research article inventory control by multiple service levels ...research article inventory control...
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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 ยฉ 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
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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) ๐: per unit marginal cost,(2) ๐๐ ,๐ก: price charged for regular service in period ๐ก,
(3) ๐๐ธ,๐ก: price charged for express service in period ๐ก,
(4) ๐๐ก= ๐๐ธ,๐ก
โ๐๐ ,๐ก: extra charge for the express service in
period ๐ก on [0, ๐],(5) ๐๐ก: random uncertainty term of demand which had a
known distribution,(6) ๐ท
๐ธ,๐ก(๐๐ ,๐ก
, ๐๐ธ,๐ก
): demand for the express service,(7) ๐ท๐ ,๐ก
(๐๐ ,๐ก
): demand for the regular service,(8) ๐ฅ๐ก: inventory level at the beginning of each period ๐ก
before ordering,(9) ๐ฆ๐ก: inventory level at the beginning of each period ๐ก
after ordering,(10) โ
๐ก(๐ผ): inventory cost (holding or backlogging) at the
end of each period ๐ก.
Assumption 1. Backlogging is allowed.
Assumption 2. Replenishment after ordering becomes avail-able instantaneously.
Traditionally, in operations research literatures, the cus-tomerโs 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 ๐ก = 1, 2, . . . , ๐, the demandfunction for regular service ๐ท
๐ ,๐ก(๐๐ ,๐ก
, ๐๐ธ,๐ก
) = ๐๐ ,๐ก
(๐๐ก), where
๐๐ ,๐ก
(๐๐ก) is given by ๐
๐ ,๐ก(๐๐ธ,๐ก
โ ๐๐ ,๐ก
), and nondecreasinglinear function in ๐
๐ก(=๐๐ธ,๐ก
โ ๐๐ ,๐ก
) โ [0, ๐]. The demandfunction for express service ๐ท
๐ธ,๐ก(๐๐ ,๐ก
, ๐๐ธ,๐ก
) = ๐๐ธ,๐ก
(๐๐ ,๐ก
, ๐๐ก),
where ๐๐ธ,๐ก
(๐๐ ,๐ก
, ๐๐ก) is given by ๐(๐
๐ ,๐ก, ๐๐ก) โ ๐ท
๐ ,๐ก(๐๐ ,๐ก
, ๐๐ธ,๐ก
) =
๐(๐๐ ,๐ก
, ๐๐ก) โ ๐๐ ,๐ก
(๐๐ก), which is nonincreasing linear function
in ๐๐ ,๐ก
โ [๐๐ , ๐๐ ] and ๐
๐กโ [0, ๐]. Moreover, ๐(๐
๐ ,๐ก, ๐๐ก) is the
possible maximum demand and nonincreasing and linearfunction of ๐
๐ ,๐กโ [๐, ๐
๐ ]. Moreover, values for ๐ and ๐
๐ are
taken such that ๐(๐๐ , ๐๐ก) โ๐ท๐ ,๐ก
(๐๐ ,๐ก
, ๐๐ธ,๐ก
) is nonnegative withprobability 1.
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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 ๐
๐กbetween 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 ๐
๐กbetween
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 ๐ก = 1, 2, . . . , ๐ isgiven by
๐๐ ,๐ก
๐ธ [๐ท๐ ,๐ก
(๐๐ ,๐ก
, ๐๐ธ,๐ก
)] + ๐๐ธ,๐ก
๐ธ [๐ท๐ธ,๐ก
(๐๐ ,๐ก
, ๐๐ธ,๐ก
)]
= ๐๐ ,๐ก
๐ธ [๐๐ ,๐ก
(๐๐ก)] + ๐
๐ธ,๐ก๐ธ [๐๐ธ,๐ก
(๐๐ ,๐ก
, ๐๐ก)]
(1)
and is finite and concave for ๐๐ ,๐ก
โ [๐๐ , ๐๐ ] and ๐
๐กโ [0, ๐],
where ๐๐ก= ๐๐ธ,๐ก
โ ๐๐ ,๐ก.
Assumption 5. ๐ป๐ก(๐ผ) = ๐ธ[โ
๐ก(๐ผ)] is convex in ๐ผ and ๐ป
๐ก(0) = 0
in each ๐ก = 1, . . . , ๐.
Assumption 6. lim๐ฆโยฑโ
๐ป๐ก(๐ฆ โ ๐ท
๐ธ,๐ก(๐๐ ,๐ก
, ๐๐ธ,๐ก
)) =
lim๐ฆโยฑโ
(๐๐ฆ+๐ป๐ก(๐ฆโ๐ท
๐ธ,๐ก(๐๐ ,๐ก
, ๐๐ธ,๐ก
))) = โ for any๐๐ ,๐ก
โ [๐๐ ,
๐๐ ] and ๐
๐กโ [0, ๐].
