# Optimal Replenishment Strategy for Inventory Mechanism with a Known Price Increase and Backordering in Finite Horizon

Authors: Lixia Zhang; Bo Feng
Journal: Mathematical Problems in Engineering
 (2022)

---

## Abstract

For the finite horizon inventory mechanism with a known price increase and backordering, based on minimizing the inventory cost, we establish two mixed integer optimization models. By buyer’s cost analysis, we present the closed-form solutions to the models, and by comparing the minimum cost of the two strategies, we provide an optimal ordering policy to the buyer. Numerical examples are presented to illustrate the validity of the model, and sensitivity analysis on major parameters is also made to show some insights to the inventory model.


---

## Body

## 1. Introduction

Generally, the prices of grain crops such as wheat, corn, and soybeans are low during the first few months after harvest, and the prices will increase thereafter till the next harvest season, and so on. In view of the situation, the buyer should decide the ordering policy so that his inventory cost is as low as possible. This problem can also be stated as that a supplier announces an impending price increase in the future, and the buyer should decide whether to purchase additional stock before the price increase. Further, since wheat, corn, and soybeans are harvested once a year, the buyer’s inventory planning time is generally one year. In this paper, we consider the inventory problem with backordering in a finite horizon in which the seller announces the price will increase permanently and the buyer can have a special order before it happens.The inventory model with known price increase has been frequently discussed in the literature. Possibly, the first work performed in this field was done by Naddor [1] and Brown [2]. They assumed that the price of goods would increase in the future and a buyer would have an opportunity to make a special order forehand. Taylor and Bradley [3] extended the model and obtained the optimal ordering strategies for situations where the price increase does not coincide within the end of an EOQ cycle. Erel [4] investigated the effects of continuous changes of the purchasing price and holding cost on the optimal order quantity and annual cost. Tersine [5] proposed an economic production quantity model under an announced price increase. Shah [6] developed a discrete-time stochastic inventory model for perishable items when the vendor announces a price increase at some future time. Huang et al. [7] considered an infinite horizon deterministic inventory model with an announced price increase which assumes that the special order is an integral multiple of the new EOQ quantity. Abad [8] considered a supply chain model in which the producer considers temporary reduction and increase in unit purchasing cost separately and the buyer places a special order in both situations. Chung et al. [9] investigated the buyer’s selling policy in response to the future price increase for deteriorating items when the buyer can make a special order before the price increase happens. Ouyang et al. [10] explored the possible effects of price increases on a retailer’s replenishment policy when the special order quantity is limited and the rate of deterioration of the goods is assumed to be constant. Wang et al. [11, 12] considered an inventory mechanism with a nondeterministic short-term price discount.For the basic EOQ model with backordering of demand during stockout periods, Ghosh [13] presented an infinite horizon deterministic inventory model that handles inventory shortages under an announced price increase. Sharma [14] developed a production inventory model with partial backordering in which the suppliers can increase or temporarily decrease the prices and the buyers can make a special order. A comprehensive survey on this was made by Pentico and Drake [15]. Taleizadeh et al. [16] considered an EOQ problem with backordering under partial delayed payment.For the finite horizon deterministic inventory model, Lev and Soyster [17] considered the inventory model on determining the optimal ordering policies based on known information about imminent price increase without shortages. Arcelus, Pakkala, and Srinivasan [18] showed the potential differences between the profit-maximizing and the cost-minimizing solutions to the deterministic finite horizon inventory problem, and Arcelus et al. [19] considered the inventory model over a finite horizon under one-time cost changes. Lev and Weiss [20] considered the extensive inventory models with cost changes for a finite horizon and an infinite horizon.Furthermore, for the basic EOQ model that considers both price changes and backordering, Taleizadeh and Pentico et al. [21] extended classic economic order quantity (EOQ) model with partial backordering in an infinite horizon, in which the supplier announces the price will increase permanently and the retailer can have a special order before it happens. Taleizadeh et al. [22] extended the situation in [21] to probabilistic replenishment intervals further. Taleizadeh et al. [23] considered the rate of demand in the model was assumed proportional to the unit purchasing cost and partial backordering was allowed as a fixed parameter. However, the length of the inventory system operating time is infinite in these articles.This paper considers the inventory model with backordering in finite horizon, in which a permanent price increase will take place at or before buyer’s next scheduled ordering time, and the buyer may have a special order before that. For this set, to maximize the buyer’s inventory profit, the buyer should make a tradeoff between enjoying the benefit of the low ordering price and bearing the increase of inventory holding cost lead by the special order. On the basis of the inventory cost analysis, we establish an optimization inventory model based on minimizing the inventory cost and derive a closed-form solution to the model. Some numerical experiments are made to illustrate the validity of the model.The remainder of the paper is organized as follows. Section2 presents the assumptions of our model and notations used in the subsequent analysis. Section 3 considers an inventory model with backordering and without price increase in a finite planning horizon and give the optimal solution which will be used in solving the proposed model. In Section 4, we derive a global optimal solution for our concerned inventory models. We derive a solving algorithm and thus provide an optimal replenishment policy to the buyer in Section 5. Numerical experiments on sensitivity analysis are given in Section 6 to show the validity of the models. The conclusions and some extensions are given in the last section.


