# Study of the Bullwhip Effect in a Multistage Supply Chain with Callback Structure considering Two Retailers

Authors: Junhai Ma; Liqing Zhu; Ye Yuan; Shunqi Hou
Journal: Complexity
 (2018)

---

## Abstract

With the purpose of researching the bullwhip effect when there is a callback center in the supply chain system, this paper establishes a new supply chain model with callback structure, which has a material supplier, a manufacture, and two retailers. The manufacture and retailers all employ AR(1) demand processes and use order-up-to inventory policy when they make order decisions. Moving average forecasting method is used to measure the bullwhip effect of each retailer and manufacture. We investigate the impact of lead-times of retailers and manufacture, forecasting precision, callback index, and marketing share on the bullwhip effect of both retailers and manufacture. Then we use the method of numerical simulation to indicate the different parameters in this supply chain. Furthermore, this paper puts forward some suggestions to help the enterprises to control the bullwhip effect in the supply chain with callback structure.


---

## Body

## 1. Introduction

Bullwhip effect is a major issue that threats the stable and smooth performance of inventory, cost, and information of different layers of participants. Generated from the changes of orders, bullwhip effect manifests the information delivery of initial demand through supply chain. Due to the failure of information sharing, lead-time, batch order, and predictive information which is not consistent with actual need, upstream enterprises cannot receive correct signals of demand. Bullwhip effect greatly influences the inventory and causes the increase of inventory cost and the cost of marketing. Value is decreased in excessive inventory, which even affects the smooth performance of supply chain. Currently, the research of bullwhip effect mainly focuses on theoretical study of general model of supply chain, while some supply chain models with practical significance have not been explored yet. This paper builds a multilevel supply chain model with callback center and quantifies the model according to the influence of callback mechanism in actual supply chain system, which innovatively introduces callback system to the supply chain model. The callback center recalls, classifies, and sends rejected products to retailers and producers. Those rejected products go back to the supply chain after being handled. In the field of reverse supply chains, many scholars have carried out relevant researches from different aspects.Angelus and Özalp studied a supply chain where 4 kinds of ordering decisions: regular, reverse, expedited, and secondary market selling can be made by each location. They developed a new method based on short and long purchases of inventory, which enables them to find the conditions for an optimal solution to be a nested echelon base stock policy [1]. Sobotka et al. took repair and demolishing works as background and studied reverse logistics of construction products and materials. The results reveal that reverse supply chains can bring significant added value to a business project [2]. Jena et al. took various scenarios for the government subsidies and fees into consideration and established four closed-loop supply chain models. The results show that an increase in subsidy will efficiently lift both total surplus and channel profit [3]. Huang and Wang analyzed the impacts of information sharing an a closed-loop supply chain and found that information sharing always lifts the profits of the manufacturer and the third party regardless the existence of technology licensing [4].Ma and Lou consider a two-level household appliance supply chain system consisting of a manufacturer with an Internet channel and a retailer with a traditional channel and an Internet channel [5]. Ma and Bao investigated the energy-efficient air conditioning supply chain, which is composed of one supplier (MIDEA) and two retailers [6]. Chen et al. quantify this effect for simple, two-stage supply chains consisting of a single retailer and a single manufacturer [7–9]. Ma and Xie discuss the impact of fluctuated demand triggered by price-discount promotion policy on the comprehensive inventory control system in which the final inventory and supply line stock are considered and weighted differently according to corresponding practical operation situations [10]. Ma and Xie consider two scenarios in which the manufacturer holds either asymmetric or symmetric channel power as the retailer and develop the dynamic game models for the two scenarios and analyze the models’ dynamic behavior [11]. Duc et al. quantify the impact of the bullwhip effect—the phenomenon in which information on demand is distorted as moving up a supply chain—for a simple two-stage supply chain with one supplier and one retailer [12].J. Ma and X. Ma show that, in order to consider the market competition, a new supply chain with one supplier and two retailers is established in this paper. Two retailers employ different AR(1) demand processes, respectively, and an order-up-to inventory policy characterizes the inventory decisions [13]. Luong develops a measure of bullwhip effect for a simple two-stage supply chain that includes only one retailer and one supplier in the environment where the retailer employs base stock policy for their inventory and demand forecast is performed through the first-order autoregressive model, AR(1) [14]. Zhang considers the impact of forecasting methods on the bullwhip effect for a simple replenishment system in which a first-order autoregressive process AR(1) describes the customer demand and an order-up-to inventory policy characterizes the replenishment decision [15]. Caplin develops a general theory of the aggregate implications of (S,s) inventory policies. It is shown that (S,s) policies add to the variability of demand, with the variance of orders exceeding the variance of sales [16]. Kahn presents a model of production decisions with demand uncertainty that incorporates nonnegativity constraints on inventories. Even with no productivity shocks, optimal behavior by the firm is consistent with this stylized fact, either if demand exhibits positive serial correlation, or if the firm can backlog excess demand [17]. Towill et al. review the dynamic operation of supply chains and reach some simple conclusions for reducing demand amplification, which consequently attenuates swings in both production rates and stock levels. The results are based on one particular supply chain, for which the use of systems simplification techniques has generated valuable insight into supply chain design [18–20].At the same time, this paper studies callback system’s impact on bullwhip effect from the aspect of the order of retailer and manufacturer and offers theoretical support to detailed control measures in multilevel supply chain. It divides the traditional demand market in line with certain market share, to probe into the mutual influence and differentiation among several retailers. The order-up-to inventory strategy and MA forecasting method are used in AR(1) and ARMA(1, 1) demand model for the quantification and analysis of bullwhip effect.The following notation will be used in the paper:Dt: total demand quantity at period tDi,t: demand quantity of retailer i at period tδ: constant of autoregressive in demand modela: first-order autocorrelation coefficient in demand modelεt: independent and identically distributed in demand modelα: marketing share index of Retailer 1θ1: callback index of retailerθ2: callback index of manufactureqt: order of total retailer at period tqi,t: order of retailer i at period tli: lead-time of retailer iL: lead-time of manufacturek: forecasting precision of retailersK: forecasting precision of manufactureσ2: variance of demand quantitySi,t: order-up-to level of retailer i at period tD^i,tli: forecast of the lead-time demand of retailer i at period tσ^i,tli: standard deviation of forecasting errors on the lead-time demand of retailer i at period t


