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Measuring the bullwhip effect in supply chains with vector autoregressive demand process
S.K. CHAHARSOOGHI, A. SADEGHI Tarbiat Modares University; Department of Industrial Engineering; College of Engineering; P.O. Box 14115-143; Tehran; Iran; Tel: +98(21) 82883345; E-mail:
[email protected];
[email protected]
ABSTRACT In this paper, we study bullwhip effect problem in a two-echelon supply chain, which includes two products according to our assumptions. The main aim of the paper is to provide a measure for bullwhip effect that enables the analysis and reduction of this phenomenon in supply chains with two products. VAR (1) time series, a suitable pattern for demand modeling in a two product supply chain is utilized. The use of simple moving average method for lead-time demand forecasting, and OUT ordering policy are due to their wide range application in the real world. A general form of bullwhip measure has been derived and demonstrated to prove that an explicit expression for bullwhip effect measure does not exist based on present approach on the bullwhip measure. However, bullwhip effect measure may still exist in limited cases. Condition that bullwhip effect can be removed is described; and finally, bullwhip effect in a two-product supply chain is demonstrated via a numerical example.
Keywords: Bullwhip Effect; Forecasting; Supply Chain; Time series
INTRODUCTION Nowadays, outsourcing has an important role in industrial environments. Consequently, manufacturers are settling in SCM (supply chain management systems) that are growing at a very high rate (exponentially). Raw material suppliers, part manufacturers, final product
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assemblers, distributors, retailers, and final customers are various sectors of supply chains. Demand amplification is a major obstacle to achieve coordination and maintain harmony within different stages of supply chains. Many companies have observed increasing fluctuation in orders while moving up from downstream sites to upstream sites. The result is a loss of supply chain profitability. The first recorded documentation of this status was by Forrester (1958). He used industrial dynamics approach to show amplification of demand variability among supply chain. After that time many researchers such as Goodman (1974), Blinder (1982 & 1986), Blanchard (1983), Kahn (1987), Baganha and Cohen (1995), Metters (1997) continued investigation about ordering variation. Sterman (1989) developed Beer game at MIT. He proposed it as an evidence for existence demand amplification in supply chains. Lee, et al. (1997), introduced five main causes of this phenomenon: demand forecast updating, order batching, price fluctuation, rationing, and non-zero lead-time (1997). Understanding these causes of the bullwhip effect can be useful for managers to find suitable solutions for haltering and controlling it. Chen, et al. (2000), quantified and derived a lower bound for the bullwhip effect in a simple supply chain. Dejonckheere, et al. (2003), proposed a control theory approach for measuring bullwhip effect and suggested a new general replenishment rule that can reduce variance amplification significantly. Disney and Towill (2003) introduced an ordering policy that results in taming bullwhip effect. Zhang (2004) considered three forecasting methods for a simple inventory control system and presented three measures for bullwhip effect based on three forecasting methods. Kim et al. (2006) investigated stochastic lead-time and investigated role of information sharing on bullwhip effect. Chandra and Grabis (2005) measured bullwhip effect when order size is calculated according to multiple step forecasts using autoregressive models. Luong (2007) investigated effects of autoregressive coefficient and lead-time on bullwhip effect when MMSE forecasting method is used. Luong and Phien
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(2007) research was based on order of autoregressive demand pattern. They showed that in high order of demand pattern, bullwhip effect could be reduced when lead-time increases. Makui and Madadi (2007) utilized the Lyapunov exponent and provided a measure for bullwhip effect. They presented useful results on the behavior of the bullwhip effect by investigating the mathematical relationships. Gaalman and Disney (2007) investigated the behavior of the proportional order up to policy for ARMA (2, 2) demand with arbitrary leadtimes. Jaksic and Rusjan (2008) demonstrated that certain replenishment policies could be inducers of the bullwhip effect. Although many researches have investigated bullwhip effect; yet, more investigations are still needed to study it, minimize its effect, and quantifying it in order to propose solutions for complex supply chains. This research considers a two-echelon supply chain consisting of one retailer and one supplier. This supply chain produces two products in which demand pattern is based on first order vector autoregressive (VAR(1)) demand process. Ordering policy for each product is: “order up to policy;” and forecasting method is: “moving average.” According to these assumptions, bullwhip effect is quantified in a two-product supply chain; then, it was demonstrated via a numerical example. SUPPLY CHAIN ASSUMPTIONS In this paper, a two-echelon supply chain consists of one retailer and one supplier. Retailer encounters market demand and orders it from the supplier according to his/her ordering policy. Supplier complies with received orders. Hence, demand information flow is from retailer to supplier; whereas, product flow is from supplier to retailer. There are two products in the supply chain; in essence, retailer has to meet demand for two products. Each product demand has an effect on demand of the other product. Therefore, we must consider a suitable pattern for demand modeling that includes relationship between products. In the next part, we explain a proposed demand model.
