inventory control by different service levels

Applied Mathematical Modelling 35 (2011) 497–505

Contents lists available at ScienceDirect

Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

Inventory control by different service levels

Jinpyo LeeSchool of Management, Hongik University, Seoul, Republic of Korea

a r t i c l e i n f o

Article history:Received 9 February 2009Received in revised form 5 July 2010Accepted 7 July 2010Available online 13 July 2010

Keywords:Inventory/productionOperating characteristicsDynamic pricingMultiple service levels

0307-904X/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.apm.2010.07.015

E-mail address: [email protected]

a b s t r a c t

This paper examines the multiple period inventory control problem of a single productwith multiple (two) prices, depending on service level, in which optimal pricing and order-ing decisions are made in each period. Traditional inventory and pricing models consideronly single products, single prices, and single service levels. However, this research paperfinds that a seller can improve inventory control and revenue by offering multiple pricesdepending on service level. This research considers a single product with multiple (two)pricing policies corresponding to service level as follows: if the customer is willing to delaythe shipment, he/she will be offered a lower regular price. Otherwise, the customer willpay the regular price plus extra charges for express service. In this paper, I show the follow-ing: (1) there is an optimal pricing and replenishment policy that can control inventory and(2) there exists a finite threshold for inventory levels such that if the inventory level at thebeginning of each period is higher than the threshold, the customer will be offered theexpress service at the regular price, without any extra charge.

� 2010 Elsevier Inc. All rights reserved.

1. Introduction

In many cases, you are offered a low price if you are willing to accept slower service. For instance, consider online storessuch as Amazon.com and Buy.com. If you are willing to accept delayed shipping of your order, you are commonly offered alower price than that for faster shipment. In other words, you pay more for the faster shipment of your order. In this paper, toidentify how to control the inventory problem herein, single products whose inventory levels and selling prices are reviewedperiodically and decided dynamically are considered as having two prices, depending on service level. The two service levelsconsist of express service and regular service. In many practical cases, a customer will respond to the offered service leveldepending on the price in the following sense. When ordering an item from a seller, some customers might be willing todelay shipment if they are offered a lower price. This provides the seller with advance demand information to apply tothe next production/ordering period so that inventory can be controlled more efficiently. Other customers require the or-dered item to be shipped as soon as possible, even if that necessitates paying an extra charge. This is called express service.Therefore, depending on the customer’s willingness to delay the shipment, service level can be classified into regular andexpress level and discriminated by price. Suppose that at period t there are two types of customers. One orders an item withregular service and the other orders an item with express service. The items will be shipped as shown in Fig. 1. The model inthis paper is similar to the model outlined in Ref. [1] except that Ref. [1] considers single products with a single pricing pol-icy, whereas the model in this paper considers single products with a multiple (two) pricing policy depending on the servicelevel. Ref. [2] shows the properties of an optimal pricing schedule depending on the customers own surplus. Ref. [2] assumesthat total amount of demand for all service levels does not depend on the price. However, this paper assumes that totalamount of demand depends on the price. Ref. [2] uses the demand model in which each customers demand level dependson their willingness-to-pay for shipment delay. This paper, however, uses that in which customers proportionally react to

. All rights reserved.

http://dx.doi.org/10.1016/j.apm.2010.07.015

mailto:[email protected]

http://dx.doi.org/10.1016/j.apm.2010.07.015

http://www.sciencedirect.com/science/journal/0307904X

http://www.elsevier.com/locate/apm

Fig. 1. Timeline of ordering and shipping.

498 J. Lee / Applied Mathematical Modelling 35 (2011) 497–505

given prices by moving dynamically between service levels: if the price for the express service increases, some portion ofexpress service customers moves to regular service. Otherwise, some portion of regular service customers moves to expressservice. Refs. [3,1] consider the dynamic inventory control model with single priced products and single service levels, show-ing the (s,S,p)-policy and the base stock policy as the optimal policies, respectively. Ref. [3] considers that the fixed order costfor each replenishment occurs, whereas Ref. [1] does not. They assume that customers will select only one service level,whereas the customer in this paper may select one of a number of service levels. Ref. [4] assumes that each customers orderwill be shipped at some future time after the order is placed. This delay is referred to as demand lead-time, and it is equiv-alent to service level in this paper. However, this demand lead-time is fixed for any given order, resulting in only one servicelevel choice. Each customer in Ref. [4] cannot select this lead-time, whereas each customer in this paper can select the lead-time depending on the price.

