Research Article Inventory Control by Multiple Service Levels ...Research Article Inventory Control by Multiple Service Levels under Unreliable Supplying Condition ByungsooNa, 1 JinpyoLee,

Research ArticleInventory Control by Multiple Service Levels underUnreliable Supplying Condition

Byungsoo Na,1 Jinpyo Lee,2 and Hyung Jun Ahn2

1Division of Business Administration, Korea University, Sejong, Republic of Korea2College of Business Administration, Hongik University, Seoul, Republic of Korea

Correspondence should be addressed to Jinpyo Lee; [email protected]

Received 28 March 2016; Accepted 31 July 2016

Academic Editor: Francisco Gordillo

Copyright © 2016 Byungsoo Na et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider an inventory system where there is random demand from customers as well as unreliable supplying capacity fromsupplier. In many real-world cases, supplier might fail to satisfy the amount of order from retailers or producers so that onlypartial proportion of order is satisfied or even fail to deliver all of the order. Moreover, recently a concern regarding unreliablesupplying capacity has been increasing since the globalizationmakes the retailer or producer face the extended supply network withcomplicated and risky supplying capacity. Also, we consider two classified customers, of which one is willing to pay extra charge forexpedited delivery service but the other is not reluctant to delay the delivery without any extra charge. We show that there exists anoptimal threshold for inventory and price for each service level in the following sense: if the inventory level is less than the predeter-mined threshold, then the retailer or producer needs to order up to the threshold level and offer threshold price corresponding to ser-vice level. Otherwise, the retailer does not need to order.The risk of stockout due to unreliable supplying capacity can bemitigated bythe dynamic pricing and inventory control with multiple service levels.

1. Introduction

Coordination of dynamic inventory and pricing control hasbeen used as main strategy for many companies such asAmazon, Dell, and J. C. Penny [1]. In many traditional worksfor inventory control problem, how to maximize the profitand control the uncertainty of demand from customers hasbeen a stark issue, in which the optimal policy has been pro-vided using either ordering quantity or pricing of each prod-uct. Even some works show the optimal inventory controlpolicy using multiple pricing on single product dependingon the service levels. However, in addition to the randomdemand from customers, it is not well addressed that mostfirms experience the unreliable supplying capacity fromsupplier, which is another randomness carried from supplier.For example, suppose that Amazon.com orders 100 books toa publisher. However, the publisher might fail to fulfill theorder and it may deliver only some of the ordered books(i.e., 80 books) for its own supplying capacity problem. Somerecent works address this unreliable supply issue which is

incorporated with pricing on the product and replenishmentdecisions but only addresses the pricing policy on a productwith a single-service level [2]. However, as shown in manyonline stores, each product is delivered with more than asingle-service level where faster service can be suggestedto the customer willing to pay an extra charge. So, how tobalance the demand and supply by incorporating multiplepricing corresponding to the service level is an importantissue faced by many firms, especially online firms. In thispaper, we show that there can be an optimal policy toan inventory control problem in which each product ispriced depending on the service level and the supplyingcapacity from the supplier is also random in addition to therandomness of demand from the customer.

2. Literatures Review

Two streams of literatures are related to this paper:(1) The inventory/production or pricing control problem

under reliable supplying capacity.

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016, Article ID 1534580, 10 pageshttp://dx.doi.org/10.1155/2016/1534580

2 Mathematical Problems in Engineering

(2) The inventory/production or pricing control problemunder unreliable supplying capacity.

There are many literatures that study the inventory orpricing control problem under reliable supplying capacity.Reference [1] gives a comprehensive survey on this problem,in which only a single-service level is considered. Refer-ences [3, 4] address a single-period problem under riskneutrality, which is addressed using Newsvendor Model.Reference [5] suggests optimal replenishment and pricingpolicy of a single-period model under risk aversion. Formultiperiod system, when total ordering cost is a linearfunction of ordering quantity, a base stock list price isshown to be optimal for single-service level in [6] and formultiple service level in [7]. References [8, 9] show that,for multiperiod system, there is an optimal inventory andpricing policy when a fixed ordering cost is considered andbacklogging is allowed. Reference [10] provides an optimalinventory and pricing policy for multiperiod system when afixed ordering cost is considered and lost-sales are allowed.Reference [11] addresses joint inventory and pricing modelwhere there is a single item with stochastic demand subjectto reference effects and the random demand is a functionof the current price and the reference price is acting as abenchmark with which customers compare the current price.Reference [12] suggests the joint pricing and inventorymodelfor a stochastic inventory system with perishable products,where the inventory system under random demand andreliable supplying capacity is modeled by a continuous-timestochastic differential equation. Reference [13] considers jointpricing and inventory replenishment decision problem overan infinite horizon where sequentially arriving customers areforward-looking to the price of a product sold by a seller. Intheir model, so-called strategic customers wait and monitorprices offered from the seller and then anticipate a lowerfuture price. Reference [14] addresses a joint pricing andproduction decision problem for perishable items sold toprice-sensitive customers, assuming that shortages are notallowed.

The other stream of literature relevant to our paper isthe inventory/production or pricing control problem underunreliable supplying capacity. Reference [15] surveys theliterature on how to quantitatively determine lot sizes whenproduction or procurement yields are uncertain. Reference[16] addresses an inventory control model for a periodicreview with unreliable production capacity, random yields,and uncertain demand but does not consider the dynamicpricing by considering the price as exogenous and thusminimizing the total discounted expected costs which areproduction, holding, and shortage costs. References [17, 18]address the joint inventory and pricing decision problemwith random demand and unreliable suppling capacity, inwhich there are no multiple service levels. Reference [19]addresses the inventory (but no pricing) decision modelfor production-inventory systems in which the stock candeteriorate with time and supplying capacity from multiplesuppliers is unreliable. Reference [20] investigates an optimalinventory strategy for a risk-averse retailer facing demanduncertainty and unreliable supply in which the price is given

and is exogenous. Reference [21] considers a risk-neutralmonopolist manufacturer ordering a key component fromseveral suppliers using a single-period model, in which somesuppliers might face risks of complete supplying disruptions,which are called unreliable suppliers. Reference [22] suggestsoptimal sourcing strategies without pricing decision whenthere are a perfectly reliable supplier and an unreliablesupplier.

