 from $4,567. Standard deviations of results are shown in parentheses. Example 2: Cell phones with four recaptured parts As a larger example, suppose instead that each phone has four recapturable parts: 1

batteries

2

printed circuit boards

3

front and back covers

4

antennas, with yield rates for the two additional parts shown in the right hand columns of Table 1.

Returned products and demand for remanufactured products occur with the same rates as in the first example. Table 3 summarises the run results for four scenarios. Same capacity assumptions are valid as in the example with two recaptured parts. Internal and external storage costs for returned and remanufactured phones are $3 and $6 per item,

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respectively, and internal, reconfiguration, and external storage costs for batteries are $3, $3, and $6; $1, $1, and $2 for the PCBs; $2, $2, and $4 for covers; and $3, $3, and $9 for antennas. Table 3 Run scenario

Example 2 inputs and outputs (Cell phones, four recaptured parts) Results Total cost ($)

Internal cost ($)

Reconfigured cost ($)

External cost ($)

Unused space (units)

1

9,264 (789)

5,877 (65)

45 (17)

3,134 ($125)

35 (15)

2

8,992 (574)

6,405 (103)

55 (29)

2,135 (138)

66 (22)

3

8,306 (353)

6,278 (112)

124 (34)

1,802 (125)

17 (7)

4

8,041 (209)

6,001 (84)

155 (6)

1,676 (153)

35 (9)

As seen by comparing the first two scenarios, an increase of 50% in finished part (C.6) inventory capacity decreases total cost by 2.9% from $9,264 to $8,992. Increasing usable part (C.2) inventory capacity from 100 to 130 decreases total cost by 7.6%. In the fourth analysis, an increase from 10 to 40 in the repairable part (C.4) capacity decreases total cost by 3.2% to $8,041. As these two examples indicate, total cost can be affected significantly by changes in inventory capacities.

5

Optimisation model

5.1 Golden section search algorithm To identify optimal storage capacities at each location for each part type, we implemented a multi-dimensional golden section search algorithm (Gerald and Wheatley, 2004) which is a technique to find the minimum or maximum (extremum) of a unimodal function over an interval [a, b] by iteratively narrowing the search interval in a manner that guarantees it will contain the minimum value. Please refer to the appendix for a general introduction to the golden section search. This routine efficiently searches for the optimal capacities of each storage and part type, within user-defined upper and lower bounds to minimise total cost over a user-specified number of time periods. These optimal capacities are achieved by reducing the interval under consideration in each step, and continuing until the difference between the lower (CapLow) and upper bounds (CapHigh) is less than some very small value α, searching across each part type, as illustrated by Figure 11. Denoting a and b to be the lower and upper bounds, respectively, we compute x1 = c × a + (1 – c) b and x2 = (1 – c) a + c × b, where c = (√5 – 1) / 2 is the constant reduction factor for the size of the interval in each step. If the total cost at x1 is less than at x2, then x2 becomes the new upper bound, x1 becomes the new value for x2, and a new x1 is computed as c × a + (1 – c) b, so the new search interval now is [a, x2]. Otherwise, x1 becomes the new lower bound, x2 becomes the new value for x1, and a new x2 is calculated (1 – c) a + c × b, so that now the new search interval is [x1, b]. This search

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procedure continues until the difference between the lower and upper bounds is less than some user specified small value α, repeating for all optimal inventory capacities for all part types. Figure 11

Flowchart of multi-dimensional golden section search algorithm

α

≥

≥

The lower and upper capacity bounds for the cell phone example with four recaptured parts are given in Table 4. As the number of parts and storage locations increase, search computation times become exponentially prohibitive, due to the nested searching. To achieve faster results, a non-optimal heuristic based on a series of sequential (un-nested) searches might instead be conducted on each capacity one-at-a-time, with all other capacities set to their current values (or their mid values on the first iteration), repeating this process several times to improve the solution on each iteration. Performance of this

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heuristic is compared to optimal results and often found to be reasonably close. This performance agrees with related work, not reported here (Topcu, 2009), in which experimental design analysis found most interactions between the various storage location capacities to be statistically negligible. Table 4

Lower and upper capacity bounds used in the cell phone example

Type of inventory

Lower bound

Upper bound

120

180

Returned product inventory space Usable part inventory space

80

170

Disposable part inventory space

10

50

Repairable part inventory space

5

30

New part inventory space

4

30

Finished part inventory space

80

200

Remanufactured product inventory space

100

200

5.2 Sensitivity analysis Tables 5 and 6 summarise sensitivity analysis to explore how optimal capacities, storage costs, and CPU times are affected by yield percentages, the number of returned products, and demand. Due to long computation times, only models with four or fewer inventory types were investigated. Table 5