## 2. Supply Chain Model

### 2.1. Demand Process

In this research, we consider a three-stage supply chain with one supplier, one manufacture, two retailers, and a callback center as shown in Figure1. The two retailers face the same customer demand at a certain rate α and send orders to manufacture. We assume that the customer demand process conforms to AR(1) process. The callback center collects the rejected products from customers and sends a part of them to Retailer 1 and the others to manufacture in Figure 1.Figure 1
A three-stage supply chain model.The customer demand conforms an AR(1) process as(1)Dt=δ+aDt-1+εt=D1,t+D2,t.In (1), a is the autoregressive coefficient of the customer demand process, while -1<a<1. And εt are independent and identically subject to a normal distribution, whose mean is 0 and variance is σ2.As we assume that the two retailers share the customer demand and the rate isα, we can derive each process of Retailer 1 and Retailer 2:(2)D1,t=αDt=αδ+aD1,t-1+αεt,D2,t=1-αDt=1-αδ+aD2,t-1+1-αεt.Furthermore, we get(3)D1,tD2,t=α1-α.And the demand process of Retailer 1 is(4)D1,t=αδ+aD1,t-1+αεt.So we know thata is the autoregressive coefficient of the customer demand process faced by Retailer 1 and εt are independent and identically subject to a normal distribution, whose mean is 0 and variance is σ2. And we can get(5)ED1,t=ED1,t-1=μ1,d=αδ1-a,var⁡D1,t=var⁡D1,t-1=σ1,d2=α2σ21-a2.Similarly, Retailer 2 faces the same demand process and its model is(6)D2,t=1-αδ+aD2,t-1+1-αεt.And we have(7)ED2,t=ED2,t-1=μ2,d=1-αδ1-a,var⁡D2,t=var⁡D2,t-1=σ2,d2=1-α2σ21-a2.As callback center sends the rejected products to Retailer 1 and manufacture at a certain rate, we get the callback model as(8)Wt=θ1+θ2Dt-1=θ1+θ2D1,t-1+D2,t-1.Here, the partθ1Dt-1 is sent to Retailer 1 for being renewed and sold. The other part θ2Dt-1 is sent to manufacture; these rejected products can be fixed and then sold to retailers.


### 2.2. Inventory Policy

Order-up-to inventory policy is widely used in many researches, so we assist using this policy in this inventory management issue. The two retailers employ the OUT inventory policy: at the beginning of periodt, Retailer 1 knows the customer demand of period t-1asD1,t-1. So Retailer 1 can calculate the OUT policy point S1,t of period t and send an order q1,t to manufacture after calculating the order. After a lead-time l1, Retailer 1 receives the products from the manufacture at the starting of period t+l1. And the OUT level S1,t can be determined as(9)S1,t=D^1,tl1+zσ^1,tl1.In this formula,D^1,tl1 is the forecast demand of Retailer 1 in the lead-time l1 which depends on forecasting method and lead-time. σ^1,tl1 is the standard deviation of demand forecast error, and z is a given service level. Although it is difficult to accurately estimate the inventory holding costs and shortage costs in practice, the method using service level, which is defined as the probability for the first inventory fulfilling lead-time demand requirement to determine OUT level, is usually more feasible. The OUT level S1,t shown in (9) is optimal, depending on total inventory cost, for inventory system, while there is no fixed ordering cost or holding costs and shortage costs are proportional to the volume of first inventory or shortage (Luong [14]).Then, Retailer 1 sends an order of quantityq1,t to the manufacture at the start of period t. The order quantity can be given as(10)q1,t=S1,t-S1,t-1+D1,t-1-θ1Dt-1.Here,θ1Wt is the rejected products received from callback center, and S1,t-S1,t-1+D1,t-1 is the normal quantity depending on OUT level.Similarly, the OUT level of Retailer 2 at periodt is(11)S2,t=D^2,tl2+zσ^2,tl2.As Retailer 2 do not have rejected products, so its order quantity is(12)q2,t=S2,t-S2,t-1+D2,t-1.


### 2.3. Forecasting Method

In this paper, we assume the two retailers both use the MA forecasting method to forecast the demand in each lead-time. In the next section, the bullwhip effect is measured via the MA forecasting technique.According to the MA method, we can establish the demand forecast model of Retailer 1 as(13)D^1,tl1=l1k∑i=1kD1,t-i.Here,k is the number of date samples for MA forecasting method.As Retailer 2 faces the same consumer and sells the same products, we can consider that the two retailers use the same parameterk.So, the forecasted demand of Retailer 2 can be established as(14)D^2,tl2=l2k∑i=1kD2,t-i.


## 2.1. Demand Process

In this research, we consider a three-stage supply chain with one supplier, one manufacture, two retailers, and a callback center as shown in Figure1. The two retailers face the same customer demand at a certain rate α and send orders to manufacture. We assume that the customer demand process conforms to AR(1) process. The callback center collects the rejected products from customers and sends a part of them to Retailer 1 and the others to manufacture in Figure 1.Figure 1
A three-stage supply chain model.The customer demand conforms an AR(1) process as(1)Dt=δ+aDt-1+εt=D1,t+D2,t.In (1), a is the autoregressive coefficient of the customer demand process, while -1<a<1. And εt are independent and identically subject to a normal distribution, whose mean is 0 and variance is σ2.As we assume that the two retailers share the customer demand and the rate isα, we can derive each process of Retailer 1 and Retailer 2:(2)D1,t=αDt=αδ+aD1,t-1+αεt,D2,t=1-αDt=1-αδ+aD2,t-1+1-αεt.Furthermore, we get(3)D1,tD2,t=α1-α.And the demand process of Retailer 1 is(4)D1,t=αδ+aD1,t-1+αεt.So we know thata is the autoregressive coefficient of the customer demand process faced by Retailer 1 and εt are independent and identically subject to a normal distribution, whose mean is 0 and variance is σ2. And we can get(5)ED1,t=ED1,t-1=μ1,d=αδ1-a,var⁡D1,t=var⁡D1,t-1=σ1,d2=α2σ21-a2.Similarly, Retailer 2 faces the same demand process and its model is(6)D2,t=1-αδ+aD2,t-1+1-αεt.And we have(7)ED2,t=ED2,t-1=μ2,d=1-αδ1-a,var⁡D2,t=var⁡D2,t-1=σ2,d2=1-α2σ21-a2.As callback center sends the rejected products to Retailer 1 and manufacture at a certain rate, we get the callback model as(8)Wt=θ1+θ2Dt-1=θ1+θ2D1,t-1+D2,t-1.Here, the partθ1Dt-1 is sent to Retailer 1 for being renewed and sold. The other part θ2Dt-1 is sent to manufacture; these rejected products can be fixed and then sold to retailers.