2. Assumptions and notations

Notations:

1. c � per unit purchase or production cost.2. pR � price charged for regular service for all periods on ½c; �pR�.3. pE,t � price charged for express service for period t.4. pt = pE,t � pR � extra charge for the express service at period t on ½0; �p�.5. �t � random term with a known distribution.6. DE,t = dE,t(pt,�t) � demand for the express service.7. DR,t = dR,t(pt) � demand for the regular service.8. xt � inventory level at the beginning of period t before replenishment.9. yt � inventory level at the beginning of period t after replenishment.

10. ht(I) � inventory (or backlogging) cost at the end of period t.11. Htðyt; pÞ � E�t ½htðyt � dR;t�1ðpt�1; �t�1Þ � dE;tðp; �tÞÞ�.12. For convenience, E�t is written as just E.

Assumption 1. Replenishment becomes available instantaneously.

Assumption 2. Excess demand is backlogged.In the area of Economics and Operations Research, the demand has been frequently assumed to be a concave and decreas-

ing function or a linear and decreasing function in the price. Moreover expected demand is assumed to be finite and strictlydecreasing in the price. Since this decreasing demand, which is either concave or linear function in price, can be negative asthe price is sufficiently large, another assumption has been made such that the set of feasible price level is confined in thefinite interval to avoid the negative demand Refs. [1–3,5]. Thus for technical reasons, the following assumption regarding thedemand function is made.

Assumption 3. For all t = 1,2, . . . ,T, the function dE,t(p,�t) = a(pR,�t) � bp is a linearly decreasing function in p 2 ½0; �p�, anddR,t(p,�t) = bp is a linearly increasing function in p, where a(pR,�t) is the possible maximum demand and a nonincreasinglinear function of pR 2 ½c; �pR�, where values for �p and �pR are taken such that að�pR; �tÞ � b�p is positive w.p.1, and b isdeterministic.

Assumption 3 is established from the following perspective. Suppose that the price for the regular service is set at pR. Totalnumber of customers including the regular service and express service in period t, DR,t(p,�t) + DE,t(p,�t), is equal to a(pR,�t)and all customers will be divided into either regular service or express service depending on the price difference p. Thisimplies that initially all customers are enticed at regular service and then decide which service to select depending onthe price difference p = pE � pR between regular and express service. The larger the price difference will be, the less the cus-

J. Lee / Applied Mathematical Modelling 35 (2011) 497–505 499

tomer would be willing to select the express service. This implies that, out of a(pR,�t) number of customers, the number ofcustomers for express service will be a(pR,�t) at zero price difference (p = 0), which implies that all the customers would se-lect the express service when there is no price difference between express and regular service. However, the number of cus-tomers for express service decrease lineally as price difference p increase and thus the number of customers for expressservice is a(pR,�t) � bp at price difference p. The rest will be customers for regular service, the number of which isDR,t(p,�t) = a(pR,�t) � DE,t(p,�t) = a(pR,�t) � (a(pR,�t) � bp) = bp. Thus, when the price for the express service is equal to theprice for the regular service (pE = pR), the extra charge, which should be paid for express service, is equal to zero (p = 0) sothat a customer can select express service at the regular service price. This suggests that all the customers for either regularand express service will be enticed to choose express service without any extra charge (p = 0) when the price for expressservice is equal to one for regular service. Moreover, as pR increases, total demand for all service levels decreases. To sumup, as the extra charge for the express service increases from zero (p > 0), some demand for express service will move tothe regular service, so that bp of customers, who are reluctant to pay for the strictly positive extra charge (p > 0), will selectthe regular service.

Assumption 4. For all t = 1, . . . ,T, ht(I) is convex in I. Moreover, ht(I) is nonincreasing for I < 0 and nondecreasing for I > 0.

Assumption 5. limy?1Ht(y,p) = limy?1(cy + Ht(y,p)) = limy?1((1 � a)cy + Ht(y,p)) =1 for any p 2 ½0; �p� and a 2 [0,1].