4. Mathematical Formulation
We consider an additive demand model, where the demanduncertainty for regular and express service is represented byan additive random noise ๐
๐ก, respectively,
๐ท๐ ,๐ก
(๐๐ ,๐ก
, ๐๐ธ,๐ก
) = ๐๐ ,๐ก
(๐๐ก) = D
๐ ,๐ก(๐๐ก) + ๐๐ก,
๐ท๐ธ,๐ก
(๐๐ ,๐ก
, ๐๐ธ,๐ก
) = ๐๐ธ,๐ก
(๐๐ ,๐ก
, ๐๐ก) = D
๐ธ,๐ก(๐๐ ,๐ก
, ๐๐ก) + ๐๐ก.
(2)
We assume that ๐๐กhas mean zero and support [๐, ๐]. For
any feasible choice of ๐๐ ,๐ก
โ [๐๐ , ๐๐ ] and ๐
๐ธ,๐กโ [๐๐ธ, ๐๐ธ],
the demand is positive with probability one and the averagedemands ๐ธ[๐ท
๐ ,๐ก(๐๐ ,๐ก
, ๐๐ธ,๐ก
)] and ๐ธ[๐ท๐ธ,๐ก
(๐๐ ,๐ก
, ๐๐ธ,๐ก
)] are finite.Given the inventory level ๐ผ
๐กin period ๐ก, the inventory level
๐ผ๐ก+1
in period ๐ก + 1 is given as follows:
๐ผ๐ก+1
= ๐ผ๐ก+ (๐ฆ๐กโ ๐ผ๐ก) โง ๐ โ ๐
๐ธ,๐ก(๐๐ ,๐ก
, ๐๐ก) , (3)
where ๐ โ [0, ๐ฆ๐กโ ๐ผ๐ก] in period ๐ก and ๐ โง ๐ = min[๐, ๐]. Let
๐๐ก(๐ผ, ๐ฅ) be the optimal profit function in period ๐ก when the
inventory level is ๐ผ and the demand from period ๐ก โ 1 is ๐ฅ.Then, the optimality equation is given by
๐๐ก (๐ผ, ๐ฅ) = max
๐ฆโฅ๐ผ,๐๐ โค๐๐ โค๐๐ ,๐โค๐โค๐
๐๐ก(๐ผ; ๐ฆ, ๐
๐ , ๐) , (4)
where๐๐ก(๐ผ; ๐ฆ, ๐
๐ , ๐)
= ๐๐ ๐ธ [๐๐ (๐)] + ๐
๐ธ๐ธ [๐๐ธ(๐๐ , ๐)]
โ ๐๐ธ [(๐ผ + ๐) โง ๐ฆ โ ๐ผ]
โ ๐ธ [๐ป ((๐ผ + ๐) โง ๐ฆ โ ๐๐ธ(๐๐ , ๐))]
+ ๐ผ๐ธ [๐๐ก+1
((๐ผ + ๐) โง ๐ฆ โ ๐๐ธ(๐๐ , ๐) , ๐
๐ (๐))] .
(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 ๐ผ and๐ฆ be defined as ๐ผโ๐ฅ and๐ฆโ๐ฅ, respectively.Then, (๐ฆโ
๐ก+ ๐ฅ, ๐
โ
๐ ,๐ก, ๐โ
๐ก) is the set of optimal solutions to (4) if
and only if (๐ฆโ๐ก, ๐โ
๐ ,๐ก, ๐โ
๐ก) is the set of optimal solutions to
G๐ก(๐ผ) = max
๐ฆโฅ๐ผ,๐๐ โค๐๐ โค๐๐ ,๐โค๐โค๐
ฮ ๐ก(๐ผ : ๐ฆ, ๐
๐ , ๐) , (6)
where
ฮ ๐ก(๐ผ : ๐ฆ, ๐
๐ , ๐)
= ๐๐ ๐ธ [๐๐ (๐)] + ๐
๐ธ๐ธ [๐๐ธ(๐๐ , ๐)]
โ ๐๐ธ [(๐ผ + ๐) โง ๐ฆ โ ๐ผ]
โ ๐ธ [๐ป ((๐ผ + ๐) โง ๐ฆ โ ๐๐ธ(๐๐ , ๐))]
+ ๐ผ๐ธ [G๐ก+1
((๐ผ + ๐) โง ๐ฆ โ ๐๐ธ(๐๐ , ๐) โ ๐
๐ (๐))] .