3. Assumptions and Notations

In this paper, the following notations are used:

(1) 𝑐: per unit marginal cost,(2) 𝑝𝑅,𝑡: price charged for regular service in period 𝑡,

(3) 𝑝𝐸,𝑡: price charged for express service in period 𝑡,

(4) 𝑝𝑡= 𝑝𝐸,𝑡

−𝑝𝑅,𝑡: extra charge for the express service in

period 𝑡 on [0, 𝑝],(5) 𝜖𝑡: random uncertainty term of demand which had a

known distribution,(6) 𝐷

𝐸,𝑡(𝑝𝑅,𝑡

, 𝑝𝐸,𝑡

): demand for the express service,(7) 𝐷𝑅,𝑡

(𝑝𝑅,𝑡

): demand for the regular service,(8) 𝑥𝑡: inventory level at the beginning of each period 𝑡

before ordering,(9) 𝑦𝑡: inventory level at the beginning of each period 𝑡

after ordering,(10) ℎ

𝑡(𝐼): inventory cost (holding or backlogging) at the

end of each period 𝑡.

Assumption 1. Backlogging is allowed.

Assumption 2. Replenishment after ordering becomes avail-able instantaneously.

Traditionally, in operations research literatures, the cus-tomer’s demand is assumed to be decreasing concave functionor simply a decreasing linear function with the price as aninput variable. Also, the expected demand value is assumed tobe finite and strictly decreasing as the price increases, which isgenerally acceptable and reasonable.This decreasing propertyof demand can be negative if the value of price is sufficientlylarge. Thus, we need to make another assumption such thata feasible price should be selected on the restricted range inorder to be nonnegative valued [6, 9, 23, 24].

Assumption 3. In each period 𝑡 = 1, 2, . . . , 𝑇, the demandfunction for regular service 𝐷

𝑅,𝑡(𝑝𝑅,𝑡

, 𝑝𝐸,𝑡

) = 𝑑𝑅,𝑡

(𝑝𝑡), where

𝑑𝑅,𝑡

(𝑝𝑡) is given by 𝑑

𝑅,𝑡(𝑝𝐸,𝑡

− 𝑝𝑅,𝑡

), and nondecreasinglinear function in 𝑝

𝑡(=𝑝𝐸,𝑡

− 𝑝𝑅,𝑡

) ∈ [0, 𝑝]. The demandfunction for express service 𝐷


, 𝑝𝐸,𝑡

) = 𝑑𝐸,𝑡

(𝑝𝑅,𝑡

, 𝑝𝑡),

where 𝑑𝐸,𝑡

(𝑝𝑅,𝑡

, 𝑝𝑡) is given by 𝑎(𝑝

𝑅,𝑡, 𝜖𝑡) − 𝐷


, 𝑝𝐸,𝑡

) =

𝑎(𝑝𝑅,𝑡

, 𝜖𝑡) − 𝑑𝑅,𝑡

(𝑝𝑡), which is nonincreasing linear function

in 𝑝𝑅,𝑡

∈ [𝑝𝑅, 𝑝𝑅] and 𝑝

𝑡∈ [0, 𝑝]. Moreover, 𝑎(𝑝

𝑅,𝑡, 𝜖𝑡) is the

possible maximum demand and nonincreasing and linearfunction of 𝑝

𝑅,𝑡∈ [𝑐, 𝑝

𝑅]. Moreover, values for 𝑝 and 𝑝

𝑅are

taken such that 𝑎(𝑝𝑅, 𝜖𝑡) −𝐷𝑅,𝑡

(𝑝𝑅,𝑡

, 𝑝𝐸,𝑡

) is nonnegative withprobability 1.

Mathematical Problems in Engineering 3

Assumption 3 came up from the following insight. Thenumber of customers using the express service woulddecrease as the price difference 𝑝

𝑡between regular and

express service increases. For example, when you try to buysome product from an online store, the higher the pricedifference between regular shipping and express shippingservice are, the more you are reluctant to select the expressshipping service. Thus, as the price difference 𝑝

𝑡between

regular and express service increases, the customers who arereluctant to select express service will become the customersfor regular service.

Assumption 4. The revenue in each period 𝑡 = 1, 2, . . . , 𝑇 isgiven by

𝑝𝑅,𝑡

𝐸 [𝐷𝑅,𝑡

(𝑝𝑅,𝑡

, 𝑝𝐸,𝑡

)] + 𝑝𝐸,𝑡

𝐸 [𝐷𝐸,𝑡

(𝑝𝑅,𝑡

, 𝑝𝐸,𝑡

)]

= 𝑝𝑅,𝑡

𝐸 [𝑑𝑅,𝑡

(𝑝𝑡)] + 𝑝

𝐸,𝑡𝐸 [𝑑𝐸,𝑡

(𝑝𝑅,𝑡

, 𝑝𝑡)]

(1)

and is finite and concave for 𝑝𝑅,𝑡


𝑡∈ [0, 𝑝],

where 𝑝𝑡= 𝑝𝐸,𝑡

− 𝑝𝑅,𝑡.

Assumption 5. 𝐻𝑡(𝐼) = 𝐸[ℎ

𝑡(𝐼)] is convex in 𝐼 and 𝐻

𝑡(0) = 0

in each 𝑡 = 1, . . . , 𝑇.

Assumption 6. lim𝑦→±∞

𝐻𝑡(𝑦 − 𝐷


, 𝑝𝐸,𝑡

)) =

lim𝑦→±∞

(𝑐𝑦+𝐻𝑡(𝑦−𝐷


, 𝑝𝐸,𝑡

))) = ∞ for any𝑝𝑅,𝑡

∈ [𝑝𝑅,

𝑝𝑅] and 𝑝

𝑡∈ [0, 𝑝].

4. Mathematical Formulation

We consider an additive demand model, where the demanduncertainty for regular and express service is represented byan additive random noise 𝜖

𝑡, respectively,

𝐷𝑅,𝑡

(𝑝𝑅,𝑡

, 𝑝𝐸,𝑡

) = 𝑑𝑅,𝑡

(𝑝𝑡) = D

𝑅,𝑡(𝑝𝑡) + 𝜖𝑡,

𝐷𝐸,𝑡

(𝑝𝑅,𝑡

, 𝑝𝐸,𝑡

) = 𝑑𝐸,𝑡

(𝑝𝑅,𝑡

, 𝑝𝑡) = D


, 𝑝𝑡) + 𝜖𝑡.