## 2. Notations, Assumptions, and Problem Formulation

First, we present the notations and assumptions used in this paper (Table1).Table 1
Notation.
ParametersDescriptionsλDemand rateAFixed order costc0Regular purchasing pricecPurchasing price after increaseh0Regular holding cost per item per unit timehHolding cost per item per unit time under increased pricewBackorder cost per unit item per unit timeTPlanning horizon of the inventory systemt0The time when purchasing price changesq0Remaining stock at timet0Q0Order size beforet0B0Maximum shortage level for a normal order at the current price beforet0Decision variablesQsSpecial order sizeQOrder size aftert0 when the special order is not placedQ′Order size after special orderBsMaximum shortage level for a special order at the current priceBMaximum shortage level for an order at the new price aftert0 when the special order is not placedB′Maximum shortage level for an order at the new price when a special order is placedOther variablesFInventory cost over intervalt0,T without special orderFsInventory cost over intervalt0,T with special orderThe followings are the assumptions imposed on the concerned inventory model.Assumption 1.
For the inventory system, we assume that(1)
The time horizon is finite;(2)
The leading time for each order is zero;(3)
Shortage and backordered are allowed except for the last replenishment cycle;(4)
The fixed ordering cost and the unit backordering cost for regular order are, respectively, the same as those for special order;(5)
The selling price does not change;(6)
The increase of purchasing price occurs at or before the retailer’s next ordering time, which means that there is no opportunity for a regular replenishment before the price increase;(7)
When price increases, it will last to the end of the inventory mechanism;(8)
The stock level is zero at the beginning and end of the inventory system.
Without loss of generality, we assume that the holding cost per item per unit time is proportional with the purchasing cost. Then in the following, we takeh=h0+0.4c−c0/c0.
From the assumptions on the concerned models, we know that the planning horizon consists of two stages0,t0 and t0,T, where the ordering price c0 in the first stage is strictly less than c>c0 in the second stage. When the supplier announces the increase of the purchasing price at t0, there are two possible strategies by the buyer: the buyer places a special order under the current lower price before it increases to cut the cost, or the buyer does not place a special order if the remaining stock level at t0 is too high or small price increase.Strategy 1.
A special order is not placed att0.
If the buyer does not place a special order att0, he will use a “new” inventory mechanism with backordering in a finite horizon based on the increased price after the current price cycles end.Strategy 2.
A special order is placed att0.
If the buyer adopts this strategy, he will use a “new” inventory mechanism with backordering in a finite horizon based on the increased price after special order cycle ends too.
Since we suppose the unit selling price does not change and the stock level is zero at the beginning and the end of the inventory system, so the method we used is computing the difference of the minimal total costs under two strategies and choosing a strategy with the least cost to save more costs.
In order to make the solution process of solving this problem clearer, we will first provide a solution method for an inventory model with backordering in finite horizon without price increase in Section3 then exploit these conclusions for giving a solution method to the problem in Section 4.


## 3. Solution Method for Inventory Model with Backordering in a Finite Horizon

In this section, we mainly consider the inventory mechanism with backordering in a finite horizon of lengthT and without purchasing price variation. Also, for the need discussed in the next section, we assume that the initial stock is q, and the stock is zero at the end of the inventory mechanism. See Figures 1 and 2.Figure 1
Ordering policy forq≥0.Figure 2
Ordering policy forq<0.For this setting, suppose thatk orders are placed in the t0,T with order sizes Q1,Q2,⋯,Qk, and the maximum shortage levels at replenishment cycles are B1,B2,⋯,Bk, respectively. Since the demand rate is invariant, shortage is fully backordered, and the stock is zero at the end of the inventory mechanism i.e.Bk=0, so(1)Q1+Q2+⋯+Qk=λT−q.From the knowledge of inventory management [24], we can obtain the total operating cost over the planning horizon T(2)Fk,Q1,…,Qk,B1,…,Bk−1=kA+c∑i=1kQi+h2λQ1+q2+h2λ∑i=2kQi−Bi−12+w2λ∑i=1k−1Bi2=cλT−q+kA+h2λQ1+q2++h2λ∑i=2kQi−Bi−12+w2λ∑i=1k−1Bi2.To minimize the buyer’s operating cost, we only need to determine the optimal order timesk, the sizes Qii=1,…,k, and the maximum shortage levels Bii=1,…,k−1, which can be formulated as the following optimization problem:(3)mink,Qi,BiFk,Q1,…,Qk,B1,…,Bk−1s.t.Q1+Q2+⋯+Qk=λT−q,Qi≥0,i=1,…,k,Bi≥0,i=1,…,k−1,kisapositive integer.This is a nonlinear mixed integer optimization problem. For this problem, we have the following conclusion.Theorem 1.
For an inventory model with backordering in finite horizon, i.e., inventory model (3), all the optimal maximum shortage levels are the same except the last one, and all the optimal ordering sizes are the same except the first one, i.e.,(4)Q2=⋯=Qk≜Q,B1=⋯=Bk−1≜B,Q1=Q−B−q.Furthermore, the optimal ordering size, maximum shortage level, and the optimal order time are, respectively,(5)Q=w+hλTkw+k−1h,B=hλTkw+k−1h,k=Twλh2Aw+h+hw+horTwλh2Aw+h+hw+h,and the minimum inventory cost within the horizon 0,T is(6)F=cλT−q+kA+whλT22kw+k−1h.Proof.
For completeness, we include the proof of the lemma in Appendix A.
We will exploit the conclusion of Theorem1 repeatedly in the remainder of this paper. For convenience, the optimal ordering policy given in Theorem 1 for the inventory mechanism with backordering in the finite horizon and without purchasing price variation is denoted by “BEOQ” in the subsequent sections.


## 4. Solution Method for Inventory Model with a Known Price Increase and Backordering in Finite Horizon

For this inventory system, there are two ordering strategies according to whether a special order is placed or not att0, as shown in Figures 3 and 4. If the buyer places a special order at t0, the special order cycle will end at t2, and the inventory cycles based on the current lower price end at t1 if the buyer does not place a special order. It is easy to calculate that t1=t0+q0+B0/λ,t2=t0+q0+Qs+Bs/λ. For convenience, we denote T0=T−t0,T1=T−t1, and T2=T−t2.Figure 3
Ordering policy forq0≥0.Figure 4
Ordering policy forq0<0.Since shortages are allowed, two possible scenarios may occur att0: the price increase for the item occurs when there is still inventory or there is a shortage, that is:Scenario 1:q0≥0, and Scenario 2: q0<0.In the subsequent analysis, we will first discuss the minimum inventory costs of the two strategies, then compare the difference in the two minimum cost, and finally, choose the optimal ordering strategy.