(7)
Proof. It is enough to show that, for all period ๐ก,
๐๐ก(๐ผ; ๐ฆ, ๐
๐ , ๐) = ฮ
๐ก(๐ผ : ๐ฆ, ๐
๐ , ๐) โ๐ผ, ๐ฅ, ๐ฆ, ๐
๐ , ๐. (8)
Since
(๐ผ + ๐) โง ๐ฆ = (๐ผ โ ๐ฅ + ๐) โง (๐ฆ โ ๐ฅ)
=
{
{
{
๐ผ โ ๐ฅ + ๐, if ๐ผ โ ๐ฅ + ๐ โค ๐ฆ โ ๐ฅ
๐ฆ โ ๐ฅ, otherwise
= (๐ผ + ๐) โง ๐ฆ โ ๐ฅ,
(9)
for all period ๐ก, the first four terms of ๐๐ก(๐ผ; ๐ฆ, ๐
๐ , ๐) andฮ
๐ก(๐ผ :
๐ฆ, ๐๐ , ๐) are equal to each other as follows:
๐๐ ๐ธ [๐๐ (๐)] + ๐
๐ธ๐ธ [๐๐ธ(๐๐ , ๐)]
โ ๐๐ธ [(๐ผ + ๐) โง ๐ฆ โ ๐ผ]
โ ๐ธ [๐ป ((๐ผ + ๐) โง ๐ฆ โ ๐๐ธ(๐๐ , ๐))]
= ๐๐ ๐ธ [๐๐ (๐)] + ๐
๐ธ๐ธ [๐๐ธ(๐๐ , ๐)]
โ ๐๐ธ [(๐ผ + ๐) โง ๐ฆ โ ๐ผ]
โ ๐ธ [๐ป ((๐ผ + ๐) โง ๐ฆ โ ๐๐ธ(๐๐ , ๐) โ ๐ฅ)] .
(10)
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4 Mathematical Problems in Engineering
In period ๐, trivially ๐๐(๐ผ, ๐ฅ) = G
๐(๐ผ) = 0. In period ๐ โ 1,
๐๐โ1
(๐ผ; ๐ฆ, ๐๐ , ๐)
= ๐๐ ๐ธ [๐๐ ,๐โ1
(๐)] + ๐๐ธ๐ธ [๐๐ธ(๐๐ ,๐โ1
, ๐)]
โ ๐๐ธ [(๐ผ + ๐) โง ๐ฆ โ ๐ผ]
โ ๐ธ [๐ป ((๐ผ + ๐) โง ๐ฆ โ ๐๐ธ,๐โ1(๐๐ , ๐) โ ๐ฅ)]
= ๐๐ ๐ธ [๐๐ ,๐โ1
(๐)] + ๐๐ธ๐ธ [๐๐ธ,๐โ1
(๐๐ , ๐)]
โ ๐๐ธ [(๐ผ + ๐) โง ๐ฆ โ ๐ผ]
โ ๐ธ [๐ป ((๐ผ + ๐) โง ๐ฆ โ ๐๐ธ,๐โ1
(๐๐ , ๐))]
= ฮ ๐โ1
(๐ผ : ๐ฆ, ๐๐ , ๐) .
(11)
So, ๐๐โ1
(๐ผ, ๐ฅ) = G๐โ1
(๐ผ)with ๐ผ = ๐ผโ๐ฅ. By induction, supposethat, for any ๐ผ, ๐ฅ with ๐ผ = ๐ผ โ ๐ฅ,
๐๐ (
๐ผ, ๐ฅ) = G๐ (๐ผ) , ๐ = ๐, ๐ โ 1, . . . , ๐ก + 1. (12)
Then
๐๐ก(๐ผ; ๐ฆ, ๐
๐ , ๐)
= ๐๐ ๐ธ [๐๐ (๐)] + ๐
๐ธ๐ธ [๐๐ธ(๐๐ , ๐)]
โ ๐๐ธ [(๐ผ + ๐) โง ๐ฆ โ ๐ผ]
โ ๐ธ [๐ป ((๐ผ + ๐) โง ๐ฆ โ ๐๐ธ(๐๐ , ๐) โ ๐ฅ)]
+ ๐ผ๐ธ [๐๐ก+1
((๐ผ + ๐) โง ๐ฆ โ ๐๐ธ(๐๐ , ๐) โ ๐ฅ, ๐
๐ (๐))]
= ๐๐ ๐ธ [๐๐ (๐)] + ๐
๐ธ๐ธ [๐๐ธ(๐๐ , ๐)]
โ ๐๐ธ [(๐ผ + ๐) โง ๐ฆ โ ๐ผ]
โ ๐ธ [๐ป ((๐ผ + ๐) โง ๐ฆ โ ๐๐ธ(๐๐ , ๐))]
+ ๐ผ๐ธ [G๐ก+1
((๐ผ + ๐) โง ๐ฆ โ ๐๐ธ(๐๐ , ๐) โ ๐
๐ (๐))] .
