An amended shipment policy and model for single-vendor single-buyer system

Transcription

1 An amended shipment policy and model for single-vendor single-buyer system Huili Yan 1, Hao Xiong 2,Qiang Wang 3, Fan Xu 4 Abstract In the existing literature, a two-phase structure of the optimal production and shipment policy for a vendor manufacturing to supply to a single buyer under the assumption that the stocholding cost of the vendor is lower than the buyer s. Actually, these policy with the structure not always be the optimal shipment. In this paper, we found that a three-phase shipment policy is more general than the two-phase policy. And a general model based on this structure is suggested and it is shown to provide a lower or equal joint total relevant cost as compared to the existing model. Finally, a particle swarm optimization (PSO) algorithm is proposed to solve the production-inventory model, which is proved to be a good algorithm by two numerical examples. Keywords inventory; production; shipment policy; vendor-buyer 1 Introduction In classical economic order quantity (EOQ) models, the vendor s and buyer s inventory problems are treated separately. This independent decision usually cannot assure that the two parties as a whole reach the optimal state. Therefore, during the past decades, many researchers pay more attention on the problem of joint replenishment that minimizes the total relevant costs for both the vendor and the buyer. The motivation for integrating vendor-buyer s production and inventory in recent years comes from better information flow and greater cooperation between companies in the production and distribution chain, particularly between a main manufacturer and its component suppliers. The problem considered here concerns a single vendor (manufacturer) supplying a single buyer with only one ind of product. The vendor manufactures, at a finite rate, in batches and incurs a batch set up cost. Each batch is dispatched to the buyer in a number of shipments and the vendor (and/or buyer) incurs a fixed order cost associated with each Foundation: National Natural Science Foundation of China (Grant No , No and No ), the Natural Science Foundation of Hainan Province (No ), the project of Hainan province philosophy and social sciences (No. HNSK(QN)15-4), and the Education fund item of Hainan province (Hny2015ZD-7,Hny2015ZD-2) @qq.com 1 Huili Yan: Associate Professor, School of Tourism, Hainan University, Haiou, , China 2 Hao Xiong: Professor, School of Economics and Management, Hainan University, Haiou, , China 3 Qiang Wang: Postgraduate Student, School of Tourism, Hainan University, Haiou, , China 4 Fan Xu: Postgraduate Student, School of Tourism, Hainan University, Haiou, , China

2 shipment. Both vendor and buyer incur time-proportional inventory holding costs, at different rates. The buyer has to meet a fixed, level external demand. The objective is to view the system as an integrated whole and determine that production and shipment schedule which minimizes the average total cost per unit time. Essentially, the decision problems facing the decision maers (vendor and buyer) are as follows: (1) the batch quantity of production for the vendor; (2) the number and sizes of shipments be sent to the buyer. Although there are many extended wor have been introduced to the vendor-buyer integrated production inventory system, such as stochastic demand (Giri and Charaborty, 2016), cooperative mechanism (Uddin et al., 2016), lead time (Lin, 2016) and capacity constraint (Giri et al., 2015). But we found that the basic production-inventory model of single vendor and single buyer system, which is lastly improved by Hill (1999) and Zhou and Wang (2007) and Xiong (2010), still has some flaw. So, we bac again to the basic vendor-buyer integrated production and inventory system and given a more general policy and model. The rest of this paper is structured as follows. In Section 2 we discuss relevant results in the literature. In Section 3 we analyze the relationship between shipment policy and system inventory. In Section 4, we introduce the general shipment policy and the model based on it. We present our numerical experiments to evaluate the quality of the proposed solutions in Section 5. We finish the paper in Section 5 with some comments and conclusions. 2. Literature review Several earlier papers on the vendor buyer integrated system, such as Goyal (1977) and Banerjee (1986), developed a lot-for-lot model in which the vendor produces each buyer shipment as a separated batch. Goyal (1988) extends Banerjee s model to the case where a production batch consists of a number of equal-sized shipments. However, his model required the production of the batch to be finished before the shipments could start. Lu (1995) relaxed the assumption of Goyal s policy (1988) about producing a batch before starting shipments and explored a policy allowed shipments to tae place during production. Goyal and Gunasearan (1995) proposed another different shipment policy could give a lower joint total cost than the equal shipment size policy. In that model, the production batch is sent to buyer as soon as the buyer is about to run out of inventory and all of the manufactured inventory made up to that point is shipped out. This new policy involves successive shipment sizes within a batch increasing by a factor equal to the ratio P/D between the vendor s production rate P and the demand rate D on the buyer. Hill (1997) presented a more general policy to determine the vendor s production batch and successive shipments sizes. He suggested that the ith shipment size should be determined by λ i-1 q1, where q1 is the first shipment size, and λ satisfies the condition: 1 λ P/D. The equal shipment size policy and Goyal s policy (1995) represent special cases of the Hill s