(2)

We assume that 𝜖𝑡has mean zero and support [𝜖, 𝜖]. For

any feasible choice of 𝑝𝑅,𝑡


𝐸,𝑡∈ [𝑝𝐸, 𝑝𝐸],

the demand is positive with probability one and the averagedemands 𝐸[𝐷


, 𝑝𝐸,𝑡

)] and 𝐸[𝐷𝐸,𝑡

(𝑝𝑅,𝑡

, 𝑝𝐸,𝑡

)] are finite.Given the inventory level 𝐼

𝑡in period 𝑡, the inventory level

𝐼𝑡+1

in period 𝑡 + 1 is given as follows:

𝐼𝑡+1

= 𝐼𝑡+ (𝑦𝑡− 𝐼𝑡) ∧ 𝑘 − 𝑑


, 𝑝𝑡) , (3)

where 𝑘 ∈ [0, 𝑦𝑡− 𝐼𝑡] in period 𝑡 and 𝑎 ∧ 𝑏 = min[𝑎, 𝑏]. Let

𝑔𝑡(𝐼, 𝑥) be the optimal profit function in period 𝑡 when the

inventory level is 𝐼 and the demand from period 𝑡 − 1 is 𝑥.Then, the optimality equation is given by

𝑔𝑡 (𝐼, 𝑥) = max

𝑦≥𝐼,𝑝𝑅≤𝑝𝑅≤𝑝𝑅,𝑝≤𝑝≤𝑝

𝜋𝑡(𝐼; 𝑦, 𝑝

𝑅, 𝑝) , (4)

where𝜋𝑡(𝐼; 𝑦, 𝑝

𝑅, 𝑝)

= 𝑝𝑅𝐸 [𝑑𝑅(𝑝)] + 𝑝

𝐸𝐸 [𝑑𝐸(𝑝𝑅, 𝑝)]

− 𝑐𝐸 [(𝐼 + 𝑘) ∧ 𝑦 − 𝐼]

− 𝐸 [𝐻 ((𝐼 + 𝑘) ∧ 𝑦 − 𝑑𝐸(𝑝𝑅, 𝑝))]

+ 𝛼𝐸 [𝑔𝑡+1

((𝐼 + 𝑘) ∧ 𝑦 − 𝑑𝐸(𝑝𝑅, 𝑝) , 𝑑

𝑅(𝑝))] .

(5)

From Lemma 7, we can expect a dynamic programmingmodel with single state as an input variable which is equiva-lent to (4). Moreover, we can see that the optimal solution tothe equivalent dynamic programmingmodel with single statecan be translated into the optimal solution to (4).

Lemma7. Let 𝐼 and𝑦 be defined as 𝐼−𝑥 and𝑦−𝑥, respectively.Then, (𝑦∗

𝑡+ 𝑥, 𝑝

∗

𝑅,𝑡, 𝑝∗

𝑡) is the set of optimal solutions to (4) if

and only if (𝑦∗𝑡, 𝑝∗

𝑅,𝑡, 𝑝∗

𝑡) is the set of optimal solutions to

G𝑡(𝐼) = max

𝑦≥𝐼,𝑝𝑅≤𝑝𝑅≤𝑝𝑅,𝑝≤𝑝≤𝑝

Π𝑡(𝐼 : 𝑦, 𝑝

𝑅, 𝑝) , (6)

where

Π𝑡(𝐼 : 𝑦, 𝑝

𝑅, 𝑝)



− 𝑐𝐸 [(𝐼 + 𝑘) ∧ 𝑦 − 𝐼]

− 𝐸 [𝐻 ((𝐼 + 𝑘) ∧ 𝑦 − 𝑑𝐸(𝑝𝑅, 𝑝))]

+ 𝛼𝐸 [G𝑡+1

((𝐼 + 𝑘) ∧ 𝑦 − 𝑑𝐸(𝑝𝑅, 𝑝) − 𝑑

𝑅(𝑝))] .

(7)

Proof. It is enough to show that, for all period 𝑡,


𝑅, 𝑝) = Π

𝑡(𝐼 : 𝑦, 𝑝

𝑅, 𝑝) ∀𝐼, 𝑥, 𝑦, 𝑝

𝑅, 𝑝. (8)

Since

(𝐼 + 𝑘) ∧ 𝑦 = (𝐼 − 𝑥 + 𝑘) ∧ (𝑦 − 𝑥)

=

{

{

{

𝐼 − 𝑥 + 𝑘, if 𝐼 − 𝑥 + 𝑘 ≤ 𝑦 − 𝑥

𝑦 − 𝑥, otherwise

= (𝐼 + 𝑘) ∧ 𝑦 − 𝑥,

(9)

for all period 𝑡, the first four terms of 𝜋𝑡(𝐼; 𝑦, 𝑝

𝑅, 𝑝) andΠ

𝑡(𝐼 :

𝑦, 𝑝𝑅, 𝑝) are equal to each other as follows:

𝑝𝑅𝐸 [𝑑𝑅(𝑝)] + 𝑝


− 𝑐𝐸 [(𝐼 + 𝑘) ∧ 𝑦 − 𝐼]

− 𝐸 [𝐻 ((𝐼 + 𝑘) ∧ 𝑦 − 𝑑𝐸(𝑝𝑅, 𝑝))]



− 𝑐𝐸 [(𝐼 + 𝑘) ∧ 𝑦 − 𝐼]

− 𝐸 [𝐻 ((𝐼 + 𝑘) ∧ 𝑦 − 𝑑𝐸(𝑝𝑅, 𝑝) − 𝑥)] .

