Effect of a secondary market on a system with random demand and uncertain costs

International Journal of Production Economics xxx (2018) 1–9

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International Journal of Production Economics journal homepage: www.elsevier.com/locate/ijpe

Effect of a secondary market on a system with random demand and uncertain costs Yücel Gürel, Refik Güllü * Bogazici University Industrial Engineering Department, Bebek 34342, Istanbul Turkey

A R T I C L E I N F O

A B S T R A C T

Keywords: Stochastic price Inventory model Price driven demand

In this paper, we consider inventory and pricing decisions for a system where the customer demand can be partitioned into two segments: a primary and a secondary market. These kinds of systems are observed, for instance, in technology intensive products or services where the primary market, being more loyal, is generally not too sensitive to the pricing of the product or service. While the primary market customer demand occurs right after the introduction of the product, the secondary market customer demand typically occurs after the product matures, and these customers are much more sensitive to changes in the sales price. The purchasing costs of technology intensive products very much depend on the spot currency exchange rate, and hence can be modelled as a stochastic process. Consequently, the sales price for the primary market customers can be assumed to be a mark-up of the spot purchasing cost of the product. On the other hand, as the secondary market customers are more sensitive to the sales price, a demand model, where the customer demand explicitly depends on the selling price would be more appropriate. We try to accomplish three objectives: (1) to model the described system, (2) to find the optimal initial quantity to stock, and (3) to determine the optimal sales price for the secondary market customers.

1. Introduction

product or service. While the primary market customer demand occurs right after the introduction of the product, the secondary market customer demand typically occurs after the product matures, and these customers are much more sensitive to changes in the sales price. The purchasing costs of technology intensive products very much depend on the spot currency exchange rate, and hence can be modelled as a stochastic process. The sales price for the primary market customers can be assumed to be a mark-up of the spot purchasing cost of the product. On the other hand, as the secondary market customers are more sensitive to the sales price, a demand model, where the customer demand explicitly depends on the selling price would be more appropriate. Our objectives are the determination of the initial purchase quantity, and the sales price of the product for the secondary market customers. Over the primary selling season, inventory levels are depleted by a demand process which is modelled as a Poisson Process. The leftover amount of stock at the end of the primary season is sold to secondary season customers. Over the secondary selling season, the firm decides on the sales price (possibly as a markdown of the end of primary season sales price), and faces demand that depends on that price. The proposed model is appropriate for a company that imports a technology intensive product or service (such as smart phones, game or movie streaming services). The

Price fluctuation is a common challenge for the companies purchasing from spot markets and/or abroad. The prices of raw materials, precious metals, agricultural products, minerals, energy resources, and electronic components, among other things, can fluctuate substantially over short periods. Variations in market conditions, the presence of new technologies, or changes in the supply and demand characteristics may cause these fluctuations. Some companies depend on spot markets for material procurement, and this dependency brings on unit cost volatility. For example, as can be seen in Fig. 1, the Bloomberg Commodity Index (which tracks prices of major agricultural commodities, energy related inputs, and precious metals), exhibits considerable variations even over very short periods of time. Likewise, fluctuations in the foreign exchange rates make planning for inventories very difficult for companies in emerging markets. This study considers the problem of purchasing a commodity with a fluctuating market price so as to maximise its profit from selling it over two selling seasons. These kinds of systems are observed, for instance, in technology intensive products or services where the primary market, being more loyal, is generally not too sensitive to the pricing of the

* Corresponding author. E-mail addresses: [email protected] (Y. Gürel), refi[email protected] (R. Güllü). https://doi.org/10.1016/j.ijpe.2018.05.010 Received 4 November 2016; Received in revised form 9 February 2018; Accepted 9 May 2018 Available online xxxx 0925-5273/© 2018 Elsevier B.V. All rights reserved.

Please cite this article in press as: Gürel, Y., Güllü, R., Effect of a secondary market on a system with random demand and uncertain costs, International Journal of Production Economics (2018), https://doi.org/10.1016/j.ijpe.2018.05.010

Y. Gürel, R. Güllü

International Journal of Production Economics xxx (2018) 1–9

Fig. 1. Bloomberg commodity index.

demand in the primary season, but would be reluctant to stock excessively, as at the end of the maturity phase any leftover can only be sold at a considerable markdown. In order to find the optimal initial quantity to stock, and the optimal markdown price for the secondary market customers, we derive the expected profits of the system over the demand periods of primary and secondary customers, respectively. As the sales

value of these products or services usually depreciates over time as new generations are introduced. The demand lifecycle of these products or services can be roughly divided into two parts: ascent and maturity phase (where primary selling season takes place), and the decline phase (where secondary selling season takes place). Naturally, the importer would be interested in stocking enough quantity that will be sufficient to satisfy the