3 policy (1997). Goyal and Nebebe (2000) further showed the shipment policy that the first shipment, q1, is followed by (n-1) equal-sized shipments of size: q1(p/d) often achieved lower cost than the shipment policy adopted in Hill s policy (1997). Goyal (2000) presented a new shipment policy that could achieve lower cost by assuming the shipment policy structure as the form (q1, βq1,, β m-1 q1,, β m-1 q1), and β=p/d. Unlie all the above-mentioned researchers finding the optimal solution from a given structure of policy, Hill (1999) derived the structure of the globally optimal policy of shipments. This optimal structure is very similar to that of the policy considered by Goyal and Szendrovits (1986). The only difference is that under Hill s optimal policy structure the size of the equal-sized shipments at the end may not be equal to the largest batch size among the unequal-sized shipments. However, Hill s (1999) model was built on the base of assumption that the buyer s inventory holding cost per unit per unit time is always bigger than the vendor s. Xiong (2010) further showed a three-phase structure policy under the condition when the vendor s unit holding is lower than the buyer s. Later, Zhou and Wang (2007) gives a very common optimal model without considering the structure of the shipment policy removing the above-mentioned assumption made by Hill (1999). That is, their model is also suited for the condition that if the vendor s unit holding cost is bigger than the buyer s. However, this general model has too many variables and constraints, which lead to that it is difficult to solve the problem. Although researchers have given several policy structures and one general model for the production and inventory system (See in the Table1). These policy structures are not certain to be the optimal solution and the general model is too difficult to get the optimal solution. Then, it is the necessary to find the more generalized structure of the optimal shipment policy. In this paper, from the general optimal model, based on the analysis of Hill (1999) and Xiong (2010), the structure of the optimal shipment policy for general situation is built. The contributions of the paper are aspects below. First, a new system inventory analyzing method is introduced, which is based on the analyzing the changes of the buyer s and vendor s time-weighted inventory when the adjacent two shipments size change. Secondly, we explore the structure of the optimal batching and shipping policy. This new structure contains all the existed structures as the special case in it. Finally, we proposed a PSO algorithm to solve the new model and illustrate it with two numerical examples. Table 1. Different shipment policies in the literature No. Policy Structure Literature 1 equal shipments after production {q1, q1,, q1} Goyal (1988) 2 equal shipments {q1, q1,, q1} Lu (1995) 3 Increasing shipments by β, β=p/d {q1, βq1,, β n q1} Goyal (1995) 4 Increasing shipments by λ, 1 λ P/D {q1, λq1,, λ n q1} Hill (1997) 5 Increasing and equal shipments {q1, βq1,, β m-1 q1, q2,, q2} Hill (1999) 6 First one followed other different equal shipments {q1, βq1,, βq1} Goyal and Nebebe (2000) 7 Increasing shipment and equal shipments {q1, βq1,, β m-1 q1,, β m-1 q1} Goyal (2000) 8 No regular pattern {q1, q2,, qi,, qn} Zhou and Wang (2007)