## 2.2. Inventory Policy

Order-up-to inventory policy is widely used in many researches, so we assist using this policy in this inventory management issue. The two retailers employ the OUT inventory policy: at the beginning of periodt, Retailer 1 knows the customer demand of period t-1asD1,t-1. So Retailer 1 can calculate the OUT policy point S1,t of period t and send an order q1,t to manufacture after calculating the order. After a lead-time l1, Retailer 1 receives the products from the manufacture at the starting of period t+l1. And the OUT level S1,t can be determined as(9)S1,t=D^1,tl1+zσ^1,tl1.In this formula,D^1,tl1 is the forecast demand of Retailer 1 in the lead-time l1 which depends on forecasting method and lead-time. σ^1,tl1 is the standard deviation of demand forecast error, and z is a given service level. Although it is difficult to accurately estimate the inventory holding costs and shortage costs in practice, the method using service level, which is defined as the probability for the first inventory fulfilling lead-time demand requirement to determine OUT level, is usually more feasible. The OUT level S1,t shown in (9) is optimal, depending on total inventory cost, for inventory system, while there is no fixed ordering cost or holding costs and shortage costs are proportional to the volume of first inventory or shortage (Luong [14]).Then, Retailer 1 sends an order of quantityq1,t to the manufacture at the start of period t. The order quantity can be given as(10)q1,t=S1,t-S1,t-1+D1,t-1-θ1Dt-1.Here,θ1Wt is the rejected products received from callback center, and S1,t-S1,t-1+D1,t-1 is the normal quantity depending on OUT level.Similarly, the OUT level of Retailer 2 at periodt is(11)S2,t=D^2,tl2+zσ^2,tl2.As Retailer 2 do not have rejected products, so its order quantity is(12)q2,t=S2,t-S2,t-1+D2,t-1.


## 2.3. Forecasting Method

In this paper, we assume the two retailers both use the MA forecasting method to forecast the demand in each lead-time. In the next section, the bullwhip effect is measured via the MA forecasting technique.According to the MA method, we can establish the demand forecast model of Retailer 1 as(13)D^1,tl1=l1k∑i=1kD1,t-i.Here,k is the number of date samples for MA forecasting method.As Retailer 2 faces the same consumer and sells the same products, we can consider that the two retailers use the same parameterk.So, the forecasted demand of Retailer 2 can be established as(14)D^2,tl2=l2k∑i=1kD2,t-i.


## 3. Measure of the Bullwhip Effect

In this section, we will measure the bullwhip effect of the order which manufacture receives from the retailers under the MA forecasting method. Furthermore, we will measure each order sent by Retailer 1 and Retailer 2 to compare their orders’ bullwhip effect.


### 3.1. The Bullwhip Effect of Retailer 1

The order quantity of Retailer 1 is(15)q1,t=S1,t-S1,t-1+D1,t-1-θ1Dt-1=D^1,tl1+zσ^1,tl1-D^1,t-1l1+zσ^1,t-1l1+D1,t-1-θ1Dt-1.Here we can prove that(16)σ^1,tl12=l11+a1-a-2a1-al11-a2+l1k2k1+a1-a-2a1-ak1-a2-2l1ka1-al11-ak1-avar⁡D1,t.The proof of (16) can be seen in Appendix.And we knowvar⁡D1,t=varD1,t-1. So that σ^1,tl1 does not change via t and has no influence on the bullwhip effect. So (15) equals(17)q1,t=D^1,tl1-D^1,t-1l1+D1,t-1-θ1Dt-1.Putting (13) into (17), we get(18)q1,t=1+l1k-θ1D1,t-1-l1kD1,t-k-1-θ1D2,t-1.Take the variance of (18) (19)var⁡q1,t=1+l1k-θ12var⁡D1,t-1+l1k2var⁡D1,t-k-1+θ12var⁡D2,t-1-21+l1k-θ1l1kcov⁡D1,t-1,D1,t-k-1-21+l1k-θ1θ1cov⁡D1,t-1,D2,t-1+2l1kθ1cov⁡D1,t-k-1,D2,t-1.Here, we can prove(20)var⁡D1,t-1=var⁡D1,t-k-1=var⁡D1,t,var⁡D2,t-1=α1-α2var⁡D1,t,cov⁡D1,t-1,D1,t-k-1=akvar⁡D1,t,cov⁡D1,t-1,D2,t-1=1-ααvar⁡D1,t,cov⁡D1,t-k-1,D2,t-1=ak1-ααvar⁡D1,t.The proof of (20) can be seen in Appendix.So we can get the variance of order quantity of Retailer 1 as(21)var⁡q1,t=1+l1k-θ12+l1k2-21+l1k-θ1l1kakvar⁡D1,t+θ121-αα2var⁡D1,t+2l1kak-1+l1k-θ1θ11-ααvar⁡D1,t.We measure the bullwhip effect by the ratio of the variance of order quantity that Retailer 1 sends to manufacture and the variance of demand that Retailer 1 faces, which has been used by many researchers. So the bullwhip effect, denoted as BWE, can be determined as(22)BWER1=var⁡q1,tvar⁡D1,t=1+l1k-θ12+l1k2-21+l1k-θ1l1kak+θ121-αα2+2l1kak-1+l1k-θ1θ11-αα.


### 3.2. The Bullwhip Effect of Retailer 2

Similarly, we get the order quantity of Retailer 2 as(23)q2,t=S2,t-S2,t-1+D2,t-1=D^2,tl2+zσ^2,tl2-D^2,t-1l2+zσ^2,t-1l2+D2,t-1.Simplify (23) in the same way and we can get(24)q2,t=D^2,tl2-D^2,t-1l2+D2,t-1=1+l2kD2,t-1-l2kD2,t-k-1.The variance of order quantity of Retailer 2 is(25)var⁡q2,t=1+l2k2var⁡D2,t-1+l2k2var⁡D2,t-k-1-21+l2kl2kcov⁡D2,t-1,D2,t-k-1=1+l2k2+l2k2-21+l2kl2kakvar⁡D2,t.The bullwhip effect of Retailer 2 is(26)BWER2=var⁡q2,tvar⁡D2,t=1+l2k2+l2k2-21+l2kl2kak.


