# Multiperiodic Procurement Problem with Option Contracts under Inflation

Authors: Nana Wan; Xu Chen
Journal: Mathematical Problems in Engineering
 (2016)

---

## Abstract

This paper studies the problem of the multiperiod replenishment decisions for the retailer under inflation. In order to manage the risks of price and demand caused by inflation, the retailer has an opportunity to order products and purchase options from the supplier in each period. We formulate the multiperiod inventory model for the retailer with option contracts and then derive his myopic ordering policy in each period and his myopic expected total discounted profit over the entire time horizon. By taking the case without option contracts as a benchmark, we explore the effect of option contracts on the retailer’s decisions and performance under inflation. We find that the application of option contracts might induce the retailer to reduce the firm order and increase the total order in each period under inflation. We also find that the application of option contracts might benefit the retailer under inflation.


---

## Body

## 1. Introduction

Since the global financial crisis of 2008, there exists a significant slowdown in the global economy growth and the inflationary pressure has been further emerging. Inflation rate in many countries has stayed above 2% target for an incredibly long time following the crisis. Over the past few years, the global food prices continue to rise and several categories, such as fruit, dairy, and beef, have experienced a dramatic surge. Not only that, but also the increase in the price level of many other commodities remains a strong momentum. Since the pace of wage growth has always lagged behind the pace of price growth, the real incomes of consumers decline and the purchasing power of consumers is eroded. More and more people switch to less-expensive substitutes or buy fewer commodities. Inflation exerts a direct impact on the market price and the consumer demand, which further influence the daily operations of the companies.On the other hand, in order to protect against various risks derived from production, demand, and price, option contracts have been extensively used in many industries such as fashion apparel industry, food processing industry, and automobile industry [1]. It is worth noting that we limit our discussion to call option contracts in this paper. Comparing with other contracts, the advantage for option contracts is that it can provide the buyer with the right, not the obligation, to reorder the items at a prenegotiated price without sacrificing the interest of the seller. In recent years, option contracts are adopted by many famous firms such as Hewlett-Packard [2] and China Telecom [3]. For example, Hewlett-Packard use option contracts for 35% procurement value of inputs. China Telecom use option contracts for more than 100 billion RMB value of goods. Hence, we have good reasons to believe that option contracts are useful for protecting against the risks of price and demand caused by inflation.Until now, inflation characterizes the new normal operating environment. Since the inflationary process takes a relatively long time span, the effect of inflation on the companies is also a long-term effect. Wan and Chen [4] study the problem of the ordering and production decisions for the retailer and the supplier under inflation. They prove that the application of option contracts benefits the supply chain members under inflation. However, they just discuss the single-period situation. Unlike the above paper, our work aims to discuss the multiperiod replenishment problem for the retailer in the presence of option contracts under inflation. We exhibit an important feature that distinguishes from the single-period problem: the retailer needs to adapt the successive orders to respond to the successively observed market demand. Several key questions are addressed in this paper:(i)
With and without option contracts, what is the structure of the optimal ordering policy for the retailer in each period under inflation?(ii)
With and without option contracts, what is the approximate value of the optimal expected total discounted profit for the retailer over the entire time horizon under inflation?(iii)
What effect do option contracts have on the retailer’s decisions under inflation?(iv)
What effect do option contracts have on the retailer’s performance under inflation?The rest of this paper is structured as follows. The related literature review is presented in Section2. Model description and assumptions are given in Section 3. A multiperiod inventory model with option contracts is proposed in Section 4. The effect of option contracts on the retailer’s decisions and performance is discussed in Section 5. A numerical study is presented in Section 6. Conclusions and the possible future work are given in Section 7.