4 3 The relationship between shipment policy and system inventory All the variables are assumed to be continuous rather than discrete in nature. No inventory shortages for the buyer may occur. All the parameters are deterministic and fixed over an infinite horizon. The objective is to determine that production and shipment schedule which minimizes the average total cost per unit time of production set-up, shipment and inventory holding. The following terminology is used: A1 fixed production set up cost, A2 fixed order/shipment cost, h1 inventoryholding cost for the vendor per unit per unit time, h2 inventoryholding cost for the buyer per unit per unit time, P (continuous) production rate for the vendor, D constant (and continuous) demand rate on the buyer, β ratio P/ D, Q size of a production batch, N number of shipments per batch production run, qi The size of the ith shipment in a batch production run ( qi = Q), q0 total inventory in the system when the production of a batch starts, C mean total cost incurred by the system per unit time. In order to find the relationship between the shipments and the system inventory, a random pair of the adjacent orderings qi and qi+1 in a decision cycle will be analyzed. These adjacent orders may both be in the production period or in the non-production period. And they also may be another special case: one order is in the production period and the other is in the non-production period. For the first two cases, we assume qi+1 exceeds qi by b. In order to observe the order quantity effect on the system inventory, we increase qi by b/2, and decrease qi+1 by b/2 to get equal order quantity. Then according to the figure1, the time-weighted inventory of the buyer will be decreased by æ DQ b = q i+1 - b ö ç b è 2ø 2D - b q i 2 D = æ q - q - b ö ç i+1 i b è 2ø 2D = b2 (1) 4D Stoc (q i+1 -b/2) b/2 q i q i Figure 1. The buyer inventory Time Then, according to the figure 2 and figure 3, the vendor s time-weighted inventory will be increased by

5 æ DQ v = q i+1 - b ö ç b è 2ø 2D - b 2 q i D = æ q - q - b ö ç i+1 i b è 2ø 2D = b2 4D (2) Stoc q i b/2 - q i+1 + (q i+1 -b/2) Time Figure 2. The vendor inventory changes during the production period Stoc q i b/2 - q i+1 + (q i+1 -b/2) Time Figure 3. The vendor s inventory changes during the non-production period In the production period and the non-production period, according to equation (1) and (2), the buyer s inventory will be decreased and the vendor s inventory will be increased as the difference between qi and qi+1 is decreased, but the system inventory will not be effected. That is, all the possible system inventory will be ept at the vendor. The buyer s inventory will be minimized if all the shipments are equal. On the contrary, all the possible system inventory will be ept at the buyer. The buyer inventory will be maximized if all the shipments have the greatest possible difference. Although the difference between two adjacent orders can mae the inventory moving between the buyer and the vendor, but the whole inventory of them (system inventory) will eep the same. However, when qi is in the production period and qi+1 is in the non-production period, the system inventory will not eep the same when the difference of these adjacent orders change. The vendor s time-weighted inventory change is composed by q - and q + b qi b qi q s (3) 2 D 2 D b b q qi1 (4) 22D Then the vendor s inventory change is:

6 æ DQ v = Dq + - Dq - ³ q i+1 - q i - b ö ç b è 2ø 2D = b2 4D (5) Stoc s q i b/2 - q i+1 + (q i+1 -b/2) Time Figure 4. The vendor inventory change crossing the production and the non-production period From equation (5) and equation (1), the change of the system time-weighted inventory can be shown as: TQ= Qv- Qb 0. That is, if the buyer eep the orders equal as far as possible, the system inventory will be increased as the storage will be moved to the vendor side. So, if the vendor s holding cost is higher than the buyer s (h1<h2), the system inventory cost will be increased too. However, if the vendor has a lower unit storage cost (h2<h1), the inventory costs can not to be told whether is reduced or increased. 4 The generalized structure and model of optimal shipment policy In order to analyze the optimal shipment structure, we temporarily assume that all the variables are determined except the shipment policy. Bases on the analysis in chapter 4, we will focus on the relationship between the shipment policy structure and the inventory cost. Obviously, when h2=h1, the whole inventory cost will do nothing with the shipment policy structure, which is only rely on the q0 (See in Hill (1999)). That means the shipments policy has no effect on the system holding cost. So, we only need to loo at two cases: h2>h1 and h2<h When h2>h1 If h2 dominates h1, in order to minimize the system holding cost, the largest possible system inventory should be moved to the vendor whose holding cost is the lower. According to the analysis of Hill (1999), there are two conflicting pressures (this is also be analyzed in chapter 4): when the shipment sizes are equal, the system inventory cost will be decreased as the buyer s inventory will be minimized whose inventory cost is higher; when the shipment sizes are increasing through the production cycle, the inventory level q0 at the time production will be decreased. Then, the total system inventory which depends on q0 will be decreased too. So, the optimal shipment policy should include both increasing shipments and equal