### 3.3. The Bullwhip Effect of Total Order Received by Manufacture

As we know the total order is(27)qt=q1,t+q2,t.In the derivation above, we have got the order quantity of each retailer, so we get the quantity of total order as(28)qt=q1,t+q2,t=1+l1k-θ1D1,t-1-l1kD1,t-k-1+1+l2k-θ1D2,t-1-l2kD2,t-k-1.Taking the variance of this formula, we get(29)var⁡qt=1+l1k-θ12var⁡D1,t-1+l1k2var⁡D1,t-k-1+1+l2k-θ12var⁡D2,t-1+l2k2var⁡D2,t-k-1-21+l1k-θ1l1kcov⁡D1,t-1,D1,t-k-1+21+l1k-θ11+l2k-θ1cov⁡D1,t-1,D2,t-1-21+l1k-θ1l2kcov⁡D1,t-1,D2,t-k-1-2l1k1+l2k-θ1cov⁡D1,t-k-1,D2,t-1+2l1kl2kcov⁡D1,t-k-1,D2,t-k-1-21+l2k-θ1l2kcov⁡D2,t-1,D2,t-k-1.Here, we can prove(30)cov⁡D1,t-1,D1,t-k-1=α2akvar⁡Dt,cov⁡D1,t-1,D2,t-1=α1-αvar⁡Dt,cov⁡D1,t-1,D2,t-k-1=akα1-αvar⁡Dt,cov⁡D1,t-k-1,D2,t-1=akα1-αvar⁡Dt,cov⁡D1,t-k-1,D2,t-k-1=α1-αvar⁡Dt,cov⁡D2,t-1,D2,t-k-1=1-α2akvar⁡Dt.The proof of (30) can be seen in Appendix.So we can get the variance of total order quantity(31)var⁡qt=1+l1k-θ12+l1k2-21+l1k-θ1l1kakα2var⁡Dt+1+l2k-θ12+l2k2-21+l2k-θ1l2kak1-α2var⁡Dt+21+l1k-θ11+l2k-θ1-21+l1k-θ1l2kak-2l1k1+l2k-θ1ak+2l1kl2kα1-αvar⁡Dt.We use the same formula to quantify bullwhip effect as above(32)BWER=var⁡qtvar⁡Dt=1+l1k-θ12+l1k2-21+l1k-θ1l1kakα2+1+l2k-θ12+l2k2-21+l2k-θ1l2kak1-α2+21+l1k-θ11+l2k-θ1-21+l1k-θ1l2kak-2l1k1+l2k-θ1ak+2l1kl2kα1-α.


### 3.4. The Bullwhip Effect of Order Sent to Supplier by Manufacture

In this section, we will find what influences the bullwhip effect of supplier. The order is sent from manufacture to supplier, which depends on the demand and forecasting method of the Manufacture. Similar to previous research on the bullwhip effect of supplier, we use the similar technique to derive the BWE of supplier.As we know, the demand of manufacture is the total quantity of orders from the retailers, so we can get the demand of manufacture from (28) as(33)DM,t=qR,t=q1,t+q2,t=1+l1k-θ1D1,t-1-l1kD1,t-k-1+1+l2k-θ1D2,t-1-l2kD2,t-k-1.Using the OUT inventory policy, we can also get the inventory level of manufacture as(34)SM,t=D^M,tL+zσ^M,tL.So, the order that manufacture sends to supplier is(35)qM,t=SM,t-SM,t-1+DM,t-1-θ2Dt-1.Similarly, we knowσ^M,tL does not change via t and has no influence on the bullwhip effect. So (35) can be simplified as(36)qM,t=SM,t-SM,t-1+DM,t-1-θ2Dt-1=D^M,tL-D^M,t-1L+DM,t-1-θ2Dt-1.We assume the forecasting method of manufacture is also MA forecasting technique, so we have(37)D^M,tL=LK∑i=1KDM,t-i.Now we know the order is(38)qM,t=SM,t-SM,t-1+DM,t-1-θ2Dt-1=LK∑i=1KDM,t-i-LK∑i=1KDM,t-i-1+DM,t-1-θ2Dt-1=LKDM,t-1-LKDM,t-K-1+DM,t-1-θ2Dt-1=LK+1DM,t-1-LKDM,t-K-1-θ2Dt-1.We measure the bullwhip effect of supplier with the same technique as measuring that of retailers, which uses the variance of order quantity and the variance of demand faced by retailers.So the order of manufacture can be determined as(39)var⁡qM,t=var⁡LK+1DM,t-1-LKDM,t-K-1-θ2Dt-1=LK+12var⁡DM,t-1+LK2var⁡DM,t-K-1+θ22var⁡Dt-1-2LK+1LKcov⁡DM,t-1,DM,t-K-1-2LK+1θ2cov⁡DM,t-1,Dt-1+2LKθ2cov⁡DM,t-K-1,Dt-1.Here we have(40)var⁡DM,t-1=BWER,MA·var⁡Dt,var⁡DM,t-K-1=BWER,MA·var⁡Dt,var⁡Dt-1=var⁡Dt,cov⁡DM,t-1,DM,t-K-1=aK·BWER,MA·var⁡Dt,cov⁡DM,t-1,Dt-1=a·BWER,MA·var⁡Dt,cov⁡DM,t-K-1,Dt-1=a·BWER,MA·var⁡Dt.The proof of (40) can be seen in Appendix.So we can get the variance of order quantity of Supplier as(41)var⁡qM,t=LK+12BWER,MA+LK2BWER,MA+θ22-2LK+1LKaK·BWER,MA-2LK+1θ2aBWER,MA+2LKθ2a·BWER,MAvar⁡Dt.Similarly, we derive the BWE of supplier as(42)BWEM=var⁡qM,tvar⁡Dt=LK+12BWER,MA+LK2BWER,MA+θ22-2LK+1LKaK·BWER,MA-2LK+1θ2aBWER,MA+2LKθ2a·BWER,MA.