### 4.1. Strategy 1: A Special Order Is Not Placed att0

For this strategy, when the inventory cycles under the current price end att1, the buyer will continue using a new BEOQ order policy till the end of the inventory system, so that the inventory cost over the interval t0,T consists of inventory cost over two intervals t0,t1 and t1,T.Scenario 1.
q0≥0
Under this scenario, the price increases when there is no shortage, and the remaining stockq0 at t0 will be postponed to the second stage. By the knowledge of inventory control, we can calculate that the inventory cost during t0,t1 is(7)f1=h02λq02+w2λB02.
Since the stock level is−B0 at t1, the inventory system will run a new “BEOQ” order policy during the horizon t1,T under the new price c, and the stock level is −B0 and zero at the beginning and end of the planning horizon, respectively; see Figures 3 and 4.
For this scenario, when the special order is not placed att0, we denote the inventory cost during this period from t0 to the end by F, which is a function of order quantity Q1,…,Qn and maximum shortage level B1,…,Bn−1 for an order after t0. Using an approach similar to that used and according to formula (2) in Section 3, then(8)gn,Q1,…,Qn,B1,…,Bn−1=nA+c∑i=1nQi+h2λQ1−B02+h2λ∑i=2nQi−Bi−12+w2λ∑i=1n−1Bi2=cλT+B0+nA+h2λQ1−B02+h2λ∑i=2nQi−Bi−12+w2λ∑i=1n−1Bi2.
So the ordering policy during the horizont1,T of the concerned model can be formulated as the following optimization model:(9)minn,Qi,Bign,Q1,…,Qn,B1,…,Bn−1,s.t.Q1+Q2+⋯+Qn=λT1+B0,Qi≥0,i=1,…,n,Bi≥0,i=1,…,n−1,nis a positive integer.
According to formulas (5) and (6) of Theorem 1 in Section 3, we know that all the optimal ordering sizes are the same except the first one, all the optimal maximum shortage levels are the same, and the optimal ordering size, maximum shortage level, and the optimal order time over the interval t1,T, respectively, are(10)Q=w+hλT1nw+n−1h,B=hλT1nw+n−1h,(11)n=T1wλh2Aw+h+hw+h≜n1orT1wλh2Aw+h+hw+h≜n2,and the minimum inventory cost during the interval t1,T is(12)g1=cλT1+B0+n1A+whλT122n1w+n1−1h,or(13)g2=cλT1+B0+n2A+whλT122n2w+n2−1h.
Ifg1≤g2, then the optimal ordering times and the minimum inventory cost during the interval t1,T are n∗=n1 and g1, respectively; otherwise, they are n∗=n2 and g2. Denote the minimum inventory cost during horizon t1,T by(14)g∗=cλT1+B0+n∗A+whλT122n∗w+n∗−1h.
In summary, under strategy 1, if the remaining inventory stock att0 is nonnegative, the minimum inventory cost over the horizon t0,T is(15)F∗=f1+g∗=cλT1+B0+n∗A+h02λq02+w2λB02+whλT122n∗w+n∗−1h.Scenario 2.
q0<0.
Under this scenario, the stock position att0 is exhausted; therefore the inventory cost during t0,t1 needs to subtract holding costs (see Figure 4), so that the inventory cost during t0,t1 is(16)f2=w2λB02−h02λq02.
Yet, if there is a shortage att0, again using an approach similar to that used in Section 3, the minimum inventory cost during the interval t1,T can still be expressed as g∗; thus the minimum inventory cost over the horizon t0,T is(17)F∗=f2+g∗=cλT1+B0+n∗A−h02λq02+w2λB02+whλT122n∗w+n∗−1h.
Obviously, bothf1 and f2 are constants and independent of decision variables; therefore, minimizing inventory cost during the period t0,T is equivalent to minimizing the buyer’s inventory cost over the interval t1,T.
Based on the discussion above, we can obtain the following two candidate optimal ordering policies in the horizont1,T under strategy 1.
Policyπ1: n∗=n1,Q1∗=wλT1/n1w+n1−1h+B0, Q∗=w+hλT1/n1w+n1−1h, B∗=hλT1/n1w+n1−1h,which applies to the case thatg1≤g2.
For this order policy, the minimum inventory cost in horizont1,T is g∗=g1.
Policyπ2: n∗=n1,Q1∗=wλT1/n1w+n1−1h+B0, Q∗=w+hλT1/n1w+n1−1h, B∗=hλT1/n1w+n1−1h,
which applies to the case thatg1<g2.
For this order policy,g∗=g2.