7 shipments. The increasing shipment series are increased by a fixed factor P/D. The equal shipment series are all equal. Obviously, the order quantity of equal series should be less than the last order of increasing series multiply the fixed factor P/D. So, the two different shipment series has the relationship showed in formulation (6), in which, is the last order of the increasing series, and +1 is the first order of the equal series. P q 1 q D (6) The first part of the optimal sequence of the shipments have the structure lie (q0,βq0,,β q0, λβ q0, λβ q0) and the 1 λ P/D, β= P/D. That is, the shipment policy during the production time includes increasing shipments increased by P/D and m equal shipments. The whole shipment time is N=+m. However, Hill (1999) didn t give a special focus on the non-production period. In fact, as we have analyzed in the above chapter 4, it is not certain whether the system inventory cost will increase or decrease when the adjacent orders are equal in the case that two adjacent orders are crossing the production period and non-production period. But the last n optimal shipments after the +m shipment must be equal size in the non-production period as the system inventory cost is assure to be decrease. So, the operational sequence of the whole production cycle has three components: increasing shipments, equal shipments and another equal shipments, as (q0, βq0, β 2 q0,, λβ q0,, λβ q0, аq0,, аq0). The first successive shipments are increased by a fixed factor β (β=p/d). After it followed two equal sequences. The size of the first equal sequence is increased by λ to the quantity of last increasing order. The size of the second equal sequence is not necessary equal to the size of the first equal sequence (See in figure 5). Stoc Increasing shipments Equal shipments Equal shipments q 0 m n Time 4.2 When h1<h2 Figure 5. The structure of the optimal policy when h 2 >h 1 If h1 dominates h2, in order to minimize the system holding cost, the largest possible system inventory should be moved to the buyer who has the lower holding cost. To mae this movement, the orders should be difference as large as possible. As the decrease order

8 policy is conflict with the minimizing of the inventory level q0 and the production condition, so the decrease shipment policy is obviously not the optimal choice. As to the increasing order policy, the system holding cost is minimized by moving the inventory from the vendor to the buyer when all the shipments are increased by P/D during the production period (P/D is the most possible difference in the production period). At the same time, the start inventory level q0 could also be minimized when the shipment sizes is increasing. So, the structure of the optimal sequence is a series of successive shipments increased by a fixed factor P/D in the production period. After the number of increasing shipments in the production period, if there is a +1 shipment in the non-production period, eeping the difference of the two adjacent order as large as possible, that is the shipment should equal to the whole on hand inventory of the vendor, could mae the system inventory cost decrease as the inventory will move to the buyer who has the lower inventory cost. So, the optimal sequence of the whole production cycle has two components as (q0, βq0, β 2 q0,, β q0, аq0), which in fact represents a special case of the three components of the optimal sequence when h2>h1 (See in figure 6). Stoc Increasing shipments Only one shipment q 0 Time Figure 6. The structure of the optimal policy when h 2 <h The generalized optimal model According to the above analysis, no matter whatever the relationship of the buyer s inventory holding cost and the vendor s, the optimal sequence of the whole production cycle has three components: times increasing shipments increased by P/D, followed by m times equal shipments with quantity λq0(p/d) and another n times equal shipments (See in the figure 7). And the optimal sequences of the case when h2<h1 is a special case with m=0, and n=1.