## 2. Literature Review

There exist three main research streams that are mostly related to our study. The first research stream is on enterprise operation management under inflation. Hsieh and Dye [5] analyze the optimal lot size and periodic pricing policies for a deterioration item inventory under inflation, in which partial backlogging is allowed and the demand is price and time dependent. Sarkar et al. [6] propose an EMQ model with a defective production process under inflation, in which the reliability parameter is considered and the demand is time dependent. Mirzazadeh [7] formulates an inventory model with partial backlogging, in which the time horizon is random and the demand is inflation dependent. Mousavi et al. [8] discuss the optimal replenishment solutions for a multi-item multiperiod inventory with constraints and discounts under inflation. Gilding [9] demonstrated that there exists a change on the optimal inventory control policy with time-dependent demand after considering the effect of inflation. Pal et al. [10] build up an EMQ model for deteriorating items under inflation in a fuzzy situation, in which time-dependent demand rate and Weibull distribution deterioration rate are considered. All of the related papers mainly focus on the inventory decision of one single company under inflation. They never consider option contracts.The second research stream is on the use of option contracts. Xu [11] proves that the partners within a supply chain are better off in the presence of option contracts, when there are uncertainties in production, demand, and price. Zhao et al. [12] study the supply chain coordination problem based on the cooperation game method, when the retailer uses option contracts to purchase from the manufacturer. Xia et al. [13] demonstrate that the buyer tends to use firm order contracts under both low and high disruption probability whereas he tends to use option contracts under the moderate disruption probability. Moon and Kwon [14] consider a Nash bargaining game problem for an advertiser-publisher system, in which the advertiser uses option contracts to pay the advertising cost to the publisher. Chen and Shen [3] discuss how to set the parameters of portfolio contracts with options to attain the channel coordination in the presence of a service requirement. Lee et al. [15] study the procurement problem in the presence of option contracts and spot market. They also explore how the capacity constraints and fixed ordering costs influence the optimal solution. Chen et al. [16] discuss the optimal ordering policy for the loss-averse retailer and the optimal production policy for the risk-neutral supplier under option contracts. They also discuss how to design option contracts to achieve the supply chain coordination. Wang and Chen [1] study the joint ordering and pricing problem of a newsvendor with option contracts. All of the related papers mainly focus on the single-period problem under the stochastic demand condition. They never consider the effect of inflation.The third research stream is on multiperiodic inventory/supply chain management. Chao et al. [17] study the joint ordering and pricing policies for a periodic-review inventory, in which random and dependent supply capacities for different periods are taken into account. Cheaitou et al. [18] use the dynamic programming approach to analyze a two-period production/inventory problem, in which slow and fast production modes as well as capacity constraints are considered. Pan et al. [19] discuss the problem of joint ordering and pricing decisions for a dominant retailer who faces a declining price environment in a two-period situation. Chiang [20] explores the optimal order expediting control policy for a continuous-review inventory, in which regular and fast replenishment modes are used. Linh and Hong [21] find that the supply chain coordination can be achieved in a two-period newsboy problem by properly setting the parameters of revenue-sharing contracts, in which the situations of one and two ordering opportunities are discussed, respectively. Chen and Xiao [22] investigate the effect of the midlife and end-of-life policies on the coordination issue of a two-period supply chain, in which the retailer has a single ordering opportunity over the whole selling cycle and the price declines in the middle of the selling cycle. Cheaitou et al. [23] propose a two-period inventory control model with dual supply sources and demand forecast updating. Wang et al. [24] investigate how to coordinate the two-period supply chain through properly setting revenue-sharing contracts in a declining price environment. All of the related papers never consider option contracts. They also never consider the effect of inflation.The published papers that are most closely related to our work are those of Li et al. [25], Chen et al. [26], and Wan and Chen [4], respectively. Li et al. [25] deal with the problem of optimizing the multiperiod ordering and production decisions with a supplementary supply-order opportunity in each period. They obtain the optimal stationary solution for the Nash equilibrium over the planning horizon. Chen et al. [26] propose a multiperiod inventory model, in which two different types of orders, normal and expediting orders, are considered and the demand depends on the inventory level after ordering in that period. They find that the optimal replenishment strategy in each period is of base-stock type and of s,S type without and with fixed order costs. However, the above two papers never consider option contracts and the effect of inflation. Wan and Chen [4] explore the effect of option contracts on the optimal ordering and production decisions for the retailer and the supplier under inflation. They prove that option contracts benefit two members and achieve an efficient channel under inflation. However, the above paper only discusses the single-period problem. Different from the above studies, the main intension of our study is to formulate a multiperiod inventory model in the presence of option contracts under inflation, which can characterize the feature of the long-lasting duration with respect to inflation.


## 3. Model Description and Assumptions

Consider one retailer that manages the periodic-review inventory system over a finite planning horizon withn period under inflation. Let ti denote the length of period i (i=1,2,…,n). As we know, in the classic periodic-review inventory model, the ordering decisions are made at fixed time points and so the length of each period is deterministic. However, this assumption is not realistic in a stochastic environment owing to several reasons such as the variation of costs and changes in technology. In reality, the period length is always stochastic and sometimes exogenous. In this paper, the length of period i is assumed to be an exogenous random variable with probability density gi· and cumulative distribution Gi· within the interval 0,Ti. Thereby, it is concerned with the size of the order in that period.Owing to the effect of inflation, both the retail price and the market demand will change with time during each period. Similar to Jaggi and Khanna [27], we assume that the retail price in period i, denoted by Pi(ti), is represented as Pi(ti)=pierti. Here, pi represents the initial retail price in period i and r represents the inflation rate over the entire time horizon. Moreover, similar to Xiao et al. [28], we assume that the market demand in period i, denoted by diti,ϵi, is represented as diti,ϵi=xiti+ϵi. Here, xiti represents a decreasing function of the period length ti and ϵi represents a stochastic error with probability density fi· and cumulative distribution Fi· within the interval 0,+∞. F0=0, Ex=μ, and F¯x=1-Fx denotes the tail distribution.We describe the sequence of the event as follows: before the beginning of periodi, the retailer has an opportunity to order Qi1 products at unit wholesale price wi and purchase qi1 options at unit purchase price oi according to the initial inventory level before ordering yi1 and the market demand in that period. If the two types of orders are placed, the retailer can obtain the products through the firm order and instantly replenish the inventory level up to zi1 at the beginning of period i. During period i, the retailer can exercise option contracts to obtain some additional items at unit exercise price ei when the realized market demand in that period exceeds the inventory level zi1. If only the options order is placed, the retailer can first use the initial inventory level, namely, the leftover inventory level in the previous period, to meet the market demand in that period. During period i, the retailer can exercise option contracts when the realized market demand in that period exceeds the inventory level yi1. If the two types of orders are not placed, the retailer can only use the initial inventory level to meet the market demand in that period. At the end of period i, the unsatisfied market demand becomes lost sale and the unsold products are held in the inventory at unit holding cost hi. The retailer carries the unsold products to meet the market demand in the next period. At the end of the last period, each unsold product is salvaged at unit salvage value v.We list the notations throughout this paper in the Notations.The retailer is assumed to be in a competitive marketplace and all the market price is exogenous. The lead time of either the firm or options order is assumed to be zero. The retailer has no opportunity to place a supplementary order during each period. Moreover, we make the following assumptions. (1) The first assumption is as follows: hi<αhi+1, ei<αei+1, oi<αoi+1, wi<αwi+1, and pi<αpi+1. These conditions ensure that unit holding cost, unit exercise and purchase price of option, unit wholesale price, and unit initial retail price in period i+1 are higher than those in period i under inflation. They indicate that there exists an increase in the market prices in different periods due to the effect of inflation. (2) The second assumption is as follows: pi>oi+ei>wi>oi. This condition ensures the profit for the retailer in the presence of option contracts in each period. They indicate that the retailer can realize the incomes by exercising option contracts in each period. (3) The third assumption is as follows: hi+wi>αwi+1. This condition ensures that the retailer never orders more products than necessary in period i in order to meet the market demand in period i+1. (4) The fourth assumption is as follows: hn+wn>αv. This condition ensures the retailer never orders infinite products more than necessary in the last period in order to obtain the terminal value.


