Managing high-end ex-demonstration product returns

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and influences of time-value on strategic, structural or operational decisions in the reverse supply chain, and have also applied different modelling approaches. Atasu and Çetinkaya (2006) and Ruiz-Benítez et al. (2014) focus on inventory control policies to coordinate collections of returns between a collection point and a remanufacturer. Guide et al. (2006) and Difrancesco et al. (2018) apply queuing network models and investigate the impact of system delays and value erosion for different reverse supply chain designs (Guide et al., 2006)), or to gain insight in return window decisions for online fashion retailers (Difrancesco et al., 2018). Queuing theory based models are well-suited to deal with system delays and different value decay rates (typically, modelled through exponential functions), yet they focus on steady state behavior so that dynamic aspects (e.g., time varying demand and decisions) are more difficult to account for. These models are also descriptive in nature: it is up to the decision modeler to propose and experiment with various alternatives. Discrete-time, multiple period linear (LP) and integer programming (IP) models have also been employed (e.g., Guide et al., (2005); Fang et al. (2016)).
These models do have the capability to deal with various time varying features (costs, demands and decisions), but they are less flexible in their ability to deal with delays (lead times and delays must be integer multiples of the base planning period). With LP and IP models it may also be more difficult to uncover more general insights, especially if conclusions are to be derived from a rather limited set of computational experiments.
A key difference between the previous studies and the study we present here is that we consider a finite lifecycle model (see e.g., Atasu and Çetinkaya (2006) and Geyer et al. (2007)). This is of particular importance for high-end equipment (e.g. IT, medical and industrial applications) considered here where the product lifecycle is short and the number of reuse cycles are limited. We investigate this problem in continuous time and we model it as special type of a Separated Continuous Linear Program (SCLP) (see e.g., Pullan (1996)), which we can convert (as shown in Appendix) into a Continuous time Transportation Problem (CTP). To the best of our knowledge, we are not aware of other SCLP applications in the area of reverse logistics. SCLP is an elegant modelling approach, especially for dynamic problems. These models, however, are less popular than their approximate multiple period linear programming analogues, derived by time discretization. The solution approach we present (as well as to SCLPs more generally, see e.g. Pullan (1996)) changes at discrete points in time over the planning horizon. Our attention is devoted to uncover and explain where and why these switch points occur in the product lifecycle. This analysis enables stronger and more generalizable conclusions compared to analyzing our problem in discrete time.
We use a case example in Section 2 to describe and analyze the generic features of a demonstration service environment for high-end equipment that needs to be captured in a decision model. Section 3 presents a generic, deterministic, continuous time, finite lifecycle model that captures the key trade-offs in the high-end equipment environment where a demonstration product is collected and reused for a further demo request, or its residual value is salvaged in the secondary market. The objective is to maximize the revenue from selling ex-demo products in the secondary market while making sure that all demo requests are met at minimum cost. Value erosion is modelled as a nonincreasing secondary market revenue function. We demonstrate that a zero-inventory policy holds and that the model can be transformed into a Continuous time Transportation Problem. We derive two cost/revenue signals that enable us to distinguish between fast and slow value erosion. We show that the fast/slow erosion decision is dynamic and depends on the rate of value erosion and the length of the demonstration time. Sections 4 and 5 present the optimal demonstration pool strategies for both slow and fast erosion respectively. The focus in our analysis is on identifying the different time epochs over the product lifecycle where the solution structure changes. We show that in the case of fast erosion, it may be better to postpone the reuse activities until later in the lifecycle even if this means that more exdemonstration equipment will be scrapped later. In Section 6, we illustrate the application of the model to derive the optimal strategy for an example, and show that for a specific case company substantial savings can be made. The model and modelling approach has applicability for the design and management of closed-loop supply chains for high-end equipment and machinery products that are demonstrated before sale. Our conclusions and areas for further research follow in Section 7.