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Demand pattern Considering relationship between products, first order vector autoregressive process, VAR (1) is taken into consideration for demand modeling. Because of VAR (1) properties, it can be used not only for demand modeling in a two product supply chain, but also in multi-product supply chains. In our VAR (1) model, demand of each product affects demand of the other one as follows: demand of each product in every period depends on demand of the same product and demand of the other product in the last period. For example, consider supply chain of dairy products in which two products: cheese and butter are produced. In this supply chain, demand of cheese in period (t) is dependent on demand of cheese in period (t-1); as well as, demand of butter in period (t-1). This situation also exists for demand of butter in each period. Suppose that ( Dti ) is demand of (ith) product in period (t); hence, VAR (1) process for demand of two products can be determined by: Dt1 11 Dt11 12 Dt21 at1 2 1 2 2 Dt 21 Dt 1 22 Dt 1 at
[1]
where, a ti (i=1,2 and t=1,2,…) is forecast error of (ith) product for period (t) and is i.i.d. normally distributed with mean zero and variance ii . Relationship between demands of two products is clarified in equation [1]. In order for demand process to be stationary, the following relationship must hold: (11 22 ) (11 22 ) 2 412 21 2
1
It can be shown that variance of each product (i.e. 11 , 22 ) can be derived by: 11
11 [(1 11 22 )(1 222 ) 12 21 (1 222 )] 212 12 [11 (1 222 ) 12 21 22 ] 122 22 [1 12 21 11 22 ] [2] 1 (11 22 12 21 1)(11 22 12 21 112 222 ) (12 21 11 22 ) 3
22
212 11 [1 11 22 12 21 ] 2 21 12 [ 22 (1 112 ) 1112 21 ] 22 [(1 11 22 )(1 112 ) 12 21 (1 112 )] [3] 2 1 (11 22 12 21 1)(11 22 12 21 112 22 ) (12 21 11 22 ) 3
Moreover, covariance between two products (i.e. 12 ) can be determined by equation [4]:
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12
21 11 [11 (1 222 ) 12 21 22 ] 12 [(1 112 )(1 222 ) 122 212 ] 12 22 [ 22 (1 112 ) 1112 21 ] [4] 1 (11 22 12 21 1)(11 22 12 21 112 222 ) (12 21 11 22 ) 3
Ordering policy In this research, we consider an order up to policy (OUT) for retailer inventory control system. The order up to policy (OUT) is a standard ordering algorithm in many MRP systems (Gilbert 2005). OUT policy is easy to understand and is often utilized by companies to coordinate orders from suppliers where setup costs may be reasonably ignored (Gaalman & Disney 2007). In OUT system, level of inventory is reviewed periodically and an order is placed to bring inventory position to a predefined level. In considered inventory control system, at the beginning of each period, inventory position is observed and in order to raise the inventory level to ( S t ), an order ( Qt ), is placed. After the order is placed, customer demand ( Dt ) occurs. This sequence is consistent with equation [5]: Qt S t S t 1 Dt 1
[5]
Using base stock policy, order up to level, ( S t ) at the beginning of period (t) can be determined by equation [6]: St Dˆ tL z.ˆ tL
[6]
Where Dˆ tL is lead-time demand forecast and ˆ tL is standard deviation of lead time demand forecast error. Moreover, z is normal z score and can be determined by normal table based on the favorable service level of the inventory system. Substituting from equation [6], for stock values (St) and (St-1), into equation [5]; the result would be equation [7]: Qt Dˆ tL Dˆ tL1 z(ˆ tL ˆ tL1 ) Dt 1
]7]
Since i.i.d., then each of the products is ordered independently; thus, equation [7] can be used for both of them separately according to their parameters.
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Forecasting method Because of the lead-time between placing order and receiving the products into stock, we need to forecast demand (Gaalman & Disney 2007). Among the forecasting methods, moving average and exponential smoothing models have been used widely in the real world in different industrial factories because of their ease of use, flexibility, and robustness in dealing with non-linear demand processes (Silver et al., 2000). In this research, we suggest that retailer uses moving average forecasting method to forecast lead-time demand. By definition, we have: p
Dt
D
t j
j 1
]8]
p
Therefore, lead-time demand estimation can be expressed as follows: Dˆ tL L.Dt
]9]
Substituting for moving average demand (Dt) from equation [8] into equation [9] would result in: p Dt j j 1 L ˆ Dt L p
]10]
equation [10] can be used for lead-time demand forecasting for both of products.