(10)


In period 𝑇, trivially 𝑔𝑇(𝐼, 𝑥) = G

𝑇(𝐼) = 0. In period 𝑇 − 1,

𝜋𝑇−1

(𝐼; 𝑦, 𝑝𝑅, 𝑝)

= 𝑝𝑅𝐸 [𝑑𝑅,𝑇−1

(𝑝)] + 𝑝𝐸𝐸 [𝑑𝐸(𝑝𝑅,𝑇−1

, 𝑝)]

− 𝑐𝐸 [(𝐼 + 𝑘) ∧ 𝑦 − 𝐼]

− 𝐸 [𝐻 ((𝐼 + 𝑘) ∧ 𝑦 − 𝑑𝐸,𝑇−1(𝑝𝑅, 𝑝) − 𝑥)]

= 𝑝𝑅𝐸 [𝑑𝑅,𝑇−1

(𝑝)] + 𝑝𝐸𝐸 [𝑑𝐸,𝑇−1

(𝑝𝑅, 𝑝)]

− 𝑐𝐸 [(𝐼 + 𝑘) ∧ 𝑦 − 𝐼]

− 𝐸 [𝐻 ((𝐼 + 𝑘) ∧ 𝑦 − 𝑑𝐸,𝑇−1

(𝑝𝑅, 𝑝))]

= Π𝑇−1

(𝐼 : 𝑦, 𝑝𝑅, 𝑝) .

(11)

So, 𝑔𝑇−1

(𝐼, 𝑥) = G𝑇−1

(𝐼)with 𝐼 = 𝐼−𝑥. By induction, supposethat, for any 𝐼, 𝑥 with 𝐼 = 𝐼 − 𝑥,

𝑔𝑛 (

𝐼, 𝑥) = G𝑛 (𝐼) , 𝑛 = 𝑇, 𝑇 − 1, . . . , 𝑡 + 1. (12)

Then


𝑅, 𝑝)



− 𝑐𝐸 [(𝐼 + 𝑘) ∧ 𝑦 − 𝐼]

− 𝐸 [𝐻 ((𝐼 + 𝑘) ∧ 𝑦 − 𝑑𝐸(𝑝𝑅, 𝑝) − 𝑥)]

+ 𝛼𝐸 [𝑔𝑡+1

((𝐼 + 𝑘) ∧ 𝑦 − 𝑑𝐸(𝑝𝑅, 𝑝) − 𝑥, 𝑑

𝑅(𝑝))]



− 𝑐𝐸 [(𝐼 + 𝑘) ∧ 𝑦 − 𝐼]

− 𝐸 [𝐻 ((𝐼 + 𝑘) ∧ 𝑦 − 𝑑𝐸(𝑝𝑅, 𝑝))]

+ 𝛼𝐸 [G𝑡+1

((𝐼 + 𝑘) ∧ 𝑦 − 𝑑𝐸(𝑝𝑅, 𝑝) − 𝑑

𝑅(𝑝))] .

(13)

The second equality holds since (𝐼+𝑘)∧𝑦−𝑑𝐸(𝑝𝑅, 𝑝)−𝑑

𝑅(𝑝) =

(𝐼+𝑘)∧𝑦−𝑑𝐸(𝑝𝑅, 𝑝)−𝑥−𝑑

𝑅(𝑝). Therefore, the result holds.

4.1. Optimal Inventory Control Policy. Now, we will show thefollowing optimal inventory control policy: if the inventorylevel at each period 𝑡 before ordering is less than a predeter-mined level, then we need to order such that the inventorylevel is increased up to the predetermined level. Otherwise,no ordering will be made.

Lemma 8. Suppose that G𝑡+1

(⋅) is concave function and, forgiven 𝑝

𝑅and 𝑝, let 𝑦∗(𝑝

𝑅, 𝑝) be the maximizer of

𝑝𝑅𝑑𝑅(𝑝) + 𝑝

𝐸𝑑𝐸(𝑝𝑅, 𝑝) − 𝑐 (𝑦 − 𝐼)

− 𝐻 (𝑦 − 𝑑𝐸(𝑝𝑅, 𝑝))

+ 𝛼G𝑡+1

(𝑦 − 𝑑𝐸(𝑝𝑅, 𝑝) − 𝑑

𝑅(𝑝)) ,

(14)

where 𝑝𝐸= 𝑝𝑅+ 𝑝. Then, 𝑦∗(𝑝

𝑅, 𝑝) is also the maximizer of



− 𝑐𝐸 [(𝐼 + 𝑘) ∧ 𝑦 − 𝐼]

− 𝐸 [𝐻 ((𝐼 + 𝑘) ∧ 𝑦 − 𝑑𝐸(𝑝𝑅, 𝑝))]

+ 𝛼𝐸 [G𝑡+1

((𝐼 + 𝑘) ∧ 𝑦 − 𝑑𝐸(𝑝𝑅, 𝑝) − 𝑑

𝑅(𝑝))]

(15)

for any 𝐼 and, moreover,


𝐸𝐸 [𝑑𝐸(𝑝𝑅, 𝑝)] − 𝑐𝐸 [(𝐼 + 𝑘)

∧ (𝑦∗(𝑝𝑅, 𝑝) ∨ 𝐼) − 𝐼] − 𝐸 [𝐻 ((𝐼 + 𝑘)

∧ (𝑦∗(𝑝𝑅, 𝑝) ∨ 𝐼) − 𝑑

𝐸(𝑝𝑅, 𝑝))]

+ 𝛼𝐸 [G𝑡+1

((𝐼 + 𝑘) ∧ (𝑦∗(𝑝𝑅, 𝑝) ∨ 𝐼)

− 𝑑𝐸(𝑝𝑅, 𝑝) − 𝑑

𝑅(𝑝))]

(16)

is jointly concave in (𝐼, 𝑝𝑅, 𝑝), where 𝑎 ∨ 𝑏 = max[𝑎, 𝑏].