### 4.2. Strategy 2: A Special Order Is Placed att0

By the inventory management theory, if we place a one-time special order at the current price before the price increases for both scenarios ofq0≥0 and q0<0, the inventory cost in the special replenishment cycle can be expressed as(18)fs=c0Qs+A+h02λQs+q02+w2λBs2.Since the special order is exhausted att2 (see Figures 3 and 4), the next order will be placed at t2; hence, the inventory system will run a new BEOQ inventory mechanism during the period t2,T, and the shortage is Bs at t2.Suppose that the order sizesQ1′,…,Qm′ and the maximal shortage levels B1′,…,Bm−1′ are made after the special order; again using a similar discussion in Section 3, we can obtain the inventory cost in horizon t2,T(19)gsm,Q1′,…,Qm′,B1′,…,Bm−1′=cλT2+Bs+mA+h2λQ1′−Bs2+h2λ∑i=2mQi′−Bi−1′2+w2λ∑i=1m−1Bi′2=cλT0−q0−Qs+mA+h2λQ1′−Bs2+h2λ∑i=2mQi′−Bi−1′2+w2λ∑i=1m−1Bi′2.Accordingly, the total inventory cost during the periodt0,T is(20)Fsm,Qs,Bs,Q1′,…,Qm′,B1′,…,Bm−1′=fs+gsm,Q1′,…,Qm′,B1′,…,Bm−1′=cλT0−cq0−c−c0Qs+m+1A+h02λQs+q02+w2λBs2+h2λQ1′−Bs2+h2λ∑i=2mQi′−Bi−1′2+w2λ∑i=1m−1Bi′2.Thus, the problem of determining an optimal ordering policy of strategy 2 can be formulated as the following optimization problem:(21)minm,Qs,Bs,Qi′,Bi′Fsm,Qs,Bs,Q1′,…,Qm′,B1′,…,Bm−1′s.t.Qs+Q1′+Q2′+⋯+Qm′=λT0−q0Qs≥0,Bs≥0,Qi′≥0,i=1,…,m,Bi′≥0,i=1,…,m−1,mis a positive integer.For problem (21), we have the following conclusion.Theorem 2.
For inventory model (21) with q0≥0 or q0<0, assume m is fixed; after the special order, all optimal maximum shortage levels are the same, and the optimal ordering sizes are the same. Furthermore, the optimal special ordering size is(22)Qs=λT0−q0−mλw+hαmh0w+h+wh,ifα>0andβ≤0,λT0−q0,orα>0,β>0andm<δ,otherwise,the ordering size after special order is(23)Q′=λT0−q0−λw+hαmh0w+h+wh,ifα>0andβ≤0,0,orα>0,β>0andm<δ,otherwise,and the maximal shortage level is(24)Bs=B′=λhαmh0w+h+wh,ifα>0andβ≤0,0,orα>0,β>0andm<δ,otherwise,where α=h0T0+c0−c,β=q0h0+λc0−c,δ=λT0−q0wh/w+hβ.Proof. See Appendix B.From the proof of Theorem2, if α≥0andβ≤0, or α≥0,β>0andm<δ hold, then(25)Fs=cλT0+cq0+c0−cQs+m+1A+h02λQs+q02+mh2λQ′−B′2+mw2λB′2=c0λT0−c0q0+m+1A+h0λT022−mλw+hα22mh0w+h+wh.Otherwise,(26)m=0,Qs=λT0−q,Q′=0,B′=0,(27)Fs=c0λT0−c0q0+A+h0λT022.Now, we consider the optimal ordering time under condition (58) or (59).For the functionFs given by (25), (28)Fs′′m=w2hλw+hα2mh0w+h+wh3>0,so that Fsm is a convex function in m, and its minimum can be obtained at the stationary point of the function Fs, which yields(29)m=1h0wλh2Aw+hα−2Awhλw+h.Denoteγ=2Awh/λw+h and m1=⌊m⌋,m2=⌈m⌉. Taking the fact m is a positive integer into consideration, we know that the optimal ordering time is m=m1 when Fsm1≥Fsm2; otherwise it is m=m2.Based on the above discussion, and ifα≤γ, the optimal ordering time is m∗=0. We obtain the following three candidate optimal order policies in horizon t0,T under strategy 2.Policyπ3:(30)m∗=m1,Qs∗=λT0−q0−m1λw+hαm1h0w+h+wh,(31)Q′∗=λw+hαm1h0w+h+wh,B′∗=λhαm1h0w+h+wh.It applies to(32)α>γ,β≤0,Fsm1≤Fsm2,(33)orα>γ,β>0,m2<δandFsm1≤Fsm2,(34)orα>γ,β>0,m1<δ≤m2.For policyπ3, the minimum inventory cost over the interval t0,T is Fs∗=Fsm1.Policyπ4:(35)m∗=m2,Qs∗=λT0−q0−m2λw+hαm2h0w+h+wh,(36)Q′∗=λw+hαm2h0w+h+wh,B′∗=λhαm2h0w+h+wh,which applies to (37)α>γ,β≤0,andFsm1>Fsm2,or(38)orα>γ,β>0,m2<δandFsm1>Fsm2.For this policy, the minimum inventory cost over the intervalt0,T is Fs∗=Fsm2.Policyπ5:(39)m∗=0,Qs∗=λT1−q0,Q′∗=B′∗=0,i.e., no order is made after the special order; the minimum inventory cost over the interval horizon t0,T is(40)Fs∗=c0λT0−c0q0+A+h0λT022.According to the analysis above, the difference of the minimal total costs under two strategies isG∗=F∗−Fs∗. Certainly, placing a special order or not is determined by the sign of G∗. If G∗≥0, then the buyer placing a special order can save costs than not placing special orders, and the candidate optimal policy is π3, π4, or π5 after the special order.On the contrary, ifG∗<0, the buyer will not place a special order, and the candidate optimal policy is π1 or π2 after the regular order replenishment cycle.Considering all the ordering policies discussed above, we can present our algorithm for our problem in this paper.


## 4.1. Strategy 1: A Special Order Is Not Placed att0

For this strategy, when the inventory cycles under the current price end att1, the buyer will continue using a new BEOQ order policy till the end of the inventory system, so that the inventory cost over the interval t0,T consists of inventory cost over two intervals t0,t1 and t1,T.Scenario 1.
q0≥0
Under this scenario, the price increases when there is no shortage, and the remaining stockq0 at t0 will be postponed to the second stage. By the knowledge of inventory control, we can calculate that the inventory cost during t0,t1 is(7)f1=h02λq02+w2λB02.
Since the stock level is−B0 at t1, the inventory system will run a new “BEOQ” order policy during the horizon t1,T under the new price c, and the stock level is −B0 and zero at the beginning and end of the planning horizon, respectively; see Figures 3 and 4.
For this scenario, when the special order is not placed att0, we denote the inventory cost during this period from t0 to the end by F, which is a function of order quantity Q1,…,Qn and maximum shortage level B1,…,Bn−1 for an order after t0. Using an approach similar to that used and according to formula (2) in Section 3, then(8)gn,Q1,…,Qn,B1,…,Bn−1=nA+c∑i=1nQi+h2λQ1−B02+h2λ∑i=2nQi−Bi−12+w2λ∑i=1n−1Bi2=cλT+B0+nA+h2λQ1−B02+h2λ∑i=2nQi−Bi−12+w2λ∑i=1n−1Bi2.
So the ordering policy during the horizont1,T of the concerned model can be formulated as the following optimization model:(9)minn,Qi,Bign,Q1,…,Qn,B1,…,Bn−1,s.t.Q1+Q2+⋯+Qn=λT1+B0,Qi≥0,i=1,…,n,Bi≥0,i=1,…,n−1,nis a positive integer.
According to formulas (5) and (6) of Theorem 1 in Section 3, we know that all the optimal ordering sizes are the same except the first one, all the optimal maximum shortage levels are the same, and the optimal ordering size, maximum shortage level, and the optimal order time over the interval t1,T, respectively, are(10)Q=w+hλT1nw+n−1h,B=hλT1nw+n−1h,(11)n=T1wλh2Aw+h+hw+h≜n1orT1wλh2Aw+h+hw+h≜n2,and the minimum inventory cost during the interval t1,T is(12)g1=cλT1+B0+n1A+whλT122n1w+n1−1h,or(13)g2=cλT1+B0+n2A+whλT122n2w+n2−1h.
Ifg1≤g2, then the optimal ordering times and the minimum inventory cost during the interval t1,T are n∗=n1 and g1, respectively; otherwise, they are n∗=n2 and g2. Denote the minimum inventory cost during horizon t1,T by(14)g∗=cλT1+B0+n∗A+whλT122n∗w+n∗−1h.
In summary, under strategy 1, if the remaining inventory stock att0 is nonnegative, the minimum inventory cost over the horizon t0,T is(15)F∗=f1+g∗=cλT1+B0+n∗A+h02λq02+w2λB02+whλT122n∗w+n∗−1h.Scenario 2.
q0<0.
Under this scenario, the stock position att0 is exhausted; therefore the inventory cost during t0,t1 needs to subtract holding costs (see Figure 4), so that the inventory cost during t0,t1 is(16)f2=w2λB02−h02λq02.
Yet, if there is a shortage att0, again using an approach similar to that used in Section 3, the minimum inventory cost during the interval t1,T can still be expressed as g∗; thus the minimum inventory cost over the horizon t0,T is(17)F∗=f2+g∗=cλT1+B0+n∗A−h02λq02+w2λB02+whλT122n∗w+n∗−1h.
Obviously, bothf1 and f2 are constants and independent of decision variables; therefore, minimizing inventory cost during the period t0,T is equivalent to minimizing the buyer’s inventory cost over the interval t1,T.
Based on the discussion above, we can obtain the following two candidate optimal ordering policies in the horizont1,T under strategy 1.
Policyπ1: n∗=n1,Q1∗=wλT1/n1w+n1−1h+B0, Q∗=w+hλT1/n1w+n1−1h, B∗=hλT1/n1w+n1−1h,which applies to the case thatg1≤g2.
For this order policy, the minimum inventory cost in horizont1,T is g∗=g1.
Policyπ2: n∗=n1,Q1∗=wλT1/n1w+n1−1h+B0, Q∗=w+hλT1/n1w+n1−1h, B∗=hλT1/n1w+n1−1h,
which applies to the case thatg1<g2.
For this order policy,g∗=g2.