QUANTIFYING THE BULLWHIP EFFECT Many investigations on bullwhip effect are developed by ratio of order quantity that is ordered to supplier and variance of market demand that is seen by retailer. This definition for bullwhip effect measurement is due to its nature: amplification of demand while moving from upstream to downstream in supply chains. Therefore, the ratio expressed in equation [11] is a direct translation for measuring the bullwhip effect phenomenon:
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BE
Var (Qt ) Var ( D)
]11]
Thus, the above relationship can provide a measurement of the bullwhip effect. In addition, according to equation [11], it is sufficient to determine variance of order and variance of demand for quantifying bullwhip effect. However, let us introduce the following proposition that would help in determining variance of orders. Proposition 1: Standard deviation of lead-time demand forecast error for each product is constant during periods and does not depend on (t). Equation [12] can represent it as: (ˆ tL ) 2 ( L
L p L 1 L2 L L p 1 ) 2 ( L i ) (i ) 2( )[( )( ( p i ) (i )) (i ) (i )] p p p i 1 i 1 i 1
]12]
In which is covariance of each product; and (i) Cov( Dt , Dt i ) . In fact, the above proposition implies that ˆ tL ˆ tL1 . Proof: To simplify calculations, we did not consider each product separately in equation [12] as the result can be used for both of them. By variance definition, we have: ˆ L ) Var(D L ) Var(D ˆ L ) 2Cov(D L , D ˆ L) (ˆ tL ) 2 Var(DtL D t t t t t
in which: L 1
DtL Dt Dt 1 ... Dt L 1 Dt i
[13]
i 0
where, Dˆ tL has been derived before in equation [10]. For appointment (ˆ tL ) 2 we must determine three relationships for Var ( DtL ) and Var( Dˆ tL ) and Cov( DtL , Dˆ tL ) . Derivation of: Var ( DtL ) By definition, we have Cov( Dt , Dt k ) (k ) . Moreover, based on stationary conditions:
Kuwait J. Sci. Eng. 36 (1B) pp. 211-230, 2009 Var( DtL ) ... 2 (1) 2 (2) ... 2 ( L 1) 2 (1) 2 (2) ... 2 ( L 2) 2 (1) 2 (2) ... 2 ( L 3) ... 2 (1) L 2[ (1) (1) ... (1) (2) ... (2) ... ( L 1)] L 2[(L 1) (1) ( L 2) (2) ... ( L 1)] L 1
L 2[ ( L i ) (i)] i 1
Derivation of: Var( Dˆ tL ) Consider equation [10] where it shows lead-time demand forecast. According to variance calculation rules, we can provide variance of lead-time demand forecast as follows: p L L Var( Dˆ tL ) ( ) 2 Var( Dt i ) ( ) 2 Var( Dt 1 Dt 2 ... Dt p ) p p i 1 L ( ) 2 [ ... 2 (1) 2 (2) ... 2 ( p 1) p 2 (1) 2 (2) ... 2 ( p 2)
2 (1) 2 (2) ... 2 ( p 3) ... 2 (1)] L ( ) 2 [ p 2( (1) (1) ... (1) (2) ... (2) ... ( p 1))] p L ( ) 2 [ p 2(( p 1) (1) ( p 2) (2) ... ( p 1))] p p 1 L ( ) 2 [ p 2( ( p i ) (i ))] p i 1
Derivation of: Cov(DtL , Dˆ tL ) Using definition of DtL and Dˆ tL that are mentioned before, we have: L Cov( DtL , Dˆ tL ) Cov[(Dt Dt 1 ... Dt L 1 ), ( )(Dt 1 Dt 2 ... Dt p )] p L ( )[Cov( Dt , Dt 1 ) Cov( Dt , Dt 2 ) ... Cov( Dt , Dt p ) p Cov( Dt 1 , Dt 1 ) Cov( Dt 1 , Dt 2 ) ... Cov( Dt 1 , Dt p ) ... Cov( Dt L 1 , Dt 1 ) Cov( Dt L 1 , Dt 2 ) ... Cov( Dt L 1 , Dt p )] L ( )[ (1) (2) ... ( p ) (2) (3) ... ( p 1) ... p ( L) ( L 1) ... ( L p 1)] L ( )[ (1) 2 (2) ... p ( p ) ... ( L p) ( L p)] p L L p ( )[ (i ) (i )] p i 1
Now, consider again previous variance relationship:
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(ˆ tL ) 2 Var( DtL ) Var(Dˆ tL ) 2Cov( DtL , Dˆ tL )
Substituting three derived variances in above equation, we have: p 1 L 1 L L L p L 2[ ( L i ) (i )] ( ) 2 [ p 2( ( p i ) (i ))] 2( )[ (i ) (i )] p p i 1 i 1 i 1
(L
L p L 1 L2 L L p 1 ) 2 ( L i ) (i ) 2( )[( )( ( p i ) (i )) (i ) (i )] p p p i 1 i 1 i 1
Thus, it is clear that above relationship does not depend on (t) and merely depends on problem parameters ( L, p, ) .