Product demonstrations at company X
Company X's European Demo Service Center (or demo store) is co-located with the factory where new and remanufactured IT products are produced. The company's product portfolio includes high-end, mid-range and entry-level servers, network storage systems and industry standard servers. The demo store provides and manages demonstration equipment for sales and marketing support activities including participations in exhibitions and tradeshows, demonstrations and evaluations at resellers and end-customers, assignments to demo rooms or benchmark centers, and internal training of the sales force. Company X generates, in Europe, revenues in excess of $200M each year from selling remanufactured products. About 65% of these products involved ex-demonstration equipment, 20% were customer and partner returns and 15% were inventory excesses. All these products were less than three years old, but involved up to five generations of technology. Figure 1 schematically presents the product flows through the demo service center. New products and components are ordered in the store (arc 1) based on demo requests by sales representatives for potential customers. Typically, each demo is customized and built to match specific customer needs. The complete demo solution may comprise products from different product lines, e.g., an IT server solution in combination with a storage system. When products arrive in the store, they are forwarded and installed at the demo site (arc 2) where the client tests the equipment for a certain time the demo loan period (in this case a maximum 3 months). During the loan period the client may decide to purchase the equipment, in which case the equipment leaves the system (buyout) (arc 3).
Alternatively, the demo is collected from the client and returned to the store (arc 4). A demo returned at the store can be reused for a second or a third installation at a client site, or it can be retired and sent to the returns center (arc 5). Here it is disassembled, tested, refurbished and sold in the secondary market (arc 6); or scrapped and recycled. The returns center also receives product returns from outside the demo process (arc 7). These products are handled in the returns center in a similar way. Finally, it is also possible to order products from the returns center into the demo store (arc 8) and use these (instead of new products) to satisfy demo requests. The demo pool managers at company X focused on the technical aspects of the equipment, rather than taking a supply chain perspective and looking at the entire system in terms of flows and bottlenecks. The demo process was controlled by a set of simple rules (such as maximum loan period and maximum residence time of a product in the demo process before retirement), with the intention to use each newly ordered product in two or three demo installations.
We analyzed the flows (as shown in Figure 1) for a selection of mid-range servers. Different server configurations may exist, but their cost and price are mainly determined by a few core components (processor, server base unit, memory modules), which is typical in many types of high-end equipment. The commonality between servers is high when we focus on the core components. The analysis below concentrates on the server processors. These are a key component in every server; they are expensive (up to several $10,000), and are lifecycle defining. New generations of processors are introduced into the market every nine to ten months. A server's sales price erodes constantly over time (values of 1-3% per month are not uncommon), and when new generations of processors are introduced, the sales price of the older technology drops sharply. Figure 2 shows the demand and pricing information for a specific processor (processor 1) over its lifecycle (we note that the numbers have been disguised). The profile of the new products demand curve is almost triangular. The profile of the demo requests curve exhibits a very similar pattern with a (less pronounced) peak near October-November 2015. In this case the demo services seem well coordinated with the new product sales. The third curve, near the base of the graph, corresponds to the sales of refurbished products in the secondary market. These volumes are much lower and lag (typically 12 to 15 months) the shipments to the primary market. This may be due to the limited availability of returned and refurbished products earlier in the lifecycle, or an undeveloped secondary market, or a strategic decision regarding when to start selling in the secondary market so as to avoid cannibalization effects on new product sales. The curves on the right in Figure 2 show typical processor sales prices over their lifecycle.
Processors 1 and 2 were introduced in the market in November 2014. Processor 2 is more powerful, hence the higher price. In August 2015, a new generation processor became available, and consequently the prices for both processors 1 and 2 dropped considerably. Processor 1's active lifecycle finishes around August 2016, while processor 2's lifecycle continues for a few more months.  The curves show the cumulative volumes over time of demo requests (arc 2 in Figure 1); new processors ordered in the store (arc 1 in Figure 1); collections from the client sites back to the store (arc 4 in Figure   1); retirements to the returns center (arc 5 in Figure 1); and buyouts (arc 3 in Figure 1). We note that all the demo requests in Figure 3 are satisfied: the earliest requests are always fulfilled by using new products (the new products curve in Figure 3 almost coincides with the requests curve is rather flat, but the requests curve is still increasing). The horizontal displacement between the requests curve and the collections curve (adjusted for buyouts) is an indication for the actual demo loan time. The actual time that the product stayed at the clients is several months, and usually much longer than the maximum target of three months. Processor 2 is demonstrated over a longer period and most of its requests are satisfied by ordering new products in the store, even when collected / retired products were available. The buy-out rates are low and rather insignificant before August 2015. is above the maximum target of 3 months. The returns from the demo process make up a significant fraction of the total returns at the returns center, typically more than 65%. This ratio demo returns/other returns may decline over time. Most demo returns arrive in the second half and near the end of the active lifecycle of the processor. When new processor generations are introduced, clients can upgrade their servers, or trade-up the older processors for newer technology. Upgrade and trade-up programs generate a second stream of returns, which may arrive much later, long after the active lifecycle of the old processor has finished. These products have lost most of their value so that re-introduction in the secondary market is almost impossible. (In Table 1, the 'other returns' for processor 4 are much higher compared with other processors, and a significant fraction of these was due to upgrade and trade-up campaigns). The last two rows show the volumes sold in the secondary market (refurbished products) and in the primary market (new product sales). Company X manages to sell most of its demo returns in the secondary market, but there is a practice of delaying and spreading out sales over a rather long period (see Figure 2) to avoid cannibalization with new product sales.
Effective demo pool management is not an easy task. There are several characteristics and control parameters which may affect the overall profitability or cost effectiveness of the operations.
These include the shape of the demo request profile over the product's lifecycle, the demo loan period, the buyout rate, the cost of new products, the sales price of refurbished products in the secondary market and its erosion over time, the time window in which to sell refurbished products, the secondary market capacity or potential, and the availability of other returns which can be used for demonstration.
Company X used simple decision rules to control the demo store. These rules may work well under certain conditions but may need modification or refinement in other situations.
Our aim in this paper is to develop a model that captures the core aspects of a generic class of high-end equipment problems that require demonstration as part of the sales process, and to derive optimal policies for effective decision making in terms of reuse and/or resale in the secondary market of ex-demonstration product returns. As in Guide et al., (2005) by discretizing the lifecycle or planning horizon into time buckets, multiple-period linear programming models that incorporate all the abovementioned features can be developed. LP models are easy to solve and flexible in terms of handling various extensions. On the other hand, it is difficult to obtain general insights through experimentation with LPs. Therefore, we investigate the high-end equipment demonstration problem in continuous time, focusing our attention on the cost/profitability signals that drive the optimal solution and the moments in time in the planning horizon when the solution structure changes. We present our model together with the assumptions in Section 3.