2

Y. Gürel, R. Güllü

International Journal of Production Economics xxx (2018) 1–9

Whitin (1955) and Young (1978) are early examples of inventory models where, the demand is a function of the price. The interest in price-driven inventory models revived with the seminal paper of Petruzzi and Dada (1999), where extension of the newsvendor problem is modelled in which inventory quantity and selling price are determined simultaneously. Petruzzi and Dada (1999) show that simultaneous price setting and quantity optimisation improve the performance of the model. We refer the reader to Kocabiyikoglu and Popescu (2011) for a recent review of models where the inventory and pricing decisions are jointly given. Cetinkaya and Parlar (2010) and Zhu and Cetinkaya (2015) examine the optimal disposal of excess inventory before the planning horizon starts. Such models are related to ours, as our modelling of the secondary market customers (by marking down the sales price as a function of the available stock) is similar to the treatment of pre-season excess inventory (by disposing of it at possibly an unfavorable price). The optimal timing of the end-of-season sales is investigated in Feng and Gallego (1995). We refer the reader to Elmaghraby and Keskinocak (2003) for a review of inventory models under dynamic pricing. Readers are referred to Khouja (1999), Khouja (2000), and Zhang et al. (2008) for examples of joint determination of pricing and inventory control decisions.

quantity for the secondary market customers is bounded by the product availability at the end of primary sales period, the initial purchase quantity would have an impact on the sales price for the secondary market customers. We also carry out a computational study to highlight important managerial insights. Our results reveal important information regarding the impact of joint determination of secondary market price and initial purchase quantity, and the effect of cost volatility on inventory and pricing decisions. In conclusion, our model provides original analytical findings, together with important managerial insight for a system with dual customer base, and with demand and cost uncertainty. The rest of this paper is organized as follows: we review the related literature in Section 2. Section 3 presents major notation, assumptions of the model and the mathematical formulation. Section 4 presents our analytical derivations for the secondary market phase of the model. In Section 5 we discuss our numerical findings, and finally conclude the paper in Section 6. 2. Literature survey Our literature survey essentially follows two directions: inventory models with cost uncertainty, and inventory models with price dependent demand. Inventory models with price uncertainty started in the 1950s (see, for example, Fabian et al. (1959), Scarf (1960), and Kalymon (1971)), but became popular only recently. Gavirneni (2004) models the periodic review inventory control problem with fluctuating purchasing costs and shows the optimality of an order-up-to policy. Gaur and Seshadri (2005) examine mitigating the risk in carrying inventory for a short life cycle or seasonal item when there is a correlation between its demand and the price of a financial asset. Goel and Gutierrez (2012) investigate procurement policies of the stochastic inventory system by integrating commodity markets in their model. They show that, incorporating spot and futures price information in the procurement decision making process reduce the inventory related costs. Secomandi and Kekre (2014) analyze the effective use of spot and forward markets in energy procurement management and show the value of the forward procurement option on realistic natural gas instances and argue that procuring the demand forecast in the forward market captures essential benefits of an optimal strategy. Xie et al. (2013) examine procurement policies for Chinese oil refineries with the aim of reducing the cost associated with fluctuating oil prices in the international spot market. Chen et al. (2014) investigate the impact of input price variability and correlation in an inventory system with stochastic demand and stochastic input prices. They show the concavity of the expected cost function in the input price and prove that the higher input price variability leads to lower expected costs. Yang and Xia (2009) consider a system where the demand follows a compound Poisson process whereas the price process is modelled as a Markov Chain. Following similar modelling assumptions but a different analysis, Berling and Martinez-de-Albeniz (2011) aim to minimise the expected discounted cost including the purchase costs, the out of pocket holding costs and the backordering costs. They identify the optimal base stock level by the single-unit decomposition approach. As the computation of the optimal policy is involved, Berling and Xie (2014) propose an approximation. Secomandi (2010) consider optimal purchasing policies for a traded commodity under capacitated storage. Gaur et al. (2015) study the optimal timing of inventory ordering decisions with price uncertainty. They develop a continuous time inventory model where demand and price are observed at some maturity date, but the stocking decision can be made at any time until maturity. Inderfurth and Kelle (2011) examine the use of both capacity reservation contracts and purchasing through spot market in order to satisfy a random demand. They propose a simple and easy-to-implement capacity reservation base stock policy. A survey of the literature on procurement planning in spot markets is given in Haksoz and Seshadri (2007). Secomandi and Johnson (2007), on the other hand, present a survey on pricing decisions in operations management literature.

3. Description of the model In this section we develop our mathematical model and present our analytical results for the primary selling season. Although we introduce notation as the need arises, a summary of major notation is also presented in Table 1. Consider a company that stocks a certain commodity for selling it over a horizon of length CL. This horizon can be thought of as the lifecycle of the product, spanning from the point that it is introduced in the market until the time that the business gain through its sales revenue diminishes. We decompose CL into two sub-cycles: the first part, where the primary customer demand occurs, and the second part, where the secondary customer demand is observed. Let τ 2 ð0; CLÞ be the time of switch from one sub-cycle to the next, and accordingly, let ½0; τ, and ðτ; CL be the first and second sub-cycles within the planning horizon, respectively. The company makes a purchase of Q items at a unit cost of Pð0Þ at the beginning of the planning horizon. Let fPðtÞ; t  0g be the stochastic process describing the evolution of the unit cost. In our derivations we will assume that PðtÞ follows a Geometric Brownian Motion: dPðtÞ ¼ μPðtÞdt þ σ P PðtÞdWP ðtÞ;

(1)

where μ and σ P are the drift and volatility parameters of PðtÞ, and WP ðtÞ is a standard Brownian motion. Then PðtÞ can be represented as a lognormal random variable PðtÞ ¼ Pð0Þeðμð1=2Þσ P Þtþσ P WP ðtÞ ; 2

(2)

where WP ðtÞ  Nð0; tÞ. Although we assume a simple unit cost process (due to this process's common usage in quantitative finance), other more complicated processes (Schwartz (1997), Schwartz and Smith (2000))

Table 1 Major notation used. Notation

τ λ Q PðtÞ

σP μ r

α h

3

The The The The The The The The The

length of the primary selling season demand rate within the primary selling season initial purchase quantity spot price of the commodity at time t volatility of the price process drift parameter of the price process interest rate mark-up factor for the primary selling season holding cost per unit value per unit time

Y. Gürel, R. Güllü

International Journal of Production Economics xxx (2018) 1–9

Fig. 2. The order of events.

