ANGELUS, Alexandar

Institutional Knowledge at Singapore Management University

Singapore Management University Institutional Knowledge at Singapore Management University Research Collection Lee Kong Chian School Of Business Lee Kong Chian School of Business 6-2010 Optimal and Heuristics Policies for a Multiechelon Inventory Problem with Secondary Market Sales Alexandar ANGELUS Singapore Management University, angelus@smu.edu.sg Follow this and additional works at: https://ink.library.smu.edu.sg/lkcsb_research  Part of the Operations and Supply Chain Management Commons, and the Sales and Merchandising Commons Citation ANGELUS, Alexandar. Optimal and Heuristics Policies for a Multiechelon Inventory Problem with Secondary Market Sales. (2010). MSOM Annual Conference, 27-29 June 2010, Haifa, Israel. 1-4. Research Collection Lee Kong Chian School Of Business. Available at: https://ink.library.smu.edu.sg/lkcsb_research/3057 This Conference Paper is brought to you for free and open access by the Lee Kong Chian School of Business at Institutional Knowledge at Singapore Management University. It has been accepted for inclusion in Research Collection Lee Kong Chian School Of Business by an authorized administrator of Institutional Knowledge at Singapore Management University. For more information, please email libIR@smu.edu.sg.  1   Optimal and Heuristic Policies for a Multiechelon  Inventory Problem with Secondary Market Sales   Alexandar Angelus angelus@smu.edu.sg Lee Kong Chian School of Business Singapore Management University 50 Stamford Road  Abstract Most research on supply chain management deals with settings where firms do not voluntarily get rid of inventory in the system. Since voluntary inventory reductions are often observed in practice, in this paper, we propose a (multi-echelon) model where the firm can dispose of excess stock through sales in the secondary market at each stage in the supply chain. What are called nested echelon base stock policies are shown to be optimal. Secondary market sales complicate the structure of the system, so that the classical Clark and Scarf (1960) approach no longer applies. Nevertheless, we identify features of the optimal policy that significantly reduce the state space and simplify the computation, by providing conditions under which it is optimal for the firm not to both dispose of stock and order additional inventory. Finally, we formulate a class of disposal policies that achieves the Clark-Scarf decomposition, and reduces the optimal inventory policy to the classical echelon base stock policy. A numerical study demonstrates the performance of these policies as heuristics, illustrates the value of secondary markets to the supply chain, and provides managerial insights.   2Traditional research on supply chain management tends to deals with settings where firms do not voluntarily get rid of excess inventory in the system. At the same time, because uncertainty in customer demand and other factors impacting the firm often results in a surplus of inventories throughout the supply chain, their reduction by means of secondary market sales can readily be observed in practice. In this paper we propose a finite-horizon, multi-echelon model in which each stage in a series system can dispose of surplus stock through a secondary market mechanism. The decision to sell-off (i.e., dispose of) excess inventory results in the immediate (i.e., next period) and permanent removal of that inventory from the supply chain. Within the context of this model, our primary objectives are to establish the form (and special features) of the optimal policy, and to propose heuristic policies that capture the essential features of secondary-market sales while simplifying the structure of the problem. Our secondary objectives are to use numerical studies to both assess the performance of those heuristic policies, and to provide insights into the value of secondary markets to supply chains.  To the best of our knowledge, the only other paper to explore the impact of secondary markets on supply chain management is Lee and Whang (2002), which considers a two period, two-stage distribution system – one manufacturer and multiple retailers. In their paper, the secondary market exists only at the downstream stage, among the retailers themselves, which allows for the derivation of the secondary market equilibrium price, as well as the optimal order and market sale quantities for the retailers. By comparison, we address a pure series system, with multiple stages and periods, and allow secondary markets to exist at every stage in the system. Our secondary markets, on the other hand, are exogenous – market demand is abundant, and market prices are takes as given. Our focus is on optimal and practical heuristic policies. Following Clark and Scarf (1960), we identify stages for the minimum number of periods that a product is away from reaching the (stochastic) customer demand at the most downstream stage. When a unit is first ordered, it is one period away from entering the supply chain. At the beginning of the next period, this unit is at the most upstream stage in the system. The unit can be moved down one stage each period, until it reaches the most downstream stage. The progress of a unit can be delayed (if necessary) at each  3stage, by not moving it downstream. We also allow a unit to be disposed of at any stage in the system, through a sale in the secondary market at that stage.  