(13)
The second equality holds since (๐ผ+๐)โง๐ฆโ๐๐ธ(๐๐ , ๐)โ๐
๐ (๐) =
(๐ผ+๐)โง๐ฆโ๐๐ธ(๐๐ , ๐)โ๐ฅโ๐
๐ (๐). 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 ๐ก 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 G๐ก+1
(โ ) is concave function and, forgiven ๐
๐ and ๐, let ๐ฆโ(๐
๐ , ๐) be the maximizer of
๐๐ ๐๐ (๐) + ๐
๐ธ๐๐ธ(๐๐ , ๐) โ ๐ (๐ฆ โ ๐ผ)
โ ๐ป (๐ฆ โ ๐๐ธ(๐๐ , ๐))
+ ๐ผG๐ก+1
(๐ฆ โ ๐๐ธ(๐๐ , ๐) โ ๐
๐ (๐)) ,
(14)
where ๐๐ธ= ๐๐ + ๐. Then, ๐ฆโ(๐
๐ , ๐) is also the maximizer of
๐๐ ๐ธ [๐๐ (๐)] + ๐
๐ธ๐ธ [๐๐ธ(๐๐ , ๐)]
โ ๐๐ธ [(๐ผ + ๐) โง ๐ฆ โ ๐ผ]
โ ๐ธ [๐ป ((๐ผ + ๐) โง ๐ฆ โ ๐๐ธ(๐๐ , ๐))]
+ ๐ผ๐ธ [G๐ก+1
((๐ผ + ๐) โง ๐ฆ โ ๐๐ธ(๐๐ , ๐) โ ๐
๐ (๐))]
(15)
for any ๐ผ and, moreover,
๐๐ ๐ธ [๐๐ (๐)] + ๐
๐ธ๐ธ [๐๐ธ(๐๐ , ๐)] โ ๐๐ธ [(๐ผ + ๐)
โง (๐ฆโ(๐๐ , ๐) โจ ๐ผ) โ ๐ผ] โ ๐ธ [๐ป ((๐ผ + ๐)
โง (๐ฆโ(๐๐ , ๐) โจ ๐ผ) โ ๐
๐ธ(๐๐ , ๐))]
+ ๐ผ๐ธ [G๐ก+1
((๐ผ + ๐) โง (๐ฆโ(๐๐ , ๐) โจ ๐ผ)
โ ๐๐ธ(๐๐ , ๐) โ ๐
๐ (๐))]
(16)
is jointly concave in (๐ผ, ๐๐ , ๐), where ๐ โจ ๐ = max[๐, ๐].
Proof. Given๐๐ and๐,๐
๐ ๐๐ (๐)+๐
๐ธ๐๐ธ(๐๐ , ๐) is just constant
and thus wewill not consider them for a while. Since๐ป(โ ) andG๐ก+1
(โ ) are concave,
๐๐ ๐๐ (๐) + ๐
๐ธ๐๐ธ(๐๐ , ๐) โ ๐๐ฆ โ ๐ป (๐ฆ โ ๐
๐ธ(๐๐ , ๐))
+ ๐ผG๐ก+1
(๐ฆ โ ๐๐ธ(๐๐ , ๐) โ ๐
๐ (๐))
(17)
is jointly concave in (๐ฆ, ๐๐ , ๐). Now, take any ๐ฟ > 0 such that
๐ฆ+๐ฟ < ๐ฆโ(๐๐ , ๐) and we have ๐ฆโง (๐ผ+๐) โค (๐ฆ+๐ฟ)โง (๐ผ+๐) <
๐ฆโ(๐๐ , ๐). So,
โ ๐ (๐ฆ โง (๐ผ + ๐) โ ๐ผ) โ ๐ป((๐ผ + ๐) โง ๐ฆ โ ๐๐ธ(๐๐ , ๐))
+ ๐ผ (G๐ก+1
((๐ผ + ๐) โง ๐ฆ โ ๐๐ธ(๐๐ , ๐) โ ๐
๐ (๐)))
โค โ๐ ((๐ฆ + ๐ฟ) โง (๐ผ + ๐) โ ๐ผ) โ ๐ป((๐ผ + ๐) โง (๐ฆ
+ ๐ฟ) โ ๐๐ธ(๐๐ , ๐)) + ๐ผ (G
๐ก+1((๐ผ + ๐) โง (๐ฆ + ๐ฟ)
โ ๐๐ธ(๐๐ , ๐) โ ๐
๐ (๐)))
(18)
for any ๐. Thus,
๐ธ [โ๐ (๐ฆ โง (๐ผ + ๐) โ ๐ผ) โ ๐ป((๐ผ + ๐) โง ๐ฆ
โ ๐๐ธ(๐๐ , ๐)) + ๐ผG
๐ก+1((๐ผ + ๐) โง ๐ฆ โ ๐
๐ธ(๐๐ , ๐)
โ ๐๐ (๐))] โค ๐ธ [โ๐ ((๐ฆ + ๐ฟ) โง (๐ผ + ๐) โ ๐ผ)
โ ๐ป((๐ผ + ๐) โง (๐ฆ + ๐ฟ) โ ๐๐ธ(๐๐ , ๐))
+ ๐ผG๐ก+1
((๐ผ + ๐) โง (๐ฆ + ๐ฟ) โ ๐๐ธ(๐๐ , ๐)
โ ๐๐ (๐))] .