JðQÞ ¼ V1 ðQÞ þ erτ ED1ðτÞ;PðτÞ ½V2 ðQ  D1 ðτÞ; PðτÞÞ  
 ¼ V1 ðQÞ þ erτ ED1 ðτÞ;PðτÞ max ED2 p min ðQ  D1 ðτÞÞþ ; D2 ðpÞ ;

can be used at the expense of more involved derivations. Let fD1 ðtÞ; 0  t  τg be the demand process over the primary selling season. In order to understand the effect of price uncertainty on inventory and pricing decisions, we choose a reasonably simple yet frequently used demand process, and assume that fD1 ðtÞ; t  0g is a Poisson process with rate λ. The poisson process enables us to obtain a detailed accounting of various costs and revenues accrued over the primary selling season. For the secondary market customers, we choose a simpler demand process, and represent the aggregated demand over ðτ; CL as D2 ðpÞ, where p is the markdown sales price (to be determined subsequently) for the secondary market customers. Fig. 2 depicts a timeline, and order of events. For our derivations, we assume that the aggregate demand of the secondary market, D2 ðpÞ, follows a simple additive process: D2 ðpÞ ¼ a  bp þ ε, where ε is a zero mean random variable. In our model, the expected profit function consists of two parts; the expected profit from the primary market and the revenues obtained from the secondary market. Over the primary selling season, there are three ingredients to consider: the initial purchase cost, Pð0Þ, the expected sales revenue, and the expected holding costs. Let V1 ðQÞ be the expected profit obtained over the primary selling season, when the initial purchase quantity is Q. Let SRðQÞ, and HCðQÞ be the expected sales revenue and expected holding costs incurred over the primary selling season, respectively. Then, V1 ðQÞ ¼ Pð0ÞQ þ SRðQÞ  HCðQÞ:

where r is the continuously compounded interest rate. Note that, in (5), the outer expectation is taken over the random variables D1 ðτÞ and PðτÞ. 3.1. Expected sales revenue over the primary selling season In the primary selling season, customers arrive according to a Poisson process with rate λ. Let T1 ; T2 ; … be the arrival times of customers. The selling price at an arbitrary time t 2 ½0; τ is obtained by taking a fixed markup α  1 of the spot price of the commodity at time t: αPðtÞ. Then, the expected sales revenue is given by " SRðQÞ ¼ E

py

Q X j¼1



#

αP Tj erTj 1fTj τg ;

(6)

where 1A ¼ 1 if A is true and 1A ¼ 0, otherwise. In the next result we obtain an explicit expression for SRðQÞ provided that λ  μ þ r  0. This condition is not restrictive, as the risk neutral evaluation (please see Schwartz (1997)) of the expected profit would also require μ ¼ r. j P  Proposition 3.1. SRðQÞ ¼ αPð0Þ Qj¼1 λμλ þr PrfTj  τg, where Tj is an Erlang random variable with shape parameter j and rate λ  μ þ r (abbreviated as Erlangðj; λ  μ þ rÞ.

(3)

Over the secondary selling season, the firm tries to maximise its revenue using the leftover items at the end of the primary season. The revenue depends on the markdown price and on hand inventory remaining at the end of primary selling season. Let V2 ðx; yÞ be the maximum revenue that can be obtained in the secondary selling season when the net inventory is x and the current spot price for the commodity is y. Then 
 V2 ðx; yÞ ¼ maxED2 p min xþ ; D2 ðpÞ ;

(5)

pPðτÞ

Proof. Since fD1 ðtÞ; t  0g is a Poisson process with rate λ, Tj is an Erlang random variable with shape parameter j and rate λ. By first interchanging the summation and the expectation in (6), and then by using conditional expectations, we obtain

(4)

where xþ ¼ maxf0; xg. In (4) a markdown sales price p, less than the current spot price y is chosen in such a way that the expected revenue is maximised. The sum of the primary season profit (3) and the discounted secondary season revenue gives us the total profit, JðQÞ, obtained from the initial purchase quantity Q:

ii i h  h h 

E P Tj erTj 1fTj τg ¼ ETj EPðTj Þ P Tj erTj 1fTj τg Tj

(7)

h  ii h

¼ ETj erTj 1fTj τg EPðTj Þ P Tj Tj

(8)


 ¼ ETj erTj 1fTj τg Pð0ÞeμTj :

(9)

The inner expectation in (7) (conditioned on Tj ) is taken over PðTj Þ, whereas the outer expectation is taken over Tj . As given the value Tj , the price PðTj Þ is independent of the demand arrival process, we obtain (8).

