A Minimum Variance Control Theory Perspective on Supply Chain

Article pubs.acs.org/IECR

A Minimum Variance Control Theory Perspective on Supply Chain Lead Time Uncertainty Hua Xu and Gang Rong* State Key Laboratory of Industrial Control Technology, Institute of Cyber-systems and Control, Zhejiang University, Hangzhou, 310027, P. R. China ABSTRACT: This paper addresses the lead time uncertainty problem in supply chain systems. In our previous paper [Xu et al. Ind. Eng. Chem. Res.2010,49,8644], we have investigated the impact of demand uncertainty on supply chains. Here we adopt a two-echelon supply chain model, which is basically the same as that used in the research of demand uncertainty. But the fixed lead time setting is replaced by a Markovian lead time model. Since the lead time varies with time, the dynamic characteristics of the supply chain model are different from that used in the demand uncertainty research. We make a comparison analysis of these differences from the view of dynamic systems. On the basis of the above analysis, we adopt two fundamental lemmas of the minimum variance control theory as the foundation for replenishment rules design and analysis. Then we derive formulas of the Order-up-to policy and the generalized Order-up-to policy with time-varying lead time. Moreover, we offer the variant forms of the above strategies when the lead time information is incomplete. Given the strategies, we analyze the influence of lead time information on the order and inventory variances and corresponding costs. This work, together with our previous paper on demand uncertainty, may provide a coherent control theory based perspective on these two different types of uncertainties in a supply chain. customer demand was generalized as an AR(1) (a first order autoregressive) or ARMA(1,1) (a mixed first order autoregressive moving average) process. They exactly quantified the bullwhip effect when the supply chain members adopted the order-up-to policy and the more sophisticated minimum mean square error estimation method. Both of the above works provided closedform analysis of the order variance in supply chain systems. But they did not consider the influences of the stochastic lead time on the properties of the inventory in the supply chain. Boute et al.16 considered a two-echelon supply chain, which consisted of a retailer and a manufacturer, as a productioninventory system. The lead time of the supply chain was solely determined by the production time of the manufacturer, who dealt with the retailer’s orders in a first-come-first-out manner. This queue manner guaranteed that the retailer’s orders never crossed over in any time. In this situation, the iid lead time assumption could not be maintained. Moreover, in their work, the production lead time was set to be endogenous (i.e., the retailer could control the stream of orders at the manufacturer’s production queue thus the production lead time probability distribution). Given the order-up-to replenishment policy or the generalized order-up-to policy, the authors used the matrix analytic method to estimate the complicated lead time distribution and adopted numerical experiments to analyze the whole supply chain system’s stochastic properties. They found that the retailer’s smoothed orders did not increase the inventory variance due to the correlation between the distribution of the replenishment lead time and the distribution of the inventory.

1. INTRODUCTION One of the main tasks of supply chain management (SCM) is to cope with uncertainties arising from the demand and the supply sides of a supply chain. Issues of demand uncertainty have attracted much interest. From the viewpoints of the theory of dynamic systems and stochastic processes, researches range from theoretical analysis1−8 of the impacts of the demand uncertainty to the optimal design of SCM strategies for large scale supply chain systems under demand uncertainty.9−13 But in contrast to the heavy discussion of the demand uncertainty problem, few studies have been conducted on supply side uncertainties, one of which is the uncertainty in the supply chain lead time. A lead time is defined as the period of time from the moment a customer places an order to the moment the customer receives the order. It is typically treated as a known constant in the research of demand uncertainty. However, in most cases, the lead time varies with time and its exact value is not always available to decision makers. Thus, it is more accurate to model the supply chain lead time as a random process and it is necessary to investigate the impact of a stochastic lead time on the dynamics or stochastic properties of the whole supply chain system. Kim et al.14 extended previous works, which mainly focused on the impacts of the demand uncertainty and the constant lead time on the bullwhip effect (i.e., order variance amplification from downstream to upstream), by taking the effects of stochastic lead times into account. They assumed the supply chain lead time was a series of identical and independent distributed (iid) random variables. They derived analytical expressions of the order variance under both the iid demand and the iid lead time. Theses expressions held when supply chain members used the order-upto policy (OUT) and the moving average forecasting method. Duc et al.15 also investigated the bullwhip effect in a two-echelon supply chain system with the iid lead time. But in their work, the © 2012 American Chemical Society

Received: Revised: Accepted: Published: 9275

July 3, 2011 June 11, 2012 June 12, 2012 June 12, 2012 dx.doi.org/10.1021/ie2014185 | Ind. Eng. Chem. Res. 2012, 51, 9275−9286

Industrial & Engineering Chemistry Research

Article

As a sequel of the above work, Boute et al.17,18 analyzed two kinds of supply chain systems under the iid demand patterns. One contained a manufacturer with a flexible capacity (i.e., the manufacturer could increase its capacity to keep the lead time constant), the other contained a manufacturer with an inflexible capacity (i.e., the capacity of the manufacturer was limited thus the lead time varied with the retailer’s orders). They found that win−win solutions could be obtained in both of the supply chain systems if the retailer could smooth the orders to a specific degree. Note that in Boute’s works, the endogenous lead time came from the scenario that the manufacturer had only one retailer and the retailer’s orders totally determined the demand of the manufacturer. Also, there is another scenario: the manufacturer has many customers and the retailer’s orders make up such a small portion of the manufacturer’s total orders that the retailer’s orders could hardly influence the random fluctuations of the production lead time. This scenario leads to exogenous supply chain lead time, which can be described by a class of more analytical stochastic models. Kanplan19 developed a stochastic model without orders crossover and used this model to generate the stationary probability distribution of exogenous lead times. Later, Nahmias20 and Ehrhardt21 gave more concise and clearer explanations of this model. In Kanplan’s work, the lead time for a single order was considered as an event related to the realization of a sequence of iid random variables. These random variables characterized the number of orders which entered into the retailer’s inventory in every time instant. Song and Zipkin22 proposed a general Markovian modeling framework for exogenous lead times. Four types of lead time models could be derived from this framework: the fixed lead time model (used in the demand uncertainty research), Kaplan’s lead time model, a discrete time queue model (due to the limited processing capacity of the manufacturer), and a supply system model with machine breakdown. These authors also presented an important concept, which is called “order coverage”, for designing SCM strategies in a supply chain with a time-variable lead time. On the basis of the queue model described by Song and Zipkin, Chen and Yu23 constructed a two-echelon supply chain system and used the Markov chain to model its stochastic lead time. They considered two scenarios: the retailer knew the exact value of the current lead time and the retailer could only infer this value by historical information about order arrivals. They presented an algorithm to compute the optimal order amount that minimized current expected inventory costs. They also compared the supply chain’s performance in the above two scenarios through simulations. Liu et al.24 developed a more complicated stochastic model that could explicitly consider the shipment congestions. The lead time depended on the distribution of all the shipment across different stages. Assuming that the retailer in the supply chain used the order-up-to policy, they evaluated the retailer’s inventory performance with different tracking information. From the above literature review, we can see that researchers have provided deep insights into the lead time uncertainty problem: they have developed different stochastic lead time models; they have proposed methods to design SCM strategies under lead time uncertainty; they also have considered the impacts of the lead time information on the supply chain performance. However, due to the complicated dynamic properties of the supply chain models with stochastic lead times, few researches have provided analytical formulas of the replenishment rules under lead time uncertainty. Also, the impacts of