## 4.2. Strategy 2: A Special Order Is Placed att0

By the inventory management theory, if we place a one-time special order at the current price before the price increases for both scenarios ofq0≥0 and q0<0, the inventory cost in the special replenishment cycle can be expressed as(18)fs=c0Qs+A+h02λQs+q02+w2λBs2.Since the special order is exhausted att2 (see Figures 3 and 4), the next order will be placed at t2; hence, the inventory system will run a new BEOQ inventory mechanism during the period t2,T, and the shortage is Bs at t2.Suppose that the order sizesQ1′,…,Qm′ and the maximal shortage levels B1′,…,Bm−1′ are made after the special order; again using a similar discussion in Section 3, we can obtain the inventory cost in horizon t2,T(19)gsm,Q1′,…,Qm′,B1′,…,Bm−1′=cλT2+Bs+mA+h2λQ1′−Bs2+h2λ∑i=2mQi′−Bi−1′2+w2λ∑i=1m−1Bi′2=cλT0−q0−Qs+mA+h2λQ1′−Bs2+h2λ∑i=2mQi′−Bi−1′2+w2λ∑i=1m−1Bi′2.Accordingly, the total inventory cost during the periodt0,T is(20)Fsm,Qs,Bs,Q1′,…,Qm′,B1′,…,Bm−1′=fs+gsm,Q1′,…,Qm′,B1′,…,Bm−1′=cλT0−cq0−c−c0Qs+m+1A+h02λQs+q02+w2λBs2+h2λQ1′−Bs2+h2λ∑i=2mQi′−Bi−1′2+w2λ∑i=1m−1Bi′2.Thus, the problem of determining an optimal ordering policy of strategy 2 can be formulated as the following optimization problem:(21)minm,Qs,Bs,Qi′,Bi′Fsm,Qs,Bs,Q1′,…,Qm′,B1′,…,Bm−1′s.t.Qs+Q1′+Q2′+⋯+Qm′=λT0−q0Qs≥0,Bs≥0,Qi′≥0,i=1,…,m,Bi′≥0,i=1,…,m−1,mis a positive integer.For problem (21), we have the following conclusion.Theorem 2.
For inventory model (21) with q0≥0 or q0<0, assume m is fixed; after the special order, all optimal maximum shortage levels are the same, and the optimal ordering sizes are the same. Furthermore, the optimal special ordering size is(22)Qs=λT0−q0−mλw+hαmh0w+h+wh,ifα>0andβ≤0,λT0−q0,orα>0,β>0andm<δ,otherwise,the ordering size after special order is(23)Q′=λT0−q0−λw+hαmh0w+h+wh,ifα>0andβ≤0,0,orα>0,β>0andm<δ,otherwise,and the maximal shortage level is(24)Bs=B′=λhαmh0w+h+wh,ifα>0andβ≤0,0,orα>0,β>0andm<δ,otherwise,where α=h0T0+c0−c,β=q0h0+λc0−c,δ=λT0−q0wh/w+hβ.Proof. See Appendix B.From the proof of Theorem2, if α≥0andβ≤0, or α≥0,β>0andm<δ hold, then(25)Fs=cλT0+cq0+c0−cQs+m+1A+h02λQs+q02+mh2λQ′−B′2+mw2λB′2=c0λT0−c0q0+m+1A+h0λT022−mλw+hα22mh0w+h+wh.Otherwise,(26)m=0,Qs=λT0−q,Q′=0,B′=0,(27)Fs=c0λT0−c0q0+A+h0λT022.Now, we consider the optimal ordering time under condition (58) or (59).For the functionFs given by (25), (28)Fs′′m=w2hλw+hα2mh0w+h+wh3>0,so that Fsm is a convex function in m, and its minimum can be obtained at the stationary point of the function Fs, which yields(29)m=1h0wλh2Aw+hα−2Awhλw+h.Denoteγ=2Awh/λw+h and m1=⌊m⌋,m2=⌈m⌉. Taking the fact m is a positive integer into consideration, we know that the optimal ordering time is m=m1 when Fsm1≥Fsm2; otherwise it is m=m2.Based on the above discussion, and ifα≤γ, the optimal ordering time is m∗=0. We obtain the following three candidate optimal order policies in horizon t0,T under strategy 2.Policyπ3:(30)m∗=m1,Qs∗=λT0−q0−m1λw+hαm1h0w+h+wh,(31)Q′∗=λw+hαm1h0w+h+wh,B′∗=λhαm1h0w+h+wh.It applies to(32)α>γ,β≤0,Fsm1≤Fsm2,(33)orα>γ,β>0,m2<δandFsm1≤Fsm2,(34)orα>γ,β>0,m1<δ≤m2.For policyπ3, the minimum inventory cost over the interval t0,T is Fs∗=Fsm1.Policyπ4:(35)m∗=m2,Qs∗=λT0−q0−m2λw+hαm2h0w+h+wh,(36)Q′∗=λw+hαm2h0w+h+wh,B′∗=λhαm2h0w+h+wh,which applies to (37)α>γ,β≤0,andFsm1>Fsm2,or(38)orα>γ,β>0,m2<δandFsm1>Fsm2.For this policy, the minimum inventory cost over the intervalt0,T is Fs∗=Fsm2.Policyπ5:(39)m∗=0,Qs∗=λT1−q0,Q′∗=B′∗=0,i.e., no order is made after the special order; the minimum inventory cost over the interval horizon t0,T is(40)Fs∗=c0λT0−c0q0+A+h0λT022.According to the analysis above, the difference of the minimal total costs under two strategies isG∗=F∗−Fs∗. Certainly, placing a special order or not is determined by the sign of G∗. If G∗≥0, then the buyer placing a special order can save costs than not placing special orders, and the candidate optimal policy is π3, π4, or π5 after the special order.On the contrary, ifG∗<0, the buyer will not place a special order, and the candidate optimal policy is π1 or π2 after the regular order replenishment cycle.Considering all the ordering policies discussed above, we can present our algorithm for our problem in this paper.