Storage cost assumptions for sensitivity analysis Type of cost

Product related Part related

Scenario 2

Scenario 3

Internal storage

$3

$3

$3

External storage

$6

$6

$9

Internal storage

$3a; $1b

$3; $1b

$3a; $2b

Within part-type reconfiguration

$3a; $1b

$6; $2b

$6a; $3b

Within inventory-type reconfiguration

$3a; $1b

$6; $2b

$6a; $3b

External storage

$6a; $2b

$6; $2b

$9a; $6b

$6

$6

$6

Opportunity cost of unused internal space a

Scenario 1

b

Notes: battery; PCB

As shown in Table 6, run times are significantly affected by the number of possible inventory locations, but not by specific cost assumptions. As shown in Table 7, search computation time using the heuristic for a four-location, two-part problem is reduced by roughly 97% from 2,954 minutes to 84 minutes, at most 2% increase in total cost for these examples under consideration. These results also give run times of roughly 402 days to optimally solve a problem with seven storage locations and two part types, whereas the un-nested heuristic finds a solution in only about 192 minutes.

A simulation-optimisation approach for reconfigurable inventory space Table 6

Scenario 1

2

3

Table 7

Scenario 1

2

3

6

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Optimal results for different cost assumptions Number of inventory capacities optimised

Optimal capacities Part

C.1 C.2 C.3 Return Usable Disposable

C.4 Repairable

Total cost ($)

Run time (min)

2

Battery PCB

147

131 121

-

-

5,172

122.8

3

Battery PCB

145

136 117

-

8 22

5,099

477.9

4

Battery PCB

151

145 101

14 42

8 18

5,001

2,991.6

2

Battery PCB

145

136 120

-

-

5,499

127.1

3

Battery PCB

138

131 120

-

4 19

5,415

513.2

4

Battery PCB

142

138 111

13 39

8 24

5,288

2,970.5

2

Battery PCB

156

148 131

-

-

6,561

115.7

3

Battery PCB

151

127 119

-

18 29

6,450

453.5

4

Battery PCB

158

149 120

15 49

7 23

6,328

2,899.4

Comparison of optimal and heuristic solutions based on total cost and run time Number of inventory capacities optimised

Total cost ($) Optimal

Heuristic

Run time (min) Relative

Optimal Heuristic

Relative

2

5,172

5,208

1.01

122.8

51.7

0.42

3

5,099

5,134

1.01

477.9

71.2

0.15

4

5,001

5,108

1.02

2,991.6

83.6

0.03

2

5,499

5,503

1.00

127.1

52.0

0.41

3

5,415

5,450

1.01

513.2

69.6

0.14

4

5,288

5,292

1.00

2,970.5

82.4

0.03

2

6,561

6,605

1.01

115.7

49.7

0.43

3

6,450

6,490

1.01

453.5

67.9

0.15

4

6,328

6,412

1.01

2,899.4

85.6

0.03

Conclusions

Total facility cost in remanufacturing can be significantly affected by allocated inventory capacities. A simulation-based optimisation model was developed to identify optimal capacity levels and reconfiguration decisions to minimise total long-term expected costs. While this approach is viable for a small number of inventory and part types, heuristics

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will be necessary for larger-sized problems. Using heuristics, run times were reduced by roughly 97%. The run time to find the optimal solution is computed to vary between 116 and 127 minutes with two inventory capacities. Increasing the model size to four inventory capacities increases the run time to vary between 48 and 50 hours. With the heuristic approach based on this algorithm, the computation duration is successfully reduced by 97% from almost 50 hours to 84 minutes for four inventory capacities as shown in scenario 1. While no longer guaranteeing optimality, in several cases costs tend to be less than 2% higher than the optimal results, which may be acceptable in most practical applications. Future possible work includes exploration of more advanced search algorithms, other non-greedy approaches, or use of this model to develop general guidelines, insights, and rules of thumb for practitioners. Another possible extension of this research is testing reconfiguration on systems other than remanufacturing, such as hospital and emergency room settings. A limitation of this work is that we assumed the remanufacturing system starts empty in time period 0 whereas in real settings items are most likely already present in the system.