replenishment rules on supply chain performances are only investigated through simulations. From a control theory point of view, a supply chain is essentially a dynamic material balance system. A supply chain with a stochastic lead time is a coherent extension of the traditional one with a fixed lead time, which is used in the demand uncertainty research. As there have been analytical frameworks for demand uncertainty, we can naturally raise the question whether there is a similar analytical framework for stochastic lead times, which can degenerate to the framework for demand uncertainty when the lead time is fixed. We address this issue in this paper. On the basis of a stochastic exogenous lead time model,23 we propose an analytical framework, which inherits the methodology of the minimum variance control theory, for SCM strategies design and analysis. This study is a sequel to our previous work8 on demand uncertainty. Both of the studies use the same methodology and provide a unified perspective on the lead time uncertainty and the demand uncertainty in a supply chain system.

2. DESCRIPTION OF A SUPPLY CHAIN SYSTEM AND ITS SUPPLY PROCESS Consider a two-echelon supply chain model23 depicted in Figure 1, which consists of a retailer and a manufacturer.

Figure 1. A supply chain model.

The retailer has an inventory to meet the customer demand Dt, which is a series of iid random variables with an expected value d and a standard deviation σD. At the beginning of each period, the retailer receives goods from the manufacturer. Then he observes and satisfies the customer demand. Finally, he places orders with the manufacturer. The manufacturer has a production line to produce the retailer’s replenishment orders. His production activities can be summarized in a stochastic lead time model. In this model, the exogenous lead time Lt is described by a Markov chain with state space S = {1, 2, ..., M}, whereM is a positive integer. The one step transition probability of the Markov chain is pij = p[Lt+1 = j | Lt = i]. To ensure the orders do not cross over, pij = 0 for any j < i − 1. Thus the transition matrix P has a semiupper triangular form. We can further use a discrete queueing model to generate the above lead time process. A more detailed explanation of the lead time model is presented in the Appendix.

3. DYNAMIC ANALYSIS OF THE RETAILER’S INVENTORY CONTROL MODEL First we consider the retailer’s inventory model with a fixed lead time (Lt ≡ L). The order placed in time period t will be received in time period t + L. The dynamic material balance of the retailer can be represented by the following difference equation: NSt = NSt − 1 + Ot − 1 − Dt (1) where NSt is the net inventory level in time period t and Ot is the order placed to the manufacturer in time period t. Note that NSt is the deviation from the safety stock level T, which is always 9276

dx.doi.org/10.1021/ie2014185 | Ind. Eng. Chem. Res. 2012, 51, 9275−9286


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thing when the lead time varies with time. Fortunately, we can still use the concept of the inventory position, which is not related to the lead time (eq 5), to obtain the basic dynamic material balance equation. From eq 6, we can see that the size of the current order Ot determines the net inventory level in time period t + Lt. But note that the order placed in the next period (i.e., t + 1) will be only received in time period t + 1 + Lt+1. No order arrives in the time interval [t + 1 + Lt, t + Lt+1]. Thus we can also determine the net inventory level in this time interval:

greater than zero. Detail explanations can be seen in our previous paper.8 Equation 1 is equivalent to the following equation:8 t−1

t+L

∑

NSt + L = NSt +

Oh + Ot −

h=t−L+1

∑

Dh

h=t+1

(2)

Now let IPt be the inventory position in time period t. It includes the net inventory level plus the outstanding orders (i.e., orders requisitioned but not yet received). When the lead time is fixed, t‑1 the outstanding orders are Σh=t − L+1Oh and the inventory position can be represented as

t + 1 + Lt

t−1

IPt = NSt +

∑

Dh

h=t+1

Oh

t + 2 + Lt

(3)

h=t−L+1

∑

NSt + 2 + Lt = IPt + Ot −

Then, eq 2 becomes

Dh

h=t+1

···

t+L

NSt + L = IPt + Ot −

∑

NSt + 1 + Lt = IPt + Ot −

∑

Dh

h=t+1

t + Lt + 1

(4)

NSt + Lt+1 = IPt + Ot −

IPt = IPt − 1 + Ot − 1 − Dt

∑

Dh

h=t+1

According to eq 1 and eq 3, we can derive the following equation:

(7)

We can say the current order Ot covers the time interval [t + Lt, t + Lt+1](i.e., order coverage22). Then eqs 6 and 7 become

(5)

We can also intuitively understand eq 5 from Figure 2.

t+τ

NSt + τ = IPt + Ot −

∑ h=t+1

Dhτ ∈ [Lt , Lt + 1]

(8)

According to the description of the Markov chain model in section 2, we have the relationship Lt+1 ≥ Lt − 1. If Lt+1 ≡ Lt, the supply chain lead time is fixed and eq 8 degenerates to eq 4. If Lt+1 = Lt − 1, [Lt, Lt+1] in eq 8 becomes an empty set and the size of the current order Ot does not influence the net inventory level (but it will influence the inventory position). If Lt+1 > Lt, the current order covers more than one period. Figure 3 and Figure 4

Figure 2. Fluid representation of the inventory position.