(19)
-
Mathematical Problems in Engineering 5
Therefore,
โ ๐๐ธ [(๐ผ + ๐) โง ๐ฆ โ ๐ผ]
โ ๐ธ [๐ป ((๐ผ + ๐) โง ๐ฆ โ ๐๐ธ(๐๐ , ๐))]
+ ๐ผ๐ธ [G๐ก+1
((๐ผ + ๐) โง ๐ฆ โ ๐๐ธ(๐๐ , ๐) โ ๐
๐ (๐))]
(20)
is increasing in ๐ฆ when ๐ฆ โค ๐ฆโ(๐๐ , ๐). By the similar
argument,
โ ๐๐ธ [(๐ผ + ๐) โง ๐ฆ โ ๐ผ]
โ ๐ธ [๐ป ((๐ผ + ๐) โง ๐ฆ โ ๐๐ธ(๐๐ , ๐))]
+ ๐ผ๐ธ [G๐ก+1
((๐ผ + ๐) โง ๐ฆ โ ๐๐ธ(๐๐ , ๐) โ ๐
๐ (๐))]
(21)
is decreasing in ๐ฆ when ๐ฆ โฅ ๐ฆโ(๐๐ , ๐). The first result holds.
Since, for given ๐๐ and ๐, ๐ฆโ(๐
๐ , ๐) is the maximizer of
โ ๐ ((๐ผ + ๐) โ ๐ผ) โ ๐ป((๐ผ + ๐) โง ๐ฆ โ ๐๐ธ(๐๐ , ๐))
+ ๐ผG๐ก+1
((๐ผ + ๐) โง ๐ฆ โ ๐๐ธ(๐๐ , ๐) โ ๐
๐ (๐))
(22)
for any ๐ฆ โ R, for given ๐๐ and ๐, (๐ฆโ(๐
๐ , ๐) โจ ๐ผ) โง (๐ผ + ๐) is
the maximizer of
โ ๐ ((๐ผ + ๐) โ ๐ผ) โ ๐ป((๐ผ + ๐) โง ๐ฆ โ ๐๐ธ(๐๐ , ๐))
+ ๐ผG๐ก+1
((๐ผ + ๐) โง ๐ฆ โ ๐๐ธ(๐๐ , ๐) โ ๐
๐ (๐))
(23)
for ๐ฆ โ [๐ผ, ๐ผ + ๐]. Now, take (๐ผ1, ๐๐ ,1
, ๐1), (๐ผ2, ๐๐ ,2
, ๐2), and
๐พ โ [0, 1]. Then
๐พ (โ๐ ((๐ผ1+ ๐) โง (๐ฆ
โ(๐๐ ,1
, ๐1) โจ ๐ผ1) โ ๐ผ1) โ ๐ป((๐ผ
1
+ ๐) โง (๐ฆโ(๐๐ ,1
, ๐1) โจ ๐ผ1) โ ๐๐ธ(๐๐ ,1
, ๐1))
+ ๐ผ (G๐ก+1
((๐ผ1+ ๐) โง (๐ฆ
โ(๐๐ ,1
, ๐1) โจ ๐ผ1)
โ ๐๐ธ(๐๐ ,1
, ๐1) โ ๐๐ (๐๐ ,1
, ๐1)))) + (1 โ ๐พ)
โ (โ๐ ((๐ผ2+ ๐) โง (๐ฆ
โ(๐๐ ,2
, ๐2) โจ ๐ผ2) โ ๐ผ2)
โ ๐ป((๐ผ2+ ๐) โง (๐ฆ
โ(๐๐ ,2
, ๐2) โจ ๐ผ2)
โ ๐๐ธ(๐๐ ,2
, ๐2)) + ๐ผG
๐ก+1((๐ผ2+ ๐)
โง (๐ฆโ(๐๐ ,2
, ๐2) โจ ๐ผ2) โ ๐๐ธ(๐๐ ,2
, ๐2)
โ ๐๐ (๐๐ ,2
, ๐2))) โค โ๐ (๐พ [(๐ผ
1+ ๐)
โง (๐ฆโ(๐๐ ,1
, ๐1) โจ ๐ผ1)] + (1 โ ๐พ) [(๐ผ
2+ ๐)
โง (๐ฆโ(๐๐ ,2
, ๐2) โจ ๐ผ2)] โ ๐ผ
๐พ) โ ๐ป(๐พ [(๐ผ
1+ ๐)
โง (๐ฆโ(๐๐ ,1
, ๐1) โจ ๐ผ1)] + (1 โ ๐พ) [(๐ผ