Assumption 6. 0 6 Ht(y,pt) = O(jyjc) for some integer c.

Assumption 7. E[dt(p,�t)]c <1 for all p 2 ½0; �p� for some integer c.

3. Mathematical formulation

In this section, I formulate the mathematical model maximizing expected discounted revenue, under a given discount factora 2 (0,1). The planning horizon consists of finite T periods, numbered 1,2, . . . ,T � 1,T. During the planning horizon, the price canbe increased or decreased. Since the pricecan be changed from period to period, the problemcan be formulated as a dynamic pro-gramming model with (xt,DR,t�1) as the state of the system at the beginning of period t. Let gt(xt,DR,t�1) be the maximum expecteddiscounted revenue from t to T. Thus, at period t, the seller’s maximum expected discounted revenue is

gtðxt ;DR;t�1Þ ¼maxa � ðpR þ pÞ � E½DE;t � þ a � pR � E½DR;t � � cðy� xtÞ � Htðy; pÞ þ a � E½gtþ1ðy� DR;t�1 � DE;t;DR;tÞ� ð1Þs:t: xt 6 y;

0 6 p 6 �p;

0 6 pR 6 �pR;

(1) can be written equivalently as

gtðxt ;DR;t�1Þ ¼ c � ðxt � DR;t�1Þ þmax ptðy;p; pRÞ ð2Þs:t: xt 6 y;

0 6 p 6 �p;

0 6 pR 6 �pR;

where
ptðy;p;pRÞ ¼ a � ðpR þ pÞ � E½DE;t � þ a � pR � E½DR;t � � cðy� DR;t�1Þ � Htðy;pÞ þ a � E½gtþ1ðy� DR;t�1 � DE;t ;DR;tÞ�: ð3Þ
Lemma 1 shows that there is a DP model with one state variable equivalent to (2) so that the optimal solution to the equiv-alent DP model with one state variable can be easily translated into the optimal solution to (2).

Lemma 1. Let �xt and zt defined as xt � DR,t�1 and yt � DR,t�1, respectively. Then, y�t ; p�; p�R

� �is the optimal solution to (2) if and

only if y�t � DR;t�1; p�; p�R� �

is the optimal solution to

Gtð�xtÞ ¼max Ptðz;p; pRÞ ð4Þs:t: �xt 6 z;

0 6 p 6 �p;

0 6 pR 6 �pR;

where

Ptðz;p; pRÞ ¼ a � ðpR þ p� cÞ � E½DE;t� þ a � ðpR � cÞ � E½DR;t � � ð1� aÞcz� Htðz;pÞ þ a � E½Gtþ1ðz� DE;t � DR;tÞ�; ð5Þ

where Htðz; pÞ ¼ Htðzþ DR;t�1; pÞ ¼ E�t ½htðz� dE;tðp; �tÞÞ�.


Proof. It is enough to show that (2) is equivalent to (4). By letting z = y � DR,t�1, (2) is equivalent to

gtðxt ;DR;t�1Þ ¼ c � ðxt � DR;t�1Þ þmax ptðz;p;pRÞs:t: xt � DR;t�1 6 z;

0 6 p 6 �p;

0 6 pR 6 �pR;

where

ptðz; p;pRÞ ¼ a � ðpR þ pÞ � E½DE;t � þ a � pR � E½DR;t � � c � z� Htðzþ DR;t�1;pÞ þ a � E½gtþ1ðz� DE;t;DR;tÞ�:

By letting Gt(xt,DR,t�1) � gt(xt,DR,t�1) � c(xt � DR,t�1), the above is equivalent to

Gtðxt;DR;t�1Þ ¼max ptðz;p;pRÞs:t: xt � DR;t�1 6 z;

0 6 p 6 �p;