## 5. Algorithm

The algorithm flowchart for problem is shown in Figure5. See Appendix C.Figure 5
Algorithm flowchart.To compute the remaining stockq0 at t0, we first give the optimal order policy under the current price. For an inventory mechanism, if there is not price change, this is an inventory model with backordering in the finite horizon of length T under the current price c0, and the stock level is zero at the beginning and the end of the inventory system; based on the conclusions (5) and (6) in Theorem 1, the optimal ordering size Q0, maximum shortage level B0, and the optimal order times k under the current price are, respectively, (41)Q0=w+h0λTkw+k−1h0,B0=h0λTkw+k−1h0,(42)k=Twλh02Aw+h0+hw+h0≜k1orTwλh02Aw+h0+h0w+h0≜k2.According to these, we can compute that(43)q0=λt0Q0Q0−B0−λt0.


## 6. Computational Experiments and Sensitivity Analysis

In this section, in order to show the applicability of the presented models, we will perform some numerical examples and sensitivity analysis on major parameters. Based on the algorithm above, if the basic parameters of the models are given, we can obtain an optimal ordering policy. The results of computational experiments and sensitivity analysis can be obtained by the Matlab procedures of algorithm above, so we mainly list the important results and list the simple solution process.


### 6.1. Computational Experiments

Example 1.
Consider the inventory system with the following parameters:λ=500,A=50,c0=10,c=10.5,h0=4,h=4.2,w=3,T=12,t0=2.04.
Solution: For this inventory system, first, we calculate the optimal regular order times, order size, the maximum shortage level beforet0, and the remaining stock at t0 before the price increase does not take place according to formulas (41)–(43) as follows:(44)k=36,Q0=169.35,B0=96.77,q0=68.71.
If we adopt strategy 1, then the candidate optimal policy set isπ1,π2.
However, if we adopt strategy 2, sinceα=39.34>γ=0.5916, and β=24.84>0,m2=29<λT0−q0wh/w+hβ=346.02; based on Theorem 2 in Section 4, the candidate optimal policy set is π3,π4. The numerical results by algorithm are listed in Table 2.
From Table2, we can see that F∗=54493, Fs∗=54525, and thus G∗=−32; hence the optimal ordering policy of Example 1 is π1, and the optimal order times, optimal ordering sizes, and the optimal maximum shortage level over the interval [t1,T] are, respectively,(45)m∗=29,Q1∗=167.37,Q∗=169.43,B∗=98.83.
In this way, the buyer’s inventory cost savings are 32.Table 2
Numerical results on Example1.
PolicyQsm\nQ\Q′B\B′F\Fsπ1—29169.4398.8354493π2—30163.6795.4754494π369.4428172.92100.8754525π466.8729167.0597.4554525Table 3
Numerical result on Example2.
PolicyQsm\nQ\Q′B\B′F\Fsπ1—28169.4398.8352596π2—29163.6795.4752597π3129.1827170.8499.65652580π4126.5528164.8496.1552581Example 2.
For the inventory system considered in Example1, set t0=2.5 and other parameters remain unchanged.
For this inventory system, the optimal regular order times, order size, and the maximum shortage level are the same as in Example1, and the remaining stock at t0 is q0=8.06.
Similar to Example1, since α=39.34>γ=0.5916, and β=−375.16<0, if we apply strategy 2, the candidate optimal policy set is π3,π4. The numerical results are listed in Table 3, from which we can see that the cost of applying π3 is minimal, so the optimal ordering policy of Example 2 is π3, and the buyer’s inventory cost saving of placing a special order compared to not placing special order is 16.Example 3.
For the inventory system considered in Example1, set t0=2.33 and other parameters remain unchanged.
For this example, the remaining stock att0 is q0=−76.29.
Similar to Example1, since α=38.18>γ=0.5916, and β=−555.16<0, if a special order is placed, the candidate optimal policy set is π3,π4. The numerical results are listed in Table 4, from which we can see that the optimal ordering policy is π4, and the buyer’s inventory cost saving of placing a special order compared to not placing special order is 91.
In the numerical experiments, we see that the remaining stockq0 at t0 significantly affects the buyer’s ordering policy. Using the same data as those in Example 1, we study the sensitivity analysis.Table 4
Numerical result on Example3.
PolicyQsm\nQ\Q′B\B′F\Fsπ1—29169.4398.8354457π2—30163.6795.4754458π3214.8927173.94101.4754368π4212.2128167.8297.9054366