Model, assumptions and notation
The problem environment modelled is shown in Figure 4. We assume that the requests d(t) for demos at any time t are known and occur over a finite lifecycle of length T, and that the demand profile is concave (or unimodal i.e., monotonically increasing for t ≤ m and monotonically decreasing for t ≥ m, with maximum at m) over [0, T). A concave, or unimodal, demand profile is a good approximation for short lifecycle products (such as IT equipment) and has also been used in previous research (e.g., Atasu and Çetinkaya (2006) and Geyer et al. (2007)). The requests must be satisfied at the time when they occur, either by new products ordered from a supplier (cost cN), or by reusing previously collected and refurbished demos. The demo loan period L (residence time at the client site) is assumed to be constant and is the only major delay in the system. That is, the time to order a product from the supplier, the time to transport/collect a demo product to/from a client site and the time for refurbishment are negligible (in practice, a few days at most) compared with the demo loan period (typically several months). It is possible to specify a buyout profile, but this only results in a modification of the original demo requests profile. It is therefore not incorporated in the model and all demo products are collected after loan period L, with a time-phased collection profile d(t-L) for L ≤ t < T+L.  There is no starting inventory of new products in the demo system: new products are ordered from the supplier when these are required to fulfil a request. For high-tech customized equipment demos are built-to-order after assessing the customer's needs, and the store does not stock new products because these are too expensive. The new products node in Figure 4 corresponds to a source (supplier) that can provide new products whenever required. Collected ex-demo products enter the returned products inventory (cost cI) at the demo store and can be refurbished (cost cR) and reused for a next demo request. Alternatively, collected ex-demo equipment can retire from the demo store and either be sold in the secondary market (after refurbishment), or scrapped (cost cS). The scrapping cost can be negative and represents a (typically small) salvage value received from selling the retired equipment to a recycling vendor or scrap broker when the option to sell in the secondary market is exhausted. At the beginning of planning period there is no inventory of collected ex-demo products in the store.
In line with our application context of high-end customized products, we also assume all costs (cN, cR, cI and cS) to be constant. These are valid assumptions since the product lifecycle is short. We could easily incorporate transportation and collection costs, but these are very low in comparison with the dominant cost in our application (the new product acquisition cost cN). In fact, because all demo requests have to be satisfied and collected, and because there is no difference in transportation and collection cost depending on whether new or refurbished products are used, both transportation and collection cost can be considered as sunk cost -incorporating these does not affect the optimal demo policy. We also note that the new product cost cN is high compared with the refurbishment cost cR and scrap cost cS. This is due to the nature of the returns. Almost all ex-demo products have been used over a rather limited time only (a few months) and are collected in excellent or like-new condition. The refurbishment operations are low-touch, i.e., inspecting, testing and dis-and re-assemble (modular) demo equipment. There is no hard repair involved.
Apart from reuse, the main value recovery mechanism in this problem environment is through sales in the secondary market. We account for value erosion or price decay in the secondary market.
We assume that the secondary market sales price erodes over time from the launch of the product (time t = 0), until the end of the planning horizon T+L. Retired demos are sold in the secondary market at time t and fetch a price ps(t). We do not pose strong restrictions on the revenue function: we only assume that ps(t) is piecewise analytical and that it is non-increasing over time (i.e., ps(t1) ≥ ps(t2) for t1 ≤ t2). Piecewise analytical means that ps(t) may have discontinuities: ps(t) is right-continuous but not necessarily left-continuous. Our results remain valid for any erosion function that fits these conditions (including continuous, linear and exponential erosion, which are used in many studies). As in most studies, in our model only revenue is affected by time. Some studies also address erosion in other components (e.g., Guide et al. (2006) account for erosion in production and remanufacturing cost) and/or discount cash in/out flows (Guide et al. (2006) and Difrancesco et al. (2018)). We have not incorporated a discount factor in this model because all cash in/out flows happen over a short lifecycle only. Our model can be extended to deal with more time varying features and in Section 7, we comment on how our results are affected in the case where cash flows are discounted.
The demand potential in the secondary market is further limited by a global market constraint or capacity, but there is no restriction (apart from the availability of ex-demo equipment) on when sales can start or when sales must end.
The objective in the model is to maximize revenue gained from selling ex-demonstration products in the secondary market, whilst making sure that all demonstration requests are satisfied at minimum cost. We model our problem as a deterministic, continuous time optimization problem. Table   2 summarizes the variables, parameters and main notation used in the model in the remainder of the paper. XN(t) = new products used to satisfy demo requests at time t XR(t) = refurbished products used to satisfy demo requests at time t XM(t) = retired ex-demo products sold in the secondary market at time t XS(t) = retired ex-demo products scrapped at time t XI(t) = inventory of collected demos in the store at time t Subject to: The objective function Eq.(1) minimizes the cost of satisfying demo product requests by either new or refurbished products, the cost of scrapping or salvaging (after refurbishment) ex-demo retirements in the secondary market, and the inventory holding cost at the demo store. Constraints (2)  and when revenue in the secondary market erodes, a decision must be made instantly: reuse, sell or scrap. There is no benefit from storing collections temporarily in inventory to fulfil later requests, or delay sales, because costs can only increase and revenue only decrease. If the demand profile d(t) has multiple peaks and troughs (i.e. d(t) and d(t-L) intersect at different points in time), it may be economically attractive to temporarily store excess collections in inventory for later reuse until demand surges. However, this does not happen with a concave (or unimodal) demand profile and it is optimal for the store to operate a zero inventory policy.
With a zero inventory policy, we have ̇( ) = ( ) = 0 for all t and we can remove the inventory cost component from the objective function. In addition, constraints (3) reduce to: If we now substitute XN(t) and XS(t) in the objective function (1) using Eq.
The first term in Eq. (7) corresponds to the cost of a base demo policy, which involves using a new product for each demo request and scrapping all products after they have been collected. The second term in Eq. (7) shows the potential improvement over the base demo strategy by reusing collected demos. We call cN + cS -cR the reuse saving. The third term in Eq. (7) shows the potential improvement over the base policy by selling retired and refurbished ex-demo products in the secondary market. We call pS(t) + cS -cR the resale saving. When both reuse and resale saving are ≤ 0 for all t, we cannot improve over the base strategy because none of the recovery options are economically attractive. We are interested in the case where both reuse and resale savings are positive and compete against each other. The optimal strategies when either the reuse saving or resale saving are negative can be derived as special, simpler, cases from the results presented in Sections 4 and 5 below.
We note that problem (7), (2), (6), (4) and (5) is a special Separated Continuous Linear Program (SCLP) (see e.g., Pullan, 1996). In the Appendix we show that our problem can be transformed into a Continuous Transportation Problem (CTP), for which strong duality results exist (Anderson and Philpott, 1984) that resemble the well-known strong duality results for static Transportation Problems.
We use these conditions to prove the optimality of the different solution structures that we derive in the next two sections. The solution to our problem changes at discrete points in time. Our analysis in the following sections focuses on identifying (using cost economic analysis) those moments in the planning horizon where these switches occur.
First, we derive two signals that drive the optimal solution to our problem. Consider a collected demo product and analyze if it is more profitable to reuse this product for a next request, or better to satisfy the next request by a new product and sell the collected ex-demo product in the secondary market. All constraints, except constraint (4), in our model are equality constraints. We distinguish two scenarios for constraint (4): a non-binding secondary market capacity constraint or a binding secondary market constraint.
When there is no restriction on the sales potential in the secondary market (constraint (4) is not binding), in the reuse option we satisfy the new request at time t by refurbishing and reusing the available collected ex-demo product (cost cR). After the demo loan period L, this product is collected, refurbished (cost cR) and sold in the secondary market, at price pS(t+L). The total cost for the reuse option is 2cR -pS(t+L). In the no-reuse option, we use a new product to satisfy the request at time t (cost cN). Both the available collected ex-demo product at time t and the new product collected at time t+L will be refurbished and sold in the secondary market, at prices pS(t) and pS(t+L). The total cost for the no-reuse scenario is cN + 2cR -pS(t) -pS(t+L). The reuse scenario is preferred over the no-reuse scenario as long as pS(t) ≤ cN. With non-increasing pS(t), the point in time when pS(t) drops below cN is a signal to start reuse. We refer to the case where pS(t) ≥ cN for all t as the case where selling in the secondary market is 'profitable'. When pS(t) drops below cN selling in the secondary market becomes unprofitable.
When the secondary market constraint becomes binding, we assume that we can still sell one more product in the secondary market. The cost for the reuse option is as before 2cR -pS(t+L). In the no-reuse scenario, the only difference is that a new product that was used to satisfy the demo request at time t, will be scrapped (cost cS) after collection at time t+L. The total cost for the no-reuse option is in the sales revenue between selling now and later (i.e. larger than the reuse saving) is an indicator to retire and sell ex-demo equipment in the secondary market early and postpone reuse until later in the lifecycle (even if this involves scrapping the later retirements). We denote the case where pS(t) ≤ cN + cS -cR as the low erosion case, and the case where pS(t) > cN + cS -cR as the high value erosion case.
Using these two indicators, we consider four scenarios for further analysis: We derive the optimal demo strategy for the low value erosion cases in Section 4. The optimal strategies for the high value erosion cases are presented in Section 5. In the Appendix we prove the optimality of the proposed solution structures using strong duality and reduced cost results for the CTP.