Proposition (1) implies that standard deviation of lead time demand for each product does not impress bullwhip effect and only is needed for determination of ( S t ( in OUT (order up to policy) for each product in every period. According to proposition (1) result, equation [7] can be reduced to equation [14] that is used to derive variance of orders. After summarizing order quantity relationship, we have: Qt Dˆ tL Dˆ tL1 Dt 1
(14)
Variance of order quantity To provide variance of order we should have an explicit expression for order measure. So substituting equation [10] in equation [14] would result in equation [15]: Qt (1
L L ) Dt 1 ( ) Dt p 1 p p
[15]
Equation [15] provides order quantity of a product based on order up to policy when retailer uses moving average forecasting method for lead-time demand estimation. The above relationship can be used for both products if we replace relevant parameters in equation [15] as follows: Qti (1
Li i L ) Dt 1 ( i ) Dti p 1 pi pi
and i=1,2.
Specifying variance of order quantity, we have: Var (Qt ) (1
L 2 L L L ) ( ) 2 2(1 )( ) ( p) p p p p
]16]
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Thus, to determine Var (Qt ) it is necessary to calculate ( p) . Substituting from [16] in [11] results in a relationship for bullwhip effect measurement that can be used for both of products. This general form of bullwhip effect measurement is defined by equation [17]:
BE
BE [(1
Var (Qt ) Var ( D)
(1
L 2 L L L ) ( ) 2 2(1 )( ) ( p ) p p p p
L 2 L L L ( p) ) ( ) 2 ] 2(1 )( )[ ] p p p p
]17]
The following is to demonstrate conditions in which bullwhip effect exists in our two-product supply chain. Problem: When does bullwhip effect not exist in a two-product supply chain according to proposed model assumptions? Equation [18] may be viewed as a special case of equation [17], similar to Chen et al. (2000) in a single product supply chain: BE 1 (
2L 2L2 ( p) 2 )(1 ) p p
]18]
We know if BE 1, then a steady state exists between demand and orders in the supply chain; hence, we do not have bullwhip effect in supply chain. Therefore, we must have: 1 (
( p) 2L 2L2 ( p) 2L 2 L2 ) 0 . In our supply chain, we can prove that 2 )(1 ) 1 or ( 2 )(1 p p p p
( p) ( p) (see appendix A) resulting in (1 ) 0 . Because L 0 and p 0 ; therefore, (
( p) 2L 2L2 2 )(1 ) 0 ; or BE 1. In addition, if p then: p p
(
( p) 2 L 2 L2 2 )(1 ) 0 : or BE 1 . Accordingly, we can conclude: p p
In a two-product supply chain base on our assumptions, bullwhip effect always exists except status in which number of observations in moving average calculations increases
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as much as possible. Consequently, bullwhip effect can be removed from the supply chain by increasing number of observation, p.
If we consider equation [17] to quantify bullwhip effect in proposed supply chain. Obviously, equation [17] does not submit an explicit expression of bullwhip effect because there is no specific relationship for ( p) . However, knowing p, we can compute ( p) ; hence, bullwhip effect can be calculated. In this research, we derived ( p) for some specific values of p as bullwhip effect relationships are needed for analytical purposes in the fourth section of this paper. To provide a relationship for bullwhip effect measurement for both products, let us consider BEip to be the bullwhip effect of (ith) product; (i=1, 2). In addition, we will use L i i
and p i to show lead-time, and number of observations in forecasting method for each of products (i.e. L and p), respectively. In addition, we consider ii ( p i ) as covariance between two measures of i th product demand at lag p i . Moreover, ii represents demand variance of i th product. Accordingly, substituting aforementioned terms in equation [17] results in equation [19]; i.e., bullwhip effect for each product will be: BE ipi [(1
Li 2 L L L (p ) ) ( i ) 2 ] 2(1 i )( i )[ ii i ] pi pi pi pi ii
i=1, 2
[19]
Derivation of: ii ( p i ) 9 Covariance matrix function for VAR (1) process is as follows: (1)1 ( p i ) pi ( pi 1)1 (0)(1)
pi 0
[20]
pi 1
It is clear that we need to (0) and 1 p to calculate ii ( p i ) with pi 1 . In the last section, we i
described that we do not have any explicit relationship for ii ( p i ) . Indeed, equation [20] shows that due to non-existence of any determined form of 1 p when 1 is a [2x2] matrix i
and p i is greater than one, we can not provide a general formula for ii ( p i ) .