## 4. Model with Option Contracts

In this section, we plan to propose a multiperiod ordering model with option contracts under inflation. With option contracts, the retailer’s expected profit in periodi, denoted by πri1(zi1,qi1), is(1)πri1zi1,qi1=Epiertimin⁡zi1+qi1,diti,ϵi-wizi1-yi1-oiqi1-eiEmin⁡qi1,diti,ϵi-zi1+-hiEzi1-diti,ϵi+.The first term is the revenue realized by the sales of products in period i. The last four terms are the costs of placing the firm order, placing the options order, exercising the options order, and holding the unsold products in period i, respectively. Let li1=zi1+qi1. Note that determining (zi1,qi1) is equivalent to determining (zi1,li1). Then, (2)πri1zi1,li1=Epiertimin⁡li1,diti,ϵi-wizi1-yi1-oili1-zi1-eiEmin⁡li1-zi1,diti,ϵi-zi1+-hiEzi1-diti,ϵi+.With option contracts, the retailer’s expected total discounted profit over the entire time horizon, denoted by Πr1z11,…,zn1;l11,…,ln1, is (3)Πr1z11,…,zn1;l11,…,ln1=∑i=1nαi-1πri1zi1,li1+αnvyn+11.Thereby, with option contracts, the optimization problem for the retailer can be described as(4)maxli1≥zi1≥yi1,i=1,2,…,n⁡Πr1z11,…,zn1;l11,…,ln1.DefineVi1y1 as the retailer’s maximum expected total discounted profit with option contracts over periods i,…,n, providing yi1=y1 in period i. The terminal value is Vn+11y1=vy1. Since the recursion yi+11=zi1-diti,ϵi holds for the entire time horizon, we can develop the dynamic formulation as follows:(5)Vi1y1=wiy1+maxl1≥z1≥y1⁡Ji1z1,l1,where Vn+11y1=vy1 (y1≥0) and(6)Ji1z1,l1=Hi1z1,l1+αEVi+11z1-diti,ϵi+,Hi1z1,l1=Epiertimin⁡l1,diti,ϵi-wiz1-oil1-z1-eiEmin⁡l1-z1,diti,ϵi-z1+-hiEz1-diti,ϵi+.Here, Ji1z1,l1 represents the retailer’s expected total discounted profit with option contracts over periods i,…,n, when the inventory level is replenished up to z1 after receiving the firm order and the inventory level is replenished up to l1 after receiving the firm order and exercising the options order in period i. Hi1z1,l1 represents the retailer’s expected profit with option contracts in period i, when there is no initial inventory and there is no salvage value.By using the same method, Bellman’s principle of optimality, as Li et al. [25] and Chen et al. [26], we characterize the optimal ordering policy for the retailer with option contracts in each period. Define wn+1=v and Vi1+y1=Vi1y1-wiy1. Then, the following recursion is equivalent to (5)-(6):(7)Vi1+y1=maxl1≥z1≥y1⁡Ji1z1,l1,where Vn+11+y1=0(y1≥0) and(8)Ji1z1,l1=Hi1+z1,l1+αEVi+11+z1-diti,ϵi+,(9)Hi1+z1,l1=Epiertimin⁡l1,diti,ϵi-wiz1-oil1-z1-eiEmin⁡l1-z1,diti,ϵi-z1+-hi-αwi+1Ez1-diti,ϵi+.Here, Hi1+z1,l1 represents the retailer’s expected profit with option contracts in period i, when there is no initial inventory.Lemma 1.
With option contracts, there exist the optimal ordering policies for the retailer over the entire time horizon under inflation.Proof.
See the Appendix.From the above lemma, we find that with option contracts the optimal solution forzi1 in period i, denoted by zi1∗, is characterized by(10)zi1∗=si1∗ifyi1≤si1∗,yi1ifyi1>si1∗and the optimal solution for li1 in period i, denoted by li1∗, is characterized by(11)li1∗=Si1∗ifzi1≤Si1∗,zi1∗ifzi1>Si1∗.Here, si1∗ represents the minimum value of z1 maximizing Ji1z1,l1 and Si1∗ represents the minimum value of l1 maximizing Ji1z1,l1. At this moment, si1∗ is an optimal policy for the retailer’s firm order with option contracts in period i and Si1∗ is an optimal policy for the retailer’s total order with option contracts in period i. That is to say, with option contracts, the optimal firm order policy for the retailer in period i is up to the optimal base-stock level si1∗ and the optimal total order policy for the retailer in period i is up to the optimal base-stock level Si1∗. From (9), we derive that ∂2Ji1z1,l1/∂z1∂wi<0, ∂2Ji1z1,l1/∂z1∂oi>0, and ∂2Ji1z1,l1/∂z1∂ei>0. Then, we conclude that Ji1z1,l1 is submodular with respect to wi and z1, supermodular with respect to oi and z1, and supermodular with respect to ei and z1. Thereby, si1∗ is decreasing in wi and increasing in oi,ei. Similarly, we derive that ∂2Ji1z1,l1/∂l1∂oi<0 and ∂2Ji1z1,l1/∂l1∂ei<0. Then, we conclude that Ji1z1,l1 is submodular with respect to oi and l1 and submodular with respect to ei and l1. Thereby, Si1∗ is decreasing in oi,ei. This proceeding demonstrates that when unit