Since E½PðtÞ ¼ Pð0Þeμt , we have EPðTj Þ ½PðTj Þ Tj  ¼ Pð0ÞeμTj , which yields (9). Finally, taking expectation with respect to Tj we have 4

Y. Gürel, R. Güllü

International Journal of Production Economics xxx (2018) 1–9


 E P Tj erTj 1fTj τg ¼ Pð0Þ

Z

τ

eðμrÞx

0


 E erTn Lðt  Tn Þ ¼

Q Z X

τ

eðμrÞx

0

j¼1

λj xj1 eλx dx: ðj  1Þ!

After straightforward manipulation, we obtain SRðQÞ ¼ αPð0Þ

Q  X

λ

j Z

λμþr

j¼1

τ

ðλ  μ þ rÞj xj1 eðλμþrÞx dx; ðj  1Þ!

0

where Z

τ 0

o n ðλ  μ þ rÞj xj1 eðλμþrÞx dx ¼ Pr Tj  τ ðj  1Þ!

3.2. Expected holding costs over the primary selling season

HCðQÞ ¼

Let h be the holding cost per unit value per unit time. We assume that the holding cost is charged for the initial unit purchase price Pð0Þ. This is justified, if holding costs are accounted for “out-of-pocket” costs, rather than accounting for the cost of lost opportunity. Hence, a unit sold at a future time Ti will be charged erTi hPð0Þ as the discounted holding cost. Let Mði; tÞ be the expected discounted inventories carried, when we have i units on hand, and when there are t time units until the end of primary selling season, with the understanding that Mði; tÞ ¼ 0 whenever t  0 or i ¼ 0. Then Pð0ÞhMðQ; τÞ is the desired expected discounted inventory holding costs. For notational convenience we let LðtÞ : ¼ Mð1; tÞ. When there is one unit on hand, the expected discounted inventories carried for this unit can be obtained as: LðtÞ ¼ E4

Z

minðt;T1 Þ

e

" i  Q1 Q1 X X  hPð0Þ λ ðQ  iÞ Pr Ti  erτ ðQ Qþ λþr λ þ r i¼1 i¼0 #  iÞPrðNðτÞ ¼ iÞ :

(12)

Combining the relevant cost and revenue terms, the expected total profit over the primary selling season becomes: V1 ðQÞ ¼ Pð0ÞQ þ SRðQÞ  HCðQÞ " j n Q  o hPð0Þ X λ Pr Tj  τ  Q ¼ Pð0ÞQ þ αPð0Þ λμþr λþr j¼1

# i  Q1 Q1 X X   λ r τ ðQ  iÞ Pr Ti  e ðQ  iÞPrðNðτÞ ¼ iÞ ; þ λþr i¼1 i¼0

3 rx

ery

where Tn  Erlangðn; λÞ; NðtÞ  PoissonðλtÞ. ▫ The expected holding cost HCðQÞ, is then obtained by inserting the expression in Proposition 3.2 into equation (11), and by multiplying with hPð0Þ:

as desired. ▫

2

t

0

and SRðQÞ ¼ αPð0Þ

Z

1  1  eðλþrÞðtyÞ fn ðyÞ dy λþr Z t Z t 1 1 ¼ ery fn ðyÞ dy  eðλþrÞt eλy fn ðyÞ dy λþr 0 λþr 0  n Z t ðλþrÞy n1 1 λ e y ðλ þ rÞn dy ¼ ðn  1Þ! λþr λþr 0 Z eðλþrÞt λn t yn1 dy  λþr 0 ðn  1Þ!  n  eðλþrÞt λn t n λ ¼ Pr Tn < t  λþr λ þ r n! n    λ ert PrðNðtÞ ¼ nÞ; Pr Tn < t  ¼ λþr λþr

λj xj1 eλx dx; ðj  1Þ!

where Tj  Erlangðj; λ  μ þ rÞ and Ti  Erlangði; λ þ rÞ.

dx5

0

Z tZ

y

¼ 0

erx dxλeλy dy þ

0

Z t

∞

Z

t

4. Analysis of the secondary market customers erx dxλeλy dy

0

As described in equation (5), the leftover quantity (if any) from the primary market customers, are aimed to be sold to the secondary market customers, using an optimised markdown price p. In this section, we first present a general formulation for the secondary selling season problem (Section 4.1), then analyse a special case where all the leftover items can be sold in the secondary market at a fixed markdown price p ¼ sPðτÞ, where s  1  α and PðτÞ is the spot price at the beginning of the secondary selling season (Section 4.2), and subsequently we consider the more general price optimisation problem under a linear demand model with additive randomness (Section 4.3).

 1
1  eðλþrÞt ; ¼ λþr noting that T1 is exponentially distributed with rate λ. When the on hand inventory is i, each one of these units will be carried proportional to LðtÞ until the first demand occurs (at time T1 ), at which time the remaining on hand inventory and the remaining time until the end of primary season will be updated. Hence,
 Mði; tÞ ¼ iLðtÞ þ E erT1 Mði  1; t  T1 Þ :

(10)

Writing MðQ; τÞ recursively using (10) we obtain MðQ; τÞ ¼ QLðτÞ þ

4.1. A general mathematical model for the secondary market

Q1 X


 ðQ  iÞE erTi Lðτ  Ti Þ ;

At the end of the primary selling season, with the realisations of the current price PðτÞ, and the primary customer demand D1 ðτÞ, the price optimisation of the secondary market problem becomes (see equations (4) and (5)):

(11)

i¼1

as Ti is distributed as the sum of i independent and identically distributed copies of T1 . The following result obtains an explicit expression for the computation of MðQ; τÞ.