References Alinovi, A., Bottani, E. and Montanari, R. (2012) ‘Reverse logistics: a stochastic EOQ-based inventory control model for mixed manufacturing/remanufacturing systems with return policies’, International Journal of Production Research, Vol. 50, No. 5, pp.1243–1264. Benjaafar, S., Heragu, S.S. and Irani, S.A. (2002) ‘Next generation factory layouts: research challenges and recent progress’, Interfaces, Vol. 32, No. 6, pp.58–76. Benneyan, J.C., Cullinane, T.P. and Topcu, A. (2010) ‘Stochastic programming recourse models for reconfigurable multi-period storage allocation in remanufacturing facilities’, International Journal of Operations Research, Vol. 7, No. 4, pp.11–26. Bienstock, C.C., Amini, M. and Retzlaff-Roberts, D. (2011) ‘Reengineering a reverse supply chain for product returns services’, International Journal of Business Performance and Supply Chain Modelling, Vol. 3, No. 4, pp.335–352. Canon Inc (2012) Sustainability Report [online] http://www.canon.com/csr/report/pdf/Canon_Sustainability_Report_2012_e.pdf (accessed 27 September 2012). Chanintrakul, P., Mondragon, A.E.C., Lalwani, C. and Wong, C.Y. (2009) ‘Reverse logistics network design: a-state-of-the-art literature review’, International Journal of Business Performance and Supply Chain Modelling, Vol. 1, No. 1, pp.61–81. Dowlatshahi, S. (2012) ‘A framework for the role of warehousing in reverse logistics’, International Journal of Production Research, Vol. 50, No. 5, pp.1265–1277. Eastman Kodak Company (2010) Sustainability Report [online] http://www.kodak.com/ek/uploadedFiles/Content/About_Kodak/Global_Sustainability/Govern ance/Global_Sustainability_Reports/EBP_Environmental_5.11.11.pdf (accessed 27 September 2012). Elmaraghy, H.A. (2008) ‘Flexible and reconfigurable manufacturing systems paradigms’, International Journal of Flexible Manufacturing Systems, Vol. 17, No. 4, pp.261–276. Ferrari, E., Pareschi, A., Regattieri, A. and Persona, A. (2003) ‘Plant layout computerised design: logistics and relayout program (LRP)’, International Journal of Advanced Manufacturing Technology, Vol. 21, No. 12, pp.917–922.

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109

Franke, C., Basdere, B., Ciupek, M. and Seliger, S. (2006) ‘Remanufacturing of mobile phones – capacity, program and facility adaptation planning’, Omega, Vol. 34, No. 6, pp.562–570. Gerald, C.F. and Wheatley, P.O. (2004) Applied Numerical Analysis, 7th ed., Addison-Wesley, Reading, Massachusetts. Gupta, S.M., Erbis, E. and McGovern, S.M. (2004) ‘Disassembly sequencing problem: a case study of a cell phone’, Proceedings of the SPIE Conference on Environmentally Conscious Manufacturing IV, Bellingham, WA, pp.43–52. Kekre, S., Rao, U.S., Swaminathan, J.M. and Zhang, J. (2003) ‘Reconfiguring a remanufacturing line at Visteon, Mexico’, Interfaces, Vol. 33, No. 6, pp.30–43. Kiefer, J. (1953) ‘Sequential minimax search for a maximum’, Proceedings of the American Mathematical Society, Vol. 4, No. 3, pp.502–506. Kusumastuti, R.D., Piplani, R. and Lim, G.H. (2009) ‘Designing reverse logistics network for product recovery’, International Journal of Business Performance and Supply Chain Modelling, Vol. 1, No. 4, pp.257–289. Lee, G.H. (1997) ‘Reconfigurability consideration design of components and manufacturing systems’, International Journal of Advanced Manufacturing Technology, Vol. 13, No. 5, pp.376–386. Lim, H-H. and Noble, J.S. (2006) ‘The impact of facility layout on overall remanufacturing system performance’, International Journal of Industrial and Systems Engineering, Vol. 1, No. 3, pp.357–371. Mathews, J.J. and Fink, K.D. (2004) Numerical Methods: Using Matlab, 4th ed., Prentice-Hall Pub. Inc., Upper Saddle River, NJ. Mehrabi, M.G., Ulsoy, A.G. and Koren, Y. (2000) ‘Reconfigurable manufacturing systems: key to future manufacturing’, Journal of Intelligent Manufacturing, Vol. 11, No. 4, pp.403–419. Mellor, C. (2002) ‘Quick-change artists: why techs must get ready for reconfigurable manufacturing’, The Ontario Technologist, January/February, Vol. 44, No. 1, pp.12–15. Meng, G., Heragu, S.S. and Zijm, H. (2004) ‘Reconfigurable layout problem’, International Journal of Production Research, Vol. 42, No. 22, pp.4709–4729. Meyer, H. (1999) ‘Many happy returns’, Journal of Business Strategy, Vol. 20, No. 4, pp.27–31. Nenes, G. and Nikolaidis, Y. (2012) ‘A multi-period model for managing used product returns’, International Journal of Production Research, Vol. 50, No. 5, pp.1360–1376. Piplani, R. and Saraswat, A. (2012) ‘Robust optimization approach to the design of service networks for reverse logistics’, International Journal of Production Research, Vol. 50, No. 5, pp.1424–1437. Sharp Corporation (2012) ‘Sustainability report’ [online] http://www.sharp.co.jp/corporate/eco/csr_report/pdf/esr2012e.pdf (accessed 27 September 2012). Subramanian, P., Ramkumar, N. and Narendran, T.T. (2010) ‘Mathematical model for multi-echelon, multi-product, single time-period closed loop supply chain’, International Journal of Business Performance and Supply Chain Modelling, Vol. 2, Nos. 3–4, pp.216–236. Toffel, M.W. (2003) ‘The growing strategic importance of end-of-life product management’, California Management Review, Vol. 45, No. 3, pp.102–131. Topcu, A. (2009) A Heuristic Approach Based on Golden Section Simulation-Optimization for Reconfigurable Remanufacturing Inventory Space Planning, Doctoral dissertation, Northeastern University, Boston, MA. Topcu, A. and Cullinane, T. P. (2005) ‘Understanding facilities design parameters for a remanufacturing system’, Proceedings of the SPIE Conference on Environmentally Conscious Manufacturing V, pp.163–173, Boston, MA.