We assume full backlogging of unsatisfied demand at the most downstream stage, with a unit backlogging cost, and inventory carryover, with a unit holding cost at each stage. Each unit ordered into a stage incurs an ordering cost, while each unit sold in the secondary market generates revenue. We allow all model parameters and stochastic demands to vary with time. Our model is a finite-horizon dynamic program, whose objective is to minimize the expected discounted net present value of costs (backlogging + inventory holding + ordering) less revenues incurred from the secondary market sales, over the time horizon.  In arriving at results, we first reformulate the model using echelon-type variables. Next, we show that the form of the optimal policy is a bottom-up nested echelon base stock policy: disposal decisions are made first, from the bottom-up, starting with the most downstream stage and moving upstream; order decisions, nested within adjacent sets of disposal decisions, are made next. Secondary market sales complicate the structure of the system (through the complexity of the boundary of the feasible region at each stage) so that the classical Clark and Scarf (1960) decomposition, and the additive convexity of the objective function, no longer hold, in general.  To help deal with the dimensionality inherent in the problem, we first identify features of the optimal policy that simplify the computation. In particular, we introduce some mild restrictions on cost and revenues parameters under which it is optimal for each stage in the system not to both dispose of excess inventory and order additional stock from the upstream stage; the dimensionality of the optimization space, and the corresponding calculation time, are thus (roughly) halved. Next, we identify a special class of policies that renders our model more tractable, while embodying the features of secondary market sales observed in practice. In particular, we introduce a disposal saturation policy, which, for any stage in the system, considers disposing of inventory at that stage only if there is no stock available for disposal at any upstream stage. A disposal saturation policy thus “saturates” disposal at each stage, before initiating any disposals at the next stage downstream. We show that:   4i)  This class of policies achieves the Clark-Scarf decomposition and the additive convexity of the objective function;  ii)  The optimal disposal policy is determined by only two critical numbers;  iii) The resulting optimal ordering policy follows the classical (state-independent) echelon basestock policy.  A disposal saturation policy thus allows disposal at each stage in the system, considerably simplifies the problem, and is straightforward to implement in practice. We conduct numerical studies for a two-stage and three-stage system to evaluate the performance of a disposal saturation policy as a heuristic for the original problem. Our results indicate that disposal saturation policies provide a very good approximation to the optimal policy (within 1% of the optimal objective function value) across a wide range of model parameters. Moreover, while the calculation of the optimal policy for a three-stage system and twenty periods took about 6 days to compute (for each model run), finding the optimal disposal saturation policy took only a couple of minutes.  Finally, we conduct a numerical study to assess the value of secondary markets to the supply chain performance (for a two- and a three-stage system). The results of the study indicate that this value is increasing in unit revenues (also shown analytically), and decreasing in unit order costs. Interestingly, the value of having secondary markets is neither uniformly increasing nor uniformly decreasing in inventory holding costs; instead, it appears to first increase, as unit holding costs increase, and then decrease, as those costs continue to grow.  Additional research is needed to shed more light on this relationship, as well as provide more insight into the optimality of disposal saturation policies.  References Clark, A. H., H. Scarf. 1960. Optimal policies for the multi-echelon inventory problem. Management Sci. 6, 475-490. Lee, H., S. Whang. 2002. The impact of the secondary market on the supply chain. Management Sci. 48, 719-731. 