Derivation of the optimal product demo policies with low value erosion 4.1 Selling in the secondary market is always profitable (pS(t) ≥ cN)
When pS(t) ≥ cN it is profitable to sell ex-demo products in the secondary market and we should exploit this option to its full potential. This condition also implies that the resale saving pS(t) + cS -cR is larger than the reuse saving cN + cS -cR. Resale is the preferred value recovery option at any time. The optimal amount of products sold in the secondary market is equal to the secondary market potential: Q We consider two situations.
The retirement profile is r(t) = d(t-L) for L ≤ t < tf and r(t) = Max{0; d(t-L)-d(t)} for tf ≤ t < T+L. All retired ex-demo equipment is sold in the secondary market and there is no scrapping. This policy is illustrated in the right-hand diagrams, Figure 5(b). We show the same characteristics as in Figure 5(a) and the diagram is for tf = 11.5. We note that, in the diagram on the top, the new products XN(t) follow

Selling in the secondary market becomes unprofitable at t * (pS(t) < cN)
As long as the resale saving pS(t) + cS -cR ≥ 0, some value is recovered from selling in the secondary market. Because we assume low erosion (pS(t) ≤ cN + cS -cR) and pS(t) is higher than cN (for t < t * ), the resale saving pS(t) + cS -cR is always positive (i.e., when cR ≥ cS, which is a reasonable assumption).
At t * , the preferred value recovery option switches to reuse. It is straightforward to modify the optimal strategies from Section 4.1 to the case when resale becomes unprofitable at time t * . We consider the same two scenarios: When Nmin(L) ≥ fD(T), we need more new products to satisfy all demo requests than we could possibly sell in the secondary market. Whether pS(t) drops below cN will not affect the optimal policy and the best strategy is as explained in Section 4.1 (Figure 5  When Nmin(L) ≤ fD(T), it is important to know if selling in the secondary market becomes unprofitable early in the lifecycle (t * < L), or later (t * ≥ L). In the case where t * < L, every retired ex-demo product sold in the secondary market yields a loss. We want to exploit the reuse option as much as possible and use only N = Nmin(L) new products to satisfy all requests (the reuse saving is larger than the resale saving). All Nmin(L) products will be sold in the secondary market after collection and when they are not needed anymore. This policy can also be visualized from Figure 5.1.(a), with the only difference that there is no tf and selling in the secondary market continues until T+L. The secondary market capacity constraint is not binding and there is no scrapping.
In the case where t * ≥ L, the optimal policy is to use more than Nmin(L) new products to satisfy all demo product requests. As in Section 4.1, we can determine a market saturation time tf through Eq.(10). When tf < t * , i.e., we use N = fD(T) new products in total to satisfy all demo requests and only start reuse after tf, which is the same policy as in case 4.1.(b), Figure 5(b). When tf > t * , however, it is better to start reuse just after time t * and we do not saturate the entire secondary market. The optimal strategy is therefore to postpone reuse until Min {tf; t * }.