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Now consider (0) 11 12 in which 11 and 22 are demand variance of the first and 12 22
second product respectively and 12 is covariance between demands of two products. In addition 1 is a coefficient matrix in a VAR (1) process for a two-product supply chain based on equation [1] and can be specified by 1 11 12 . So if we know p i then we can obtain 21 22 ii ( p i ) by equation [20]. In the next section, ii ( p i ) is determined for pi 1,2,3 and i=1, 2;
then, bullwhip effect measurements for both products are calculated.
Derivation of: BE11 and BE12 When we use only the last observation in moving average forecasting method (i.e. pi
1 ),
matrices (1) and 1 are needed for specifying bullwhip effect. Consequently, (1) (0)1 and (0) 11 12 . Hence, (1) 11 12 11 21 or (1) 11 11 12 12 11 21 12 22 . 12 22 1211 2212 12 21 22 22 12 22 12 22
Thus 11 (1) 1111 1212 and 22 (1) 12 21 22 22 . Therefore, using equation [19] bullwhip effect for the first product can be provided by: BE11 [(1 L1 ) 2 ( L1 ) 2 ] 2(1 L1 )( L1 )[11 (
12 )12 ] 11
[21]
Moreover, for second product we have: BE12 [(1 L2 ) 2 ( L2 ) 2 ] 2(1 L2 )( L2 )[ 22 (
12 ) 21 ] 22
[22]
Derivation of: BE 21 and BE 22 To determine 11 (2) and 22 (2) we need to square of transposed coefficient matrix, 1 . Based 2 on matrices rules we have 1 2 11 2112
21 (11 22 ) : hence, by equation [20] we have: 2 12 (11 22 ) 22 2112
(2) (0)(1) 2 . Consequently, 11 (2) 11 (112 2112 ) 1212 (11 22 ) and
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22 (2) 12 21 (11 22 ) 22 ( 222 2112 ) . Substituting for 11 (2) and 22 (2) in equation [19]
results in a bullwhip effect relationship for the first product: BE 21 [(1
L1 2 L L L ) ( 1 ) 2 ] 2(1 1 )( 1 )[(112 2112 ) 12 (11 22 )( 12 )] 2 2 2 2 11
[23]
Bullwhip effect measure for the second product can be provided too: BE 22 [(1
L2 2 L L L ) ( 2 ) 2 ] 2(1 2 )( 2 )[( 222 2112 ) 21 (11 22 )( 12 )] 2 2 2 2 22
[24]
In fact, equations [23] and [24] show relationships for quantifying bullwhip effect when retailer uses only the last two observations for lead-time demand forecasting for both of products.
Derivation of: BE 31 and BE 32 To provide 11 (3) and 22 (3) we need to calculate 1 3 . The steps are similar to previous sections; thus, it will result in: 113 2112 (211 22 ) 21 (112 2112 1122 222 ) 13 2 2 223 2112 (11 222 ) 12 (11 1122 2112 22 ) BE 31 [(1
L1 2 L L L ) ( 1 ) 2 ] 2(1 1 )( 1 )[113 2112 (211 22 ) 12 (112 11 22 2112 222 )( 12 )] 3 3 3 3 11
[25] BE 32 [(1
L2 2 L L L ) ( 2 ) 2 ] 2(1 2 )( 2 )[ 223 2112 (11 2 22 ) 21 (112 2112 11 22 222 )( 12 )] 3 3 3 3 22
[26] Because it is necessary to determine BE 41 , BE 42 , BE 51 and BE 52 for our analytical approach in the next section, we have provided them in appendix B.
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Kuwait J. Sci. Eng. 36 (1B) pp. 211-230, 2009 p p Note: If we represent 1 to the power of p by 1 p 1 p (1,1) 1 p (1,2) considering past 1 (2,1) 1 (2,2)
relationships about bullwhip effect, we can provide general forms of bullwhip effect for two products as follow: BE 1p1 (1
L1 2 L L L ) ( 1 ) 2 2(1 1 )( 1 )[1 p1 (1,1) ( 12 )1 p1 (2,1)] p1 p1 p1 p1 11
BE p22 (1
L2 2 L L L ) ( 2 ) 2 2(1 2 )( 2 )[1 p2 (2,2) ( 12 )1 p2 (1,2)] p2 p2 p2 p2 11
These relationships can be used for quantifying bullwhip effect when p is large and we intend to obtain matrix 1 p by mathematical software. NUMERICAL EXAMPLE In this section, an analytical discussion about bullwhip effect behavior in a two-product supply chain is represented, using a numerical example. Consider that demand process of two products in a supply chain is defined by: 0.7 0.6 Dt1 0.7 Dt11 0.6 Dt21 at1 . In this demand process, we have: 1 2 and according to 1 2 2 0.2 0.5 Dt 0.2 Dt 1 0.5Dt 1 at 2 stationary condition, we must have: (11 22 ) (11 22 ) 41221 1 .