wholesale price in period i tends to increase, the retailer might order more options in that period. When unit purchase and exercise price of option in period i tend to increase, the retailer might order fewer options in that period.As the previous analysis, we find that with option contracts the optimal ordering policy for the retailer in periodi is characterized by two optimal base-stock levels si1∗ and Si1∗. It is not easy to compute the exact value of si1∗ and Si1∗. However, we can calculate the approximate value of them. Define si1+ as the minimum value of z1 maximizing Hi1+z1,l1 and Si1+ as the minimum value of l1 maximizing Hi1+z1,l1. The corresponding base-stock policy maximizes the current profit while ignoring the future, so we call it the myopic policy. Hence, up-to-level si1+ is regarded as a myopic policy for the retailer’s firm order with option contracts in period i and up-to-level Si1+ is regarded as a myopic policy for the retailer’s total order with option contracts in period i. Note that si1∗≤si1+ and Si1∗≤Si1+. With some algorithm, we get the following proposition.Proposition 2.
With option contracts, the myopic firm order quantity for the retailer in periodi, denoted by Qi1+, is (12)Qi1+=si1+-yi1ifyi1≤si1+,0ifyi1>si1+.And the myopic options quantity for the retailer in period i, denoted by qi1+, is (13)qi1+=Si1+-si1+ifyi1≤si1+,Si1+-yi1ifsi1+<yi1≤Si1+,0ifyi1>Si1+,where ∫0TiFi[si1+-xiti]gitidti=oi+ei-wi/ei+hi-αwi+1 and ∫0Tipierti-eiF¯iSi1+-xitigitidti=oi.Proof.
See the Appendix.Note thatSi1+>si1+ is equivalent to oi<wi+hi-αwi+1(∫0Tipierti-eigi(ti)dti/∫0Tipiert+hi-αwi+1gi(ti)dti). This inequality implies that if unit purchase price of option in each period is too high, the retailer will never purchase any number of options.From the above proposition, we find that with option contracts the myopic ordering policy for the retailer in each period depends on the initial inventory level before ordering as well as the two myopic base-stock levelssi1+ and Si1+ in that period. When yi1≤si1+, there exist the myopic solutions with Qi1+=si1+-yi1 and qi1+=Si1+-si1+. This means that it is necessary for the retailer to place the two types of orders before the beginning of period i. In this case, the retailer can instantly replenish the inventory level up to si1+ by receiving the firm order at the beginning of period i. The retailer can obtain the extra items by exercising the options order when the market demand in that period surpasses si1+. At this moment, the myopic firm order quantity in period i is a variable, dependent on the initial inventory level in that period, whereas the myopic options order quantity in period i is a fixed value, independent of the initial inventory level in that period. When si1+<yi1≤Si1+, there exist the myopic solutions with Qi1+=0 and qi1+=Si1+-yi1. This means that it is necessary for the retailer to place the options order, not the firm order, before the beginning of period i. In this case, the retailer first uses the initial inventory level to meet the market demand in that period and then obtain the extra items by exercising the options order when the market demand in that period exceeds yi1. At this moment, the myopic options order quantity in period i is a variable, dependent on the initial inventory level in that period. When yi1>Si1+, there exist the myopic solutions with Qi1+=0 and qi1+=0. This means that it is not necessary for the retailer to place the two types of orders before the beginning of period i. In this case, the retailer can only use the initial inventory level to meet the market demand in that period.Similarly, it is not easy to compute the exact value ofΠr1z11∗,…,zn1∗;l11∗,…,ln1∗. However, the above myopic approach provides a good approximation. The approximate value is precise especially when si1+ and Si1+ are close to si1∗ and Si1∗. With option contracts, the retailer’s myopic expected total discounted profit over the entire time horizon, denoted by Πr1s11+,…,sn1+;S11+,…,Sn1+, is(14)Πr1s11+,…,sn1+;S11+,…,Sn1+=∑i=1nαi-1Hi1+si1+,Si1+.Note that Πr1z11∗,…,zn1∗;l11∗,…,ln1∗≥Πr1s11+,…,sn1+;S11+,…,Sn1+. Then,(15)Πr1s11+,…,sn1+;S11+,…,Sn1+=∑i=1nαi-1Si1+∫0Tipierti-oi-eigitidti-∫0Ti∫0Si1+-xitipierti-eiFiϵigitidϵidtisi1+oi+ei-wi-∫0Ti∫0si1+-xitiei+hi-αwi+1Fiϵigitidϵidti.