Proof. Given𝑝𝑅and𝑝,𝑝

𝑅𝑑𝑅(𝑝)+𝑝

𝐸𝑑𝐸(𝑝𝑅, 𝑝) is just constant

and thus wewill not consider them for a while. Since𝐻(⋅) andG𝑡+1

(⋅) are concave,

𝑝𝑅𝑑𝑅(𝑝) + 𝑝

𝐸𝑑𝐸(𝑝𝑅, 𝑝) − 𝑐𝑦 − 𝐻 (𝑦 − 𝑑

𝐸(𝑝𝑅, 𝑝))

+ 𝛼G𝑡+1

(𝑦 − 𝑑𝐸(𝑝𝑅, 𝑝) − 𝑑

𝑅(𝑝))

(17)

is jointly concave in (𝑦, 𝑝𝑅, 𝑝). Now, take any 𝛿 > 0 such that

𝑦+𝛿 < 𝑦∗(𝑝𝑅, 𝑝) and we have 𝑦∧ (𝐼+𝑘) ≤ (𝑦+𝛿)∧ (𝐼+𝑘) <

𝑦∗(𝑝𝑅, 𝑝). So,

− 𝑐 (𝑦 ∧ (𝐼 + 𝑘) − 𝐼) − 𝐻((𝐼 + 𝑘) ∧ 𝑦 − 𝑑𝐸(𝑝𝑅, 𝑝))

+ 𝛼 (G𝑡+1

((𝐼 + 𝑘) ∧ 𝑦 − 𝑑𝐸(𝑝𝑅, 𝑝) − 𝑑

𝑅(𝑝)))

≤ −𝑐 ((𝑦 + 𝛿) ∧ (𝐼 + 𝑘) − 𝐼) − 𝐻((𝐼 + 𝑘) ∧ (𝑦

+ 𝛿) − 𝑑𝐸(𝑝𝑅, 𝑝)) + 𝛼 (G

𝑡+1((𝐼 + 𝑘) ∧ (𝑦 + 𝛿)

− 𝑑𝐸(𝑝𝑅, 𝑝) − 𝑑

𝑅(𝑝)))

(18)

for any 𝑘. Thus,

𝐸 [−𝑐 (𝑦 ∧ (𝐼 + 𝑘) − 𝐼) − 𝐻((𝐼 + 𝑘) ∧ 𝑦

− 𝑑𝐸(𝑝𝑅, 𝑝)) + 𝛼G

𝑡+1((𝐼 + 𝑘) ∧ 𝑦 − 𝑑

𝐸(𝑝𝑅, 𝑝)

− 𝑑𝑅(𝑝))] ≤ 𝐸 [−𝑐 ((𝑦 + 𝛿) ∧ (𝐼 + 𝑘) − 𝐼)

− 𝐻((𝐼 + 𝑘) ∧ (𝑦 + 𝛿) − 𝑑𝐸(𝑝𝑅, 𝑝))

+ 𝛼G𝑡+1

((𝐼 + 𝑘) ∧ (𝑦 + 𝛿) − 𝑑𝐸(𝑝𝑅, 𝑝)

− 𝑑𝑅(𝑝))] .

(19)


Therefore,

− 𝑐𝐸 [(𝐼 + 𝑘) ∧ 𝑦 − 𝐼]

− 𝐸 [𝐻 ((𝐼 + 𝑘) ∧ 𝑦 − 𝑑𝐸(𝑝𝑅, 𝑝))]

+ 𝛼𝐸 [G𝑡+1

((𝐼 + 𝑘) ∧ 𝑦 − 𝑑𝐸(𝑝𝑅, 𝑝) − 𝑑

𝑅(𝑝))]

(20)

is increasing in 𝑦 when 𝑦 ≤ 𝑦∗(𝑝𝑅, 𝑝). By the similar

argument,

− 𝑐𝐸 [(𝐼 + 𝑘) ∧ 𝑦 − 𝐼]

− 𝐸 [𝐻 ((𝐼 + 𝑘) ∧ 𝑦 − 𝑑𝐸(𝑝𝑅, 𝑝))]

+ 𝛼𝐸 [G𝑡+1

((𝐼 + 𝑘) ∧ 𝑦 − 𝑑𝐸(𝑝𝑅, 𝑝) − 𝑑

𝑅(𝑝))]

(21)

is decreasing in 𝑦 when 𝑦 ≥ 𝑦∗(𝑝𝑅, 𝑝). The first result holds.

Since, for given 𝑝𝑅and 𝑝, 𝑦∗(𝑝

𝑅, 𝑝) is the maximizer of

− 𝑐 ((𝐼 + 𝑘) − 𝐼) − 𝐻((𝐼 + 𝑘) ∧ 𝑦 − 𝑑𝐸(𝑝𝑅, 𝑝))

+ 𝛼G𝑡+1

((𝐼 + 𝑘) ∧ 𝑦 − 𝑑𝐸(𝑝𝑅, 𝑝) − 𝑑

𝑅(𝑝))

(22)

for any 𝑦 ∈ R, for given 𝑝𝑅and 𝑝, (𝑦∗(𝑝

𝑅, 𝑝) ∨ 𝐼) ∧ (𝐼 + 𝑘) is

the maximizer of

− 𝑐 ((𝐼 + 𝑘) − 𝐼) − 𝐻((𝐼 + 𝑘) ∧ 𝑦 − 𝑑𝐸(𝑝𝑅, 𝑝))

+ 𝛼G𝑡+1

((𝐼 + 𝑘) ∧ 𝑦 − 𝑑𝐸(𝑝𝑅, 𝑝) − 𝑑

𝑅(𝑝))

(23)

for 𝑦 ∈ [𝐼, 𝐼 + 𝑘]. Now, take (𝐼1, 𝑝𝑅,1

, 𝑝1), (𝐼2, 𝑝𝑅,2

, 𝑝2), and

𝛾 ∈ [0, 1]. Then

𝛾 (−𝑐 ((𝐼1+ 𝑘) ∧ (𝑦

∗(𝑝𝑅,1

, 𝑝1) ∨ 𝐼1) − 𝐼1) − 𝐻((𝐼

1

+ 𝑘) ∧ (𝑦∗(𝑝𝑅,1

, 𝑝1) ∨ 𝐼1) − 𝑑𝐸(𝑝𝑅,1

, 𝑝1))

+ 𝛼 (G𝑡+1

((𝐼1+ 𝑘) ∧ (𝑦

∗(𝑝𝑅,1

, 𝑝1) ∨ 𝐼1)

− 𝑑𝐸(𝑝𝑅,1

, 𝑝1) − 𝑑𝑅(𝑝𝑅,1

, 𝑝1)))) + (1 − 𝛾)

⋅ (−𝑐 ((𝐼2+ 𝑘) ∧ (𝑦

∗(𝑝𝑅,2

, 𝑝2) ∨ 𝐼2) − 𝐼2)

− 𝐻((𝐼2+ 𝑘) ∧ (𝑦

∗(𝑝𝑅,2

, 𝑝2) ∨ 𝐼2)

− 𝑑𝐸(𝑝𝑅,2

, 𝑝2)) + 𝛼G

𝑡+1((𝐼2+ 𝑘)