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Topcu, A., Benneyan, J.C. and Cullinane, T.P. (2007) ‘Facility and storage space design issues in remanufacturing’, in Gupta, S.M. and Lambert, A.J.D. (Eds.): Environment Conscious Manufacturing, pp.413–444, CRC Press, Boca Raton, FL. Topcu, A., Benneyan, J.C. and Cullinane, T.P. (2008) ‘Stochastic programming recourse models for reconfigurable multi-period storage allocation in remanufacturing pull facilities’, Proceedings of the 2008 Industrial Engineering Research Conference, pp.1718–1723, Vancouver, BC, Canada. U.S. Environment Protection Agency (1994) ‘EPA proposes to reduce air pollutants from municipal waste incinerators’ [online] http://yosemite.epa.gov/opa/admpress.nsf/621960f958b27cf18525701c005d8428/1eb9b464ebf c3ba58525644c002fd60d!OpenDocument (accessed 27 September 2012). Xerox Corporation (2011) ‘Environment, health and safety report’ [online] http://www.xerox.com/corporatecitizenship/2011/pdf/Environment_Health_Safety_Report_2011.pdf (accessed 27 September 2012).

Nomenclature Symbol

Description

λU p ,t

Usable part yield rate

λD p ,t

Disposable part yield rate

λD1p ,t

Disposable part after inspection yield rate

λD 2 p ,t

Disposable part after repair yield rate

λLp ,t

Repairable part yield rate

λW p ,t

Repaired part yield rate

λAp ,t

Acceptable part yield rate

XR, t

Number of returned products

X U p ,t

Number of usable parts

X D p ,t

Number of disposable parts

X D1p ,t

Number of disposable parts after inspection

X D 2 p ,t

Number of disposable parts after repair

X Lp ,t

Number of repairable parts

ER, t

Number of spare returned products

EU p ,t

Number of spare usable parts

EDp ,t

Number of spare disposable parts

ELp ,t

Number of spare repairable parts

EFp ,t

Number of spare finished parts

EF 1p ,t

Number of spare finished parts after demand is met

E N p ,t

Number of spare new parts

EM, t

Number of spare remanufactured products

TR, t

Total number of returned products

A simulation-optimisation approach for reconfigurable inventory space

Nomenclature (continued) TU p ,t

Total number of usable parts

TD p ,t

Total number of disposable parts

TLp ,t

Total number of repairable parts

TW p ,t

Total number of repaired parts

TAp ,t

Total number of acceptable parts

TFp ,t

Total number of finished parts

TN p ,t

Total number of new parts

TO p ,t

Total number of ordered parts

TM, t

Total number of remanufactured products

SR, t

Unused space in returned product inventory

SU p ,t

Unused space in usable part inventory

S D p ,t

Unused space in disposable part inventory

S L p ,t

Unused space in repairable part inventory

S Fp ,t

Unused space in finished part inventory

S N p ,t

Unused space in new part inventory

SM,t

Unused space in remanufactured product inventory

RU p ,t

Usable parts stored in reconfigured space

RD p ,t

Disposable parts stored in reconfigured space

RLp ,t

Repairable parts stored in reconfigured space

RLU p ,t

Repairable parts stored in reconfigured usable part space

RFp ,t

Finished parts stored in reconfigured space

RN p ,t

New parts stored in reconfigured space

RNFp ,t

New parts stored in reconfigured finished part space

ϑ R p ,t

Capacity of returned product storage space

ϑU p ,t

Capacity of usable part storage space

ϑD p ,t

Capacity of disposable part storage space

ϑLp ,t

Capacity of repairable part storage space

ϑFp ,t

Capacity of finished part storage space

ϑ N p ,t

Capacity of new part storage space

ϑM, t

Capacity of remanufactured product storage space

Dp, t

Demand for refurbished parts

TCp, t

Total cost

ICp, t

Internal cost

RCp, t

Reconfigured cost

ECp, t

External cost

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Nomenclature (continued) Unused space

USp, t cU p ,internal

Unit cost of internal storage for usable parts