9 Stoc Production period Production cycle C Non-production period increasing shipments m equal shipments n equal shipments t q 0 T 1 T +m-1 T +m T +m+n Figure 7. Graph of inventory against time for a typical production cycle Time So, the general structure of the optimal shipment policy includes three possible stages. And this three-stage structure includes all the other structures have ever been analyzed in other literatures. Their relationships could be showed in table1 2 as follow: Table 2. The relationship between the present policy and the others No. Value of Equivalent policy literature 1 =0, m 0, n 0 {q 1, q 1,, q 1 } Lu (1995) 2 0, m=0, n=0 {q 1, βq 1,, β n q 1 } Goyal (1995) 3 0, m=0, n 0 {q 1, βq 1,, β m-1 q 1, q 2,, q 2 } Hill (1999) 4 =1, m 0, n=0 {q 1, βq 1,, βq 1 } Goyal and Nebebe (2000) 5 0, m 0, n=0 {q 1, βq 1,, β m-1 q 1,, β m-1 q 1 } Goyal (2000) Figure 7 illustrates the pattern of inventory against time within a typical production cycle for vendor inventory, buyer inventory and total system inventory. The solid line gives the vendor inventory, the narrow-dash line is the buyer inventory and the wide-dash line (where this differs from the narrow-dash line) is the overall inventory. The mean total cost per unit time taes the following form without considering the shipment policy structure is: q A1NA2 D P DQ i1 C h1 q0 h2 h1 Q 2P 2Q Consequently, by substituting β=p/d, qi=q0β i (i=1,2,..., ), qi=λq0β (i=+1,2,...,m), and qi=αq0 (i = +m+1,2,...,+m+n), into (7), the average total cost of the integrated system, C(, m, n, q0, λ, а), is given by N 2 i (7)

10 A1 m n A2D P D Q C, m, n, q0,, а h1 q0 h2 h1 B Q 2P q0 2 m n q0 1 1 B 2 1 q0 m n q0 1 1 Q q m n q (8) (9) (10) Consider now the constraints to which the variables should be subject. The whole production batch should be equal to the whole shipments quantities. So, one easily nows In which, t is the production time in the interval between the last shipment in the production period and the first shipment in the non-production period. Since the production period contains +m shipments, t must satisfy 0<t<T+m. So from (11) one has the following inequalities: (11) (12) By substituting β=p/d, qi=q0β i (i=1,2,, ), qi=λq0β (i=+1,2,,m), and qi=αq0 (i= +m+1,2,, +m+n), into (12), one can obtain that the variables (, m, n, λ, α) should satisfy: 1 m n (13) n m1 1 (14) The second constraint is that the ith shipment size (i=2,3,..., +m) in the production period cannot exceed the amount of items the vendor has just before the ith shipment is sent, i.e., the following inequalities should be satisfied (15) Noting qi=q0β i (i=1,2,, ), qi=λq0β (i=+1,2,,m), and qi=αq0 (i= +m+1,2,, +m+n), one can rewrite the above inequalities as

11 1 m 1 (16) Thus, our problem is to determine the optimal values of (, m, n, λ, α) that minimize the average total cost (8) subject to the constraints (9), (10), (13), (14), (16). A1 m n A2D P D Q C, m, n, q0,, а h1 q0 h2 h1 B Q 2P 1 q m n q 1 1 B 2 1 q0 m n q Q q m n q m n n m1 1 1 m 1 In the whole model, there are only six variables:, m, n, q0, λ, α. And the other notation are all belong to the input. 5 Algorithm and numerical examples 5.1. The solution procedure Considering the fact that the proposed model is a mixed integer non-linear formulation, due to the complexity of the problem, we can present its numerical solution through PSO (Particle Swarm Optimization) method. PSO is the one of the population based optimization algorithms, which was presented by Kennedy and Eberhart (1995). First, the set of particles is placed in the response space and starts moving with initial velocity. Then, these particles move in response space and the movements are guided by their own best nown position in the search-space as well as the entire swarm's best nown position in each stage. Over time, these particles accelerate with specified velocity towards other particles available in their communication group in multidimensional search space, which have higher fitness value. Any position of particles shows a solution for the problem. Let f be the cost function which must be minimized. S be the number of particles in the swarm, each having a position xi R n in the search-space and a velocity vi R n. Let pi be the best nown position of particle i and let g be the best nown position of the entire swarm. c1 and c2 are the fixed coefficient to control the impact of pi and pg. φ1 and