Also note that eq 5 comes from the fixed lead time model. But it holds when the lead time is time-variable. From Figure 2, we can see that if we consider the outstanding orders (note that the sizes of outstanding order are influenced by the time-variable lead time) and the actual net inventory level (note that this is also influenced by the time-variable lead time) as a generalized “inventory level” IPt, the generalized inventory level is irrelevant to the time-variable lead time. We can determine the inventory position IPt without considering variations of the lead time. Thus eq 5 is true no matter whether the lead time is fixed or time-variable. Now we consider the situation when the inventory model has a time-variable lead time. When the lead time varies with time, the order placed in time period t will be received in the time period t + Lt. Because the orders are processed in a first-come-firstserviced manner, all the orders placed before the time period t will be received before the time period t + Lt. Thus, eq 4 remains true and we can modify this equation as follows:

Figure 3. The sequence diagram of the outstanding orders with a fixed lead time.

show the sequence diagrams of the outstanding orders with a fixed lead time and a time-variable lead time. In Figure 3, the current order exactly determines one net inventory level. In Figure 4, we can see that the current order determines net inventory levels in a time interval. In summary, eq 5 and eq 8 describe the dynamic characteristics of the retailer’s inventory model with a time-variable lead time. Under the conditions implied by eq 8, if the retailer wants to determine the size of the current order Ot to keep the net inventory level, the retailer needs to know the exact values of

t + Lt

NSt + Lt = IPt + Ot −

∑ h=t+1

Dh

(6)

Note that we can obtain an explicit expression of the outstanding orders when the lead time is fixed while we cannot do the same 9277



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We can obtain some basic equations: AINVt+ = (|AINVt | + AINVt )/2 AINVt− = (|AINVt | − AINVt )/2 AINVt+ − AINVt− = AINVt

(11)

From the above equations, we can get the following equation: E[AINVt+] − E[AINVt ] = E[AINVt−]

(12)

Thus we can rewrite eq 10 as follows: E[C INV ] = h × E(AINVt+) + b × E(AINVt−) =hE[AINVt+] + b(E[AINVt+] − E[AINVt ]) = (h + b)E[AINVt+] Figure 4. The sequence diagram of the outstanding orders with a timevariable lead time.

Notice the following inequality: ⎛ |AINVt | + AINVt ⎞ ⎟ E[AINVt+] = E⎜ ⎠ ⎝ 2 1 = (E[|AINVt |] + E[AINVt ]) 2 1 1 = E(|AINVt |) ≤ (E(AINVt 2))1/2 2 2 (Cauchy−Schwartz Inequality)

Lt and Lt+1. However, such information is not always available. Later we will discuss the SCM strategy design and analysis with the complete lead time information and the incomplete lead time information.

4. OPERATIONS COST STRUCTURE When the lead time varies with time, an order precisely covers a time interval. Here we average the net inventory levels in this time interval and let a new variable AINVt denote the result. The expression of the new variable is as follows:

=

E[C INV ] ≤

∑τ =t+L1 NSt + τ t

Lt + 1 − Lt + 1

when Lt+1 = Lt − 1, define AINVt = 0. We use AINVt to calculate the expected inventory cost of the retailer. Note that the inventory cost is calculated at the end of the time interval between two consecutive arrived orders. If AINVt > 0, a holding cost h per unit is incurred. If AINVt < 0, a backlog cost b per unit shortage is incurred. The expected inventory cost can be represented as follows: AINV+t

(14)

(h + b) σAINV 2

(15)

Note that the closeness of the above inequality depends on the specific probability distribution of AINVt. If we assume that AINVt follows the normal distribution, we can derive the exact expression:8

(9)

E[C INV ] = h × E(AINVt+) + b × E(AINVt−)

1 σAINV 2

Finally, we can get the expected inventory cost:

L

AINVt =

(13)

E[C INV ] =

(h + b) σAINV 2π

Then the inequality turns to an exact expression and we can evaluate how close the inequality is. Also eq 15 indicates that the upper bound of the expected inventory cost is proportional to the standard deviation. Thus it is reasonable to control the variance of AINVt to reduce the expected inventory cost. The same considerations hold for the expected order costs. We can reduce the expected order cost directly by controlling the variance of Ot.

(10)

AINV−t

where = max(AINVt, 0), = max(−AINVt, 0). Note that the proposed inventory cost measure (eq 10) is an intuitive generalization of the cost function for the fixed lead time model. When the lead time is fixed (Lt+1 = Lt = L), eq10 becomes hE(NS+t+L) + bE(NS−t+L), which is the well- known newsboy model. Further, if NSt follows a normal distribution N(0,σNS), the expected inventory cost can be explicitly represented by the expression E[C INV ] = ((h + b)/ (2π)1/2)σNS, which indicates that the expected inventory cost is proportional to the standard deviation.8 Thus, we can directly use the minimum variance control strategy to control the inventory variance and the inventory cost. However, when the lead time is time varying, it may be not true to assume that AINVt follows the normal distribution. Now we aim to obtain a simple but reasonable approximation of the expected inventory costs when we do not know the specific probability distribution of AINVt. Suppose that we know the expected value and the variance of the random variable AINVt: E[AINVt] = 0, E[AINV2t ] = σ2AINV. We begin to calculate eq 10.

5. REPLENISHMENT RULES UNDER COMPLETE LEAD TIME INFORMATION 5.1. Preliminaries. Now we consider replenishment rules design and analysis under lead time uncertainty. Although the lead time varies randomly, we can assume the retailer exactly knows Lt and Lt+1 in time period t. In this case, the retailer has “complete lead time information”. If the retailer is not able to obtain exact values of Lt or Lt+1 in time period t and the retailer needs to estimate the unknown value from other information sources, we say that the retailer has “incomplete lead time information”. To systematically deal with our problem, we adopt two fundamental lemmas of the minimum variance control theory. These lemmas clearly show the difference between the cases of complete information and incomplete information. 9278



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Lemma 1: Complete State Information.25 Assume u is a function of x,y and the function g(x,y,u) has a unique minimum with respect to u. Then min E[g (x , y , u)] = E[min g (x , y , u)]

u(x , y)

Thus, the optimal order quantity can be represented as follows: d − IPt (23) 2 Substituting eq 5 into eq 23, we can get the following equation: Ot = (Lt + 1 + Lt )

(16)

u(x , y)