 V2 ðQ  D1 ðτÞ; PðτÞÞ ¼ max ED2 pmin ðQ  D1 ðτÞÞþ ; D2 ðpÞ : pPðτÞ

rt

λ  e Proposition 3.2. E½erTn Lðt  Tn Þ ¼ ðλþrÞ nþ1 PrðTn < tÞ  λþr PrðNðtÞ ¼ nÞ, where Tn is distributed as an Erlang random variable with shape parameter n and rate λ þ r, and NðtÞ is a Poisson random variable with mean λt. n

For notational simplification, let Rðx; pÞ ¼ pED2 ½minfx; D2 ðpÞg; for x  0, and assume that Rðx; pÞ is concave in p. This assumption does not automatically hold, and requires conditions on the secondary season demand D2 ðpÞ. Petruzzi and Dada (1999), and Kocabiyikoglu and

Proof. Let fn ðyÞ be the density function of Tn ( Erlangðn; λÞ). Then

5

Y. Gürel, R. Güllü

International Journal of Production Economics xxx (2018) 1–9

4.2. Special case: all the leftovers can be sold at a fixed discount price

Popescu (2011) present conditions under which Rðx; pÞ is concave in p. Consequently, let p ðxÞ be the maximiser of Rðx; pÞ over p  0. Due to concavity of Rðx; pÞ,

If all the leftover items can be sold at fixed price sPðτÞ, then equation

(    R ðQ  D1 ðτÞÞþ ; p ðQ  D1 ðτÞÞþ if p ðQ  D1 ðτÞÞþ  PðτÞ;   þ þ R ðQ  D1 ðτÞÞ ; PðτÞ if p ðQ  D1 ðτÞÞ > PðτÞ:

V2 ðQ  D1 ðτÞ; PðτÞÞ ¼

We define gðyÞ as the probability density function of PðτÞ, the price at the end of the primary selling season. Note that lnðPðτÞÞ is normally  distributed with mean μτ ¼ lnðPð0ÞÞ þ μ  12σ P τ and variance σ 2P τ. The

(4) simplifies to

expected revenue associated with the secondary selling season becomes:

and hence

Z þ

V2 ðQ  D1 ðτÞ; PðτÞÞ ¼ sPðτÞðQ  D1 ðτÞÞþ ;


 E½V2 ðQ  D1 ðτÞ; PðτÞÞ ¼ E sPðτÞðQ  D1 ðτÞÞþ
þ μτ ¼ sPð0Þe E ðQ  D1 ðτÞÞ Q1 X ðQ  iÞPrðD1 ðτÞ ¼ iÞ; ¼ sPð0Þeμτ


ED1 ðτÞ;PðτÞ V2 ðQ  D1 ðτÞ; PðτÞÞ 3 ∞ p ðQiÞ

7 RðQ  i; p ðQ  iÞÞgðyÞdy5PrfD1 ðτÞ ¼ ig

i¼0

2 Z Q X 4 ¼ i¼0

p ðQiÞ

(13)

where D1 ðτÞ  PoissonðλτÞ. Therefore, for this special simplified case, the total expected profit JðQÞ becomes

RðQ  i; yÞgðyÞdy

0



JðQÞ ¼ E V1 ðQÞ þ erτ E V2 ðQ  D1 ðτÞ; PðτÞÞ " j n Q  o hPð0Þ X λ  ¼ Pð0ÞQ þ αPð0Þ Pr Tj  τ  Q λμþr λþr j¼1

   lnðp ðQ  iÞÞ  μτ pffiffiffi þ 1Φ RðQ  i; p ðQ  iÞÞ PrfD1 ðτÞ ¼ ig;

σP τ

where ΦðxÞ is the cumulative distribution function of a normal random variable with mean zero and variance one. The second equality above follows since (1) RðQ  i; p ðQ  iÞÞ does not depend on y, and (2) lnðPðτÞÞ is normally distributed. By combining these observations, a computational procedure for the expected revenue of the secondary selling season is provided in Algorithm 1.

þ

þsPð0ÞeðμrÞτ

p ðQiÞ

Proof.

RðQ  i; yÞgðyÞdy

þsPð0ÞeðμrÞτ

0

i¼0 wi ðQÞ

Q X

PrfD1 ðτÞ ¼ jg

j¼1

# " i Q  Q X X   hPð0Þ λ r τ Pr Ti  τ  e PrðNðτÞ ¼ iÞ 1þ  λþr λþr i¼1 i¼0

σP τ

PQ

The first difference of JðQÞ with respect to Q is:

ΔJðQÞ ¼ JðQ þ 1Þ  JðQÞ Qþ1 n  o λ  Pr TQþ1 τ ¼ Pð0Þ þ αPð0Þ λμþr

  lnðp ðQ  iÞÞ  μτ pffiffiffi RðQ  i; p ðQ  iÞÞ þ 1Φ Return E½V2 ððQ  D1 ðτÞ; PðτÞ ¼

ðQ  jÞPrfD1 ðτÞ ¼ jg:

Proposition 4.1. When all the leftover stock can be sold at a fixed price sPðτÞ, then JðQÞ is concave in Q for μ  r.