Managerial insights
The analysis of the low erosion case reveals that several problem parameters are key in order to derive and formulate an optimal demo strategy. Time related and/or time varying characteristics are of particular importance to capture the dynamic nature of the policies and to identify the exact moments in time when policy switches occur. The length of the demo loan period L relative to the length of the product's lifecycle T are two clear problem characteristics that impact the value recovery options. When L is long relative to T there is less opportunity for reuse and more new products are required to satisfy all demo requests. With eroding secondary market sales prices pS(t), less revenue is recovered the longer demo products reside at the client sites.
The demo loan period L can have a strong impact on Nmin(L) -the minimum number of new products required to satisfy all requests (see Eq. (8)). Nmin(L) is a critical policy and cost driver in this problem environment and it is important to understand its sensitivity to changes in L and in the demand profile d(t). Nmin(L) is larger for demo request profiles with a more pronounced peak. We illustrate this in Figure 6, which shows four request profiles for 0 ≤ t < 20 and the sensitivity of Nmin(L) to changes in The relations between revenue pS(t) and cost cN and between the sales potential in the secondary market fD(T) and Nmin(L) are the two another policy defining factors. These define the possible switching point t * , where resale in the secondary market becomes unattractive, and a market saturation time tf. Each of these factors interact and may influence each other. It is therefore not possible to define an optimal demo strategy based just on partial information (e.g., simply comparing ps(t) and cN while ignoring fD(T) and Nmin(L)). All characteristics have to be considered as well as the positions of different switching points (tS, t * , tf) in relation to L and T.

Selling in the secondary market is always profitable (pS(t) ≥ cN)
As in Section 4.1, when pS(t) ≥ cN, it is profitable to sell in the secondary market and this recovery option should be exploited to its full potential. The total number of ex-demo products sold in the secondary market is Q = fD(T). In addition, because the revenue profile pS(t) exhibits large price drops, it may be better to sell collected ex-demo products early in the lifecycle and use new products to satisfy subsequent demo requests, even when these products have to be scrapped after collection. In other words, it can be optimal to use more than Max{Nmin(L); fD(T)} new products to satisfy all demo requests. We analyze the same two scenarios as before: The left hand side in Eq. (11) comprises of the collections in [L, L+t) that can possibly be reused; the right hand side in Eq. (11) are excess collections in [tf -tf,, tf) that can only be sold in the market or be scrapped. We call the left integral in Eq.(11), ( + ∆ ), that is, the extra number of new products (in addition to Nmin(L)) that will be used to satisfy all demo requests.
A convenient way to find the values for t (and tf ) is to consider the function pS(t) = pS(t) -pS(t+L), which compares the sales revenues at times t and t+L. If pS(t) is smaller than the reuse saving cN + cS -cR, value erosion is low (see Section 4). Otherwise, when pS(t) > cN + cS -cR for some values of t, we look for the largest value of t (L ≤ t ≤ T) for which pS(t) > cN + cS -cR. Let us denote this t-value by tR. We can set tR = L+t as the upper limit in the left integral in Eq.(11), and find the corresponding tf value from Eq.(11). This approach ignores the secondary market capacity constraint.
We will never use more extra new products than we could possibly sell in the market (i.e., fD(T)). The limiting value for this constraint ( + ∆ ) = ( ) thus defines another candidate upper limit tC for L+t. Therefore, we can set L+t = Min{tR, tC} and require pS(L+t) -pS(tf -tf) ≥ cN + cS -cR.
The optimal demo retirement profile is   that can only be sold in the market or be scrapped. We denote the left integral in Eq.(12) by E(tf +tf,).
As before, the values for tf and t can be found from the function pS(t) = pS(t) -pS(t+L). One candidate point for tf +tf, is the largest t-value (called tR) for which pS(t) > cN + cS -cR. The other candidate point tC is determined by the limit value of a 'reduced' secondary market capacity constraint

E(tf +tf,) = fD(t)-E(tf) = Nmin(L).
Note that the market capacity must be reduced by the early sales E(tf), which generate the most revenue. We can then set tf +tf = Min{tR, tC}, and the value for t can be calculated from Eq.(12).
The optimal ex-demo retirement profile is

Selling in the secondary market becomes unprofitable at t * (pS(t) < cN)
As in Section 4.2 pS(t) dropping below cN at time t * is a signal to start reuse. We can modify the optimal policies in Section 5.1 in a similar fashion as we modified the optimal strategies in Section 4.1.