## 5. Discussion

### 5.1. Model without Option Contracts

To begin with, we plan to propose a multiperiod ordering model without option contracts under inflation. This model can be used as a benchmark to compare with the case of option contracts. We describe the sequence of the event as follows: before the beginning of periodi, the retailer has an opportunity to order Qi0 products according to the initial inventory level before ordering yi0 and the market demand in that period. If the firm order is placed, the retailer can obtain the products through the firm order and instantly replenish the inventory level up to zi0 at the beginning of period i. If the firm order is not placed, the retailer can only use the initial inventory level to meet the market demand in that period.Without option contracts, the retailer’s expected profit in periodi, denoted by πri0zi0, is(16)πri0zi0=Epiertimin⁡zi0,diti,ϵi-wizi0-yi0-hiEzi0-diti,ϵi+.The first term is the revenue realized by the sales of products in period i. The last two terms are the costs of placing the firm order and holding the unsold products in period i, respectively. Without option contracts, the retailer’s expected total discounted profit over the entire time horizon, denoted by Πr0z10,…,zn0, is(17)Πr0z10,…,zn0=∑i=1nαi-1πri0zi0+αnvyn+10.Thereby, without option contracts, the optimization problem for the retailer can be described as(18)maxzi0≥yi0,i=1,2,…,n⁡Πr0z10,…,zn0.DefineVi0y0 as the retailer’s maximum expected total discounted profit without option contracts over periods i,…,n, providing yi0=y0 in period i. The terminal value is Vn+10y0=vy0 and the recursion is yi+10=zi0-diti,ϵi. We develop the dynamic formulation as follows:(19)Vi0y0=wiy0+maxz0≥y0⁡Ji0z0,where Vn+10y0=vy0(y0≥0) and(20)Ji0z0=Hi0z0+αEVi+10z0-diti,ϵi+,Hi0z0=Epiertimin⁡z0,diti,ϵi-wiz0-hiEz0-diti,ϵi+.Here, Ji0z0 represents the retailer’s expected total discounted profit without option contracts over periods i,…,n, when the inventory level is replenished up to z0 after receiving the firm order in period i. Hi0z0 represents the retailer’s expected profit without option contracts in period i, when there is no initial inventory and there is no salvage value.Similarly, definewn+1=v and Vi0+y0=Vi0y0-wiy0. Then, the following recursion is equivalent to (19)-(20):(21)Vi0+y0=maxz0≥y0⁡Ji0z0,where Vn+10+y0=0(y0≥0) and (22)Ji0z0=Hi0+z0+αEVi+10+z0-diti,ϵi+,(23)Hi0+z0=Epiertimin⁡z0,diti,ϵi-wiz0-hi-αwi+1Ez0-diti,ϵi+.Here, Hi0+z0 represents the retailer’s expected profit without option contracts in period i, when there is no initial inventory.Lemma 3.
Without option contracts, there exist the optimal ordering policies for the retailer over the entire time horizon under inflation.Proof.
See the Appendix.From the above lemma, we find that without option contracts the optimal solution forzi0 in period i, denoted by zi0∗, is characterized by(24)zi0∗=si0∗ifyi0≤si0∗,yi0ifyi0>si0∗.Here, si0∗ represents the minimum value of z0 maximizing Ji0z0. At this moment, si0∗ is an optimal policy for the retailer’s firm order without option contracts in period i. That is to say, without option contracts, the optimal firm order policy for the retailer in period i is up to the optimal base-stock level si0∗. From (23), we derive that ∂2Ji0z0/∂z0∂wi<0. Then, we conclude that Ji0z0 is submodular with respect to wi and z0. Thereby, si0∗ is decreasing in wi. This proceeding demonstrates that when unit wholesale price in period i tends to increase, the retailer might order fewer products in that period. When unit wholesale price in period i tends to decrease, the retailer might order more products in that period.Similarly, definesi0+ as the minimum value of z0 maximizing Hi0+z0. Hence, up-to-level si0+ is regarded as a myopic policy for the firm order without option contracts in period i. Note that si0∗≤si0+. With some algorithm, we get the following proposition.Proposition 4.
Without option contracts, the myopic firm order quantity for the retailer in periodi, denoted by Qi0+, is (25)Qi0+=si0+-yi0ifyi0≤si0+,0ifyi0>si0+,where ∫0Ti(pierti+hi-αwi+1)F¯i[si0+-xiti]gitidti=wi+hi-αwi+1.Proof.
See the Appendix.From the above proposition, we find that without option contracts the myopic ordering policy for the retailer in each period depends on the initial inventory level before ordering as well as the myopic base-stock levelsi0+ in that period. When yi0≤si0+, there exists the myopic solution with Qi0+=si0+-yi0. This means that it is necessary for the retailer to place the firm order before the beginning of period i. In this case, the retailer can instantly replenish the inventory level up to si0+ by receiving the firm order at the beginning of period i. At this moment, the myopic firm order quantity in period i is a variable, dependent on the initial inventory level in that period. When yi0>si0+, there exists the myopic solution with Qi0+=0. This means that it is not necessary for the retailer to place the firm order before the beginning of period i. In this case, the retailer can only use the initial inventory level to meet the market demand in that period.Without option contracts, the retailer’s myopic expected total discounted profit over the entire time horizon, denoted byΠr0s10+,…,sn0+, is(26)Πr0s10+,…,sn0+=∑i=1nαi-1Hi0+si0+.Note that Πr0z10∗,…,zn0∗≥Πr0s10+,…,sn0+. Then,(27)Πr0s10+,…,sn0+=∑i=1nαi-1si0+∫0Tipierti-wigitidti-∫0Ti∫0si0+-xitipierti+hi-αwi+1Fiϵigitidϵidti.