0 6 pR 6 �pR;

where

ptðz;p; pRÞ ¼ a � ðpR þ pÞ � E½DE;t � þ a � pR � E½DR;t� � c � z� Htðzþ DR;t�1;pÞ þ a � E½gtþ1ðz� DE;t ;DR;tÞ�;¼ a � ðpR þ pÞ � E½DE;t � þ a � pR � E½DR;t� � c � z� Htðzþ DR;t�1;pÞ þ a � E½Gtþ1ðz� DE;t ;DR;tÞ þ cðz� DE;t � DR;tÞ�;¼ a � ðpR þ p� cÞ � E½DE;t � þ a � ðpR � cÞ � E½DR;t � � ð1� aÞcz� Htðzþ DR;t�1;pÞ þ a � E½Gtþ1ðz� DE;t ;DR;tÞ�;¼ a � ðpR þ p� cÞ � E½DE;t � þ a � ðpR � cÞ � E½DR;t � � ð1� aÞcz� Htðzþ DR;t�1;pÞ þ a � E½Gtþ1ðz� DE;t ;DR;tÞ�:

Again, the above problem can be rewritten as the following equivalent problem. Using �xt ¼ xt � DR;t�1 as the state of the sys-tem at the beginning of period t,

Gtð�xtÞ ¼ max Ptðz; p;pRÞs:t: �xt 6 z;

0 6 p 6 �p;0 6 pR 6 �pR;

where,

Ptðz;p; pRÞ ¼ a � ðpR þ p� cÞ � E½DE;t� þ a � ðpR � cÞ � E½DR;t � � ð1� aÞcz� Htðz; pÞ þ a � E½Gtþ1ðz� DE;t � DR;tÞ�;

where Htðz; pÞ ¼ Htðzþ DR;t�1; pÞ � E�t ½htðz� dE;tðp; �tÞÞ�. h

Now, it is enough to find the optimal solution to (4) since it is easy to determine the optimal solution to (2) using Lemma 1.

3.1. Optimal policy

In this section, the following policy is shown to be optimal: if the inventory level at the beginning of period t is less thansome threshold level, then it is increased to the threshold level. Otherwise, nothing is ordered. Lemma 2 and Proposition 1 arethe general versions of the results in [1] in the following sense: [1] considers single product with single pricing as correspond-ing to a single service level. This paper, however, considers single product with multiple (two) pricing depending on servicelevel. Thus, if p is fixed at zero, the express service level would be that selected by all customers according to Assumption 3.

Lemma 2. For any t 2 {1,2, . . . , T � 1,T}, Pt(z,p,pR) is a jointly concave function of (z,p,pR) and Gtð�xtÞ is a nonincreasing concavefunction of �xt.

Proof. Let’s start at t = N; For the first and second term of (5), fix �t. Since demand function is linearly nonincreasing in p andpR by Assumption 3,

a � ðpR þ p� cÞ � DE;N þ a � ðpR � cÞ � DR;N;

¼ a � ðpR þ p� cÞ � ðaðpR; �tÞ � bpÞ þ a � ðpR � cÞ � bp;

¼ a � p � ðaðpR; �tÞ � bpÞ þ a � aðpR; �tÞ � ðpR � cÞ;

is jointly concave in (p,pR) so that a � (pR + p � c) � E[DE,N] + a � (pR � c) � E[DR,N] is jointly concave in (p,pR). The third term islinear in z. With the Assumption 4, the forth term is jointly concave since for any z1 – z2 and p1 – p2


HNz1 þ z2

2;pR1þ pR2

2;p1 þ p2

2

� �¼ E hN

z1 þ z2

2� dE;t

pR1þ pR2

2;p1 þ p2

2; �t

� �� ;

¼ E hNz1 þ z2

2� 1

2dE;tðpR1

; p1; �tÞ þ12

dE;tðpR2;p2; �tÞ

� �� ;

¼ E hN12ðz1 � dE;tðpR1

;p1; �tÞÞ þ12ðz2 � dE;tðpR2

; p2; �tÞÞ� ��

;

612

E½hNðz1 � dE;tðpR1; p1; �tÞ� þ

12

E½hNðz2 � dE;tðpR2;p2; �tÞ�:

The second equality holds due to the linearity of demand function and the inequality holds by Assumption 4. Thus, GNð�xNÞ isconcave and nonincreasing in �xN . Suppose that Pt+1(�, �, �) is jointly concave and that Gtþ1ð�xtþ1Þ is concave and nonincreasing.Then Pt(z,p,pR) is jointly concave: the first four terms of (5) are verified in the same way as above. Now to verify the last termin (5), need to show that for any given value of �t, Gt+1(z � DE,t � DR,t) is jointly concave in z and p. Since Gt+1(�) is nonincreas-ing and concave,