Case 5.2.(a): Nmin(L) > fD(T)
When t * < L, we already need more new products to satisfy all demo demands than we could possibly sell in the secondary market. When selling in the secondary market becomes unprofitable very early in the lifecycle before any collections are available (t * < L), the reuse recovery option is always preferred.
The optimal strategy is to use only N = Nmin(L) new products to satisfy all demo requests. As long as the resale saving pS(t) + cN -cR is positive, we will sell retired ex-demo products in the secondary market until the market is saturated. In case the resale saving becomes negative at some time tN before the market saturation time tf (see Figure 5(a)), secondary market sales will be halted and all retirements after tN will be scrapped.
When t * ≥ L, t can be determined as explained in Case 5.1.(a). Note that in Section 5.1 reuse is postponed until time Lt. Because the reuse option is preferred over the resale option at time t * , we can modify the Section 5.1 policy so as to start reuse at time Min{Lt; t * }. The optimal number of new products to satisfy all demo requests is N = Nmin(L) + E(Min{L+t; t * }). With respect to the resale revenue in the secondary market, we halt sales in the secondary market as soon as the resale saving becomes negative or when the market is saturated, whichever occurs earliest.

Case 5.2.(a): Nmin(L) ≤ fD(T)
When t * < L, the optimal strategy is to use N = Nmin(L) new products to satisfy all demo requests. We stop selling in the secondary market as soon as the resale saving becomes negative.
When t * ≥ L, tf is determined as in Case 5.1.(b). Because the reuse option is preferred over the resale option at time t * , we can modify the Section 5.1 policy so as to start reuse at time Min{tf+tf; t * }. The optimal number of new products used to satisfy all demo requests is given by N = Nmin(L)+E(Min{tf+tf; t * }). When L ≤ t * < tf; reuse starts right at time t * , i.e. earlier than tf (see Figure   7(b)); when L ≤ tf+tf ≤ t * < T, reuse is as in Figure 7(b); when L ≤ tf ≤ t * < tf+tf reuse starts at t * between tf and tf+tf. With respect to the resale revenue in the secondary market, sales in the secondary market are halted as soon as the resale saving becomes negative (time tN) or when the market is saturated, whichever occurs earliest.

Managerial insights
High value erosion may occur when the drop in resale revenue over the demo loan period pS(t) = pS(t) -pS(t+L) is larger than the reuse saving cN + cS -cR and when the secondary market capacity constraint is binding. From an economic perspective it is better to salvage available collections in the secondary market and to keep satisfying demo requests with new products. Reuse is postponed until later, i.e., until pS(t) drops below cN + cS -cR, or until the early sold collections have saturated the secondary market. The high erosion case is less intuitive and more complex than the low erosion case. The key piece of information needed to establish the presence of high value erosion is the pS(t)-function, which is defined or relevant in time interval [L, T). For constant value erosion (e.g., linear price decay), pS(t) is constant. For more complex price decay (e.g., exponential or piecewise linear decay with discrete drops in sales price) pS(t) is dynamic and changes over time. The decision of whether high erosion applies may thus change during the lifecycle. This makes sense because product returns early in the lifecycle (e.g., before the price drops in Figure 2) may be prone to high erosion whereas product returns near the end of the lifecycle (e.g. after the price drops in Figure 2) may have lost most of their value or may have missed the high revenue time window in the secondary market. Further delays for these last returns may have only limited financial impact so that low erosion applies. pS(t) also depends on the demo loan period L: the longer L, the higher pS(t) for the same value of t. However, pS(t) is defined over a longer time interval [L, T) when L is lower. Short or medium demonstration times L could make products at some time during the lifecycle prone to high value erosion, whereas (very) long demonstration times could mean that the products have lost most of their resale value after demonstration, and should be controlled by a low erosion strategy.
The high erosion case introduces additional, potential policy switching points (tR, tC, and tN).
The point tN is the time when resale revenue becomes negative and sales in the secondary market should stop. The point tR is the largest time for which pS(t) > cN + cS -cR and tC is a market saturation time for selling early collections. The minimum of tR and tC define a reuse postponement time (t or tf).