∧ (𝑦∗(𝑝𝑅,2

, 𝑝2) ∨ 𝐼2) − 𝑑𝐸(𝑝𝑅,2

, 𝑝2)

− 𝑑𝑅(𝑝𝑅,2

, 𝑝2))) ≤ −𝑐 (𝛾 [(𝐼

1+ 𝑘)

∧ (𝑦∗(𝑝𝑅,1

, 𝑝1) ∨ 𝐼1)] + (1 − 𝛾) [(𝐼

2+ 𝑘)

∧ (𝑦∗(𝑝𝑅,2

, 𝑝2) ∨ 𝐼2)] − 𝐼

𝛾) − 𝐻(𝛾 [(𝐼

1+ 𝑘)

∧ (𝑦∗(𝑝𝑅,1

, 𝑝1) ∨ 𝐼1)] + (1 − 𝛾) [(𝐼

2+ 𝑘)

∧ (𝑦∗(𝑝𝑅,2

, 𝑝2) ∨ 𝐼2)] − 𝑑

𝐸(𝑝𝑅,𝛾

, 𝑝𝐸,𝛾

))

+ 𝛼G𝑡+1

(𝛾 [(𝐼1+ 𝑘) ∧ (𝑦

∗(𝑝𝑅,1

, 𝑝1) ∨ 𝐼1)] + (1

− 𝛾) [(𝐼2+ 𝑘) ∧ (𝑦

∗(𝑝𝑅,2

, 𝑝2) ∨ 𝐼2)]

− 𝛾 [𝑑𝐸(𝑝𝑅,1

, 𝑝1) − 𝑑𝑅(𝑝𝑅,1

, 𝑝1)] − (1 − 𝛾)

⋅ [𝑑𝐸(𝑝𝑅,2

, 𝑝2) − 𝑑𝑅(𝑝𝑅,2

, 𝑝2)]) = −𝑐 (𝛾 [(𝐼

1+ 𝑘)

∧ (𝑦∗(𝑝𝑅,1

, 𝑝1) ∨ 𝐼1)] + (1 − 𝛾) [(𝐼

2+ 𝑘)

∧ (𝑦∗(𝑝𝑅,2

, 𝑝2) ∨ 𝐼2)] − 𝐼

𝛾) − 𝐻(𝛾 [(𝐼

1+ 𝑘)

∧ (𝑦∗(𝑝𝑅,1

, 𝑝1) ∨ 𝐼1)] + (1 − 𝛾) [(𝐼

2+ 𝑘)

∧ (𝑦∗(𝑝𝑅,2

, 𝑝2) ∨ 𝐼2)] − 𝑑

𝐸(𝑝𝑅,𝛾

, 𝑝𝐸,𝛾

))

+ 𝛼G𝑡+1

(𝛾 [(𝐼1+ 𝑘) ∧ (𝑦

∗(𝑝𝑅,1

, 𝑝1) ∨ 𝐼1)] + (1

− 𝛾) [(𝐼2+ 𝑘) ∧ (𝑦

∗(𝑝𝑅,2

, 𝑝2) ∨ 𝐼2)] − 𝑑

𝐸(𝑝𝑅,𝛾

,

𝑝𝛾) − 𝑑𝑅(𝑝𝑅,𝛾

, 𝑝𝛾)) ≤ −𝑐 ((𝐼

𝛾+ 𝑘)

∧ (𝑦∗(𝑝𝑅,𝛾

, 𝑝𝛾) ∨ 𝐼𝛾) − 𝐼𝛾) − 𝐻((𝐼

𝛾+ 𝑘)

∧ (𝑦∗(𝑝𝑅,𝛾

, 𝑝𝛾) ∨ 𝐼𝛾) − 𝑑𝐸(𝑝𝑅,𝛾

, 𝑝𝛾))

+ 𝛼G𝑡+1

((𝐼𝛾+ 𝑘) ∧ (𝑦

∗(𝑝𝑅,𝛾

, 𝑝𝛾) ∨ 𝐼𝛾)

− 𝑑𝐸(𝑝𝑅,𝛾

, 𝑝𝛾) − 𝑑𝑅(𝑝𝑅,𝛾

, 𝑝𝛾)) ,

(24)

where 𝐼𝛾

= 𝛾𝐼1+ (1 − 𝛾)𝐼

2, 𝑝𝑅,𝛾

= 𝛾𝑝𝑅,1

+ (1 − 𝛾)𝑝𝑅,2

, and𝑝𝛾

= 𝛾𝑝1+ (1 − 𝛾)𝑝

2. The first inequality holds due to the

concavity of𝐻𝑡andG

𝑡+1, the second equality holds due to the

linearity of demand functions, and the last inequality holdssince, for given 𝑝

𝑅,𝛾and 𝑝

𝛾, 𝑦 = (𝐼

𝛾+ 𝑘) ∧ 𝑦

∗(𝑝𝑅,𝛾

, 𝑝𝛾) ∨ 𝐼𝛾

is the maximizer of

− 𝑐 (𝑦 − 𝐼) − 𝐻 (𝑦 − 𝑑𝐸(𝑝𝑅, 𝑝))

+ 𝛼G𝑡+1

(𝑦 − 𝑑𝐸(𝑝𝑅, 𝑝) − 𝑑

𝑅(𝑝))

(25)

for 𝑦 ∈ [𝐼𝛾, 𝐼𝛾+ 𝑘]. So, we need to verify that

𝛾 [(𝐼1+ 𝑘) ∧ (𝑦

∗(𝑝𝑅,1

, 𝑝1) ∨ 𝐼1)]

+ (1 − 𝛾) [(𝐼2+ 𝑘) ∧ (𝑦

∗(𝑝𝑅,2

, 𝑝2) ∨ 𝐼2)]

(26)

is within [𝐼𝛾, 𝐼𝛾+ 𝑘].

𝛾 [(𝐼1+ 𝑘) ∧ (𝑦

∗(𝑝𝑅,1

, 𝑝1) ∨ 𝐼1)]

+ (1 − 𝛾) [(𝐼2+ 𝑘) ∧ (𝑦

∗(𝑝𝑅,2

, 𝑝2) ∨ 𝐼2)]

≥ 𝛾 [(𝐼1+ 𝑘) ∧ 𝐼

1] + (1 − 𝛾) [(𝐼

2+ 𝑘) ∧ 𝐼

2]

= 𝛾𝐼1+ (1 − 𝛾) 𝐼

2= 𝐼𝛾,


Table 1: Parameters for numerical analysis.