### 5.2. The Effect of Option Contracts

Now, we analyze the effect of option contracts on the retailer’s decisions under inflation and get the following proposition.Proposition 5.
For alli(i=1,2,…,n), si1+<si0+<Si1+.Proof.
See the Appendix.From the above proposition, we find that the application of option contracts has a significant effect on the myopic base-stock levels under inflation. Without option contracts, the retailer needs to guarantee that the inventory level after receiving the firm order in each period is not less thansi0+. With option contracts, the retailer needs to guarantee that the inventory level after receiving the firm order in each period is not less than si1+. In addition, the retailer also guarantees that the inventory level after receiving the firm order and excising the option order is not less than Si1+. Thereby, we conclude that the application of option contracts might prompt the retailer to increase the total order and reduce the firm order in each period under inflation.Now, we analyze the effect of option contracts on the retailer’s performance under inflation and get the following proposition.Proposition 6.
For alli(i=1,2,…,n), Πr1s11+,…,sn1+;S11+,…,Sn1+>Πr0s10+,…,sn0+.Proof.
See the Appendix.From the above proposition, we find that the application of option contracts has a significant effect on the retailer’s myopic expected total discounted profit over the entire time horizon under inflation. We observe that the myopic expected total discounted profit of the retailer over the entire time horizon is greater with option contracts than without under inflation. Thereby, we conclude that the application of option contracts might help the retailer improve the performance under inflation.


## 5.1. Model without Option Contracts

To begin with, we plan to propose a multiperiod ordering model without option contracts under inflation. This model can be used as a benchmark to compare with the case of option contracts. We describe the sequence of the event as follows: before the beginning of periodi, the retailer has an opportunity to order Qi0 products according to the initial inventory level before ordering yi0 and the market demand in that period. If the firm order is placed, the retailer can obtain the products through the firm order and instantly replenish the inventory level up to zi0 at the beginning of period i. If the firm order is not placed, the retailer can only use the initial inventory level to meet the market demand in that period.Without option contracts, the retailer’s expected profit in periodi, denoted by πri0zi0, is(16)πri0zi0=Epiertimin⁡zi0,diti,ϵi-wizi0-yi0-hiEzi0-diti,ϵi+.The first term is the revenue realized by the sales of products in period i. The last two terms are the costs of placing the firm order and holding the unsold products in period i, respectively. Without option contracts, the retailer’s expected total discounted profit over the entire time horizon, denoted by Πr0z10,…,zn0, is(17)Πr0z10,…,zn0=∑i=1nαi-1πri0zi0+αnvyn+10.Thereby, without option contracts, the optimization problem for the retailer can be described as(18)maxzi0≥yi0,i=1,2,…,n⁡Πr0z10,…,zn0.DefineVi0y0 as the retailer’s maximum expected total discounted profit without option contracts over periods i,…,n, providing yi0=y0 in period i. The terminal value is Vn+10y0=vy0 and the recursion is yi+10=zi0-diti,ϵi. We develop the dynamic formulation as follows:(19)Vi0y0=wiy0+maxz0≥y0⁡Ji0z0,where Vn+10y0=vy0(y0≥0) and(20)Ji0z0=Hi0z0+αEVi+10z0-diti,ϵi+,Hi0z0=Epiertimin⁡z0,diti,ϵi-wiz0-hiEz0-diti,ϵi+.Here, Ji0z0 represents the retailer’s expected total discounted profit without option contracts over periods i,…,n, when the inventory level is replenished up to z0 after receiving the firm order in period i. Hi0z0 represents the retailer’s expected profit without option contracts in period i, when there is no initial inventory and there is no salvage value.Similarly, definewn+1=v and Vi0+y0=Vi0y0-wiy0. Then, the following recursion is equivalent to (19)-(20):(21)Vi0+y0=maxz0≥y0⁡Ji0z0,where Vn+10+y0=0(y0≥0) and (22)Ji0z0=Hi0+z0+αEVi+10+z0-diti,ϵi+,(23)Hi0+z0=Epiertimin⁡z0,diti,ϵi-wiz0-hi-αwi+1Ez0-diti,ϵi+.Here, Hi0+z0 represents the retailer’s expected profit without option contracts in period i, when there is no initial inventory.Lemma 3.
Without option contracts, there exist the optimal ordering policies for the retailer over the entire time horizon under inflation.Proof.
See the Appendix.From the above lemma, we find that without option contracts the optimal solution forzi0 in period i, denoted by zi0∗, is characterized by(24)zi0∗=si0∗ifyi0≤si0∗,yi0ifyi0>si0∗.Here, si0∗ represents the minimum value of z0 maximizing Ji0z0. At this moment, si0∗ is an optimal policy for the retailer’s firm order without option contracts in period i. That is to say, without option contracts, the optimal firm order policy for the retailer in period i is up to the optimal base-stock level si0∗. From (23), we derive that ∂2Ji0z0/∂z0∂wi<0. Then, we conclude that Ji0z0 is submodular with respect to wi and z0. Thereby, si0∗ is decreasing in wi. This proceeding demonstrates that when unit wholesale price in period i tends to increase, the retailer might order fewer products in that period. When unit wholesale price in period i tends to decrease, the retailer might order more products in that period.Similarly, definesi0+ as the minimum value of z0 maximizing Hi0+z0. Hence, up-to-level si0+ is regarded as a myopic policy for the firm order without option contracts in period i. Note that si0∗≤si0+. With some algorithm, we get the following proposition.Proposition 4.
Without option contracts, the myopic firm order quantity for the retailer in periodi, denoted by Qi0+, is (25)Qi0+=si0+-yi0ifyi0≤si0+,0ifyi0>si0+,where ∫0Ti(pierti+hi-αwi+1)F¯i[si0+-xiti]gitidti=wi+hi-αwi+1.Proof.
See the Appendix.From the above proposition, we find that without option contracts the myopic ordering policy for the retailer in each period depends on the initial inventory level before ordering as well as the myopic base-stock levelsi0+ in that period. When yi0≤si0+, there exists the myopic solution with Qi0+=si0+-yi0. This means that it is necessary for the retailer to place the firm order before the beginning of period i. In this case, the retailer can instantly replenish the inventory level up to si0+ by receiving the firm order at the beginning of period i. At this moment, the myopic firm order quantity in period i is a variable, dependent on the initial inventory level in that period. When yi0>si0+, there exists the myopic solution with Qi0+=0. This means that it is not necessary for the retailer to place the firm order before the beginning of period i. In this case, the retailer can only use the initial inventory level to meet the market demand in that period.Without option contracts, the retailer’s myopic expected total discounted profit over the entire time horizon, denoted byΠr0s10+,…,sn0+, is(26)Πr0s10+,…,sn0+=∑i=1nαi-1Hi0+si0+.Note that Πr0z10∗,…,zn0∗≥Πr0s10+,…,sn0+. Then,(27)Πr0s10+,…,sn0+=∑i=1nαi-1si0+∫0Tipierti-wigitidti-∫0Ti∫0si0+-xitipierti+hi-αwi+1Fiϵigitidϵidti.