cLp ,internal

Unit cost of internal storage for repairable parts

cD p ,internal

Unit cost of internal storage for disposable parts

cFp ,internal

Unit cost of internal storage for finished parts

cN p ,internal

Unit cost of internal storage for new parts

cU p ,recon

Unit cost of reconfigured storage for usable parts

cL p ,recon

Unit cost of reconfigured storage for repairable parts

cD p ,recon

Unit cost of reconfigured storage for disposable parts

cFp ,recon

Unit cost of reconfigured storage for finished parts

cN p ,recon

Unit cost of reconfigured storage for new parts

cU p ,external

Unit cost of external storage for usable parts

cL p ,external

Unit cost of external storage for repairable parts

cD p ,external

Unit cost of external storage for disposable parts

cFp ,external

Unit cost of external storage for finished parts

cN p ,external

Unit cost of external storage for new parts

cp,urecon

Unit cost of reconfiguring space for repairable parts in usable part inventory

cp,frecon

Unit cost of reconfiguring space for new parts in finished part inventory

cr,internal

Unit cost of storing returned products internally

cr,external

Unit cost of storing returned products externally

cm,internal

Unit cost of storing remanufactured products internally

cm,external

Unit cost of storing remanufactured products externally

cunused

Opportunity cost of unused space

Appendix Golden section search The golden section search is an approach used to find the maximum or minimum of a unimodal function in a given interval. It was introduced by the American statistician, Jack Kiefer (1953). This approach receives its name from the fact that the algorithm maintains the function values for triples of points whose distances form a golden ratio. To be able to use this method, a special condition must be met to ensure that there is a proper minimum in the given interval. This condition is that the function has to be unimodal meaning that only one minimum or maximum exists in the given interval. Definition of unimodal function: The function, f(x), is unimodal on I = [a, b], if there exists a unique number p є I such that

A simulation-optimisation approach for reconfigurable inventory space

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f(x) is decreasing on [a, p] and f(x) is increasing on [p, b] One approach to replace the interval with a subinterval on which the function takes on its minimum value is to select two interior points, such as x1 and x2, where x1 is less than x2. Please note that having a good strategy to determine where the function is to be evaluated reduces the number of evaluations. The condition that the function is unimodal guarantees that the function values at x1 and x2, f(x1) and f(x2), are less than the function values at a and b, f(a) and f(b). Figure 12

The decision process for the golden section search (see online version for colours)

f(x2) f(x1)

x1

f(x1)

x1

x2

f(x2)

x2

If f(x1) ≤ f(x2), the minimum must occur in the subinterval [a, x2], and we replace b with x2 and continue the search in the new subinterval [a, x2]. If f(x2) < f(x1), then the minimum must occur in the interval [x1, b], and we replace a with x1 and continue the search in the new subinterval [x1, b]. These choices are shown in Figure 12. The golden section search requires the interior points x1 and x2 to be constructed in a way that the resulting subintervals [a, x1] and [x2, b] are symmetrical in [a, b]. If not, the

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run of bad luck could lead to the wider interval to be selected which slows down the rate of convergence. The reader can read more on constructing these subintervals in Mathews and Fink (2004). This iterative process terminates once the difference between points, a and b, is less than some very small value.