⎡ ⎤ ⎡ ⎤ d d Ot − Ot − 1 = ⎢(Lt + 1 + Lt ) − IPt ⎥ − ⎢(Lt + Lt − 1) − IPt − 1⎥ ⎣ ⎦ ⎣ ⎦ 2 2

25

Lemma 2: Incomplete State Information. Assume u is a function of y and the function f(y,u) = E[g(x,y,u)|y] has a unique minimum with respect to u. Then min E[g (x , y , u)] = E{min E[g (x , y , u)|y]} u(y)

y

u(y)

(17)

= (Lt + 1

where E[•|y] denotes the conditional mean given y and Ey[•] denotes the mean value with respect to the probability distribution of y. Note that the above two lemmas offer basic ideas to deal with a stochastic optimization problem with a single random variable (Lemma 1) and multiple random variables (Lemma2). These lemmas also illustrate the importance of specifying the information pattern or data available for the decision. Here we divide the future customer demand into two categories:8 predictable and unpredictable parts. Dt + k = d − εt + kk ≥ 1 τ

(24)

Thus we obtain an equivalent form of eq 23: d + Dt (25) 2 Note that if the lead time is fixed, eq 25 becomes Ot = Dt. We can also obtain this result from previous works about demand uncertainty when we assume the customer demand is iid. Given the optimal policy, the average net inventory levels in the order coverage AINVt can be expressed as follows: Ot = (Lt + 1 − Lt − 1)

(18)

Lt + 1

τ

h=1

AINVt =

(19)

h=1

where {εt} is a series of iid random variables with expected value 0 and the standard deviation σD. In this case, the one-step minimum mean square error estimator is the expected value d. 5.2. Replenishment Rules Design. (1) Minimum variance control: the order-up-to (OUT) policy. We directly select the variance of AINVt as the performance index: 2

J = E(AINVt ) = σAINV

τ

∑ ∑

∑ Dt + h = τd − ∑ εt + h

2

d − (IPt − IPt − 1) 2 d − Lt − 1) − (Ot − 1 − Dt ) 2

= (Lt + 1 − Lt − 1)

εt + i

τ = Lt i = 1

Lt + 1 − Lt + 1

(26)

(2) Minimum variance control where the order variance is restricted: the generalized order-up-to policy. In order to control the variance amplification (i.e., the bullwhip effect) in the supply chain, we need to take the order variance into consideration. The objective function is modified as: J *1 = min E(AINVt 2 + r02Ot2)

(27)

Ot

where r0 ≥ 0is a penalty factor, which can be adjusted arbitrarily. Expanding eq 27, we can obtain the following equation:

(20)

We expand eq 20 using eq 8, eq 9 and eq 19. We obtain the following equation:

J1* = min E[AINVt 2 + r0 2Ot 2] Ot

2 ⎧⎡ τ L ∑τ =t +L1 ∑i = 1 εt + i ⎤ ⎪ d t ⎥ = min E⎨⎢(IPt + Ot ) − (Lt + 1 + Lt ) + Ot 2 (Lt + 1 − Lt + 1) ⎥⎦ ⎪⎢⎣ ⎩ ⎫ ⎪ 2 2 + r0 Ot ⎬ ⎪ ⎭ 2 ⎧ ⎧⎡ ⎫⎫ d⎤ = E⎨min⎨⎢(IPt + Ot ) − (Lt + 1 + Lt ) ⎥ + r0 2Ot 2⎬⎬ ⎣ ⎦ 2 ⎭⎭ ⎩ Ot ⎩ ⎡ (∑Lt + 1 ∑τ ε )2 ⎤ τ = Lt i=1 t+i ⎥ + E⎢ ⎢ (L − L + 1)2 ⎥ t ⎣ t+1 ⎦

Notice the equation Cov(A,B) = 0 (Cov means the covariance of two random variables). According to Lemma1, we can obtain the following equation: 2

⎪

We can directly calculate the optimal rule from the following equation:

Ot

2 ⎧ ⎡ d⎤ ⎫ = E⎨min⎢(IPt + Ot ) − (Lt + 1 + Lt ) ⎥ ⎬ 2⎦ ⎭ ⎩ Ot ⎣ ⎤ ⎡ Lt + 1 τ ⎢ ( ∑ ∑ εt + i)2 ⎥ ⎥ ⎢ τ=L i=1 t ⎥ + E⎢ 2 ⎢ (Lt + 1 − Lt + 1) ⎥ ⎥ ⎢ ⎦ ⎣ ⎪

⎪

⎪

(28)

2

J * = min E(AINVt ) = E{min(AINVt )} Ot

⎪

d 2 ∂ ⎡⎣(IPt + Ot ) − (Lt + 1 + Lt ) 2 ⎤⎦ + r0 2Ot 2

{

⎪

∂Ot

} =0

(29)

We obtain the generalized order-up-to policy: ⎡ ⎤ d Ot = β ⎢(Lt + 1 + Lt ) − IPt ⎥ ⎣ ⎦ 2

(22) 9279

(30)



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where β = (1/1 + r02) and 0 < β < 1. Combining eq 5 and eq 30, we get the following equation: ⎡ ⎤ d Ot = (1 − β)Ot − 1 + β ⎢(Lt + 1 − Lt − 1) + Dt ⎥ ⎣ ⎦ 2

E[AINVt 2] ⎤ ⎡ Lt + 1 τ ⎢ ( ∑ ∑ εt + i)2 ⎥ ⎥ ⎢ τ=L i=1 t ⎥ = E⎢ 2 ⎢ (Lt + 1 − Lt + 1) ⎥ ⎥ ⎢ ⎦ ⎣

(31)

Under this policy, AINVt can be represented as follows: Lt + 1

∑ ∑ AINVt =

= E⎡⎣ (Lt + 1 − Lt + 1)(ε1 + ... + εLt ) + (Lt + 1 − Lt )εLt + 1 + (Lt + 1 − Lt − 1)εLt + 2 + ... + εLt + 1 2 /((Lt + 1 − Lt + 1))⎤⎦

τ

(

εt + i

⎛ 1⎞ + ⎜1 − ⎟Ot − Lt + 1 ⎝ β⎠

)

τ = Lt i = 1

Lt + 1

(32)