Gtþ1z1 þ z2

2� dE;t

p1 þ p2

2; �t

� � dR;t

p1 þ p2

2; �t

� � ;

¼ Gtþ1z1 þ z2

2� 1

2dE;tðp1; �tÞ þ

12

dE;tðp2; �tÞ� �

� 12

dR;tðp1; �tÞ þ12

dR;tðp2; �tÞ� ��

;

¼ Gtþ112

z1 � dE;tðp1; �tÞ � dR;tðp1; �tÞð Þ þ 12

z2 � dE;tðp2; �tÞ � dR;tðp2; �tÞð Þ� �

;

P12

Gtþ1 z1 � dE;tðp1; �tÞ � dR;tðp1; �tÞð Þ þ 12

Gtþ1 z2 � dE;tðp2; �tÞ � dR;tðp2; �tÞð Þ:

So, Pt(z,p,pR) is jointly concave in (z,p,pR). Therefore, Gtð�xtÞ is nonincreasing and concave. h

Proposition 1. Suppose that at period t the inventory level is xt and the demand for regular service from period t � 1 is DR,t�1.There exists a finite solution pair y�t ; p

�t ; p

�R

� �such that: if xt 6 y�t , it is optimal to order up to y�t and to charge the prices p�Rt

þ p�tfor the express service and p�Rt

for the regular service. Otherwise, it is optimal not to place an order.

Proof. The proof is similar to that of [1] but should consider one more variable. First, need to find the optimal solution to (4).Then, we can equivalently find the optimal solution to (2) by Lemma 1. For any t 2 {1,2, . . . ,T � 1,T}, need to show thatPt(z,p,pR) = O(jzjc), Gtð�xtÞ ¼ Oðj�xt jcÞ, and Pt(z,p,pR) have a finite maximizer which is here denoted by z�t ; p

�t ; p

�Rt

� . By induc-

tion, PN(z,p,pR) = O(jzjc) for some integer c by Assumption 6. By Lemma 2 and Assumption 5, PN has a finite maximizer.Now, suppose that Pt+1(z,p,pR) = O(jzjc) and that Pt+1 has a finite maximizer, z�tþ1; p

�tþ1; p

�Rtþ1

� . Consequently,

Gtþ1ð�xÞ ¼ Oðj�xjcÞ and thus there exists some constant K 2 (0,1) such that

Gtþ1ðz� DE;t � DR;tÞ 6 Kjz� DE;t � DR;tjc;6 K jzj þ DE;t þ DR;tð Þð Þc;

6 KXc

n¼0

c

n

� �jzjn max

p2½0;�p� pR2½c;�pR �DE;t þ DR;tð Þc�n

;

6 KXc

n¼0

cn

� �jzjc max

p2½0;�p� pR2½c;�pR �DE;t þ DR;tð Þc�n

;

by the Binomial expansion. Thus, with Assumption 6, Pt(z,p,pR) = O(jzjc). Since Pt+1 has a finite maximizer z�tþ1; p�tþ1; p

�Rtþ1

� ;

Gtþ1ðz� DE;t � DR;tÞ 6 Gtþ1 z�tþ1

� �¼ O z�tþ1

c� <1 for any z 2 R. So, by (5) and Assumption 5, limjzj?+1Pt(z,p,pR) = �1 for any

p 2 ½0; �p� and pR 2 ½c; �pR�. Now, with Lemma 2, Pt(z,p,pR) has a finite maximizer. Thus, with Lemma 2, if �xt ¼ xt�DR;t�1 6 z�t ; z�t ; p

�t ; p

�Rt

� is optimal decision. Otherwise, choose z = xt � DR,t�1 by the joint concavity of Pt. Now, by Lemma 1,

we can take y�t ¼ z�t þ DR;t�1. Then, if xt 6 y�t ; y�t ; p�t ; p

�Rt

� is the optimal solution to (2). Therefore, the result holds. h

Now, suppose that pR is fixed for all periods. This implies that the seller offers a pre-selected price for the regular servicefor all periods and attempts to control the extra charge for the express service to attract customers and maximize revenue.This generates interesting and useful results. Occasionally, the seller offers an extra charge of zero to the customer, meaningall customers are attracted to the express service. For example, when buying a book at an online bookstore, from time to timeyou are offered the express shipping service at the price for the regular shipping service and are willing to take this offer. Thisgives the seller the opportunity to reduce the holding inventory level more quickly by shipping all the ordered items in thesame period as in the ordered period. Proposition 2 shows that, under a sufficient condition, there exists a threshold forinventory level at the beginning of each period such that if the inventory level before replenishment is higher than thethreshold, then it is optimal to offer the zero extra charge for the express service.