12 φ2 is the uniform random number between [0,1]. w control the intensification and diversification. At any iteration, velocity and position of particles are updated according to Eq. (17) and Eq. (18): v t wv t 1 c p x t 1 c p x t 1 (17) i i 1 1 i i 2 2 g i 1 x t x t v t (18) i i i In PSO, primary population is considered 50 in each repetition. The maximum number of repetitions at any time of the algorithm execution is considered Values of φ1 and φ2 are both considered as 1.5. Personal and ultimate learning coefficients of c1 and c2 are as fixed coefficients of φ1 and φ2, respectively. w is considered Here, stop condition is achieved when algorithm reaches the maximum number of repetitions. Notation of the algorithm are introduced in Table 3. Table 3. Notations of PSO algorithm Notations Value p g the best nown position of the entire swarm - p i the best nown position of particle i - w: intensification and diversification coefficient 0.73 φ 1 and φ 2 : uniform random number between [0,1] rand c 1 and c 2 : fixed coefficient to control the impact of p i and p g 1.5 S: the primary population 50 The maximum number of repetitions Numerical examples For the sae of comparison, we use the same example as ever used by Hill (1999) and Hoque and Goyal (2000). Example 1. The parameters of the model are listed below. A1=400; A2=25; h1=4; h2=5; P=3200; D=1000 After using the proposed solution procedure, we can obtain the optimal production and shipment policy of the integrated system: q0=7.39, =2, m=1, n=1, α=3.03. So, the optimal shipment number is N=4 and the batch sizes of 4 shipments are respectively q1=23.6, q2=75.6, q3=229.2, q4=229.3, and total cost is This result compare to the results of other policy The computed results are shown in Table 3. Example 2. If h2 is increased from 5 to 7 as Hill (1995) did. After using the proposed solution procedure, we can obtain the optimal production and shipment policy of the integrated system: q0=9.72, =2, m=1, n=2, α=1.38. So, the optimal shipment number is N=5 and the batch sizes of 5 shipments are respectively q1=31.1, q2=99.5, q3=137, q4=136.96, q4=136.96, and total cost is This result is compared to the results of other policy. And the computed results are shown in Table 4.

13 Examples Case 1 Case 2 Table 4. Results of two examples Policies Total Order cost times Order quantity Goyal (1995) ,116,370 Lu (1995) ,111,111,111,111 Hill (1997) ,68,142,310 Hill (1999) ,75.6,229.3,229.3 Goyal and Nebebe (2000) ,166,166,166 Goyal (2000) ,99,211,211 Shengdong Wang ,75.5,228.9,228.9 Three-Phase policy ,75.6,229.2,229.3 Goyal (1995) ,101,322 Lu (1995) ,91,91,91,91,91 Hill (1997) ,72,99,131,177 Hill (1999) ,99.5,136.96,136.96, Goyal and Nebebe (2000) ,128,128,128,128 Goyal (2000) ,123,123,124,124 Shengdong Wang ,99.5,136.9,136.9,136.9 Three-Phase policy ,99.5,1396,136.96, Conclusions In this paper, the vendor buyer integrated system is reconsidered and a generalized structure of the optimal shipment policy is built. The relationship between the shipment policy and the system inventory can be revealed by observing the changes of the system inventory during the production time and the nonproduction time as the quantities of the buyer s adjacent orderings changed. These relationship showed that, in the optimal shipment policy, there may has another independent equal series of the shipments after the Hill s (1999) increasing and equal series shipments. The generalized structure of the optimal shipment policy should include three stages: increasing orders and equal orders in production time and another series equal orders in nonproduction time. Based on this policy structure, an optimization model is built,which is more simple than Wang and Zhou s (2007). Unlie the existing vendor-buyer integrated models, the present model and the structure of shipment policy needn t assume the vendor s unit holding cost less than the buyer s. Numerical experiments have shown that the model can always generate the lowest average total cost among the hitherto existing vendor-buyer integrated models. Acnowledgements This wor was supported by National Natural Science Foundation of China (Grant No , No and No ), the Natural Science Foundation of Hainan Province (No ), the project of Hainan province philosophy and social sciences (No. HNSK(QN)15-4), and the Education fund item of Hainan province (Hny2015ZD-7, Hny2015ZD-2).

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