⎡ (L − Lt )(2Lt + 1 − 2Lt + 1) ⎤ 2 ⎥σD = ⎢Lt + t + 1 6(Lt + 1 − Lt + 1) ⎣ ⎦

Note that we can use the parameter β (0 < β < 1) in the generalized policy to balance the inventory variance and the order variance. A detail explanation of the generalized policy can be seen in previous papers.7,8,18 5.3. Statistical Properties of the Orders and the Inventory. Now we analyze the stochastic properties of the supply chain when the replenishment rules are given. First we need to calculate the autocovariance of the stochastic process Lt, which is described by a Markov chain in section 2. We can use the following equation:

(38)

Note that E[AINVt ] = ELt{E[AINVt |Lt]}. We assume that Lt follows the stationary distribution of the Markov chain and obtain the value of E[AINVt2|Lt]. Finally we can get the variance of AINVt. (2) Analysis of the generalized order-up-to policy. Here we introduce a shift operator z (i.e., z−1Xt = Xt−1) and rewrite eq 31 as follows: 2

Cov(Lk , L1)

Ot =

= E[Lk L1] − E[Lk ]E[L1] =

= E{E[Lk L1|L1]} − (E[L1]) L1

= E{L1E[Lk |L1]} − (E[L1]) L1

=

(33)

where E[Lk|L1]can be calculated by using the k − firstep transition matrix P(k−1). We also force the Markov chain to take its stationary distribution as the starting state distribution. It means that L1 follows the stationary distribution of the Markov chain. Define Ut = Lt+1 − Lt−1 as the lead time estimator. On the basis of eq 33, we can calculate its autocovariance. (1) Analysis of the order-up-to policy. The expected value of the orders can be represented as follows: d d E[Ut ] + E[Dt ] = (L − L) + d = d 2 2

∞

k=1

k=1

(39)

d 2

∞

∑ β(1 − β)k− 1E(Ut− k+ 1)+ k=1

∞

∑ β(1 − β)k− 1E(Dt− k+ 1) = d k=1

(40)

The variance of the order is (34)

∞

Var(Ot ) =

d2 Var[∑ β(1 − β)k − 1Ut − k + 1] 4 k=1 ∞

+Var[∑ β(1 − β)k − 1Dt − k + 1]

(35)

k=1 ∞

2

=

d Var[∑ β(1 − β)k − 1Ut − k + 1] 4 k=1 ∞

+ ∑ β 2(1 − β)2k − 2 Var(Dt ) k=1

(36)

2

From eq 36, we can see that the bullwhip effect is unavoidable when the supply chain lead time varies with time. If the lead time fluctuates widely (i.e., Var(Ut) is large), the order-up-to policy sends a highly variable orders to the manufacturer. The stochastic properties of AINVt can be obtained as follows: ⎤ ⎡ Lt + 1 τ ⎢ ( ∑ ∑ εt + i) ⎥ ⎥ ⎢ τ=L i=1 t ⎥=0 E[AINVt ] = E⎢ ⎢ (Lt + 1 − Lt + 1) ⎥ ⎥ ⎢ ⎦ ⎣

∞

∑ β(1 − β)k − 1Ut− k + 1+ ∑ β(1 − β)k − 1Dt− k + 1

E(Ot ) =

Note that Cov(Ut, Dt) = 0. The lead time and the demand process are uncorrelated. We can calculate the bullwhip effect of the order-up-to policy: d2 G0 = Var(Ot ) − Var(Dt ) = Var(Ut ) > 0 4

d 2

The expected value of the order is

The variance of the orders is d2 Var(Ut ) + Var(Dt ) 4

⎛d ⎞ Ut − k + 1 + Dt − k + 1⎟ ⎠ 2

∑ β(1 − β)k − 1⎝⎜ k=1

2

Var(Ot ) =

⎡d ⎤ β Ut + Dt ⎥ −1 ⎢ ⎣ ⎦ 2 1 − (1 − β)z ∞

2

E[Ot ] =

2

=

∞

d Var[∑ β(1 − β)k − 1Ut − k + 1] 4 k=1 β2 + Var(Dt ) 1 − (1 − β)2 ∞

(37)

=

β d2 Var[∑ β(1 − β)k − 1Ut − k + 1]+ Var(Dt ) β − 4 2 k=1

=

β d2 Var(K t ) + Var(Dt ) 4 2−β (41)

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where Kt = Σk∞= 1β(1 − β)k−1Ut−k+1.We use the following formula to approximate the infinite series Kt: N1

∞

Kt =

lead time information. The generalized order-up-to policy can be derived by analogy. Since the retailer only knows Ht in the time period t, we can not directly calculate eq 20. According to Lemma 2, we should rewrite it as follows:

∑ β(1 − β)k− 1Ut− k+ 1≈ ∑ β(1 − β)k− 1Ut− k+ 1 k=1

k=1

J = E(AINVt 2) = E[E(AINVt 2|Ht )]

= βUt + β(1 − β)Ut − 1 + β(1 − β)2 Ut − 2 + ... + β(1 − β)N1 − 1Ut − N1 + 1

J * = min E(AINVt 2) = E[min E(AINVt 2|Ht )]

(42)

Ot

where N1 is a positive integer. Let ci = β(1 − β)i−1. Then the variance of Kt can be written as follows: N1

Var(K t ) ≈

(46)

Ht

Ht

(47)

Ot

Consider the computation of E(AINVt |H t). It can be represented as follows: 2

N1

∑ ∑ cicjCov(Ui , Uj) (43)

i=1 j=1

Now we focus on the stochastic properties of AINVt under the generalized order-up-to policy.The expected value is based on eq 32: ⎛ Lt+1 τ ⎞ ⎜ ∑ ∑ εt + i ⎟ ⎜ τ = Lt i = 1 ⎟ ⎛ 1⎞ E(AINVt ) = E⎜ ⎟ + ⎜1 − ⎟E(Ot ) β⎠ ⎜ Lt + 1 − Lt + 1 ⎟ ⎝ ⎜ ⎟ ⎝ ⎠ ⎛ 1⎞ = ⎜1 − ⎟d β⎠ ⎝

We can obtain the optimal replenishment rules from the formulaA1. The formula A1 can be rewritten as

(44)

Note that 0 < β < 1 and the expected value of AINVt is a negative value.The variance can be expressed as follows:

Note that the probability distribution ofLtis determined by a realization ofHt. First we compute the inner conditional expected value B2. LetLt = aand Ht = h.We can obtain the following equation:

⎛ Lt+1 τ ⎞ ⎜ ∑ ∑ εt + i ⎟ 2 ⎜ τ = Lt i = 1 ⎟ ⎛ 1⎞ Var(AINVt ) = Var ⎜ ⎟ + ⎜1 − ⎟ Var(Ot ) β⎠ ⎜ Lt + 1 − Lt + 1 ⎟ ⎝ ⎜ ⎟ ⎝ ⎠ ⎡ (L − Lt )(2Lt + 1 − 2Lt + 1) ⎤ 2 ⎥σD = ⎢Lt + t + 1 6(Lt + 1 − Lt + 1) ⎣ ⎦ ⎛ 1⎞ + ⎜1 − ⎟ Var(Ot ) β⎠ ⎝

2 ⎧⎡ ⎫ d⎤ B2 (a) = E⎨⎢(IPt + Ot ) − (Lt + 1 + Lt ) ⎥ Lt = a , Ht = h⎬ 2⎦ ⎭ ⎩⎣ 2 ⎡ (2a − 1)d ⎤ = pa , a − 1 ⎢(IPt + Ot ) − ⎥ ⎣ ⎦ 2 2 ⎡ (2m)d ⎤ + pa , a ⎢(IPt + Ot ) − ⎥ + ... ⎣ 2 ⎦

2

2 ⎡ (m + M )d ⎤ + pa , M ⎢(IPt + Ot ) − ⎥ ⎣ ⎦ 2

(45)

By comparing eq 45 to eq 38, we can find the variance of AINVt increases under the generalized order-up-to policy.

M

=

6. REPLENISHMENT RULES UNDER INCOMPLETE LEAD TIME INFORMATION Now we consider the following scenario that the retailer does not know the exact values of Lt or Lt+1, but some related information is available (e.g., the lead time of the most recent received order23). We assume that the retailer knows the stochastic lead time process. Then if he obtains the exact value of Lt, he can acquire the probability distribution of Lt+1 by using the transition probability pij = p[Lt+1 = j|Lt = i]. Also we assume there is an information source Ht with the sample space S = {1, 2, ..., N}. The retailer obtains Ht = h in the time period t.Then he can acquire the probability distribution of Lt by using the known conditional probability p′ht = p[Li = i|Ht = h]. 6.1. Replenishment Rules Design. Here we mainly focus on the design of the order-up-to policy with the incomplete

2 ⎛ (a + i)d ⎞ ⎟ pa , i ⎜IPt + Ot − ⎠ ⎝ 2 i=a−1

∑

(50)

Then we calculate the outer conditional expected value: M

J2 (h) = E {B2 (a)|Ht = h} = Lt (Ht ) M

∑ pha′ B2(a) a=1

M

2 ⎛ (a + i )d ⎞ ′ ∑ pai ⎜IPt + Ot − = ∑ pha ⎟ ⎝ ⎠ 2 a=1 i=a−1

(51)

Thus we can obtain the optimal rule from the following equation: ∂J2 ∂Ot

M

= IPt + Ot −

d (∑ p′ a + 2 a = 1 ha

M

M

∑ pha′ ∑ a=1

pa , i i) = 0

i=a−1

(52) 9281



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Finally we obtain the order-up-to policy with incomplete lead time information: M

Ot =

=

=

d (∑ p′ a + 2 a = 1 ha

M

M

∑ pha′ ∑ a=1

Please note that there are some relationships.

pai i) − IPt

i=a−1 M

d (E[Lt |Ht = h] + 2

∑ pha′ E[Lt + 1|Lt = a]) − IPt

d (E[Lt |Ht = h] + 2

∑ pha′ E[Lt + 1|Lt = a , Ht = h]) − IPt

a=1 M a=1

d (E[Lt |Ht = h] + E[Lt + 1|Ht = h]) − IPt 2 d = (Lt̂ | Ht + Lt̂ + 1 | Ht ) − IPt 2 =

(53)

Equation 59 indicates that the one step optimal prediction error is a series of iid random variables. The key point is that the formula A2 satisfies the following relationship:

where L̂ t|Ht = E[Lt|Ht] and L̂ t+1|Ht = E[Lt+1|Ht]. They are the minimum mean square error estimators for Lt and Lt+1. By comparing eq 23 to eq 53, we can find that the structure of the order-up-to policy does not change when the lead time information becomes incomplete. The difference between the cases of incomplete information and complete information is that optimal estimators substitute for the exact values. This result implies that the optimal replenishment rules can be separated into two parts. One is the optimal lead time estimator and the other one is the control law when the lead time is known exactly. It is essentially the version of the well-known separation theorem in stochastic control theory. Combining eq 5 and eq 53, we obtain the following equation: Ot =

d ̂ (Lt + 1 | Ht + Lt̂ | Ht − Lt̂ | Ht−1 − Lt̂ − 1 | Ht−1) + Dt 2

E(La + 1|L 2) = E(E(La + 1|La)|L 2) = E(Lâ + 1 | a|L 2)

Another relationship can be expressed as follows: Cov(Ut , et ) − Cov(et , et ) = E[(Lt + 1 − Lt − 1)(Lt + 1 − Lt̂ + 1 | t )] − E[(Lt + 1 − Lt̂ + 1 | t )(Lt + 1 − Lt̂ + 1 | t )] = −E[Lt − 1(Lt + 1 − Lt̂ + 1 | t )] + E[Lt̂ + 1 | t (Lt + 1 − Lt̂ + 1 | t )] = − E {Lt − 1[E(Lt + 1|Lt − 1) − E(Lt̂ + 1 | t |Lt − 1)]} Lt − 1

(54)

+ E{Lt̂ + 1 | t [E(Lt + 1|Lt ) − E(Lt̂ + 1 | t |Lt )]} Lt

According to eq 8, eq 9, and eq 53, we can obtain the expression of AINVt: AINVt =

=0 (61)

d ̂ (Lt + 1 | Ht + Lt̂ | Ht − Lt + 1 − Lt ) 2 Lt + 1

+

According to eq 59 and eq 61, we can calculate the variance of Vt:

τ

∑ ∑

Var(AINVt )

εt + i

= Var(Ut ) + Var(et ) + Var(et − 1) − 2Cov(Ut , et ) + 2Cov(Ut , et − 1) − 2Cov(et , et − 1)

τ = Lt i = 1

Lt + 1 − Lt + 1

(55)