Proposition 2. Suppose that pR is fixed for all periods and that for period t there exists some finite number jZtj <1 such that

lim supp!0

E½htðZt � aðpR; �tÞ þ bpÞ� � E½htðZt � aðpR; �tÞÞ�bp

>a � E½aðpR; �tÞ�

b;

holds. Then, there exists some finite threshold for the inventory level at the beginning of each period such that if the inventory levelbefore replenishment is greater than the threshold, then the optimal extra charge p�t for the express service is zero.

Proof. For zt = Zt,

PtðZt; p;pRÞ ¼ a � ðpR þ p� cÞ � E½DE;t � þ a � ðpR � cÞ � E½DR;t� � ð1� aÞcZt � E½htðZt � aðpR; �tÞ þ bpÞ�þ a � E½Gtþ1ðZt � DE;t � DR;tÞ�;¼ a � ðpR þ p� cÞ � E½aðpR; �tÞ � bp� þ a � ðpR � cÞ � E½bp� � ð1� aÞcZt

� E½htðZt � aðpR; �tÞ þ bpÞ� þ a � E½Gtþ1ðZt � ðaðpR; �tÞ � bpÞ � bpÞ�;¼ a � p � E½aðpR; �tÞ � bp� þ a � ðpR � cÞ � E½aðpR; �tÞ� � ð1� aÞcZt

� E½htðZt � aðpR; �tÞ þ bpÞ� þ a � E½Gtþ1ðZt � ðaðpR; �tÞ � bpÞ � bpÞ�;PtðZt;0;pRÞ ¼ a � ðpR þ 0� cÞ � E½DE;t � þ a � ðpR � cÞ � E½DR;t� � ð1� aÞcZt � E½htðZt � aðpR; �tÞ þ b0Þ�

þ a � E½Gtþ1ðZt � DE;t � DR;tÞ�;¼ a � ðpR � cÞ � E½aðpR; �tÞ� � ð1� aÞcZt � E½htðZt � aðpR; �tÞÞ� þ a � E½Gtþ1ðZt � aðpR; �tÞÞ�:

Then,

PtðZt; p;pRÞ �PtðZt; 0;pRÞ ¼ a � p � E½aðpR; �tÞ � bp� � E½htðZt � aðpR; �tÞ þ bpÞ� þ E½htðZt � aðpR; �tÞÞ�;

¼ pa � p � E½aðpRÞ � bp�

p� E½htðZt � aðpR; �tÞ þ bpÞ� � E½htðZt � aðpR; �tÞÞ�

p

� �;

¼ p a � E½aðpR; �tÞ � bp� � b � E½htðZt � aðpR; �tÞ þ bpÞ� � E½htðZt � aðpR; �tÞÞ�bp

� �:

Since E[ht(I)] is convex by Assumption 4, for all p > 0


P lim supp!0


>a � E½aðpR; �tÞ�

b:

Also, for all p > 0

a � E½aðpR; �tÞ � bp� 6 lim supp!0

a � E½aðpR; �tÞ � bp� ¼ a � E½aðpR; �tÞ�:

Thus, for all p > 0

a � E½aðpR; �tÞ � bp� � b � E½htðZt � aðpR; �tÞ þ bpÞ� � E½htðZt � aðpR; �tÞÞ�bp

;

6 lim supp!0

a � E½aðpR; �tÞ � bp� � b � E½htðZt � aðpR; �tÞ þ bpÞ� � E½htðZt � aðpR; �tÞÞ�bp

� �;

¼ a � E½aðpR; �tÞ� � b � lim supp!0


< 0:

Thus, p�t is zero for zt = Xt. Now, it is enough to show that the optimal extra charge p�t for express service is nonincreasing in zt.First note that when pR is fixed for all periods, Ptðzt ; pt ; pRt

Þ is the jointly concave function of (zt,pt) and Gtð�xtÞ is still nonin-creasing concave function of �xt . By Topkis’ Theorem in [6], it is enough to show that Ptðzt ; pt ; pRt

Þ is submodular function of(zt,pt). Note that the sum of submodular functions is submodular. The first, second and third term are submodular since theyare linear zt or pt. To prove the submodularity of the forth term, consider any pair of (z1,z2) and (p1,p2) with z1 > z2 and p1 > p2.