2
Substituting coefficients will result in: (0.7 0.5) (0.7 0.5) 2 4(0.6)(0.2) 0.96 ; and 2
(0.7 0.5) (0.7 0.5) 2 4(0.6)(0.2) 0.23 . 2
Therefore, stationary condition is satisfied. Now, we can obtain bullwhip effect measure for different values of L and limited measures of p (p=1,2,3,4,5) for two products by previous determined relationships. For simplicity in mathematical calculations, we assume that error terms have standard normal distribution and are uncorrelated. This hypothesis does not affect generality of the problem. Table [1(a)] and table [1(b)] contain various values of bullwhip effect for the first and second product, respectively.
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Kuwait J. Sci. Eng. 36 (1B) pp. 211-230, 2009 Table [1(a)] Bullwhip effect values for p1
L1
1 2 3 4 5 6
2
1.215 1.644 2.287 3.145 4.218 5.505
1.142 1.377 1.708 2.132 2.651 3.265
4
1.116 1.291 1.524 1.814 2.164 2.571
1.103 1.248 1.434 1.661 1.93 2.24
Table [1(b)] Bullwhip effect values for p2 1
2
p2 3
4
5
1.73 3.191 5.383 8.305 11.96 16.34
1.374 1.997 2.869 3.99 5.36 6.979
1.255 1.638 2.148 2.786 3.551 4.444
1.198 1.476 1.832 2.268 2.783 3.378
1.165 1.386 1.661 1.992 2.378 2.819
5 1.095 1.222 1.381 1.571 1.793 2.047
L2
1
p1 3
1 2 3 4 5 6
(a)
(b)
The following figures depict bullwhip effect behavior while L and p are varying for each of products separately. It is clear that bullwhip effect is related to lead-time directly and is relevant to number of observations in moving average calculations (p), reversely. Fig. [1(a)] shows that bullwhip effect of the first product increases when its lead-time increases. A steady rise can be seen for p=2, 3, 4, 5; when p=1 bullwhip effect curve dramatically increases. Difference between bullwhip effects when p changes from 1 to 2 indicates that only the last observation is not sufficient for lead-time demand forecasting and causes huge demand fluctuation. Fig. [1(b)] shows bullwhip effect of the second product with respect to its lead-time. All above notes about the bullwhip effect of first product is valid for the second product but the difference between bullwhip effects for p=1 and p=2 is more than the first product. Nevertheless, a considerable rise in bullwhip effect curve is evident while we are using last observation alone for lead-time demand forecasting. 6
16 11
4
BE2
BE1
5
3
6
2 1
1 1 P1=1
2 P1=2
3
4 P1=3
(a)
5 P1=4
6
1
L1 P1=5
P2=1
2 P2=2
3
4 P2=3
(b)
Fig. [1(a)] Bullwhip effect variation with respect to L1; Fig. [1(b)] Bullwhip effect variation with respect to L2.
5 P2=4
6
L2 P2=5
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6
16
4
BE2
BE1
5
3
11 6
2 1
1
1
2
3
L1=1 L1=4
4 L1=2 L1=5
5 L1=3 L1=6
(a)
p1
1
2
3
L2=1 L2=4
4
L2=2 L2=5
5
p2
L2=3 L2=6
(b)
Fig. [2(a)] Bullwhip effect variation with respect to p1; Fig. [2(b)] Bullwhip effect variation with respect to p2.
