Competitive analysis of the online inventory problem

doi:10.1016/j.ejor.2010.05.019

European Journal of Operational Research

Volume 207, Issue 2, 1 December 2010, Pages 685-696

https://doi.org/10.1016/j.ejor.2010.05.019 Get rights and content

Abstract

We consider a real-time version of the inventory problem with deterministic demand in which decisions as to when to replenish and how much to buy must be made in an online fashion without knowledge of future prices. We suggest online algorithms for each of four models for the problem and use competitive analysis to obtain algorithmic upper and lower bounds on the worst-case performance of the algorithms compared to an optimal offline algorithm. These bounds are closely related to the tight $\sqrt{M / m}$ -bound obtained for the simplest of the models, where M and m are the upper and lower bounds on the price fluctuation.

Introduction

We consider an inventory problem where the procurement price of a commodity is uncertain. In particular, we consider a set-up where the price fluctuates on a day to day basis, and decisions as to when and how much to buy have to be made in an online fashion, i.e., without any knowledge of future prices. Such problems often arise in dealing with raw materials. Our goal is to construct online algorithms to make decisions in such an environment and to prove bounds on their worst-case performance. We compare our algorithms to an optimal offline algorithm using competitive analysis. In this way, we get a measure of the cost obtained by an algorithm compared to the optimal cost that could have been obtained if we had known all future prices in advance. This can be thought of as measuring the value of the information of future prices.

Deterministic inventory models and models with stochastic demand or stochastic lead time have been extensively studied in the literature; see [26], [32] for an overview. Models that takes into account the uncertainty of the various cost parameters are more rare. We give examples of such models below.

Ben-Daya and Hariga [3], Chaudhuri and Ray [9], and Horowitz [20] take inflation into account. Akőz et al. [1] and Petrović et al. [27], among others, use fuzzy sets to model uncertainty of various cost parameters. Chaouch [8], Gurnani [18], and Moinzadeh [25] study models with two prices, where the lower price occurs at random points of time. Gurnani and Tang [19] consider a nested news vendor model with price uncertainty. Goel and Gutierrez [14] use Brownian motions to model two stochastic price streams based on different markets, such as spot markets and future contracts.

A number of papers study the problem of stochastic procurement price. Golabi [16] and Berling [4] study the problem where the price is drawn from a known distribution function. Wang [30] studies a set-up where prices are stochastically decreasing over time and demand is stochastic. Kingsman [23] studies the problem when demand is known and Kalymon [21] studies the problem with price dependency on previous prices where also demand is uncertain. Kouvelis and Li [24] analyze supply contracts in a model with stochastic prices.

In the next section, we introduce four inventory management models. Then we discuss the analytic methods we are using and present our results. In the following four sections, we treat the different models, after which we conclude.

Section snippets

Inventory management models

In a deterministic setting with constant price and constant withdrawal from the inventory, it is well known that the Economic Order Quantity (EOQ) is an optimal strategy [31]. The optimal order size is obtained by balancing the inventory storage costs on one side and the costs of placing orders on the other side. Doing that, it can be verified that the optimal order size is $Q = \sqrt{\frac{2 \times order cost \times demand}{inventory cost per item per time unit}} .$

We consider models which can be seen as variations of the EOQ model. We

Analytic methods and results

The aim is to design online algorithms with the best possible worst-case guarantees. We use the well-established technique of competitive analysis [17], [22], [29] to evaluate our design. An introduction to the technique can be found in [5]. Intuitively the idea is to compare the result obtained by an online algorithm to the result that could have been obtained if one had known all future prices in advance, where the latter scenario is represented by an optimal offline algorithm.

The basic idea

Bounded Storage Model

The following algorithm is inspired by the simple reservation-price-policy from the introduction and results in a significantly better competitive ratio than obtained through the naive strategy.Bounded Storage Algorithm (BA). Let $p^{*} = \sqrt{mM}$ . If at time t, we have p_t ⩽ p∗, order Q_t = U − L_t items. Otherwise, if p_t > p∗ and L_t = 0, order one item.

Thus, when the price is better than p∗, we fill up to capacity, and otherwise we buy just enough to satisfy the removal from the inventory at the current time.

Fig. 2

Bounded storage order cost model

Now, we add the order cost parameter to the previous model.

Let $k = \frac{2 - \frac{\sqrt{\frac{M}{m}}}{U} + \sqrt{\frac{M}{{mU}^{2}} + \frac{4 \sqrt{\frac{M}{m}}}{U} + 8}}{2 + \frac{4 \sqrt{\frac{M}{m}}}{U}}$ and $b = \frac{U}{k \sqrt{\frac{M}{m}}}$ .Bounded storage order cost algorithm (BOA). Assume that $U ⩾ (\sqrt{2} - 1) \sqrt{\frac{M}{m}}$ . Let $p^{*} = \sqrt{Mm}$ . If at time t, we have p_t ⩽ p∗ and L_t ⩽ U − b, order Q_t = U − L_t, i.e., fill up to capacity. If p_t > p∗ and L_t < 1, order up to level b.