Case example
We now apply our generic analysis above to develop optimal demo policies for case company X noted in Section 2. Figure 8 (left) shows the demo request profile of processor 1 over its 20 months lifecycle.
In total 298 requests were received with an average demo loan period around 4.5 months (132 days see Table 1). We base our calculations on an approximation by an isosceles triangular profile (d(t) = 3.5t for t < 10 and d(t) = 35 -3.5(t-10) for 10 ≤ t ≤ 20). This approximation provides a close fit for the peak in the real demo requests profile but overestimates the total demo demand (350 units in the approximation vs. 298 in real life). Figure 8 (right) depicts the price erosion of the processor in the secondary market: pS(t) = $6000 -$100t when t < 11 and pS(t) = $2900 -$100(t-11) when t ≥ 11. The price drop of $2000 at t = 11 corresponds to the introduction of a new generation processor. We assume cN = $2400, cS = $0 and cR = $0 and that the secondary market sales capacity is limited to 180 units.  or $1200 (L = 12 months); the situations where L = 4.5 months or L = 6.5 months are high erosions cases (pS(t) reaches maximum values of $2450 and $2650, which is higher than the $2400 reuse saving). Before we discuss each of the four cases in more detail, we present a summary policy map in Figure 9. The diagram shows the optimal demo policy as a function of the demo loan period L and the secondary market sales potential f for the triangular demand profile and for the given price erosion profile pS(t). The dotted line in Figure 9 shows the Nmin(L) = fD(T) boundary. Points above this line correspond to scenarios where Nmin(L) is less than the market capacity fD(T); points below the line correspond to cases where Nmin(L) is larger than the market capacity fD(T) and thus, where scrapping applies. By considering pS(t) vs. the reuse saving cN + cS -cR, two limit values for the demo loan period are found.
Demo loan period values below L = 4 months and above L = 11 months correspond to low value erosion; L-values between 4 and 11 months define high erosion cases. We can also consider the relationship ps(t) vs. cN. In this example, selling in the secondary market becomes unattractive after t = 16. Unless necessary (e.g., when L > 16), we will not use new products after t = 16 to satisfy demo requests. We have indicated these three boundaries in Figure 9, and we also show our four cases: (a) L = 2.5 months,  Table 3 summarizes key characteristics of the optimal strategies for the 4 different cases. The   = $432k). The last two columns in Table 3 report the profit of two alternative policies: the max reuse policy uses only Nmin(L) new products to satisfy all requests and sells excess collections, up to the market capacity, as soon as possible (i.e., between tS and T+L, see the second part of the XM(t) curve in Figure   10, case (a)). The no reuse policy satisfies each request by a new product (new product cost $2.4*350 = $840k), and sells the fD(T) = 180 earliest collections. Both these policies are quite different from the optimal case (a) policy and generate a loss (-$4.8k and -$33.8).   9)). For case (b), the maximum tf value is found from the pS(t) function for L = 4.5, where t = 11 still shows a value larger than the reuse saving. We set tf = 11-9.29 = 1.71. Also, for case (c), the maximum t value is determined by t = 11 in the pS(t) function for L = 6.5. In this case, we set and L+t = 11 (case (b)) and does not impact on the policies.
The flows over time for cases (b) and (c) are shown in the middle section of Figure 10. The curves for case (b) and (c) show a similar but somewhat shifted pattern. We note that for both cases reuse XR(t) is postponed until t =11. New products are ordered until tS = 12.25 (case (b)) and tS = 13.25 (case (c)); market sales are clearly higher earlier in the lifecycle and also start earlier in case (b); there are no market sales between t = 11 and tS. Both scenarios use more new products than the market capacity and these products are scrapped after the market is saturated. The scrap flows XS(t) are not shown on the left diagrams, but they follow the collection curve near the end of the lifecycle and start when market sales have dropped to zero. From Table 3, we observe that the optimal policy for case ( Table 3 shows that case (d) yields a loss (-$362.5k) and that the no-reuse policy is worse. The optimal flow patterns in Figure 10 show that even for relatively simple demand profiles, product flows over the lifecycle can be quite different and are heavily influenced by lead time. This information should appeal to managers. The summary of policies in Figure 9 provide a clear informative guide to managers on when different policies should be used. Furthermore, constructing flow diagrams like in Figure 10 provide a visual aid, which managers can use to better forecast product returns (collections) and plan and coordinate reuse and resale operations. These diagrams show that we need to evaluate and consider all the relevant characteristics and information to identify an optimal policy. Making a judgement based on partial information (e.g., only considering pS(t) vs. cN and ignoring factors such a demo loan time L, market capacity f and the shape of the demo request profile d(t)), will result in incorrect decisions, which can be very costly for high-end IT product demonstrations. Due to confidentiality, we cannot report exact estimates of savings by the case company in implementing the optimal strategies. However, for products like processor 1, life cycle savings well in excess of $100k can be easily achieved with very modest changes (e.g., limiting the demonstration time to 3 months), whereas implementing an optimal strategy could generate savings in the region of $200k-$250k.
Processor 1 is only one component from the case companies' product portfolio, and from  Table 3, was however the worst performing policy.
Finally, managers can combine the policy map in Figure 9 with a profit/cost surface plot in Figure 11. This can reveal for different demand and price erosion profiles where the key sensitivities in profit are, and what policy changes are required to either drive performance improvement or to identify 'safe zones' (see Figure 11 left). Managers should aim to operate well within a safe zone where small deviations in L and f would still generate a healthy profit.