Fig. [2(a)] depicts bullwhip effect variations of the first product with respect to number of observations in lead-time demand forecasting. It is clear that more observations results less bullwhip effect. In addition, a sharp fall can be seen when number of observations changes from 1 to 2. This is the same result of Fig. [1]. and it means that using only the last observation in lead-time demand forecasting is not satisfactory. Moreover, Fig. [2(a)] shows that slope of the bullwhip effect curve decreases dramatically when lead-time increases from 1 to 6. In the other words if lead-time is large, (i.e. 6); then, increasing p from 1 to 2 can reduce bullwhip effect considerably. Finally, Fig. [2(b)] shows bullwhip effect curve for the second product with respect to number of observations, p. It is clear that bullwhip effect decreases when number of observations increases. A dramatic fall can be seen when lead-time of the second product increases from 1 to 2. Slope of curves between bullwhip effects of two products differs in Fig. [2(a)] and Fig.[2(b)]. Comparison between these two figures shows that slant of bullwhip effect graph for the first product is less than second one. As a result, we can conclude by the above graphs as well as table [1(a)] and table [1(b)] that decreasing lead-time along with increasing number of observations in lead-time demand forecasting based on moving average forecasting method can reduce bullwhip effect of each product. Accordingly, in our example, minimum measure of bullwhip effect occurs when L=1
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and p=5 and its measure is 1.0952 for the first product and is equal to 1.1653 for the second one. In addition, when lead-time is long and number of observations is small, we encounter maximum value of bullwhip effect. In fact, longer L and smaller p results larger bullwhip effect for both of products. Therefore, maximum value of bullwhip effect for two products is 5.5051 and 16.339; when L=6 and p=1, respectively. Another interpretation of bullwhip effect variation while there are two products in supply chain. It is clear that 12 and 21 represent effects of demand for each product on demand of another one as demonstrated in equation [1]. Thus, we need to determine effects of variation of 12 and 21 on the bullwhip effect of each product. In our example, when 11 , 21 , 22 are constant then 12 must be .05 12 .75 based on stationary conditions. In addition, for the second product we must have .016 21 .25 when 11 , 12 , 22 are constant. Fig. [3(a,b)]
4
4
3
3 BE
BE
show bullwhip effect of two products according to coefficients variation.
2 1
2 1
0 0
0.1
0.2
0.3
B E1
0.4
0.5
0.6 B E2
(a)
0.7
12
0 0
0.05
0.1
B E1
0.2 21
0.15 B E2
(b)
Fig. [3(a)] Bullwhip effect variation for 12 : when 11 0.7 , 21 0.2 , 22 0.5 , L1=2, and p1=3; Fig. [3(b)] Bullwhip effect variation for 21 : when 11 0.7 , 12 0.6 , 22 0.5 , L2=2, and p2=3.
It is clear that when 12 and 21 increases bullwhip effect of two products decreases; in essence, dependence between products has an important role on the bullwhip reduction. In other words when we do not have any relation between products (i.e. 12 0 or 21 0 ), bullwhip effect has maximum value; whereas, when dependence rate is high (i.e. 12 and 21 are as much as possible), bullwhip effect has minimum value. Moreover, we can continue our
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investigations on bullwhip effect measure with respect to 11 and 22 variation. According to stationary conditions, we can find limitations to these coefficients that implies: 11 0.76 when other coefficients are constant; and, 0.93 22 0.6 when others are constant. Fig. [4(a)] depicts bullwhip effect variation while 12 varies in its range. 6
4
5 4
BE
BE
3 2
3 2
1
1
0 -0.5
-0.2
0
0.2
B E1
0.4
0.6
0.75
11
0 -0.9
B E2
-0.5
-0.2
0
B E1
(a)
0.2
0.4
0.5
22
B E2
(b)
Fig. [4(a)] Bullwhip effect variation for 11 : when 12 0.6 , 21 0.2 , 22 0.5 , L1=2, and p1=3 Fig.[4(b)] Bullwhip effect variation for 22 when 11 0.7 , 12 0.6 , 21 0.2 , L2=2, and p2=3.
Obviously, we can see that bullwhip effect of two products can be reduced by increasing 11 . It means that if we increase dependence of each product on demand of the last period of the same product, not only bullwhip effect of the same product can be reduced but also bullwhip effect of the another one can be decreased. This condition is valid for 22 and is clear in Fig.[4(b)], as 22 increases bullwhip effect of two products decreases.
CONCLUSION In this research, we have investigated bullwhip effect in a two echelon supply chain consisting of two products. Demand of each product was correlated to demand of the other one. This relationship is described by VAR (1) model. Retailer used OUT policy for ordering for both of products. Order of first product does not depend on the order of the second product. We assumed retailer simple moving average method to forecast lead-time demand of each product independently. After description of model, we derived a general expression for bullwhip effect and mentioned that it is not possible to provide an explicit equation for
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bullwhip effect of two products when we use covariance function of VAR (1) process in order to quantify bullwhip effect. Then we provided bullwhip effect measure of each product in limited cases for better analysis. In addition, we determined conditions that bullwhip effect could be removed from supply chain without affecting other calculations. Finally, in the last section we analyzed behavior of the bullwhip effect by a numerical example. We showed that lead-time reduction, as well as, increasing number of observation in lead-time demand forecasting can reduce bullwhip effect for both products, simultaneously. In other words, if retailer uses more historical information in forecasting lead-time demand, bullwhip effect can be decreased. In addition, if manufacturer decreases lead-time then the phenomenon can be removed from the supply chain. After that, we provided an analytical approach on bullwhip effect based on dependence of demand of each product on demand of the same product on the last period and demand of the other product in the last period. It was clear that increasing of dependence between products has an important role in bullwhip reduction of both of products. This research would be incomplete if we did not mention its drawbacks. Our aim was to provide mathematical relationships for quantifying bullwhip effect; yet, a more analytical approach is still needed. In this paper, our assumptions is based on Chen et al. (2000a), therefore, disadvantages of that paper (such as cost considerations, ordering policy, and forecasting method) exist in current articles. Also, similar to Zhang (2004), a study on different forecasting methods, as well as, more analytical approaches on conditions that bullwhip effect exists in our supply chain can be accomplished. It is more interesting when retailer uses different forecasting methods for two products. This study is the first basic investigation on bullwhip effect in more than one product supply chains and continuing the research in more complex supply chains is strongly recommended.