## 5.2. The Effect of Option Contracts

Now, we analyze the effect of option contracts on the retailer’s decisions under inflation and get the following proposition.Proposition 5.
For alli(i=1,2,…,n), si1+<si0+<Si1+.Proof.
See the Appendix.From the above proposition, we find that the application of option contracts has a significant effect on the myopic base-stock levels under inflation. Without option contracts, the retailer needs to guarantee that the inventory level after receiving the firm order in each period is not less thansi0+. With option contracts, the retailer needs to guarantee that the inventory level after receiving the firm order in each period is not less than si1+. In addition, the retailer also guarantees that the inventory level after receiving the firm order and excising the option order is not less than Si1+. Thereby, we conclude that the application of option contracts might prompt the retailer to increase the total order and reduce the firm order in each period under inflation.Now, we analyze the effect of option contracts on the retailer’s performance under inflation and get the following proposition.Proposition 6.
For alli(i=1,2,…,n), Πr1s11+,…,sn1+;S11+,…,Sn1+>Πr0s10+,…,sn0+.Proof.
See the Appendix.From the above proposition, we find that the application of option contracts has a significant effect on the retailer’s myopic expected total discounted profit over the entire time horizon under inflation. We observe that the myopic expected total discounted profit of the retailer over the entire time horizon is greater with option contracts than without under inflation. Thereby, we conclude that the application of option contracts might help the retailer improve the performance under inflation.