Initialize: For a given value of Q For i ¼ 0; 1; …; Q 1. Find p ðQ  iÞ maximising RðQ  i; pÞ over p. 2. Evaluate

Z

Q1 X j¼1

Algorithm 1 An Algorithm for the Expected Revenue of the Secondary Market.

wi ðQÞ :¼

# i  Q1 X  λ ðQ  iÞ Pr Ti  τ  erτ ðQ  iÞPrðNðτÞ ¼ iÞ λþr i¼1 i¼0

Q1 X

eλτ ðλτÞQ Q!

The second difference is given by: Δ2 JðQÞ ¼ ΔJðQ þ 1Þ  ΔJðQÞ " # Qþ2 n Qþ1 n o  λ o λ   Pr TQþ2 τ  Pr TQþ1 τ ¼ αPð0Þ λμþr λμþr

The specific implementation of Algorithm 1 depends on the form of Rðx; pÞ (or on the demand function D2 ðpÞ for the secondary market customers). In the subsequent subsections we present two specific cases. We first assume that all the leftover items from the primary selling season can be sold. Essentially, this is equivalent to assuming that D2 ðpÞ is large enough so that the customers purchase every leftover item at a fixed discount price. Although this case is not too realistic, it enables us to analytically derive optimality condition for the initial order quantity Q. This, in turn can be used as an upper bound on the optimal order quantity of a system where D2 ðpÞ is finite. Such a system is analysed later by assuming D2 ðpÞ ¼ a  bp þ ε, where ε is a random variable.

þsPð0ÞeðμrÞτ PrfD1 ðτÞ ¼ Q þ 1g þ  Qþ1 n o hPð0Þ λ  Pr TQþ1 τ :  λþr λþr

hPð0Þ rτ e PrfD1 ðτÞ ¼ Q þ 1g λþr

The cumulative distribution function of an Erlang random variable Pk1 1 γx with shape parameter k and rate γ is 1 ðγxÞn ¼ n¼0 n!e P∞ 1 γx n ðγxÞ , and the probability mass function of the Poisson distrin¼k n!e

6

Y. Gürel, R. Güllü

International Journal of Production Economics xxx (2018) 1–9 λx λk

bution with rate lambda is e

k!

. Using these identities in Δ2 JðQÞ yields:

Table 2 Optimal order quantity: λ ¼ 10, μ ¼ 0:1, σ P ¼ 1.

 Qþ1 X  ∞ rμ λ eðλμþrÞτ ððλ  μ þ rÞτÞn Δ2 JðQÞ ¼ αPð0Þ λμþr λμþr n! n¼Qþ2

v ¼ a=3

eðλμþrÞτ ðλτÞQþ1 eðλμþrÞτ ðλτÞQþ1 þ sPð0Þ αPð0Þ ðQ þ 1Þ! ðQ þ 1Þ!  Qþ1 X ∞ ðλþrÞτ hPð0Þ λ e ððλ þ rÞτÞn ;  λþr λþr n! n¼Qþ2

τ ¼1 τ ¼2 τ ¼3

v ¼ a=5

τ ¼1 τ ¼2

and we can conclude that for μ  r (and α  s), Δ2 JðQÞ  0 so that the expected profit function is concave. ▫ In this case, the optimal order quantity is obtained as

τ ¼3

Q* ¼ minfQ : ΔJðQÞ  0g:

α

h ¼ 0:01

h ¼ 0:03

h ¼ 0:05

1.01 1.05 1.01 1.05 1.01 1.05 1.01 1.05 1.01 1.05 1.01 1.05

4 6 8 13 10 21 4 6 8 13 10 21

2 5 3 12 3 17 2 5 3 12 3 17

1 5 2 9 2 10 1 5 2 9 2 10

Table 3 Optimal profit: λ ¼ 10, μ ¼ 0:1, σ ¼ 1.

Hence, the smallest value of Q satisfying Qþ1 n  o λþhþr λ  α Pr TQþ1 τ λþr λμþr X  Q h PrfD1 ðτÞ ¼ ig þerτ seμτ þ λ þ r i¼0 i Q    h X λ Pr Ti  τ  λ þ r i¼1 λ þ r

v ¼ a=3

τ ¼1 τ ¼2 τ ¼3

v ¼ a=5

τ ¼1 τ ¼2

is optimal.

τ ¼3

α

h ¼ 0:01

h ¼ 0:03

h ¼ 0:05

1.01 1.05 1.01 1.05 1.01 1.05 1.01 1.05 1.01 1.05 1.01 1.05

0.022 0.212 0.044 0.504 0.047 0.764 0.023 0.219 0.044 0.507 0.047 0.766

0.011 0.180 0.012 0.346 0.012 0.412 0.011 0.183 0.012 0.348 0.012 0.413

0.005 0.151 0.005 0.231 0.005 0.236 0.005 0.153 0.005 0.231 0.005 0.236

4.3. A linear demand model with additive random error term Otherwise, if pB ðxÞ  ða  v  xÞ=b, then pB ðxÞ ¼ ða  v  xÞ=b, and if pB ðxÞ  ða þ v  xÞ=b, then pB ðxÞ ¼ ða þ v  xÞ=b. Region C: On p  ða þ v  xÞ=b the demand is (for sure) smaller than the available quantity, and Rðx; pÞ ¼ RC ðx; pÞ, where