Note that we must have b ⩾ 1, such that any given order will contain at least one item in order for the algorithm to be well-defined. This is equivalent to requiring that $\frac{\sqrt{\frac{M}{m}}}{U} ⩽ \frac{1}{k}$ . We consider the following

Unbounded Storage Model

We replace the maximum inventory level by an inventory holding cost per item per time unit, h.Unbounded storage algorithm (UA). Let $p^{*} = \sqrt{(m + \frac{h}{2}) (M + \frac{h}{2})} + \frac{h}{2}$ and for any time t, let $Q_{t}^{'} = 2 \frac{p^{*} - p_{t}}{h} - 2 L_{t} + 1$ and order $Q_{t} = \{\begin{matrix} Q_{t}^{'}, & if Q_{t}^{'} > 0 and L_{t} + Q_{t}^{'} ⩾ 1, \\ 1 - L_{t}, & if L_{t} + \max {0, Q_{t}^{'}} < 1, \\ 0, & otherwise . \end{matrix}$

The intuition behind UA is as follows. UA replenish in two situations: When the price and the inventory level is relatively low, such that the average cost of items bought are below a pre-specified threshold value, and when it is necessary

Unbounded storage order cost model

We add an order cost parameter to the model of the previous section.Unbounded storage order cost algorithm (UOA). Let $b = \frac{\sqrt{\frac{S}{h}}}{\sqrt{\frac{M}{m}}} + 1, p^{*} = \sqrt{Mm} + h (\frac{\sqrt{\frac{S}{h}}}{\sqrt{\frac{M}{m}}} + 1)$ , and for any time t, let $Q_{t}^{'} = 2 \frac{p^{*} - p_{t}}{h} - 2 L_{t} + 1$ and order $Q_{t} = \{\begin{matrix} Q_{t}^{'}, & if Q_{t}^{'} ⩾ b, \\ b - L_{t}, & if Q_{t}^{'} < b and L_{t} < 1, \\ 0, & otherwise . \end{matrix}$

The algorithm is illustrated in Fig. 7. In the algorithm there are two situations: In Situation 1, we replenish because it is attractive to do so due to the current price and inventory level. In Situation 2, we replenish because we are forced to do so due to

Concluding remarks

This is a first attempt at establishing results for an online version of the inventory problem. We have obtained tight results on the competitive ratio of online algorithms for the simpler model whereas the gap between the established upper and lower bounds of the algorithms grows with the complexity of the models.

In analyzing the algorithms involving order cost, because of the complexity of the calculations, we are bounding the competitive ratios for order and item costs separately and then

Acknowledgments

The authors thank the anonymous referees for many insightful and constructive comments which have improved the presentation of the results. This online inventory problem was first brought up at the 4th NOGAPS, and we would like to thank the participants in that meeting for interesting initial discussions.

References (32)

J. Boyar et al.
The maximum resource bin packing problem
Theoretical Computer Science
(2006)
H. Gurnani
Optimal ordering policies in inventory systems with random demand and random deal offerings
European Journal of Operational Research
(1996)
I. Horowitz
EOQ and inflation uncertainty
International Journal of Production Economics
(2000)
Y. Wang
The optimality of myopic stocking policies for systems with decreasing purchasing prices
European Journal of Operational Research
(2001)
G.Y. Tűtűncű et al.
Continuous review inventory control in the presence of fuzzy costs
International Journal of Production Economics
(2008)
Baruch Awerbuch, Yossi Azar, Amos Fiat, Tom Leighton, Making commitments in the face of uncertainty: how to pick a...
M. Ben-Daya et al.
Optimal time varying lot-sizing models under inflationary conditions
European Journal of Operational Research
(1996)
P. Berling
The capital cost of holding inventory with stochastically mean-reverting purchase price
European Journal of Operational Research
(2007)
A. Borodin et al.
Online Algorithms and Competitive Analysis
(1998)
J. Boyar et al.
The seat reservation problem
Algorithmica
(1999)

B.A. Chaouch

Inventory control and periodic price discounting campaigns

Naval Research Logistics

(2006)

K.S. Chaudhuri et al.

An EOQ model with stock-dependent demand, shortage, inflation and time discounting

International Journal of Production Economics

(1997)

R. El-Yaniv

Competitive solutions for online financial problems

ACM Computing Surveys

(1998)

R. El-Yaniv, A. Fiat, R.M. Karp, G. Turpin, Competitive analysis of financial games, in: IEEE Symposium on Foundations...

R. El-Yaniv et al.

Optimal search and one-way trading online algorithms

Algorithmica

(2001)

L. Epstein et al.

Online scheduling of splittable tasks in peer-to-peer networks

ACM Transactions on Algorithms

(2006)

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¹: Supported in part by the Danish Natural Science Research Council. Part of this work was carried out while this author was visiting the University of California, Irvine.

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Production, Manufacturing and LogisticsCompetitive analysis of the online inventory problem

Abstract

Introduction

Section snippets

Inventory management models

Analytic methods and results

Bounded Storage Model

Bounded storage order cost model

Unbounded Storage Model

Unbounded storage order cost model

Concluding remarks

Acknowledgments

Theoretical Computer Science

European Journal of Operational Research

International Journal of Production Economics

European Journal of Operational Research

Continuous review inventory control in the presence of fuzzy costs

International Journal of Production Economics

Optimal time varying lot-sizing models under inflationary conditions

European Journal of Operational Research

The capital cost of holding inventory with stochastically mean-reverting purchase price

European Journal of Operational Research

Online Algorithms and Competitive Analysis

The seat reservation problem

Algorithmica

Inventory control and periodic price discounting campaigns

Naval Research Logistics

An EOQ model with stock-dependent demand, shortage, inflation and time discounting

International Journal of Production Economics

Competitive solutions for online financial problems

ACM Computing Surveys

Optimal search and one-way trading online algorithms

Algorithmica

Online scheduling of splittable tasks in peer-to-peer networks

ACM Transactions on Algorithms

Production, Manufacturing and Logistics
Competitive analysis of the online inventory problem