## 6. Numerical Example

In this section, we plan to use several groups of numerical examples to show the effect of inflation and the effect of option contracts on the retailer’s decisions and performance in a multiperiod situation. Similar to Wan and Chen [4], we assume that the function of the nonstochastic part xiti declines exponentially; that is, xiti=λie-δiti. Here, λi is the initial market demand in period i and δi is the demand contraction factor in period i. We study the two-period case and the corresponding parameters are as follows: λ1=250, λ2=150, t1~U0,10, t2~U0,10, ϵ1~U0,20, ϵ2~U0,40, p1=10, p2=15, w1=4, w2=5.5, o1=0.8, o2=2.3, e1=3.5, e2=5, h1=2, h2=2.8, v=4.5, and α=0.8.We show the effect ofγ on the retailer’s decisions in Table 1.Table 1
The effect ofγ on the retailer’s decisions.
γ
s
1
1
+
S
1
1
+
s
2
1
+
S
2
1
+
s
1
0
+
s
2
0
+
0.0030
254.209
266.339
165.653
179.503
264.618
176.240
0.0035
254.209
266.347
165.653
179.535
264.630
176.272
0.0040
254.209
266.354
165.653
179.567
264.642
176.304
0.0045
254.209
266.362
165.653
179.598
264.654
176.336
0.0050
254.209
266.370
165.653
179.630
264.666
176.368From Table1, we find that when there exists an obvious increase in γ, the myopic base-stock levels S11+ and S21+ tend to increase while the myopic base-stock levels s11+ and s21+ tend to remain unchanged. This means that with option contracts the increase in γ might cause the retailer to raise the total order and maintain the firm order in the first and second period under inflation. When there exists an obvious increase in γ, the myopic base-stock levels s10+ and s20+ also tend to increase. This means that without option contracts the increase in γ might cause the retailer to raise the total order, namely, the firm order, in the first and second period under inflation.We show the effect ofγ on the retailer’s performance in Table 2.Table 2
The effect ofγ on the retailer’s performance.
γ
Π
r
1
s
1
1
+
,
s
2
1
+
;
S
1
1
+
,
S
2
1
+
Π
r
0
s
1
0
+
,
s
2
0
+
0.0030
3165.048
3158.180
0.0035
3178.048
3171.140
0.0040
3191.098
3184.150
0.0045
3204.178
3197.200
0.0050
3217.318
3210.300From Table2, we also find that when there exists an obvious increase in γ, the myopic expected total discounted profits Πr1s11+,s21+;S11+,S21+ and Πr0s10+,s20+ tend to increase. This means that with and without option contracts the increase in γ might cause the retailer to obtain more profit under inflation.We show the effect ofδ1 on the retailer’s decisions in Table 3.Table 3
The effect ofδ1 on the retailer’s decisions.
δ
1
s
1
1
+
S
1
1
+
s
2
1
+
S
2
1
+
s
1
0
+
s
2
0
+
0.0010
254.209
266.339
165.653
179.503
264.618
176.240
0.0015
253.589
265.715
165.653
179.503
263.994
176.240
0.0020
252.971
265.092
165.653
179.503
263.372
176.240
0.0025
252.355
264.472
165.653
179.503
262.752
176.240
0.0030
251.742
263.854
165.653
179.503
262.135
176.240From Table3, we find that when there exists an obvious increase in δ1, the myopic base-stock levels S11+ and s11+ tend to decrease. This means that with option contracts the increase in δ1 might cause the retailer to reduce the total order and the firm order simultaneously in the first period under inflation. When there exists an obvious increase in δ1, the myopic base-stock level s10+ tends to decrease. This means that without option contracts the increase in δ1 might cause the retailer to reduce the total order, namely, the firm order, in the first period under inflation.We show the effect ofδ1 on the retailer’s performance in Table 4.Table 4
The effect ofδ1 on the retailer’s performance.
δ
1
Π
r
1
s
1
1
+
,
s
2
1
+
;
S
1
1
+
,
S
2
1
+
Π
r
0
s
1
0
+
,
s
2
0
+
0.0010
3165.048
3158.180
0.0015
3161.288
3154.220
0.0020
3157.498
3150.220
0.0025
3153.678
3146.190
0.0030
3149.838
3142.130From Table4, we also find that when there exists an obvious increase in δ1, the myopic expected total discounted profits Πr1s11+,s21+;S11+,S21+ and Πr0s10+,s20+ tend to decrease. This means that with and without option contracts the increase in δ1 might cause the retailer to obtain less profit under inflation.We show the effect ofδ2 on the retailer’s decisions in Table 5.Table 5
The effect ofδ2 on the retailer’s decisions.
δ
2
s
1
1
+
S
1
1
+
s
2
1
+
S
2
1
+
s
1
0
+
s
2
0
+
0.0020
254.209
266.339
165.653
179.503
264.618
176.240
0.0025
254.209
266.339
165.283
179.131
264.618
175.869
0.0030
254.209
266.339
164.915
178.760
264.618
175.498
0.0035
254.209
266.339
164.548
178.391
264.618
175.130
0.0040
254.209
266.339
164.182
178.022
264.618
174.762From Table5, we find that when there exists an obvious increase in δ2, the myopic base-stock levels S21+ and s21+ tend to decrease. This means that with option contracts the increase in δ2 might cause the retailer to reduce the total order and the firm order simultaneously in the second period under inflation. When there exists an obvious increase in δ2, the myopic base-stock level s20+ tends to decrease. This means that without option contracts the increase in δ2 might cause the retailer to reduce the total order, namely, the firm order, in the second period under inflation.We show the effect ofδ2 on the retailer’s performance in Table 6.Table 6
The effect ofδ2 on the retailer’s performance.
δ
2
Π
r
1
s
1
1
+
,
s
2
1
+
;
S
1
1
+
,
S
2
1
+
Π
r
0
s
1
0
+
,
s
2
0
+
0.0020
3165.048
3158.180
0.0025
3162.038
3154.490
0.0030
3159.038
3150.810
0.0035
3156.038
3147.110
0.0040
3153.028
3143.420From Table6, we also find that when there exists an obvious increase in δ2, the myopic expected total discounted profits Πr1s11+,s21+;S11+,S21+ and Πr0s10+,s20+ tend to decrease. This means that with and without option contracts the increase in δ2 might cause the retailer to obtain less profit under inflation.From all the tables, we find thats10+ is always higher than s11+ but lower than S11+, except that s20+ is always higher than s21+ but lower than S21+. This means that the application of option contracts might prompt the retailer to reduce the firm order and increase the total order under inflation. We also find that Πr1s11+,…,sn1+;S11+,…,Sn1+ is always greater than Πr0s10+,…,sn0+. This means that the application of option contracts might help the retailer obtain more profit under inflation.


## 7. Conclusions and Suggestions for Further Research

This paper studies the multiperiod inventory problem for one retailer, who operates the inventory under a periodic review over a finite planning horizon, in the presence of option contracts under inflation. Owing to the effect of inflation, the retailer has to face the rising price and the shrinking demand. To manage the risks of price and demand caused by inflation, the retailer has an opportunity to order products and purchase options from the supplier in each period. To the best of our knowledge, our study is among the first efforts to integrate option contracts and the effect of inflation into a multiperiod modeling framework. This work provides several interesting observations.Observation 1.
We demonstrate that there exist the optimal ordering policies for the retailer with and without option contracts under inflation. By characterizing the relationship between the initial inventory level and the optimal base-stock levels in each period, we also exhibit the expressions of the optimal ordering policy in each period with and without option contracts under inflation.Observation 2.
We derive the myopic ordering policy for the retailer in each period with and without option contracts under inflation. We also derive the myopic expected total discounted profit of the retailer over the entire time horizon with and without option contracts under inflation. Our analysis provides scientific suggestion for companies on how to make appropriate replenishment decisions to maximize the profit under inflation.Observation 3.
Compared with the case without option contracts, we prove that the application of option contracts might cause the retailer to reduce the firm order and increase the total order in each period under inflation. We also prove that the application of option contracts might help the retailer improve the performance under inflation. Our analysis provides scientific suggestion for the companies on how to select the appropriate contract type to enhance the profit under inflation.In the near future, we plan to extend this study in the following directions: (1) we plan to consider the case of price-dependent demand and analyze the joint optimal ordering and pricing policies in a multiperiod setting. (2) We plan to formulate this problem as a multi-inventory game with different market power, such as supplier Stackelberg, retailer Stackelberg, and vertical Nash [29, 30]. (3) We plan to consider the risk preference of the retailer, such as the case of loss aversion.



---

*Source: 4041871-2016-03-07.xml*