𝑎 𝑏𝑅

𝑏𝐸

Marginal cost Holding cost Backlog cost Salvage price Range for regular price (𝑝𝑅) Range for extra charge (𝑝)

250 5 6 30 0.3 28 20 25–65 0–30

𝛾 [(𝐼1+ 𝑘) ∧ (𝑦

∗(𝑝𝑅,1

, 𝑝1) ∨ 𝐼1)]

+ (1 − 𝛾) [(𝐼2+ 𝑘) ∧ (𝑦

∗(𝑝𝑅,2

, 𝑝2) ∨ 𝐼2)]

≤ 𝛾 (𝐼1+ 𝑘) + (1 − 𝛾) (𝐼

2+ 𝑘) = 𝐼

𝛾+ 𝑘.

(27)

By taking the expectation, we can obtain the second result.

Proposition 9. For all 𝑡 = 1, 2, . . . , 𝑇, G𝑡(⋅) is concave

function.

Proof. Since we consider the 𝑇−period problem, G𝑇+1

(𝐼) =

0 and G𝑇+1

(𝐼) is concave in 𝐼. By induction, suppose thatG𝑡+1

(⋅) is concave. Let 𝑦∗(𝑝𝑅, 𝑝) be defined as in Lemma 8.

By Assumption 4 and Lemma 8,

Π𝑡(𝐼 : 𝑦∗(𝑝𝑅, 𝑝) ∨ 𝐼, 𝑝

𝑅, 𝑝) = 𝑝

𝑅𝐸 [𝑑𝑅(𝑝)]

+ 𝑝𝐸𝐸 [𝑑𝐸(𝑝𝑅, 𝑝)] − 𝑐𝐸 [(𝐼 + 𝑘) ∧ (𝑦

∗(𝑝𝑅, 𝑝)

∨ 𝐼) − 𝐼] − 𝐸 [𝐻 ((𝐼 + 𝑘) ∧ (𝑦∗(𝑝𝑅, 𝑝) ∨ 𝐼)

− 𝑑𝐸(𝑝𝑅, 𝑝))] + 𝛼𝐸 [G

𝑡+1((𝐼 + 𝑘)

∧ (𝑦∗(𝑝𝑅, 𝑝) ∨ 𝐼) − 𝑑

𝐸(𝑝𝑅, 𝑝) − 𝑑

𝑅(𝑝))]

(28)

is jointly concave in (𝐼, 𝑝𝑅, 𝑝). So,

G𝑡(𝐼) = max

𝑝𝑅≤𝑝𝑅≤𝑝𝑅,𝑝≤𝑝≤𝑝

Π𝑡(𝐼 : 𝑦∗(𝑝𝑅, 𝑝) ∨ 𝐼, 𝑝

𝑅, 𝑝) (29)

is concave in 𝐼. Therefore, for all 𝑡 = 1, 2, . . . , 𝑇, G𝑡(⋅) is

concave.

Proposition 10. Suppose that, in period 𝑡, the inventory levelis 𝐼𝑡and the advanced demand from period 𝑡 − 1 which is for

regular service is 𝑥𝑡−1

.Then, in period 𝑡, there is some finite pair(𝑦∗

𝑡, 𝑝∗

𝑡, 𝑝∗

𝑅,𝑡) such that if 𝐼

𝑡− 𝑥𝑡−1

≤ 𝑦∗

𝑡, it is optimal to place

positive number (𝑦∗𝑡−𝐼𝑡+𝑥𝑡−1

) of order and to charge the prices𝑝∗

𝑅,𝑡+𝑝∗

𝑡for the express service and 𝑝∗

𝑅,𝑡for the regular service.

Otherwise, no ordering will take place.

Proof. As seen in Lemma 8, the optimal solution pair(𝑦∗

𝑡, 𝑝∗

𝑅,𝑡, 𝑝∗

𝑡) does not depend on 𝐼

𝑡and 𝑥

𝑡−1but 𝑦∗𝑡depends

on (𝑝∗𝑅, 𝑝𝑡) which is 𝑦∗

𝑡= 𝑦∗

𝑡(𝑝∗

𝑅,𝑡, 𝑝∗

𝑡). So, if 𝐼


≤ 𝑦∗

𝑡,

it is optimal to place 𝑦∗𝑡− 𝐼𝑡+ 𝑥𝑡−1

of orders. Otherwise, theoptimal order up to level is 𝐼


which is not to order.Now, we need to verify that there exists finite solution pair ineach period 𝑡. In each period 𝑡, the feasible sets for 𝑝

𝑅,𝑡and 𝑝

𝑡

are bounded so that the optimal pricing 𝑝∗𝑅,𝑡

and 𝑝∗𝑡are finite.

𝑦∗

𝑡= 𝑦∗

𝑡(𝑝∗

𝑅,𝑡, 𝑝∗

𝑡) is solution to

𝑝∗

𝑅,𝑡𝑑𝑅(𝑝∗

𝑡) + (𝑝

∗

𝑅,𝑡+ 𝑝∗

𝑡) 𝑑𝐸(𝑝∗

𝑅,𝑡, 𝑝∗

𝑡) − 𝑐 (𝑦 − 𝐼)

− 𝐻(𝑦 − 𝑑𝐸(𝑝∗

𝑅,𝑡, 𝑝∗

𝑡))

+ 𝛼G𝑡+1

(𝑦 − 𝑑𝐸(𝑝∗

𝑅,𝑡, 𝑝∗

𝑡) − 𝑑𝑅(𝑝∗

𝑡)) .