In this section, we turn back to the analysis of (4), and apply the general framework presented in Section 4.1. In order to find the profit maximising markdown price, we assume the demand model D2 ðpÞ ¼ a  bp þ ε, where ε is distributed uniformly over ½v; v and a;b > 0, v < a. In order to ensure that the demand is non-negative, we restrict the feasible set of prices to ½0; ða  vÞ=b. In what follows, for notational convenience we let x ¼ ðQ  D1 ðτÞÞþ be the number of leftover items at the end of the primary selling season. The expected secondary market revenue Rðx; pÞ will be analysed over three regions of p: Region A: On p  ða  v  xÞ=b the demand is (for sure) larger than the available quantity, and Rðx;pÞ ¼ RA ðx;pÞ ¼ px. RA ðx; pÞ is maximised at pA ðxÞ ¼ ða  v  xÞ=b, and Rðx; pA ðxÞÞ ¼ pA ðxÞx. Region B: On p 2 ðða  v  xÞ=b; ða þ v  xÞ=bÞ, Rðx; pÞ ¼ RB ðx; pÞ, where Z RB ðx; pÞ ¼ p

xðabpÞ

v

dz ða  bp þ zÞ þ p 2v

Z

RC ðx; pÞ ¼ pE½D2 ðpÞ ¼ pa  bp2 : RC ðx; pÞ is maximised at pC ðxÞ ¼ a=2b. Then, the maximiser over Region C becomes: pC ðxÞ ¼ maxfða þ v  xÞ=b; a=2bg. It can easily be verified that Rðx; pÞ is concave over p  0, and the optimal prices found over the three regions above completely characterise the pricing problem of the secondary selling season. Let J 2 fA; B; Cg be the index for which the maximum expected revenue is achieved. That is       RJ x; pJ ðxÞ ¼ max RA x; pA ðxÞ ; RB x; pB ðxÞ ; RC x; pC ðxÞ : Then, the global maximising price of the secondary phase is given by pJ ðxÞ. Finally, in equation (13), and in the computation of wi ðQÞ in Algorithm 1, if the current spot price y ¼ PðτÞ is larger than p ðQ  iÞ, then we use RðQ  i; p ðQ  iÞÞ, and otherwise we use RðQ  i; yÞ

v

dz x xðabpÞ 2v

 1 2 3 ¼ b p þ 2bða  v  xÞp2 þ 4vx  ða  v  xÞ2 p : 4v

5. Numerical findings

Then,   dRB ðx; pÞ 1
¼ 3b2 p2 þ 4bða  v  xÞp þ 4vx  ða  v  xÞ2 dp 4v

Our numerical study serves two objectives: (1) we would like to understand the impact of unit cost uncertainty on the optimal order quantity and the resulting optimal profit, and (2) we would like to assess if there is any significant loss in ignoring the volatility of the unit cost. Throughout our analysis, we fix λ ¼ 10, and the economic life of the item is taken as CL ¼ 5. The yearly interest rate, r is set to 0.1, and the initial purchase price, Pð0Þ is set to 1. Other parameters are varied as: σ P 2 f0; 0:5; 0:8; 1g, τ 2 f1; 2; 3; 4; 4:5g, μ 2 f0:09; 0:1; 0:15g, h 2 f0:01; 0:03; 0:05g and α 2 f1; 1:01; 1:05g. We want to relate the parameters of the secondary selling season demand to the demand and unit cost parameters of the primary selling season. The average price of the commodity over the primary selling

 d2 RB ðx; pÞ 1
6b2 p þ 4bða  v  xÞ < 0: ¼ 4v dp2 The inequality is due to the fact that a  v  x < bp and this ensures concavity of RB ðx; pÞ over p. A tedious but straightforward derivation yields that RB ðx; pÞ is maximised at pB ðxÞ ¼

 1=2  1  2ða  v  xÞ þ ða  v  xÞ2 þ 12vx : 3b

If pB ðxÞ 2 ðða  v  xÞ=b; ða þ v  xÞ=bÞ, then let pB ðxÞ ¼ pB ðxÞ. 7

Y. Gürel, R. Güllü

International Journal of Production Economics xxx (2018) 1–9

“more active”. Consequently, any excess inventory can only be solved at a less desirable price, and this leads to a possible decrease in optimal order quantities. Finally, in Table 6, we present the percent improvement using the volatility information: the improvement in the expected profit, when in fact the order quantity is chosen by taking the unit cost volatility into account as opposed to operating the system as if there is no unit cost volatility. Table 6 presents percent improvement in the profit when the unit cost volatility is σ P ¼ 1. As it can be observed, the penalty for ignoring unit cost volatility can be substantial, especially when the drift parameter of the unit cost, and the unit holding cost are high. We also observe that, the improvement one can gain by taking the volatility information into account would be much higher when the length of the primary selling season is relatively short as compared to the length of the secondary season (that is, when τ=CL is small).