### 6.2. Sensitivity Analysis

To better understand the effect ofq0 on the ordering policy, we conduct a sensitivity analysis of the models by varying the parameter q0 while keeping the other parameters fixed as same as in Example 1. The numerical results are presented in Table 5 and Figure 6.Table 5
Impact of parameterq0 on the retailer’s ordering policy, c=10.5.
t0q0π1π2π3π4G∗QsOptimal policy2.00−80.655633356334562415624093216.53π42.0563.7154490544925452054519−29—π12.1038.7154480544825449254492−12—π12.1513.715447654476544655446510123.60 or 121.06π3orπ42.20−11.29544745447554437¯5443836148.21π32.25−36.29544705447254410¯5441260172.83π32.30−61.2954463544645438454383¯80197.44π42.35−86.295445254453543565435696224.73 or 222.06π3orπ42.4058.0652609526115263552635−26—π12.4533.0652600526025260852608−8—π12.508.06525965259752580¯5258116129.18π32.55−16.94525955259652553¯5255442153.78π32.60−41.9452590525925252752525¯65178.38π42.65−66.9452582525845250052498¯84202.98π42.70−91.9452570525725247252471¯99227.58π42.7552.4250728507305075150750−22—π12.8027.4250720507225072350723−3—π12.852.425071750719506965069611134.76 or 132.03π3orπ42.90−22.58507165071750668¯5067048159.34π32.95−47.5850710507125064350641¯69183.93π43.00−72.5850701507035061550614¯87208.51π4The underlined values indicate the ordering cost under the optimal ordering strategy.Figure 6
The changes of theq0 and G with the price growth rate.Figure6 exhibits that G≥0 when q0≤24 , the buyer should place a special order at t0 for this scenario, and the optimal special order quantity decreases with the increase of q0. However, when q0>24, the buyer makes a special order more cost-saving than not making a special order, and his optimal replenishment policy is one of the candidate optimal policy sets π1,π2.In general, whether the buyer places a special order not only relates toq0, but also relates to the extent of price increase. In order to discuss this issue, we conduct a sensitivity analysis of the model by varying the parameters c and h while keeping the other parameters fixed as same as in Example 1. The numerical results are presented in Table 6 and Figure 7.Table 6
Impact of price increase ratio parameter on the retailer’s ordering policy,t0=2.04.
Δc/c0 (%)hπ1π2π3π4G∗QsOptimal policy14.0452506525055255552554−49—π224.0853003530025304953049−47—π234.1253501534995354353542−43—π244.1653996539985403553034−38—π154.2054493544945452554525−32—π164.2454990549915501555014−24—π174.2855487554885550355502−15—π184.3255983559845598955989−6—94.36564805648256475564746117.24π4104.45697756977569585695818132.40 or 129.82π3orπ4114.445747357474574415744132144.98 or 142.39π3orπ4124.485797057970579225792248157.55 or 154.96π3orπ4134.525846658466584025840264170.12 or 167.53π3orπ4144.565896358963588815888182182.69 or 180.09π3orπ4154.6059459594595935859358101195.25 or 192.65π3orπ4164.6459995599955983459834121207.81 or 205.21π3orπ4174.6860451604516030860309143220.36π3184.7260948609476078260782165232.91 or 230.30π3orπ4194.7661444614436125461254189245.452 or 242.85π3orπ4204.8061940619396172461725215258.99π3The underlined values indicate the ordering cost under the optimal ordering strategy.Figure 7
The cost difference between the two strategies with the price growth rate.From Table6 and Figure 7, it is not difficult to see that when q0 is fixed, a special order is not required when the price increase rate is small. When the price increases to a certain level, placing a special replenishment is the buyer’s optimal replenishment strategy. The buyer placing a special replenishment can save costs when the price increase rate is more than 8.5%.


## 6.1. Computational Experiments

Example 1.
Consider the inventory system with the following parameters:λ=500,A=50,c0=10,c=10.5,h0=4,h=4.2,w=3,T=12,t0=2.04.
Solution: For this inventory system, first, we calculate the optimal regular order times, order size, the maximum shortage level beforet0, and the remaining stock at t0 before the price increase does not take place according to formulas (41)–(43) as follows:(44)k=36,Q0=169.35,B0=96.77,q0=68.71.
If we adopt strategy 1, then the candidate optimal policy set isπ1,π2.
However, if we adopt strategy 2, sinceα=39.34>γ=0.5916, and β=24.84>0,m2=29<λT0−q0wh/w+hβ=346.02; based on Theorem 2 in Section 4, the candidate optimal policy set is π3,π4. The numerical results by algorithm are listed in Table 2.
From Table2, we can see that F∗=54493, Fs∗=54525, and thus G∗=−32; hence the optimal ordering policy of Example 1 is π1, and the optimal order times, optimal ordering sizes, and the optimal maximum shortage level over the interval [t1,T] are, respectively,(45)m∗=29,Q1∗=167.37,Q∗=169.43,B∗=98.83.
In this way, the buyer’s inventory cost savings are 32.Table 2
Numerical results on Example1.
PolicyQsm\nQ\Q′B\B′F\Fsπ1—29169.4398.8354493π2—30163.6795.4754494π369.4428172.92100.8754525π466.8729167.0597.4554525Table 3
Numerical result on Example2.
PolicyQsm\nQ\Q′B\B′F\Fsπ1—28169.4398.8352596π2—29163.6795.4752597π3129.1827170.8499.65652580π4126.5528164.8496.1552581Example 2.
For the inventory system considered in Example1, set t0=2.5 and other parameters remain unchanged.
For this inventory system, the optimal regular order times, order size, and the maximum shortage level are the same as in Example1, and the remaining stock at t0 is q0=8.06.
Similar to Example1, since α=39.34>γ=0.5916, and β=−375.16<0, if we apply strategy 2, the candidate optimal policy set is π3,π4. The numerical results are listed in Table 3, from which we can see that the cost of applying π3 is minimal, so the optimal ordering policy of Example 2 is π3, and the buyer’s inventory cost saving of placing a special order compared to not placing special order is 16.Example 3.
For the inventory system considered in Example1, set t0=2.33 and other parameters remain unchanged.
For this example, the remaining stock att0 is q0=−76.29.
Similar to Example1, since α=38.18>γ=0.5916, and β=−555.16<0, if a special order is placed, the candidate optimal policy set is π3,π4. The numerical results are listed in Table 4, from which we can see that the optimal ordering policy is π4, and the buyer’s inventory cost saving of placing a special order compared to not placing special order is 91.
In the numerical experiments, we see that the remaining stockq0 at t0 significantly affects the buyer’s ordering policy. Using the same data as those in Example 1, we study the sensitivity analysis.Table 4
Numerical result on Example3.
PolicyQsm\nQ\Q′B\B′F\Fsπ1—29169.4398.8354457π2—30163.6795.4754458π3214.8927173.94101.4754368π4212.2128167.8297.9054366