Optimal and Heuristics Policies for a Multiechelon Inventory Problem with Secondary Market Sales

Singapore Management University 
Institutional Knowledge at Singapore Management University 
Research Collection Lee Kong Chian School Of 
Business Lee Kong Chian School of Business 
6-2010 
Optimal and Heuristics Policies for a Multiechelon Inventory 
Problem with Secondary Market Sales 
Alexandar ANGELUS 
Singapore Management University, angelus@smu.edu.sg 
Follow this and additional works at: https://ink.library.smu.edu.sg/lkcsb_research 
 Part of the Operations and Supply Chain Management Commons, and the Sales and Merchandising 
Commons 
Citation 
ANGELUS, Alexandar. Optimal and Heuristics Policies for a Multiechelon Inventory Problem with 
Secondary Market Sales. (2010). MSOM Annual Conference, 27-29 June 2010, Haifa, Israel. 1-4. Research 
Collection Lee Kong Chian School Of Business. 
Available at: https://ink.library.smu.edu.sg/lkcsb_research/3057 
This Conference Paper is brought to you for free and open access by the Lee Kong Chian School of Business at 
Institutional Knowledge at Singapore Management University. It has been accepted for inclusion in Research 
Collection Lee Kong Chian School Of Business by an authorized administrator of Institutional Knowledge at 
Singapore Management University. For more information, please email library@smu.edu.sg. 
 1
 
 
 
Optimal and Heuristic Policies for a Multiechelon  
Inventory Problem with Secondary Market Sales 
 
 
Alexandar Angelus 
angelus@smu.edu.sg 
Lee Kong Chian School of Business 
Singapore Management University 
50 Stamford Road 
 