## 3.1. The Bullwhip Effect of Retailer 1

The order quantity of Retailer 1 is(15)q1,t=S1,t-S1,t-1+D1,t-1-θ1Dt-1=D^1,tl1+zσ^1,tl1-D^1,t-1l1+zσ^1,t-1l1+D1,t-1-θ1Dt-1.Here we can prove that(16)σ^1,tl12=l11+a1-a-2a1-al11-a2+l1k2k1+a1-a-2a1-ak1-a2-2l1ka1-al11-ak1-avar⁡D1,t.The proof of (16) can be seen in Appendix.And we knowvar⁡D1,t=varD1,t-1. So that σ^1,tl1 does not change via t and has no influence on the bullwhip effect. So (15) equals(17)q1,t=D^1,tl1-D^1,t-1l1+D1,t-1-θ1Dt-1.Putting (13) into (17), we get(18)q1,t=1+l1k-θ1D1,t-1-l1kD1,t-k-1-θ1D2,t-1.Take the variance of (18) (19)var⁡q1,t=1+l1k-θ12var⁡D1,t-1+l1k2var⁡D1,t-k-1+θ12var⁡D2,t-1-21+l1k-θ1l1kcov⁡D1,t-1,D1,t-k-1-21+l1k-θ1θ1cov⁡D1,t-1,D2,t-1+2l1kθ1cov⁡D1,t-k-1,D2,t-1.Here, we can prove(20)var⁡D1,t-1=var⁡D1,t-k-1=var⁡D1,t,var⁡D2,t-1=α1-α2var⁡D1,t,cov⁡D1,t-1,D1,t-k-1=akvar⁡D1,t,cov⁡D1,t-1,D2,t-1=1-ααvar⁡D1,t,cov⁡D1,t-k-1,D2,t-1=ak1-ααvar⁡D1,t.The proof of (20) can be seen in Appendix.So we can get the variance of order quantity of Retailer 1 as(21)var⁡q1,t=1+l1k-θ12+l1k2-21+l1k-θ1l1kakvar⁡D1,t+θ121-αα2var⁡D1,t+2l1kak-1+l1k-θ1θ11-ααvar⁡D1,t.We measure the bullwhip effect by the ratio of the variance of order quantity that Retailer 1 sends to manufacture and the variance of demand that Retailer 1 faces, which has been used by many researchers. So the bullwhip effect, denoted as BWE, can be determined as(22)BWER1=var⁡q1,tvar⁡D1,t=1+l1k-θ12+l1k2-21+l1k-θ1l1kak+θ121-αα2+2l1kak-1+l1k-θ1θ11-αα.


## 3.2. The Bullwhip Effect of Retailer 2

Similarly, we get the order quantity of Retailer 2 as(23)q2,t=S2,t-S2,t-1+D2,t-1=D^2,tl2+zσ^2,tl2-D^2,t-1l2+zσ^2,t-1l2+D2,t-1.Simplify (23) in the same way and we can get(24)q2,t=D^2,tl2-D^2,t-1l2+D2,t-1=1+l2kD2,t-1-l2kD2,t-k-1.The variance of order quantity of Retailer 2 is(25)var⁡q2,t=1+l2k2var⁡D2,t-1+l2k2var⁡D2,t-k-1-21+l2kl2kcov⁡D2,t-1,D2,t-k-1=1+l2k2+l2k2-21+l2kl2kakvar⁡D2,t.The bullwhip effect of Retailer 2 is(26)BWER2=var⁡q2,tvar⁡D2,t=1+l2k2+l2k2-21+l2kl2kak.


## 3.3. The Bullwhip Effect of Total Order Received by Manufacture

As we know the total order is(27)qt=q1,t+q2,t.In the derivation above, we have got the order quantity of each retailer, so we get the quantity of total order as(28)qt=q1,t+q2,t=1+l1k-θ1D1,t-1-l1kD1,t-k-1+1+l2k-θ1D2,t-1-l2kD2,t-k-1.Taking the variance of this formula, we get(29)var⁡qt=1+l1k-θ12var⁡D1,t-1+l1k2var⁡D1,t-k-1+1+l2k-θ12var⁡D2,t-1+l2k2var⁡D2,t-k-1-21+l1k-θ1l1kcov⁡D1,t-1,D1,t-k-1+21+l1k-θ11+l2k-θ1cov⁡D1,t-1,D2,t-1-21+l1k-θ1l2kcov⁡D1,t-1,D2,t-k-1-2l1k1+l2k-θ1cov⁡D1,t-k-1,D2,t-1+2l1kl2kcov⁡D1,t-k-1,D2,t-k-1-21+l2k-θ1l2kcov⁡D2,t-1,D2,t-k-1.Here, we can prove(30)cov⁡D1,t-1,D1,t-k-1=α2akvar⁡Dt,cov⁡D1,t-1,D2,t-1=α1-αvar⁡Dt,cov⁡D1,t-1,D2,t-k-1=akα1-αvar⁡Dt,cov⁡D1,t-k-1,D2,t-1=akα1-αvar⁡Dt,cov⁡D1,t-k-1,D2,t-k-1=α1-αvar⁡Dt,cov⁡D2,t-1,D2,t-k-1=1-α2akvar⁡Dt.The proof of (30) can be seen in Appendix.So we can get the variance of total order quantity(31)var⁡qt=1+l1k-θ12+l1k2-21+l1k-θ1l1kakα2var⁡Dt+1+l2k-θ12+l2k2-21+l2k-θ1l2kak1-α2var⁡Dt+21+l1k-θ11+l2k-θ1-21+l1k-θ1l2kak-2l1k1+l2k-θ1ak+2l1kl2kα1-αvar⁡Dt.We use the same formula to quantify bullwhip effect as above(32)BWER=var⁡qtvar⁡Dt=1+l1k-θ12+l1k2-21+l1k-θ1l1kakα2+1+l2k-θ12+l2k2-21+l2k-θ1l2kak1-α2+21+l1k-θ11+l2k-θ1-21+l1k-θ1l2kak-2l1k1+l2k-θ1ak+2l1kl2kα1-α.