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Appendix A Due to the fact that bullwhip effect is increasing function in lead-time we must have: BE 0 . L
Therefore taking first derivative of bullwhip effect with respect to L we have: ( p) in which should be positive. Because L and p are positive, we can BE 2L 2 (1 )( )(1 ) L p p
conclude that (1 ( p) ) 0 or ( p) .
Appendix B BE41 (1
L1 2 L1 2 L L ) ( ) 2(1 1 )( 1 )[114 11 2112 (311 2 22 ) 2112 ( 2112 222 ) 4 4 4 4
12 )] 11 L L L L BE42 (1 2 ) 2 ( 2 ) 2 2(1 2 )( 2 )[ 224 2112 22 (3 22 211 ) 2112 ( 2112 112 ) 4 4 4 4 21 (11 22 )(112 2 2112 222 )( 12 )] 22 12 (11 22 )(112 2 2112 222 )(
BE51 (1
L1 2 L1 2 L L ) ( ) 2(1 1 )( 1 )[115 2112 (4113 (11 22 2112 )(311 2 22 ) 223 ) 5 5 5 5
12 ( 22 (11 22 )(112 2 2112 222 ) 2112 (11 (311 2 22 ) ( 2112 222 )) 114 )( BE52 (1
12 )] 11
L2 2 L2 2 L L ) ( ) 2(1 2 )( 2 )[ 225 2112 (4 223 (11 22 2112 )(3 22 211 ) 113 ) 5 5 5 5
21 (11 (11 22 )(112 2 2112 222 ) 2112 ( 22 (3 22 211 ) ( 2112 112 )) 224 )(
12 )] 22
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Submitted: 7/10/2007 Revised: 11/3/2008 Accepted: 25/3/2008
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قیاس أثر البىلفیب في سالسل التمىین بطلب االتىرغرسیف الکمی المحدد للمرة االولی
سید کمال جهار سىقی و احمد صادقی خاهعح تشتیح الوذسسیي – کلیح الٌِذسح – قسن ٌُذسح الصٌاعاخ ص .ب .سقن -23224 -232طِشاى -ایشاى
خالصـت
في ُزا الوقال تتن الذساسح حْل سلسلح توْیي راخ هشحلتیي راخ هٌتدیي علی اساس افتشاضاخ الوْدیل. الِذف الشئیسي ُْ الحصْل علی عالقح لقیاس أثش الثْلفیة الزي یْفش لٌا اهکاًیح دساسح ّ تقلیل ُزٍ الظاُشج في السالسل راخ الوٌتدیي .یستفاد هي هْدیل االتْسغشسیف الکوی الوحذد للوشج األّلی الزي یعذّ ًوْرخاً هٌاسثاً لصٌع هْدیل الطلة في هسلسل التوْیي رّ هٌتدیي .تالٌظش الی أًَ في العالن الْاقعي یستفاد کثیشاً هي اسلْب الوعذل الوتحشک ) ّ (MAسیاسح تقذین الطلثاخ حتی الحذّد الوحذدج ) ّ (OUTیستخذم ُزاى االسلْتاى لشصذ طلة دّسج اللیذتاین ّ عشض الطلثاخ .في ُزا الوقال ًحي ًحصل علی عالقح عاهح لقیاس أثش الثْلفیة ّ ًقْم تالوزاکشج حْلَ ّ ًستٌتح تأًَ تالٌظش الی االسلْب الوستخذم ،ال تْخذ اهکاًیح الحصْل علی عالقح صشیحح لقیاس أثش الثْلفیة ّ .لکي تستحصل عالقح کویح أثش الثْلفیة للحاالخ الوحذّدج تصفح ًوْرج .تن ایضاذ الظشّف التي یوکي هي خاللِا حزف أثش الثْلفیة ّ في الٌِایح تتقذین هثال عذدي ،تتن دساسح أثش الثْلفیة في هسلسل التوْیي راخ هٌتدیي.