E½htðz1 � aðpR; �tÞ þ bp1Þ� � E½htðz2 � aðpR; �tÞ þ bp1Þ�;¼ E½htðz2 � aðpR; �tÞ þ bp1 þ ðz1 � z2ÞÞ� � E½htðz2 � aðpR; �tÞ þ bp1Þ�;6 E½htðz2 � aðpR; �tÞ þ bp2 þ ðz1 � z2ÞÞ� � E½htðz2 � aðpR; �tÞ þ bp2Þ�;¼ E½htðz1 � aðpR; �tÞ þ bp2Þ� � E½htðz2 � aðpR; �tÞ þ bp2Þ�:

The inequality holds since E[ht(I)] is convex in I. Since the sum of submodular functions is still submodular, Ptðzt ; pt ; pRtÞ is

submodular in (zt,pt). Consequently, for fixed pR, the optimal extra charge p�t for express service is nonincreasing in zt as wellas in �xt as well as in xt. Therefore, we can take xt = Zt + DR,t�1 as the threshold which is finite and the result holds. h

Table 1Parameters for Denim item.

a bR bE Purchase cost Holding cost Backlogging cost Salvage Range for regular price (pR) Range for extra charge (p)

180 3 6 30 0.3 28 24 25–50 0–50

Fig. 2. Seasonality factors.

Fig. 3. Optimal base stock levels for varying demand uncertainty (c.d.v).


In Proposition 2, the condition

lim supp!0

E½htðZt � aðpR; �tÞ þ bpÞ� � E½htðZt � aðpR; �tÞÞ�p

> a � E½aðpR; �tÞ�;

implies that the increasing rate of holding costs for a certain inventory level Zt is increasing, so that when selling a productfor the express service at a strictly positive extra charge (p > 0), the selling profit is lower than the decrease in the inventoryholding costs. Therefore, the seller would attempt to sell all the products with the express service for no extra charge in orderto reduce inventory. Intuitively, as inventory level costs increase, it is better to tempt customers to express service delivery,without any extra charge, so as to lessen the inventory level as quickly as possible.

4. Numerical study

In this section, report is provided on a numerical study conducted to obtain qualitative insights into the followings:

1. the benefits of an inventory control by different service levels compared to an inventory control by one service levelwhich has been used traditionally;

2. the sensitivity of the optimal policy, which is a combination of base stock and list prices (prices for regular and expressservice), with respect to the degree of variability in the demand uncertainty.

Fig. 4. List price for regular service for varying demand uncertainty (c.d.v).

Fig. 5. List price for express service for varying demand uncertainty (c.d.v).


As an illustration, the data from a retailer of men’s apparel is considered, especially for a denim (cotton pants). Table 1summarize all parameters for the denim item. As assumed, the demand function for express service is linearly decreasingin regular price and extra charge and given as follows:

dE;tðpR; p; �tÞ ¼ a� bRpR � bEpþ �t :

Random variable �t is independent and normally distributed, and are truncated to avoid negative demand realizations.E[�t] = 0 and Var[�t] = [(a � bRpR � bEp) � c.d.v.]2 where c.d.v. is a specified coefficient for demand variations to give a degreeof variability in demand uncertainty. Two sets of scenarios are considered, which are stationary and non-stationary. Station-ary set is the one without seasonality of demand which means that demands are independent and identically distributed.However, the non-stationary set is the one with seasonality of demand, which means that seasonality factors are consideredto gauge the impact of seasonalities in demand. The seasonality factors used in the numerical analysis are given in Fig. 2.