2+ ๐)
โง (๐ฆโ(๐๐ ,2
, ๐2) โจ ๐ผ2)] โ ๐
๐ธ(๐๐ ,๐พ
, ๐๐ธ,๐พ
))
+ ๐ผG๐ก+1
(๐พ [(๐ผ1+ ๐) โง (๐ฆ
โ(๐๐ ,1
, ๐1) โจ ๐ผ1)] + (1
โ ๐พ) [(๐ผ2+ ๐) โง (๐ฆ
โ(๐๐ ,2
, ๐2) โจ ๐ผ2)]
โ ๐พ [๐๐ธ(๐๐ ,1
, ๐1) โ ๐๐ (๐๐ ,1
, ๐1)] โ (1 โ ๐พ)
โ [๐๐ธ(๐๐ ,2
, ๐2) โ ๐๐ (๐๐ ,2
, ๐2)]) = โ๐ (๐พ [(๐ผ
1+ ๐)
โง (๐ฆโ(๐๐ ,1
, ๐1) โจ ๐ผ1)] + (1 โ ๐พ) [(๐ผ
2+ ๐)
โง (๐ฆโ(๐๐ ,2
, ๐2) โจ ๐ผ2)] โ ๐ผ
๐พ) โ ๐ป(๐พ [(๐ผ
1+ ๐)
โง (๐ฆโ(๐๐ ,1
, ๐1) โจ ๐ผ1)] + (1 โ ๐พ) [(๐ผ
2+ ๐)
โง (๐ฆโ(๐๐ ,2
, ๐2) โจ ๐ผ2)] โ ๐
๐ธ(๐๐ ,๐พ
, ๐๐ธ,๐พ
))
+ ๐ผG๐ก+1
(๐พ [(๐ผ1+ ๐) โง (๐ฆ
โ(๐๐ ,1
, ๐1) โจ ๐ผ1)] + (1
โ ๐พ) [(๐ผ2+ ๐) โง (๐ฆ
โ(๐๐ ,2
, ๐2) โจ ๐ผ2)] โ ๐
๐ธ(๐๐ ,๐พ
,
๐๐พ) โ ๐๐ (๐๐ ,๐พ
, ๐๐พ)) โค โ๐ ((๐ผ
๐พ+ ๐)
โง (๐ฆโ(๐๐ ,๐พ
, ๐๐พ) โจ ๐ผ๐พ) โ ๐ผ๐พ) โ ๐ป((๐ผ
๐พ+ ๐)
โง (๐ฆโ(๐๐ ,๐พ
, ๐๐พ) โจ ๐ผ๐พ) โ ๐๐ธ(๐๐ ,๐พ
, ๐๐พ))
+ ๐ผG๐ก+1
((๐ผ๐พ+ ๐) โง (๐ฆ
โ(๐๐ ,๐พ
, ๐๐พ) โจ ๐ผ๐พ)
โ ๐๐ธ(๐๐ ,๐พ
, ๐๐พ) โ ๐๐ (๐๐ ,๐พ
, ๐๐พ)) ,
(24)
where ๐ผ๐พ
= ๐พ๐ผ1+ (1 โ ๐พ)๐ผ
2, ๐๐ ,๐พ
= ๐พ๐๐ ,1
+ (1 โ ๐พ)๐๐ ,2
, and๐๐พ
= ๐พ๐1+ (1 โ ๐พ)๐
2. The first inequality holds due to the
concavity of๐ป๐กandG
๐ก+1, the second equality holds due to the
linearity of demand functions, and the last inequality holdssince, for given ๐
๐ ,๐พand ๐
๐พ, ๐ฆ = (๐ผ
๐พ+ ๐) โง ๐ฆ
โ(๐๐ ,๐พ
, ๐๐พ) โจ ๐ผ๐พ
is the maximizer of
โ ๐ (๐ฆ โ ๐ผ) โ ๐ป (๐ฆ โ ๐๐ธ(๐๐ , ๐))
+ ๐ผG๐ก+1
(๐ฆ โ ๐๐ธ(๐๐ , ๐) โ ๐
๐ (๐))
(25)
for ๐ฆ โ [๐ผ๐พ, ๐ผ๐พ+ ๐]. So, we need to verify that
๐พ [(๐ผ1+ ๐) โง (๐ฆ
โ(๐๐ ,1
, ๐1) โจ ๐ผ1)]
+ (1 โ ๐พ) [(๐ผ2+ ๐) โง (๐ฆ
โ(๐๐ ,2
, ๐2) โจ ๐ผ2)]
(26)
is within [๐ผ๐พ, ๐ผ๐พ+ ๐].