6.2. Replenishment Rules Analysis: A Simplified Case. In this section, we want to investigate the stochastic properties of the supply chain system. However, a comprehensive analysis will cover the details of the stochastic process Ht, which is only in an abstract form in this paper. Thus our analysis is limited to the Markov chain of Lt. We assume the retailer knows Lt exactly in every time period, but does not know the exact value of Lt+1. Thus we substitute Lt for Ht in eqs 54 and 55 and let L̂ t+1|t = E[Lt+1|Lt]. We can obtain the following equations: d Ot = (Lt + Lt̂ + 1 | t − Lt − 1 − Lt̂ | t − 1) + Dt 2 Lt + 1

= Var(Ut ) + 2Cov(Ut , et − 1)

Var(Ot ) = =

d2 Var(Vt ) + Var(Dt ) 4 d2 d2 Var(Ut ) + Var(Dt ) + Cov(Ut , et − 1) 4 2 (63) 2

From eq 63, we can see that the order variance increases (d /2) Cov(Ut,et−1) (in our numerical example, Cov(Ut,et−1) is always greater than zero) due to the loss of information of Lt+1. Thus information loss may cause a larger bullwhip effect in supply chains. The variance of AINVt can be expressed as follows:

(56)

τ

εt + i d ̂ τ = Lt i = 1 AINVt = (Lt + 1 | t − Lt + 1) + 2 Lt + 1 − Lt + 1

(62)

Thus the variance of the order can be represented as follows:

∑ ∑

d2 Var(et ) 4 ⎡ (L − Lt )(2Lt + 1 − 2Lt + 1) ⎤ 2 ⎥σD + ⎢Lt + t + 1 6(Lt + 1 − Lt + 1) ⎣ ⎦

(57)

Var(AINVt ) =

Define Vt = Lt + L̂ t+1|t − Lt−1 − L̂ t|t−1 and the prediction error et = Lt+1 − L̂ t+1|t . We also have the following equation: Vt = Ut − et + et − 1

(60)

(58) 9282

(64)



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Comparing eq 38 to eq 64, we can find that the variance of AINVt increases when the lead time information is incomplete. The prediction error of Lt+1 will cause more expected inventory costs.

7. NUMERICAL EXAMPLES To illustrate and verify our theoretical analysis, we consider the following numerical examples. 7.1. The Effects of the Lead Time on the Order Variance. We set different values of the parameter λ to construct different lead time processes (Details of the stochastic lead time process can be seen in Appendix). The lead time becomes more fluctuating when λ increases from 0 to 1 or decreases from 2 to 1. For customer demand, we set d = 4 and σD = 1. Figure 5 and Figure 6 show the fluctuations of the retailer’s orders in two different stochastic lead time settings (λ = 0.3 and

Figure 7. Comparison of demand and order variability (OUT and G_OUT policy).

Figure 5. The retailer’s orders behavior with λ = 0.3.

Figure 6. The retailer’s orders behavior with λ = 0.8.

λ = 0.8). The orders are generated by the replenishment rules eq 25. The demand patterns of the retailer are the same in both settings. We can see that the orders become more unstable when the lead time becomes more unstable. This phenomenon confirms our explanations about eq 36. 7.2. Comparison of the Order-up-to Policy and the Generalized Order-up-to Policy. Figure 7 and Figure 8 show the comparison between the order-up-to policy and the generalized order-up-to policy with β = 0.5 in the case of the complete lead time information. We set λ = 0.7. From Figure 7, we can see that orders under the OUT policy are more variable than the customer demand while the order variability under the generalized OUT policy is relatively close to the customer demand variability. However, from Figure 8, we observe that the expected value of AINVt under the generalized OUT policy obviously deviates from zero and its variance increases a little.

Figure 8. Comparison of AINV variability (OUT and G_OUT policy).

Table 1 records means and variances of the inventory and the orders under the different replenishment rules, which is helpful to 9283



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Table 1. Statistical Data of the OUT Policy and the Generalized OUT Policy variables (OUT) demand mean demand variance order mean order variance AINV mean AINV variance

theory experiment 4.0 1.0 4.0 3.9775 0 1.1580

4.0 1.0 4.0001 3.9716 −0.0021 1.1559

variables (G_OUT) demand mean demand variance order mean order variance AINV mean AINV variance

theory 4.0 1.0 4.0 1.5036 −4.0 2.6616

experiment 4.0 1.0 4.0001 1.5006 −4.0001 2.6626

quantitatively investigate the above phenomenon. The mean and variance of each variable are obtained by theoretical calculation and the statistical results of the simulation experiment. We can see that the theoretical inference is almost consistent with the experiment statistics. 7.3. Comparison of the Complete and Incomplete Lead Time Information. Figure 9 and Figure 10 show the

Figure 10. Comparison of AINV variability (complete and incomplete information).

Table 2. Statistical Data of the Order and the Inventory in Two Different Situations variables

theory

experiment

order_variance (complete information) AINV_variance (complete information) order_variance (incomplete information) AINV_variance (incomplete information) information loss: (d2/2)(Cov(Ut,et−1)) information loss: (d2/4)Var(et)

3.9775 1.1559 6.0609 2.6059 2.0832 1.4480

3.9716 1.1580 6.0501 2.6201 2.0785 1.4621

8. CONCLUSIONS In this work, we have investigated a simple two-echelon supply chain system with a Markov chain lead time model. We have proposed an analytical framework, which relies heavily on the concepts and techniques of the traditional minimum variance control theory, for retailer’s replenishment rules design and analysis under exogenous lead time uncertainty. In this framework, we can see clearly the effects of lead time uncertainty and lead time information completeness on the inventory and the order patterns. These results may help readers to understand uncertainty and the value of information in a supply chain. Also, since a supply chain often can be described by difference or differential equations and its uncertainties often can be characterized as stochastic processes, we hope that a more general stochastic control theory can be applied to analyze and cope with theses uncertainties.

Figure 9. Comparison of demand and order variability (complete and incomplete information).

performances of the order-up-to policy with complete lead time information and incomplete information. We set λ = 0.7. From these figures, we can see that the order variance and the AINVt variance increase a lot when the lead time information becomes incomplete (the exact value of Lt+1 is not known to retailers). Information loss directly leads to additional inventory and order costs. It will not only influence the individual benefits but also the whole system’s benefits through a larger bullwhip effect. Table 2 records the statistical data of the inventory and the order. We can see that the difference of the orders variances is equal to (d2/2)Cov(Ut, et−1) and the difference of the variances of AINVt is equal to (d2/4)Var(et).