## 3.4. The Bullwhip Effect of Order Sent to Supplier by Manufacture

In this section, we will find what influences the bullwhip effect of supplier. The order is sent from manufacture to supplier, which depends on the demand and forecasting method of the Manufacture. Similar to previous research on the bullwhip effect of supplier, we use the similar technique to derive the BWE of supplier.As we know, the demand of manufacture is the total quantity of orders from the retailers, so we can get the demand of manufacture from (28) as(33)DM,t=qR,t=q1,t+q2,t=1+l1k-θ1D1,t-1-l1kD1,t-k-1+1+l2k-θ1D2,t-1-l2kD2,t-k-1.Using the OUT inventory policy, we can also get the inventory level of manufacture as(34)SM,t=D^M,tL+zσ^M,tL.So, the order that manufacture sends to supplier is(35)qM,t=SM,t-SM,t-1+DM,t-1-θ2Dt-1.Similarly, we knowσ^M,tL does not change via t and has no influence on the bullwhip effect. So (35) can be simplified as(36)qM,t=SM,t-SM,t-1+DM,t-1-θ2Dt-1=D^M,tL-D^M,t-1L+DM,t-1-θ2Dt-1.We assume the forecasting method of manufacture is also MA forecasting technique, so we have(37)D^M,tL=LK∑i=1KDM,t-i.Now we know the order is(38)qM,t=SM,t-SM,t-1+DM,t-1-θ2Dt-1=LK∑i=1KDM,t-i-LK∑i=1KDM,t-i-1+DM,t-1-θ2Dt-1=LKDM,t-1-LKDM,t-K-1+DM,t-1-θ2Dt-1=LK+1DM,t-1-LKDM,t-K-1-θ2Dt-1.We measure the bullwhip effect of supplier with the same technique as measuring that of retailers, which uses the variance of order quantity and the variance of demand faced by retailers.So the order of manufacture can be determined as(39)var⁡qM,t=var⁡LK+1DM,t-1-LKDM,t-K-1-θ2Dt-1=LK+12var⁡DM,t-1+LK2var⁡DM,t-K-1+θ22var⁡Dt-1-2LK+1LKcov⁡DM,t-1,DM,t-K-1-2LK+1θ2cov⁡DM,t-1,Dt-1+2LKθ2cov⁡DM,t-K-1,Dt-1.Here we have(40)var⁡DM,t-1=BWER,MA·var⁡Dt,var⁡DM,t-K-1=BWER,MA·var⁡Dt,var⁡Dt-1=var⁡Dt,cov⁡DM,t-1,DM,t-K-1=aK·BWER,MA·var⁡Dt,cov⁡DM,t-1,Dt-1=a·BWER,MA·var⁡Dt,cov⁡DM,t-K-1,Dt-1=a·BWER,MA·var⁡Dt.The proof of (40) can be seen in Appendix.So we can get the variance of order quantity of Supplier as(41)var⁡qM,t=LK+12BWER,MA+LK2BWER,MA+θ22-2LK+1LKaK·BWER,MA-2LK+1θ2aBWER,MA+2LKθ2a·BWER,MAvar⁡Dt.Similarly, we derive the BWE of supplier as(42)BWEM=var⁡qM,tvar⁡Dt=LK+12BWER,MA+LK2BWER,MA+θ22-2LK+1LKaK·BWER,MA-2LK+1θ2aBWER,MA+2LKθ2a·BWER,MA.


## 4. The Behavior of the Bullwhip Effect Measure and Numerical Simulation

From the last section we have derived the formulation of bullwhip effect of each retailer and manufacture; we realize that it is a function with the lead-time, forecasting precision, callback index, and marketing share rate. In the following part, we use method of numerical analysis to indicate the impact of all the parameters on the bullwhip effect of retailers and manufacture in Figure2.Figure 2
The impact ofl1 on bullwhip effect.Figures2 and 3 show the behavior of BWER1 with respect to various values of l1 or k on the autocorrelation coefficient. It is observed from the presentation that the bullwhip effect of Retailer 1 decreases when the autocorrelation coefficient increases, which is a general conclusion. In Figure 2, we use α as an argument, fixed other variables, and plotted other variables in the case of l1=1, l1=2, and l1=3. It shows that we can make the bullwhip decrease by cutting down the lead-time of Retailer 1.Figure 3
The impact ofk on bullwhip effect.


### 4.1. The Impact of Parameters on the Bullwhip Effect of Retailer 1

Similarly shifting the parameterk while fixing the others in Figure 3, we can observe that if we improve forecasting precision of Retailer 1, the bullwhip effect will also decrease. And the decrease rate of bullwhip effect via k slows down when the forecasting precision increases.Figures4 and 5 show the different behaviors of bullwhip effect via various α when we shift l1 or a. From the two pictures, we can have a general conclusion about the market share index that bullwhip effect decreases fast and then increases when the market share changes from 0 to 0.7. As we observed in the presentation, bullwhip effect always has a minimum when the value of α varies between 0.1 and 0.2. In BWER1’s decreasing area, it reduces very fast while changing α, but it increases slowly in the following area.Figure 4
The impact ofα on bullwhip effect (various l1).Figure 5
The impact ofα on bullwhip effect (various a).Figure6 shows the variation of bullwhip effect when the callback index of Retailer 1 changes from 0 to 0.7. A general conclusion can be observed that when the callback index improves, BWER1 of Retailer 1 will decrease, and the rate will slow down. The same conclusion can be observed when we change other parameters such as lead-time or forecasting precision.Figure 6
The impact ofθ1 on bullwhip effect (various a).