๐พ [(๐ผ1+ ๐) โง (๐ฆ
โ(๐๐ ,1
, ๐1) โจ ๐ผ1)]
+ (1 โ ๐พ) [(๐ผ2+ ๐) โง (๐ฆ
โ(๐๐ ,2
, ๐2) โจ ๐ผ2)]
โฅ ๐พ [(๐ผ1+ ๐) โง ๐ผ
1] + (1 โ ๐พ) [(๐ผ
2+ ๐) โง ๐ผ
2]
= ๐พ๐ผ1+ (1 โ ๐พ) ๐ผ
2= ๐ผ๐พ,
-
6 Mathematical Problems in Engineering
Table 1: Parameters for numerical analysis.
๐ ๐๐
๐๐ธ
Marginal cost Holding cost Backlog cost Salvage price Range for regular price (๐๐ ) Range for extra charge (๐)
250 5 6 30 0.3 28 20 25โ65 0โ30
๐พ [(๐ผ1+ ๐) โง (๐ฆ
โ(๐๐ ,1
, ๐1) โจ ๐ผ1)]
+ (1 โ ๐พ) [(๐ผ2+ ๐) โง (๐ฆ
โ(๐๐ ,2
, ๐2) โจ ๐ผ2)]
โค ๐พ (๐ผ1+ ๐) + (1 โ ๐พ) (๐ผ
2+ ๐) = ๐ผ
๐พ+ ๐.
(27)
By taking the expectation, we can obtain the second result.
Proposition 9. For all ๐ก = 1, 2, . . . , ๐, G๐ก(โ ) is concave
function.
Proof. Since we consider the ๐โperiod problem, G๐+1
(๐ผ) =
0 and G๐+1
(๐ผ) is concave in ๐ผ. By induction, suppose thatG๐ก+1
(โ ) is concave. Let ๐ฆโ(๐๐ , ๐) be defined as in Lemma 8.
By Assumption 4 and Lemma 8,
ฮ ๐ก(๐ผ : ๐ฆโ(๐๐ , ๐) โจ ๐ผ, ๐
๐ , ๐) = ๐
๐ ๐ธ [๐๐ (๐)]
+ ๐๐ธ๐ธ [๐๐ธ(๐๐ , ๐)] โ ๐๐ธ [(๐ผ + ๐) โง (๐ฆ
โ(๐๐ , ๐)
โจ ๐ผ) โ ๐ผ] โ ๐ธ [๐ป ((๐ผ + ๐) โง (๐ฆโ(๐๐ , ๐) โจ ๐ผ)
โ ๐๐ธ(๐๐ , ๐))] + ๐ผ๐ธ [G
๐ก+1((๐ผ + ๐)
โง (๐ฆโ(๐๐ , ๐) โจ ๐ผ) โ ๐
๐ธ(๐๐ , ๐) โ ๐
๐ (๐))]
(28)
is jointly concave in (๐ผ, ๐๐ , ๐). So,
G๐ก(๐ผ) = max
๐๐ โค๐๐ โค๐๐ ,๐โค๐โค๐
ฮ ๐ก(๐ผ : ๐ฆโ(๐๐ , ๐) โจ ๐ผ, ๐
๐ , ๐) (29)
is concave in ๐ผ. Therefore, for all ๐ก = 1, 2, . . . , ๐, G๐ก(โ ) is
concave.
Proposition 10. Suppose that, in period ๐ก, the inventory levelis ๐ผ๐กand the advanced demand from period ๐ก โ 1 which is for
regular service is ๐ฅ๐กโ1
.Then, in period ๐ก, there is some finite pair(๐ฆโ
๐ก, ๐โ
๐ก, ๐โ
๐ ,๐ก) such that if ๐ผ
๐กโ ๐ฅ๐กโ1
โค ๐ฆโ
๐ก, it is optimal to place
positive number (๐ฆโ๐กโ๐ผ๐ก+๐ฅ๐กโ1
) of order and to charge the prices๐โ
๐ ,๐ก+๐โ
๐กfor the express service and ๐โ
๐ ,๐กfor the regular service.
Otherwise, no ordering will take place.