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APPENDIX: A DISCRETE QUEUEING MODEL23 Assume the manufacturer has a production line, which deals with the retailer’s orders in a first-come-first-serviced manner. The capacity of the production line is C per period. The manufacturer 9284



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receives the retailer’s orders and other orders from outside. An order from the retailer is so small in size that the lead time for the order is essentially the time the order spends waiting in the production queue. Let Xt be the total orders that the manufacturer receives in period t. Xt follows the Poisson distributions with the parameter λ: Pr(X t = k) = e−λλ k /k!

p11 = Pr(A1|B1 ∪ B2) Pr(A1 ∩ (B1 ∪ B2)) Pr(B1 ∪ B2) Pr(A1 ∩ B1) + Pr(A1 ∩ B2) = Pr(B1) + Pr(B2) Pr(A1|B1)P(B1) + Pr(A1|B2)P(B2) = Pr(B1) + Pr(B2) Pr(B1) = Pr(A1|B1) Pr(B1) + Pr(B2) Pr(B2) + Pr(A1|B2) Pr(B1) + Pr(B2) =

(A1)

Let Qt be the total length of the production queue at the beginning of the period t. We can obtain the dynamic model of the production queue. +

Q t + 1 = min{(Q t − C) + X t , N }

Consider the computation of Pr{A1|B1}:

(A2)

Pr(A1|B1) = Pr(Q t + 1 = 1|Q t = 0) + Pr(Q t + 1 = 0

where Y+ = max(Y, 0) and N is the maximum length of the production queue (the manufacturer rejects orders when the queue length reaches N). The retailer places an order at the beginning of period t. The manufacturer will be able to deliver this order when the current production queue of size Qt is cleared. Then we can determine Lt as follows: Lt = 1 + min{l: (l + 1)C ≥ Q t }

|Q t = 0)

f (2) f (3) f ̅ (3)⎤ ⎥ f (1) f (2) f ̅ (2)⎥ ⎥ f (0) f (1) f ̅ (1)⎥ ⎥ f (0) f ̅ (0)⎥⎦ 0

(A7)

Note that Qt+1 = min{Xt, 4} if Qt = 0. Thus Pr(Qt+1 = 0 | Qt = 0) = f(0) and Pr(Qt+1 = 1 | Qt = 0) = f(1). We can obtain Pr(A1 | B1) = f(0) + f(1). By analogy, we can obtain Pr(A1 | B1) = f(0) + f(1). Finally, eq A6 becomes: ⎛ Pr(B1) p11 = Pr(A1|B1)⎜ ⎝ Pr(B1) + Pr(B2)

(A3)

where l is a nonnegative integer. When C increases a lot, according to eq A3, we can see that the time-varying lead time Lt becomes a constant 1 (Lt ≡ 1). In other words, the time-varying lead time model becomes a fixed lead time model. This phenomenon is understandable. Note that C is the capacity of the production line. A production line with a large capacity can finish orders as quickly as possible. Thus the supply chain system has a minimum fixed lead time 1. In the numerical example of this paper, we set C = 1, N = 4. Thus Lt = max(1, Qt) with S = {1,2,3,4}. Let f(k) = Pr(Xt = k), f ̅(k) = Pr(Xt > k). According to eqs A1−A3, we can obtain the transition matrix of the Markov chain as follows: ⎡ f (0) + f (1) ⎢ ⎢ f (0) P=⎢ ⎢0 ⎢ ⎢⎣ 0

(A6)

+

⎞ Pr(B2) ⎟ Pr(B1) + Pr(B2) ⎠

= f (0) + f (1)

(A8)

Other pij (i ≥ 2, j ≥ i − 1) can be computed by analogy. Note that the transition probability pij is a function of λ, which represents the average number of orders received by the manufacturer in one period (E[Xt] = λ). The larger the value of λ is, the more total orders the manufacturer receives. Also note that the production capacity per period C equals one. If λ = 0, there are no input orders, thus Lt = 1 and the supply chain system has a fixed lead time. If λ < C = 1, the average order input is less than the average output (the manufacturer may finish the retailer’s orders at the current period), thus it is probable that Qt = 0, Lt = 1. If λ > C = 1, the average order input exceeds the capacity, it is quite probable that there is congestion in the production line and Lt > 1. From Figure A1, We can see the steady state distribution of the lead time process with different values of λ.

(A4)

Here we explain how to obtain the above matrix. We can see that pij = 0 if j < i − 1 since orders do not cross over. Also note that pij ≠ 0 if j = i − 1. It means that orders placed at consecutive time instants may arrive at the same time. Now we begin to calculate p11. p11 = Pr{Lt + 1 = 1|Lt = 1} = Pr{Q t + 1 = 1 ∪ Q t + 1 = 0 |Q t = 1 ∪ Q t = 0}

(A5)

Let A1 = {Qt+1 = 1 ∪ Qt+1 = 0}, B1 = {Qt = 0}, and B2 = {Qt = 1}, then p11 can be represented as follows.

Figure A1. The steady state distribution of the lead time process. 9285



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(21) Ehrhardt, R. (s,S) policies for a dynamic inventory model with stochastic lead times. Oper. Res. 1984, 32, 121−132. (22) Song, J. S.; Zipkin, P. H. Inventory control with information about supply conditions. Manage. Sci. 1996, 42, 1409−1419. (23) Chen, F.; Yu, B Quantifying the value of leadtime information in a single-location inventory system. Manuf. Serv. Oper. Manage. 2005, 7, 144−151. (24) Liu, M.; Srinivasan, M. M.; Vepkhvadze, N. What is the value of realtime shipment taking information? IIE Trans. 2009, 41, 1019−1034. (25) Åström,K. J. Introduction to Stochastic Control Theory; Academic Press: New York, 1970.

AUTHOR INFORMATION

Corresponding Author

*Tel.: 86-571-87953145. Fax: 86-571-87952277. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge the support of the National Nature Science Foundation of China (No.60421002) and the 973 Program of China (No.2012CB720500).

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A Minimum Variance Control Theory Perspective on Supply Chain

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