Fig. 3 gives the results demonstrating the optimal base stock level as a function of remaining time for varying the demanduncertainty (c.d.v). For both stationary and non-stationary cases, as the demand uncertainty (c.d.v) increases, the optimalbase stock level increases. The following can be one explanation. For the given base stock level, the expected shortage costsincrease as the coefficient for demand variations (demand uncertainty) increases. Thus, it is beneficial to increase the basestock level, thus reducing the expected lost sales (shortage costs).

Figs. 4 and 5 give the results demonstrating the optimal list price for regular and express service, respectively as a func-tion of remaining time for varying the demand uncertainty (c.d.v).

Ref. [7] reports that the dramatic improvement in forecast accuracy after observing some of initial demand suggests astrategy for reducing the cost of too much or too little inventory. By using the inventory control by different service levels,

% in

crea

se in

pro

fit

30

40

50

20

10

0c.d.v.=0.4 c.d.v.=1.0 c.d.v.=1.4c.d.v.=1.2c.d.v.=0.8c.d.v.=0.6

% in

crea

se in

pro

fit

30

40

50

20

10

0c.d.v.=0.4 c.d.v.=1.0 c.d.v.=1.4c.d.v.=1.2c.d.v.=0.8c.d.v.=0.6

Fig. 6. Percentage incresae in profit from inventory control by different service levels compared to traditional one service level.


customers’ willingness to delay shipment for regular service provides the seller with advanced demand information to applyto the next purchasing/production period so that inventory can be controlled more efficiently and the cost of too much or toolittle inventory can be reduced. Fig. 6 shows the benefits of an inventory control by different service levels compared to tra-ditional inventory control by one service level. In both stationary and non-stationary case, the result shows that the proposedmodel provides more profit than the traditional inventory control model. Moreover, at higher level of demand uncertainty(c.d.v), the percentage increase in profit by the proposed model for non-stationary case is larger than one for the stationarycase. This implies that the benefit of inventory control by different service levels can be relatively large in those settingswhere the system experiences higher demand uncertainty and seasonality in demand fluctuation: inventory control by dif-ferent service levels (multiple pricing), which gives the system advanced demand information, effectively reduces the de-mand uncertainty compared to traditional inventory control by one service level, thus giving more profit. This resultreveals that the proposed model can obtain the better solution of the inventory control problem with different service levelsand pricing.

5. Conclusion

This research was initiated by the following two practical intuitions: (1) if customers are offered a lower price, they arewilling to delay the shipment of their orders. (2) if the inventory level is over some level, it is better for the seller to offer theexpress service to customers without any extra charge. These two intuitions are verified and analyzed by the reasonable de-mand model (Assumption 3) and by the mathematical DP model where a single seller sells a single product with multiple(two) prices depending on the service level. I have shown that there is an optimal pair of threshold levels for inventoryand prices for service levels such that if the inventory level is less than the threshold, then it becomes optimal to replenishit up to the threshold level and offer threshold prices for service levels. Otherwise, it is optimal not to replenish the inven-tory. Also, I have shown that under a sufficient condition there exists a threshold for the inventory level such that if theinventory level is over the threshold, then it is optimal to offer the express service to customers without any extra charge.

Acknowledgements

This work was supported by the Hongik University new faculty research support fund.

References

[1] A. Federgruen, A. Heching, Combined pricing and inventory control under uncertainty, Oper. Res. 47 (1999) 454–475.[2] F. Chen, Market segmentation, advanced demand information, and supply chain performance, Manuf. Serv. Oper. Manage. 3 (1) (2001) 53.[3] X. Chen, D. Simchi-Levi, Coordinating inventory control and pricing strategies with random demand and fixed ordering cost: the infinite horizon case,

Math. Oper. Res. 29 (3) (2004) 698–723.[4] R. Hariharan, P. Zipkin, Customer-order information, leadtimes, and inventories, Manage. Sci. 41 (10) (1995) 1599–1607.[5] L. Li, A stochastic theory of the firm, Math. Oper. Res. 13 (3) (1988) 447–466.[6] D. Topkis, Minimizing a submodular function on a lattice, Oper. Res. 26 (2) (1978) 305–321.[7] M. Fisher, A. Raman, Reducing the cost of demand uncertainty through accurate response to early sales, Oper. Res. 44 (1) (1996) 87–99.

inventory control by different service levels

Documents