Proof. As seen in Lemma 8, the optimal solution pair(๐ฆโ
๐ก, ๐โ
๐ ,๐ก, ๐โ
๐ก) does not depend on ๐ผ
๐กand ๐ฅ
๐กโ1but ๐ฆโ๐กdepends
on (๐โ๐ , ๐๐ก) which is ๐ฆโ
๐ก= ๐ฆโ
๐ก(๐โ
๐ ,๐ก, ๐โ
๐ก). So, if ๐ผ
๐กโ ๐ฅ๐กโ1
โค ๐ฆโ
๐ก,
it is optimal to place ๐ฆโ๐กโ ๐ผ๐ก+ ๐ฅ๐กโ1
of orders. Otherwise, theoptimal order up to level is ๐ผ
๐กโ ๐ฅ๐กโ1
which is not to order.Now, we need to verify that there exists finite solution pair ineach period ๐ก. In each period ๐ก, the feasible sets for ๐
๐ ,๐กand ๐
๐ก
are bounded so that the optimal pricing ๐โ๐ ,๐ก
and ๐โ๐กare finite.
๐ฆโ
๐ก= ๐ฆโ
๐ก(๐โ
๐ ,๐ก, ๐โ
๐ก) is solution to
๐โ
๐ ,๐ก๐๐ (๐โ
๐ก) + (๐
โ
๐ ,๐ก+ ๐โ
๐ก) ๐๐ธ(๐โ
๐ ,๐ก, ๐โ
๐ก) โ ๐ (๐ฆ โ ๐ผ)
โ ๐ป(๐ฆ โ ๐๐ธ(๐โ
๐ ,๐ก, ๐โ
๐ก))
+ ๐ผG๐ก+1
(๐ฆ โ ๐๐ธ(๐โ
๐ ,๐ก, ๐โ
๐ก) โ ๐๐ (๐โ
๐ก)) .
(30)
By Assumption 6, in each period ๐ก,
lim๐ฆโยฑโ
โ ๐๐ฆ โ ๐ป(๐ฆ โ ๐๐ธ(๐โ
๐ ,๐ก, ๐โ
๐ก)) = โโ. (31)
Thus, as ๐ฆ โ ยฑโ
(๐โ
๐ ,๐ก๐๐ (๐โ
๐ก) + (๐
โ
๐ ,๐ก+ ๐โ
๐ก) ๐๐ธ(๐โ
๐ ,๐ก, ๐โ
๐ก) โ ๐ (๐ฆ โ ๐ผ)
โ ๐ป(๐ฆ โ ๐๐ธ(๐โ
๐ ,๐ก, ๐โ
๐ก))) + ๐ผG
๐ก+1(๐ฆ
โ ๐๐ธ(๐โ
๐ ,๐ก, ๐โ
๐ก) โ ๐๐ (๐โ
๐ก))
(32)
will go to โ, and thus the solution ๐ฆโ๐ก
= ๐ฆโ
๐ก(๐โ
๐ ,๐ก, ๐โ
๐ก) 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,
๐๐ธ(๐๐, ๐) = ๐ ร ๐พ โ ๐
๐ ๐๐ โ ๐๐ธ๐ + ๐. (33)
The random term ๐ is assumed to be normally distributedwith mean 0. However, it is truncated to avoid the negativevalue of demand such that the minimum of ๐ is โ1 ร (๐ โ๐๐ ๐๐ โ๐๐ธ๐). Also, to capture the degree of demand variability,
we have used [c.v. ร (๐ โ ๐๐ ๐๐ โ ๐๐ธ๐)]2 as the variance of ๐,
-
Mathematical Problems in Engineering 7
Time
1.510.5
02 3 4 51
50
100
150
200
250
300
Opt
imal
bas
e sto
ck
(a) Nonseasonal case
0
50
100
150
200
250
300
350
400
Opt
imal
bas
e sto
ck
Time2 3 4 51
1.510.5
(b) Seasonal case
Figure 1: Optimal base stock levels for varying demand uncertainty.
30
35
40
45
50
55
List
regu
lar p
rice
Time2 3 4 51
1.510.5
(a) Nonseasonal case
30
35
40
45
50
55
60
65
70
75
80
54321
List
regu
lar p
rice
Time
1.510.5
(b) Seasonal case
Figure 2: Price for regular service for varying demand uncertainty.
where c.v. is the coefficient for the variability in demand. ๐พ 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
-
8 Mathematical Problems in Engineering
Time2 3 4 51
1.510.5
30
35
40
45
50
55Li
st ex
pres
s pric
e
(a) Nonseasonal case
30
35
40
45
50
55
60
65
70
75
80
54321
List
expr
ess p
rice
Time
1.510.5
(b) Seasonal case
Figure 3: Price for express service for varying demand uncertainty.
0
5
10
15
20
25
1.510.5(a) Nonseasonal case
0
5
10
15
20
25
1.510.5(b) Seasonal case
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
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Mathematical Problems in Engineering 9
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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