(30)

By Assumption 6, in each period 𝑡,

lim𝑦→±∞

− 𝑐𝑦 − 𝐻(𝑦 − 𝑑𝐸(𝑝∗

𝑅,𝑡, 𝑝∗

𝑡)) = −∞. (31)

Thus, as 𝑦 → ±∞

(𝑝∗

𝑅,𝑡𝑑𝑅(𝑝∗

𝑡) + (𝑝

∗

𝑅,𝑡+ 𝑝∗

𝑡) 𝑑𝐸(𝑝∗

𝑅,𝑡, 𝑝∗

𝑡) − 𝑐 (𝑦 − 𝐼)

− 𝐻(𝑦 − 𝑑𝐸(𝑝∗

𝑅,𝑡, 𝑝∗

𝑡))) + 𝛼G

𝑡+1(𝑦

− 𝑑𝐸(𝑝∗

𝑅,𝑡, 𝑝∗

𝑡) − 𝑑𝑅(𝑝∗

𝑡))

(32)

will go to ∞, and thus the solution 𝑦∗𝑡

= 𝑦∗

𝑡(𝑝∗

𝑅,𝑡, 𝑝∗

𝑡) should

be finite. Therefore, the result holds.

5. Numerical Analysis

In this section, we provide a report on a numerical analysiscarried on to obtain insights into the structure of optimalpolicies and their sensitivity and quantitative comparisonwith the traditional policy (single-pricing policy). Among themain questions, we focus on

(1) the benefits of a multiple pricing strategy comparedto a one pricing strategy in settings with continuousinventory replenishment opportunities,

(2) the sensitivity of the optimal base stock and listprices with respect to the degree of variability and theseasonality in the demands,

(3) the comparison of profit with the traditional single-pricing policy.

Our numerical study is based on data in Table 1. As men-tioned in Assumption 3, the demand function is a linearfunction of the regular price and extra charge. That is,

𝑑𝐸(𝑝𝑟, 𝑝) = 𝑎 × 𝛾 − 𝑏

𝑅𝑝𝑅− 𝑏𝐸𝑝 + 𝜖. (33)

The random term 𝜖 is assumed to be normally distributedwith mean 0. However, it is truncated to avoid the negativevalue of demand such that the minimum of 𝜖 is −1 × (𝑎 −𝑏𝑅𝑝𝑅−𝑏𝐸𝑝). Also, to capture the degree of demand variability,

we have used [c.v. × (𝑎 − 𝑏𝑅𝑝𝑅− 𝑏𝐸𝑝)]2 as the variance of 𝜖,


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Figure 1: Optimal base stock levels for varying demand uncertainty.

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Figure 2: Price for regular service for varying demand uncertainty.

where c.v. is the coefficient for the variability in demand. 𝛾 isthe randomly generated number for the demand seasonality.

Figure 1 shows the base stock for both nonseasonal andseasonal demand cases. As you can see in this figure, as thedemand variation increases, the base stock tends to increase.This might be from the fact that more demand fluctuationcan increase the possibility of inventory shortage. So, in orderto decrease the shortage cost, the base stock would increase.Figures 2 and 3 show the threshold price predetermined for

regular service and express service in each time for eachdemand variation (c.v.).

The inventory controlling strategy by multiple servicelevels can provide the retailer or producer with the one-period advanced information regarding demand since somecustomers, who select regular service, are willing to delayshipment for their order. This one-period advanced infor-mation can be useful in managing the inventory in thenext period in the sense that inventory can be controlled


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Figure 3: Price for express service for varying demand uncertainty.

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Figure 4: Profit comparison between inventory control by multiple service levels and traditional inventory control by single-service level.

more efficiently and moreover there is chance to reduce thecost which is induced from holding too much inventory orshortage inventory.

Figure 4 shows the benefits of an inventory controlthrough multiple service levels compared to traditionalinventory control by single-service level. In both nonseasonaland seasonal case, you can see that our model under unreli-able supplying capacity is more profitable than the traditionalsingle-service and single-pricing model. Moreover, at higherlevel of demand uncertainty (c.v.), the profit increase fromour model for seasonal demand case is higher than thenonseasonal case. Thus, we can see that the benefit frominventory control strategy by multiple service levels canbe relatively large in the environment where the inventorysystem experiences higher demand uncertainty and season-ality: inventory control by multiple service levels under theunreliable supplying capacity, which provides the system

with one-period advanced information regarding demand,efficiently captures the demanduncertainty in order to reducethe cost compared to traditional controlling model by single-service level, thus giving more profit. From this numericalanalysis, we can see that the proposed model provides betterstrategy for the inventory controlling problem even under theunreliable supplying capacity.

6. Conclusion

We study dynamic pricing and inventory replenishmentproblems under unreliable supplying condition.This researchwas initiated by the following practical intuition; that is,if customers are willing to expedite the service for theirorder, then they are willing to pay extra charge. In thispaper, we have verified this intuition through constructinga reasonable demand model (Assumption 3) and by the


mathematical dynamic programming model where productcan be provided to the customers with multiple (two) pricescorresponding to the service level, either express service orregular service. It is shown in this paper that there exist apair of threshold levels for inventory ordering up to leveland prices corresponding to service levels in the followingstrategy: if the inventory level before ordering in each periodwhich is subtracted by the regular demand advanced fromthe previous period is less than the predetermined threshold,then it is optimal to make ordering decision and increaseinventory level up to the predetermined threshold level andoffer threshold price for each service level. Otherwise, noordering is optimal.

We can extend our model in the following possibleresearch directions. First, we can extend the result to the caseof infinite horizon. Considering an infinite horizon modelwith stationary parameters with some practical assumptions,some optimality result similar to one from the finite horizonmodel might be extended. Second, we assume the depen-dency of demand in a period only on the price in the sameand current period. However, the demand function mightbe extended to depend not only on the current period butalso on the current and the historical prices, which are theprices in past periods. Even though it would be complex toanalyze, it would be interesting to incorporate these moregeneral demand settings into the model and examine thecorresponding optimal policies. Third, our assumption inwhich the backlogging is allowed can be modified such thatthe backlogging is not allowed but lost-sales are assumed.Finally, it would be interesting and important to incorporatea fixed ordering cost into our model. Analysis of suchextensions can be more complicated to analyze but would beworth further exploration.

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper.

Acknowledgments

This research was supported by 2014 Hongik UniversityResearch Fund and also partially supported by the NationalResearch Foundation of Korea Grant funded by the KoreanGovernment (NRF-2015R1C1A1A02037096). These researchfunds do not lead to any competing interests regarding thepublication of this manuscript.

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Research Article Inventory Control by Multiple Service Levels ...Research Article Inventory Control by Multiple Service Levels under Unreliable Supplying Condition ByungsooNa, 1 JinpyoLee,