### 4.2. The Impact of Parameters on the Bullwhip Effect of Manufacture

In this section, we will analyze the impact of lead-time, forecasting precision, callback index, and market share rate on the bullwhip effect of the order of manufacture.(a) The Impact of Retailers’ Lead-Time on Bullwhip Effect.When we research the impact of retailers’ lead-time on bullwhip effect of manufacture, we consider two situations:( 1 ) When the lead-time satisfies l1=l2, the impact of retailers’ lead-time of bullwhip when we set k=3, L=4, K=4, θ1=0.3, and θ2=0.2 is presented by Figure 7. We must point out that market share has no influence on bullwhip effect of manufacturer in this situation. Hence, market share can get any number between 0 and 1. We can see that the BWE of the manufacture goes up at first when -1<a<-0.6. And then it falls down slowly when -0.6<a<0.4. Finally, the BWE decreases fast when 0.4<a<1. We can also investigate that the BWE becomes grater when the lead-time of the retailers becomes greater.Figure 7
The bullwhip effect on different lead-time (l1=l2).Figure7 shows the bullwhip effect on different lead-time (l1=l2).Figure8 shows the bullwhip effect on different lead-time (l1≠l2).Figure 8
The bullwhip effect on different lead-time (l1≠l2).( 2 ) When l1≠l2, the bullwhip is investigated in a different way. The bullwhip will increase fast no matter which retailer’s lead-time is larger. And the picture also investigates that if the sum of retailers’ lead-times is fixed, the larger the difference of lead-times is, the bigger the bullwhip effect of manufacture will be, which means the difference in the retailers’ lead-times will also impact the bullwhip effect.(b) The Impact of Manufacture’s Lead-Time on Bullwhip Effect.Figure 9 shows the impact of manufacture’s lead-time on bullwhip when the values of l1, l2, k, a, θ1, θ2, and α are fixed. It can be observed that the bullwhip effect will increase fast when the manufacture has a larger lead-time.Figure 9
The impact of manufacture’s lead-time on bullwhip effect.(c) The Impact of Forecasting Precision on Bullwhip Effect.Figure 10 shows different bullwhip effect curves with various values of forecasting index of retailers. When the forecasting index is even, the bullwhip effect is an increasing function in the area -1<a<0 but changes to a decreasing function in the area 0<a<1. But if the forecasting index is odd, bullwhip effect is always a decreasing function when a changes from −1 to 1.Figure 10
The impact ofk on bullwhip effect.So when we draw the curves of bullwhip effect with different forecasting indexes of manufacture, we have to draw different pictures.Figure11 shows the impact of K on bullwhip effect when k is even. We can see that all the curves of bullwhip effect are similar. When a increases, the bullwhip turns bigger at first, and then comes down when a turns bigger than 0. And if the k is odd as shown in Figure 12, the bullwhip will be much bigger when K is odd too and a is negative. But the bullwhip effect is always a decreasing function when a changes from −1 to 1.Figure 11
The impact ofK on bullwhip effect (k is even).Figure 12
The impact ofK on bullwhip effect (k is odd).If we fix the other parameters as shown in Figure13 and just increase the value of forecasting index of retailers or manufacture, we find that the bullwhip effect decreases soon when the forecasting index increases from 1 to 4. The decreasing rate becomes smaller when it enlarges k or K continuously. Finally, the BWE is stable near a value around 0. So, the span of the parameters k and K is really counts for the forecasting. The BWE would become smaller when the parameters k and K becomes greater.Figure 13
The impact ofK  and  k on bullwhip effect.(d) The Impact of Callback Index on Bullwhip Effect. First we simulate different behaviors of the bullwhip effect when θ1 or θ2 changes from 0 to 0.8. The presentation is shown in Figures 14 and 15. It is observed that the bullwhip effect decreases when the callback index increases for both θ1 and θ2. But θ1 has a much bigger effect than θ2.Figure 14
The impact ofθ1 on bullwhip effect.Figure 15
The impact ofθ2 on bullwhip effect.And if we consider the situation where the sum of callback index is a fixed value, for example,θ1+θ2=0.5, the bullwhip effect curve is shown as in Figure 16.Figure 16
The impact of callback index on bullwhip effect (θ is fixed).Figure16 shows that the bullwhip effect decreases when θ1 enlarges from 0 to 0.5, which means the increase of θ2 makes the bullwhip effect bigger. So, if the total callback index is a constant number, we shall make θ1 larger.(e) The Impact of Marketing Share Index on Bullwhip Effect. When we research the impact of marketing share rate on the bullwhip effect, we must consider the lead-time of retailers. If the retailers’ lead-time is a constant, the expression of retailers’ bullwhip effect is(43)BWER,MA=var⁡qtvar⁡Dt=1+l1k-θ12+l1k2-21+l1k-θ1l1kak.It is observed from the formulation that the bullwhip has nothing to do with the marketing share rate whenl1=l2. So we can conclude that the marketing share rate has no impact on manufacture’ bullwhip effect in this case either.We further probe into the case whenl1≠l2; the curve of bullwhip effect is shown in Figure 17.Figure 17
The impact ofα on bullwhip effect (θ is fixed).Figure7 shows the impact of marketing share rate on the bullwhip effect of manufacture. The bullwhip effect increases as α increases if the lead-time of Retailer 1 is bigger than Retailer 2. But if the lead-time of Retailer 1 is smaller, the bullwhip effect will be different. It shows the two situations can be descripted as a symmetric relation.


## 4.1. The Impact of Parameters on the Bullwhip Effect of Retailer 1

Similarly shifting the parameterk while fixing the others in Figure 3, we can observe that if we improve forecasting precision of Retailer 1, the bullwhip effect will also decrease. And the decrease rate of bullwhip effect via k slows down when the forecasting precision increases.Figures4 and 5 show the different behaviors of bullwhip effect via various α when we shift l1 or a. From the two pictures, we can have a general conclusion about the market share index that bullwhip effect decreases fast and then increases when the market share changes from 0 to 0.7. As we observed in the presentation, bullwhip effect always has a minimum when the value of α varies between 0.1 and 0.2. In BWER1’s decreasing area, it reduces very fast while changing α, but it increases slowly in the following area.Figure 4
The impact ofα on bullwhip effect (various l1).Figure 5
The impact ofα on bullwhip effect (various a).Figure6 shows the variation of bullwhip effect when the callback index of Retailer 1 changes from 0 to 0.7. A general conclusion can be observed that when the callback index improves, BWER1 of Retailer 1 will decrease, and the rate will slow down. The same conclusion can be observed when we change other parameters such as lead-time or forecasting precision.Figure 6
The impact ofθ1 on bullwhip effect (various a).