Conclusions and further research
In this paper we discussed and investigated a closed-loop supply chain for providing demonstrations of expensive, high-end products, with a short lifecycle. Potential customers request demos, while the manufacturer has the option to satisfy the requests either by new products or by reusing previously collected demos. Ex-demo equipment can be sold in the secondary market, but the product's sales price erodes over time. Eroding sales prices may change the economics of reuse versus no-reuse (resale) during the lifecycle.
We derived two measures, which are necessary to develop an optimal demo strategy in a deterministic setting with concave or unimodal demo demand profile over the product lifecycle. The first indicator is whether or not the sales price pS(t) drops below the acquisition cost cN. In general, the time when the sales price drops below cN is a signal to start reuse. The second indicator is whether the drop in sales price over the demo loan time L is larger than the reuse saving cN + cS -cR. Large drops in sales prices are characteristics of fast value erosion; small drops in sales price characterize slow value erosion. Inspecting the ps(t) function provides a convenient way to identify high erosion. We show that the distinction between high and low erosion depends on the stage in the product's lifecycle and the demo loan period L. In case of fast erosion, we also show that it may be better to postpone reuse until later in the lifecycle, even if this results in more products being scrapped. The optimal strategy depends on the interactions of several indicators (i.e., if these are present) and on the availability of collected products (residence time L); the demo demand profile (d(t)) and the secondary market potential (fD(T)).
The application of the generic analysis presented in this paper led to a number of useful insights for managers at the case company. Managers could recognize the importance of the demo request profile, the value of time, and their effect on value recovery. It became evident also that not all products should be treated in the same way: company X has products in its portfolio that belong to the four categories considered, and our analysis clearly shows that different optimal policies apply.
Furthermore, by applying the generic analysis, managers became aware that more stringent controls and measures were required to manage the demo store, with respect to the demo loan period L. Most of the products examined had average loan periods in excess of the recommended 3 months. The longer L, the more new products are required to satisfy demo request and the lower the value recovery through reuse and/or resale in the secondary market.
The context of our study focuses on demonstrations of high-end IT products. Demonstrations of medical equipment (e.g., dental chair), optical, audio and measurement equipment and categories of industrial equipment and machinery share many of the characteristics above although in some cases the product's lifecycle may be longer. For products with longer lifecycles, it may be recommended to incorporate an additional discount factor to compare cash flows at different times and calculate a net present value. The indicators in the models above can be easily modified for this situation. For example, assuming that f(t) is a decreasing discount function over time and that all cash in/out flows are discounted, it is not too difficult to show that the first signal (ps(t) vs. cN) is unaffected by the presence of the discount function. Hence, all our results still hold for the low erosion case, but clearly total profit/or cost will be discounted. The second indicator, however, is affected by the discount factor and becomes: . Because 0< f(t+L)/f(t) < 1, the discount factor amplifies the revenue price drop ps(t). The effect on the reuse saving depends on the relation between cS and cR: if cS-cR < 0, also the reuse saving is amplified, otherwise it is decreased. The reuse saving part becomes time dependent, rather than static which may change the high/low erosion decision (using ps(t) function). However, the same generic approach presented in the paper can be used to further explore this situation.
Value erosion is also important other contexts: in the mobile phone sector with new generation handsets introduced every 6 months, and in the fashion/clothing industry where increased online sales go together with increased product returns. Returning, repacking, re-labelling and reselling fashion apparel in a short time window puts tremendous pressure on clothing companies, yet some stores do offer free returns. Elements of the modelling approach we present can be used to capture several key profitability drivers for closed-loop supply chains where the rotation of assets and their subsequent recovery in the secondary market are important. Certain aspects of the model are more generally applicable (e.g., rental or leasing of products), or could be further extended to incorporate revenue from rental or leasing, or other characteristics. More time varying features and/or more constraints can be considered, but the analysis will be more involved. Further extensions could also investigate more complex demand profiles (which may require inventory) and/or stochastic elements (demo loan period, demo demand, secondary market demand).
Our modelling approach as a continuous time rather than a multiple period optimization problem enabled us to focus on the features that define and drive the optimal solution structure. This modelling approach results in stronger and more general conclusions, which is of importance when delays, time value and dynamic features have to be accounted for. Our model can be transformed into a Continuous time Transportation Problem. For the more general Continuous time Minimum Cost Network Flow Problems -although NP-hard to solve (Klinz and Woeginger, 2004) -optimality and strong duality results were recently derived by Koch and Nasrabadi, (2014) under certain conditions.
These models can be explored for generalizations of our problem.

Appendix
We show how our problem can be transformed into a Continuous Transportation Problem (CTP). We also demonstrate the optimality of the solution structures derived in Sections 4 and 5, relying on the strong duality results for the CTP (see e.g., Anderson and Philpott, 1984). We present each identified structure as a solution to the CTP in a transportation tableau format. The rows and columns display the supplies (new products or reuse from store) and demands (demo requests, secondary market and the scrapping) that are active in different time epochs. Let cij(t) denote the cost function to supply demand j from supply i. Based on the cells that characterize the solution, we calculate the dual cost functions i(t) and j(t) for the supply and demand cells via i(t) + j(t) = cij(t). This resembles the reduced cost relationships in static Transportation Problems, but in the CTP the i(t) and j(t) are functions. Next, we calculate the reduced cost functions cij(t) -i(t) -j(t) for the cells that are not used in the solution. When all the reduced cost functions are ≤ 0, we conclude optimality. Figure 12: A balanced CTP Figure 12 shows our problem as a CTP. We model the Demo Store as a supply node (supply of collected ex-demo products) with a time-phased (over loan period L), time varying supply equal to the time varying demand at the Demo Client Site node. If we specify total demands fD(T) and D(T) for the Secondary Market and Scrap node, then we can set a total supply of (1+f)D(T) at the New Products node to balance the problem. The arcs connecting the New Products node with the Secondary Market and Scrap nodes carry a zero cost. Flows through these two arcs are virtual flows and do not affect the solution. The only arc with a time varying cost is the arc from the Demo Store node to the Secondary Market node, with cost cR -pS(t).
We illustrate the solution structure optimality through two examples. Example 1 (see Section 4.1 and Figure 5(a)) is for the case of low erosion and pS(t) ≥ cN over the entire planning horizon T+L.
We further assume that the minimum number of new products needed to satisfy all demo requests is larger than the secondary market potential (Nmin(L) > fD(T)). Example 2 (see Section 5.2 and Figure   7(b)) is for fast erosion (pS(t) drops below cN at time t * with L < t * < T and the resale saving (pS(t) + cS -cR) becomes negative at a time tN with T < tN < T+L -t). In Example 2 we further assume Nmin(L) ≤  Figure 13 shows the CTP for Example 1 in a transportation tableau format. We have suppressed the time dependency in the notation: the demo store supply functions are s1, s2, s3 and s4, the demo request functions d1, d2, d3 and d4 and the secondary market price functions p1, p2, p3 and p4. In the top right corner in each cell we show the relevant cost function. The cells with shaded borders fully characterize a feasible solution to the CTP. With m supply cells and n demand cells we need m+n-1 cells that do not contain a loop. In this example the CTP solution is degenerate: we need 5+6-1 = 10 Cell (3,6): No, because p is non-increasing.