Table 4 Optimal order quantity: λ ¼ 10, μ ¼ 0:1, h ¼ 0:01. v ¼ a=3

τ ¼1 τ ¼2

v ¼ a=5

τ ¼1 τ ¼2

α

σP ¼ 0

σ P ¼ 0:5

σ P ¼ 0:8

σP ¼ 1

1.01 1.05 1.01 1.05 1.01 1.05 1.01 1.05

4 6 9 13 4 6 9 14

4 6 9 13 4 6 9 13

4 6 9 13 4 6 9 13

4 6 8 13 4 6 9 13

Table 5 Optimal profit: λ ¼ 10, μ ¼ 0:1, h ¼ 0:01. v ¼ a=3

τ ¼1 τ ¼2

v ¼ a=5

τ ¼1 τ ¼2

α

σP ¼ 0

σ P ¼ 0:5

σ P ¼ 0:8

σP ¼ 1

1.01 1.05 1.01 1.05 1.01 1.05 1.01 1.05

0.024 0.225 0.045 0.516 0.026 0.238 0.045 0.527

0.024 0.223 0.045 0.513 0.025 0.233 0.045 0.519

0.023 0.216 0.045 0.508 0.024 0.223 0.045 0.512

0.022 0.212 0.044 0.504 0.023 0.219 0.044 0.507

6. Conclusions and possible research directions In this paper we considered a system where the customer demand can be partitioned into two segments: primary market customers and secondary market customers. The main characteristics of these systems are as follows:  The primary market customers, being more loyal for the product or service offered, is not too sensitive to the pricing of the product or service,  The secondary market customers, whose demand occurs after the product matures, are much more sensitive to changes in the sales price,  The unit purchasing costs of technology intensive products (where the primary/secondary market distinction is prominent) depend on the spot currency exchange rate, and hence can be modelled as a stochastic process.

Rτ season can be computed as: p ¼ E½ 0 PðtÞdt=τ ¼ Pð0Þðeμτ  1Þ=μτÞ. Then, the values of a and b for the demand model of the secondary selling season are chosen as: a ¼ 1:5ðCL  τÞλ and b ¼ 1:5ðCL  τÞλ=p. Moreover, in our analysis we choose two values for representing demand variability in the secondary season: v 2 fa=3; a=6g. Tables 2 and 3 present optimal order quantities and corresponding optimal profits for λ ¼ 10, μ ¼ 0:1, σ ¼ 1. In each table we present our results for different values of v and for τ ¼ f1; 2; 3g. Our observations are similar for other parameter settings. It can be observed in Tables 2 and 3 that the optimal quantity, and the optimal profit value are nondecreasing as one of the following three parameters increases: length of primary selling season, τ, the drift of the price process, μ (not reported in the tables), and the mark-up multiplier for sales price, α. It can also be observed that the optimal quantity, and the optimal profit value are nonincreasing as holding cost coefficient, h, increases. It can also be observed from Table 3 that the optimal profit is non-decreasing in the variability of secondary market demand v. These observations are quite intuitive and as expected. The optimal order quantity, and the optimal profit values are nonincreasing as the variance of the price, σ P increases. The latter observation is indeed expected (please see corresponding optimal profits in Table 5). However, the former observation is surprising (corresponding order quantities are presented in Table 4 for h ¼ 0:01). One generally expects larger order quantities associated with an increasing variability.  pffi Note that, PrfPðτÞ  E½PðτÞg ¼ Φ σP2 τ . Hence, as σ P increases, the

Based on these characteristics, our study aimed  To model the described system  To find the optimal initial quantity to stock  To determine the optimal sales price for the secondary market customers. In order to achieve these objectives, as the modelling method, we derived the expected profits of the system over the demand periods of primary and secondary customers, respectively. As the sales quantity for the secondary market customers is bounded by the product availability at the end of primary sales period, the initial purchase quantity would have an impact on the sales price for the secondary market customers. We also carried out a computational study to highlight important managerial insights. Our results revealed important information regarding the impact of joint determination of secondary market price and initial purchase quantity, and the effect of cost volatility on inventory and pricing decisions. In conclusion, our model provided original analytical

probability that the price at time τ will be below the mean price (the price when σ P ¼ 0) increases. Hence, the price constraint p  PðτÞ becomes

Table 6 The value of the volatility information (%). μ v

τ

h

a=3

1 1 2 2 1 1 2 2

α α α α α α α α

a=5

0.090

¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼

1:01 1:05 1:01 1:05 1:01 1:05 1:01 1:05

0.1

0.15

0.01

0.03

0.05

0.01

0.03

0.05

0.01

0.03

0.05

0.000 1.342 0.000 1.056 0.000 0.000 0.000 0.700

0.000 0.000 0.000 0.167 0.000 6.108 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.048 0.000 0.000 0.000 0.000 2.070

0.000 0.000 0.000 0.000 2.261 1.994 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.382

7.449 1.929 2.152 1.090 3.161 0.000 1.267 0.000

0.000 0.000 2.935 0.000 0.000 3.745 2.344 0.000

0.000 0.000 0.555 0.000 12.805 0.000 0.000 0.000

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International Journal of Production Economics xxx (2018) 1–9

findings, together with important managerial insight for a system with dual customer base, and with demand and cost uncertainty. There are several possibilities for extending this study. We made a number of simplifying assumptions, that can be relaxed. For instance, we may investigate the impact of correlation between unit cost and demand process. Over the secondary selling season, we assumed a linear relationship between the secondary market demand and the markdown price. More complicated demand models can be investigated. We assumed that the primary market customers occur according to a Poisson process. Other processes (such as compound Poisson) can be employed, at the expense of a more involved expected cost model. Finally, our model only considers one ordering opportunity (occurring at the beginning of the planning horizon), whereas one could have a model with several ordering opportunities.

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