Abstract 
Most research on supply chain management deals with settings where firms do not 
voluntarily get rid of inventory in the system. Since voluntary inventory reductions are 
often observed in practice, in this paper, we propose a (multi-echelon) model where the 
firm can dispose of excess stock through sales in the secondary market at each stage in 
the supply chain. What are called nested echelon base stock policies are shown to be 
optimal. Secondary market sales complicate the structure of the system, so that the 
classical Clark and Scarf (1960) approach no longer applies. Nevertheless, we identify 
features of the optimal policy that significantly reduce the state space and simplify the 
computation, by providing conditions under which it is optimal for the firm not to both 
dispose of stock and order additional inventory. Finally, we formulate a class of disposal 
policies that achieves the Clark-Scarf decomposition, and reduces the optimal inventory 
policy to the classical echelon base stock policy. A numerical study demonstrates the 
performance of these policies as heuristics, illustrates the value of secondary markets to 
the supply chain, and provides managerial insights.  
 2
Traditional research on supply chain management tends to deals with settings where 
firms do not voluntarily get rid of excess inventory in the system. At the same time, 
because uncertainty in customer demand and other factors impacting the firm often 
results in a surplus of inventories throughout the supply chain, their reduction by means 
of secondary market sales can readily be observed in practice. 
In this paper we propose a finite-horizon, multi-echelon model in which each 
stage in a series system can dispose of surplus stock through a secondary market 
mechanism. The decision to sell-off (i.e., dispose of) excess inventory results in the 
immediate (i.e., next period) and permanent removal of that inventory from the supply 
chain. Within the context of this model, our primary objectives are to establish the form 
(and special features) of the optimal policy, and to propose heuristic policies that capture 
the essential features of secondary-market sales while simplifying the structure of the 
problem. Our secondary objectives are to use numerical studies to both assess the 
performance of those heuristic policies, and to provide insights into the value of 
secondary markets to supply chains. 
 To the best of our knowledge, the only other paper to explore the impact of 
secondary markets on supply chain management is Lee and Whang (2002), which 
considers a two period, two-stage distribution system – one manufacturer and multiple 
retailers. In their paper, the secondary market exists only at the downstream stage, among 
the retailers themselves, which allows for the derivation of the secondary market 
equilibrium price, as well as the optimal order and market sale quantities for the retailers. 
By comparison, we address a pure series system, with multiple stages and periods, and 
allow secondary markets to exist at every stage in the system. Our secondary markets, on 
the other hand, are exogenous – market demand is abundant, and market prices are takes 
as given. Our focus is on optimal and practical heuristic policies. 
Following Clark and Scarf (1960), we identify stages for the minimum number of 
periods that a product is away from reaching the (stochastic) customer demand at the 
most downstream stage. When a unit is first ordered, it is one period away from entering 
the supply chain. At the beginning of the next period, this unit is at the most upstream 
stage in the system. The unit can be moved down one stage each period, until it reaches 
the most downstream stage. The progress of a unit can be delayed (if necessary) at each 
 3
stage, by not moving it downstream. We also allow a unit to be disposed of at any stage 
in the system, through a sale in the secondary market at that stage.  
We assume full backlogging of unsatisfied demand at the most downstream stage, 
with a unit backlogging cost, and inventory carryover, with a unit holding cost at each 
stage. Each unit ordered into a stage incurs an ordering cost, while each unit sold in the 
secondary market generates revenue. We allow all model parameters and stochastic 
demands to vary with time. Our model is a finite-horizon dynamic program, whose 
objective is to minimize the expected discounted net present value of costs (backlogging 
+ inventory holding + ordering) less revenues incurred from the secondary market sales, 
over the time horizon. 
 In arriving at results, we first reformulate the model using echelon-type variables. 
Next, we show that the form of the optimal policy is a bottom-up nested echelon base 
stock policy: disposal decisions are made first, from the bottom-up, starting with the most 
downstream stage and moving upstream; order decisions, nested within adjacent sets of 
disposal decisions, are made next. Secondary market sales complicate the structure of the 
system (through the complexity of the boundary of the feasible region at each stage) so 
that the classical Clark and Scarf (1960) decomposition, and the additive convexity of the 
objective function, no longer hold, in general.  
To help deal with the dimensionality inherent in the problem, we first identify 
features of the optimal policy that simplify the computation. In particular, we introduce 
some mild restrictions on cost and revenues parameters under which it is optimal for each 
stage in the system not to both dispose of excess inventory and order additional stock 
from the upstream stage; the dimensionality of the optimization space, and the 
corresponding calculation time, are thus (roughly) halved. 
Next, we identify a special class of policies that renders our model more tractable, 
while embodying the features of secondary market sales observed in practice. In 
particular, we introduce a disposal saturation policy, which, for any stage in the system, 
considers disposing of inventory at that stage only if there is no stock available for 
disposal at any upstream stage. A disposal saturation policy thus “saturates” disposal at 
each stage, before initiating any disposals at the next stage downstream. We show that:  
 4
i)  This class of policies achieves the Clark-Scarf decomposition and the additive 
convexity of the objective function;  
ii)  The optimal disposal policy is determined by only two critical numbers;  
iii) The resulting optimal ordering policy follows the classical (state-independent) 
echelon basestock policy.  
A disposal saturation policy thus allows disposal at each stage in the system, 
considerably simplifies the problem, and is straightforward to implement in practice. 
We conduct numerical studies for a two-stage and three-stage system to evaluate the 
performance of a disposal saturation policy as a heuristic for the original problem. Our 
results indicate that disposal saturation policies provide a very good approximation to the 
optimal policy (within 1% of the optimal objective function value) across a wide range of 
model parameters. Moreover, while the calculation of the optimal policy for a three-stage 
system and twenty periods took about 6 days to compute (for each model run), finding 
the optimal disposal saturation policy took only a couple of minutes.  
Finally, we conduct a numerical study to assess the value of secondary markets to 
the supply chain performance (for a two- and a three-stage system). The results of the 
study indicate that this value is increasing in unit revenues (also shown analytically), and 
decreasing in unit order costs. Interestingly, the value of having secondary markets is 
neither uniformly increasing nor uniformly decreasing in inventory holding costs; 
instead, it appears to first increase, as unit holding costs increase, and then decrease, as 
those costs continue to grow.  Additional research is needed to shed more light on this 
relationship, as well as provide more insight into the optimality of disposal saturation 
policies. 
 
References 
Clark, A. H., H. Scarf. 1960. Optimal policies for the multi-echelon inventory problem. 
Management Sci. 6, 475-490. 
Lee, H., S. Whang. 2002. The impact of the secondary market on the supply chain. 
Management Sci. 48, 719-731. 


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Optimal and Heuristics Policies for a Multiechelon Inventory Problem with Secondary Market Sales

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