## 4.2. The Impact of Parameters on the Bullwhip Effect of Manufacture

In this section, we will analyze the impact of lead-time, forecasting precision, callback index, and market share rate on the bullwhip effect of the order of manufacture.(a) The Impact of Retailers’ Lead-Time on Bullwhip Effect.When we research the impact of retailers’ lead-time on bullwhip effect of manufacture, we consider two situations:( 1 ) When the lead-time satisfies l1=l2, the impact of retailers’ lead-time of bullwhip when we set k=3, L=4, K=4, θ1=0.3, and θ2=0.2 is presented by Figure 7. We must point out that market share has no influence on bullwhip effect of manufacturer in this situation. Hence, market share can get any number between 0 and 1. We can see that the BWE of the manufacture goes up at first when -1<a<-0.6. And then it falls down slowly when -0.6<a<0.4. Finally, the BWE decreases fast when 0.4<a<1. We can also investigate that the BWE becomes grater when the lead-time of the retailers becomes greater.Figure 7
The bullwhip effect on different lead-time (l1=l2).Figure7 shows the bullwhip effect on different lead-time (l1=l2).Figure8 shows the bullwhip effect on different lead-time (l1≠l2).Figure 8
The bullwhip effect on different lead-time (l1≠l2).( 2 ) When l1≠l2, the bullwhip is investigated in a different way. The bullwhip will increase fast no matter which retailer’s lead-time is larger. And the picture also investigates that if the sum of retailers’ lead-times is fixed, the larger the difference of lead-times is, the bigger the bullwhip effect of manufacture will be, which means the difference in the retailers’ lead-times will also impact the bullwhip effect.(b) The Impact of Manufacture’s Lead-Time on Bullwhip Effect.Figure 9 shows the impact of manufacture’s lead-time on bullwhip when the values of l1, l2, k, a, θ1, θ2, and α are fixed. It can be observed that the bullwhip effect will increase fast when the manufacture has a larger lead-time.Figure 9
The impact of manufacture’s lead-time on bullwhip effect.(c) The Impact of Forecasting Precision on Bullwhip Effect.Figure 10 shows different bullwhip effect curves with various values of forecasting index of retailers. When the forecasting index is even, the bullwhip effect is an increasing function in the area -1<a<0 but changes to a decreasing function in the area 0<a<1. But if the forecasting index is odd, bullwhip effect is always a decreasing function when a changes from −1 to 1.Figure 10
The impact ofk on bullwhip effect.So when we draw the curves of bullwhip effect with different forecasting indexes of manufacture, we have to draw different pictures.Figure11 shows the impact of K on bullwhip effect when k is even. We can see that all the curves of bullwhip effect are similar. When a increases, the bullwhip turns bigger at first, and then comes down when a turns bigger than 0. And if the k is odd as shown in Figure 12, the bullwhip will be much bigger when K is odd too and a is negative. But the bullwhip effect is always a decreasing function when a changes from −1 to 1.Figure 11
The impact ofK on bullwhip effect (k is even).Figure 12
The impact ofK on bullwhip effect (k is odd).If we fix the other parameters as shown in Figure13 and just increase the value of forecasting index of retailers or manufacture, we find that the bullwhip effect decreases soon when the forecasting index increases from 1 to 4. The decreasing rate becomes smaller when it enlarges k or K continuously. Finally, the BWE is stable near a value around 0. So, the span of the parameters k and K is really counts for the forecasting. The BWE would become smaller when the parameters k and K becomes greater.Figure 13
The impact ofK  and  k on bullwhip effect.(d) The Impact of Callback Index on Bullwhip Effect. First we simulate different behaviors of the bullwhip effect when θ1 or θ2 changes from 0 to 0.8. The presentation is shown in Figures 14 and 15. It is observed that the bullwhip effect decreases when the callback index increases for both θ1 and θ2. But θ1 has a much bigger effect than θ2.Figure 14
The impact ofθ1 on bullwhip effect.Figure 15
The impact ofθ2 on bullwhip effect.And if we consider the situation where the sum of callback index is a fixed value, for example,θ1+θ2=0.5, the bullwhip effect curve is shown as in Figure 16.Figure 16
The impact of callback index on bullwhip effect (θ is fixed).Figure16 shows that the bullwhip effect decreases when θ1 enlarges from 0 to 0.5, which means the increase of θ2 makes the bullwhip effect bigger. So, if the total callback index is a constant number, we shall make θ1 larger.(e) The Impact of Marketing Share Index on Bullwhip Effect. When we research the impact of marketing share rate on the bullwhip effect, we must consider the lead-time of retailers. If the retailers’ lead-time is a constant, the expression of retailers’ bullwhip effect is(43)BWER,MA=var⁡qtvar⁡Dt=1+l1k-θ12+l1k2-21+l1k-θ1l1kak.It is observed from the formulation that the bullwhip has nothing to do with the marketing share rate whenl1=l2. So we can conclude that the marketing share rate has no impact on manufacture’ bullwhip effect in this case either.We further probe into the case whenl1≠l2; the curve of bullwhip effect is shown in Figure 17.Figure 17
The impact ofα on bullwhip effect (θ is fixed).Figure7 shows the impact of marketing share rate on the bullwhip effect of manufacture. The bullwhip effect increases as α increases if the lead-time of Retailer 1 is bigger than Retailer 2. But if the lead-time of Retailer 1 is smaller, the bullwhip effect will be different. It shows the two situations can be descripted as a symmetric relation.


## 5. Conclusion

This paper draws the following conclusions: the callback system and the market share will affect the whole supply chain and will especially influence the bullwhip effect of different layers of participants. We can see that the call back factor is very important for the bullwhip effect on a reverse supply chain. For single retailer or the total order, the decrease of lead-time and the increase of demand forecasting precision will reduce bullwhip effect effectively. In the meantime, the introduction of callback system can cut down the bullwhip effect of retailer demand, and a higher callback index relates to a lower bullwhip effect. From the perspective of marker share, retailers should take action to improve their market shares in order to ensure the drop of bullwhip effect results from introducing the callback system. As for manufacturer, retailers’ callback index and manufacturer’s callback information will decrease its bullwhip effect to some extent. The higher the callback index is, the lower the manufacturer’s bullwhip effect will be. If the comprehensive callback index is fixed, the larger the retailers’ callback amount is, the lower the bullwhip effect will be. This paper also explores the market share of retailers and finds that only if the lead-times of retailers are different, manufacturer’s bullwhip effect is affected by market share. And a bigger share has more positive significance to retailer whom has a shorter lead-time. This research may be valuable in the perspective of management.



---

*Source: 3650148-2018-04-23.xml*