## 6.2. Sensitivity Analysis

To better understand the effect ofq0 on the ordering policy, we conduct a sensitivity analysis of the models by varying the parameter q0 while keeping the other parameters fixed as same as in Example 1. The numerical results are presented in Table 5 and Figure 6.Table 5
Impact of parameterq0 on the retailer’s ordering policy, c=10.5.
t0q0π1π2π3π4G∗QsOptimal policy2.00−80.655633356334562415624093216.53π42.0563.7154490544925452054519−29—π12.1038.7154480544825449254492−12—π12.1513.715447654476544655446510123.60 or 121.06π3orπ42.20−11.29544745447554437¯5443836148.21π32.25−36.29544705447254410¯5441260172.83π32.30−61.2954463544645438454383¯80197.44π42.35−86.295445254453543565435696224.73 or 222.06π3orπ42.4058.0652609526115263552635−26—π12.4533.0652600526025260852608−8—π12.508.06525965259752580¯5258116129.18π32.55−16.94525955259652553¯5255442153.78π32.60−41.9452590525925252752525¯65178.38π42.65−66.9452582525845250052498¯84202.98π42.70−91.9452570525725247252471¯99227.58π42.7552.4250728507305075150750−22—π12.8027.4250720507225072350723−3—π12.852.425071750719506965069611134.76 or 132.03π3orπ42.90−22.58507165071750668¯5067048159.34π32.95−47.5850710507125064350641¯69183.93π43.00−72.5850701507035061550614¯87208.51π4The underlined values indicate the ordering cost under the optimal ordering strategy.Figure 6
The changes of theq0 and G with the price growth rate.Figure6 exhibits that G≥0 when q0≤24 , the buyer should place a special order at t0 for this scenario, and the optimal special order quantity decreases with the increase of q0. However, when q0>24, the buyer makes a special order more cost-saving than not making a special order, and his optimal replenishment policy is one of the candidate optimal policy sets π1,π2.In general, whether the buyer places a special order not only relates toq0, but also relates to the extent of price increase. In order to discuss this issue, we conduct a sensitivity analysis of the model by varying the parameters c and h while keeping the other parameters fixed as same as in Example 1. The numerical results are presented in Table 6 and Figure 7.Table 6
Impact of price increase ratio parameter on the retailer’s ordering policy,t0=2.04.
Δc/c0 (%)hπ1π2π3π4G∗QsOptimal policy14.0452506525055255552554−49—π224.0853003530025304953049−47—π234.1253501534995354353542−43—π244.1653996539985403553034−38—π154.2054493544945452554525−32—π164.2454990549915501555014−24—π174.2855487554885550355502−15—π184.3255983559845598955989−6—94.36564805648256475564746117.24π4104.45697756977569585695818132.40 or 129.82π3orπ4114.445747357474574415744132144.98 or 142.39π3orπ4124.485797057970579225792248157.55 or 154.96π3orπ4134.525846658466584025840264170.12 or 167.53π3orπ4144.565896358963588815888182182.69 or 180.09π3orπ4154.6059459594595935859358101195.25 or 192.65π3orπ4164.6459995599955983459834121207.81 or 205.21π3orπ4174.6860451604516030860309143220.36π3184.7260948609476078260782165232.91 or 230.30π3orπ4194.7661444614436125461254189245.452 or 242.85π3orπ4204.8061940619396172461725215258.99π3The underlined values indicate the ordering cost under the optimal ordering strategy.Figure 7
The cost difference between the two strategies with the price growth rate.From Table6 and Figure 7, it is not difficult to see that when q0 is fixed, a special order is not required when the price increase rate is small. When the price increases to a certain level, placing a special replenishment is the buyer’s optimal replenishment strategy. The buyer placing a special replenishment can save costs when the price increase rate is more than 8.5%.


## 7. Conclusion and Extensions

This study investigated the finite horizon inventory system with backordering and a known price increase, which is observed in common items like grain crops, wheat, corn, and soybeans, etc. The supplier allows the buyer to place a special order when the price increases. The objective of this study is to determine whether or not to purchase additional stock and optimal number of orders so that the inventory cost is minimum. For this mechanism, based on the retailers cost analysis, the problem is formulated as an optimization model and is solved by optimal techniques. A global optimal solution for the model was obtained through the algorithm given in the paper. From numerical studies, we show that the results of the numerical analysis are consistent with managerial implications and economical common sense. It has been shown that if those basic parameters of the models are given, we can get an optimal ordering policy, and under those basic parameters such as in Example1 being fixed, the optimal order policy changes with the change of parameters q0 and Δc, and the special replenishment size increases as q0 decreases and c increases.These results in the paper are important to the real world. The prices of goods, for example, wheat or soybeans, will increase after two or three months after harvest. Also when and how much the price increases are generally due to the interval between two normal orders. When buyers have known the price changes, they need to adjust their ordering strategy. Our model provides the decision-maker useful for this mechanism, and the algorithm can provide an optimal replenishment and stocking strategy to the buyers quickly. Numerical experiments were also carried out to illustrate the validity of the models. Sensitivity analysis has been performed in order to determine the robustness of the models presented above.To make the concerned model more practical, we may introduce the shortages and partial backordering into the model. Further, the time of price change may be stochastic. This will be considered in the future research.



---

*Source: 9935814-2022-01-12.xml*