routing problem

European Journal of Operational Research 177 (2007) 786–802 www.elsevier.com/locate/ejor

Discrete Optimization

A partition approach to the inventory/routing problem Qiu-Hong Zhao a, Shou-Yang Wang b, K.K. Lai

q

c,*

a

b

School of Economics and Management, Beihang University, Beijing 100083, China Institute of Systems Science, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100080, China c Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong Received 24 February 2003; accepted 23 November 2005 Available online 23 February 2006

Abstract In this study we focus on the integration of inventory control and vehicle routing schedules for a distribution system in which the warehouse is responsible for the replenishment of a single item to the retailers with demands occurring at a specific constant (but retailer-dependent) rate, combining deliveries into efficient routes. This research proposes a fixed partition policy for this type of problem, in which the replenishment interval of each of the retailers’ partition region as well as the warehouse is accorded the power of two (POT) principle. A lower bound of the long-run average cost of any feasible strategy for the considered distribution system is drawn. And a tabu search algorithm is designed to find the retailers’ optimal partition regions under the fixed partition policy proposed. Computational results reveal the effectiveness of the policy as well as of the algorithm. 2006 Elsevier B.V. All rights reserved. Keywords: Distribution system; Inventory control and vehicle routing schedules; Partition approach; Tabu search algorithm

1. Introduction It has been shown that important activities in a supply chain can be coordinated through effective inventory control (Thomas and Griffin, 1996). Specifically, an effective way to improve supply chain performance is for the vendor to determine the quantities that should be ordered by its downstream customers, rather than the other way around. This approach, which is known as vendor managed inventory (VMI), can reduce inventories and stock-outs by using advanced online messages and data-retrieval systems (Aviv and Federguen, 1998; Parker, 1996; Schenck and McInerney, 1998; Angulo et al., 2004). In addition, as the vendor is guided by

q The work of this study is supported by the Strategic Research Grant of City University of Hong Kong (Project no. 7001637), a research grant of the 100 Talent Program from the Chinese Academy of Sciences and National Natural Science Foundation of China (Project no. 70301001). * Corresponding author. Tel.: +852 27888563; fax: +852 27888560. E-mail address: [email protected] (K.K. Lai).

0377-2217/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2005.11.030

Q.-H. Zhao et al. / European Journal of Operational Research 177 (2007) 786–802

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mutual agreements on inventory levels, fill rates and transaction costs, trading partners can maximize their benefits (Andel, 1996). Many authors have provided theoretical analysis of the strategic aspect of VMI (Lee and Rosenblatt, 1986; Anupindi and Akella, 1993; Kohli and Park, 1994; Weng, 1995; Achabal et al., 2000; Dong and Xu, 2002; Yang et al., 2003). The benefits of VMI are well recognized by successful retail businesses such as Wal-Mart, Kmart and Dillard Department Stores (Cetinkaya and Lee, 2000; Dong and Xu, 2002). Some authors, such as Cetinkaya and Lee (2000) and Cheung and Lee (2002), address the coordination of stock replenishment with vehicle shipments. However, these papers do not provide an in-depth study of the VMI mode with vehicle routing schedule variation, which may be more costly. There are several potential models that integrate inventory and vehicle routing problems into VMI. Considering the decision domain, two different approaches can be chosen: time or frequency. The former suits asynchronous operating modes, possibly using feedback, see Federgruen and Zipkin (1984), Dror and Ball (1987), Dror and Levy (1986) and Fumero and Vercellis (1999) for reference. The frequency approach is appropriate with synchronous, periodic operations and our study suits this approach. In the following, the problem under consideration and the literature are described in Sections 2 and 3 respectively, followed by a lower bound on the cost of any feasible strategy. We present the fixed partition and power-of-two (FPPOT) policy and the corresponding algorithm in Section 5, while a tabu search algorithm used to find the optimal regional decomposition is proposed in Section 6. Computational results are discussed in Section 7. Finally, a conclusion is given in Section 8. 2. The problem The problem under study can be described as follows: a single warehouse serves retailers which are geographically dispersed in a given area; denote N = {1, 2, . . . , n} as the retailer set, let di be the distance from retailer i to the warehouse. Each retailer faces a deterministic, retailer-specific demand rate Di for a single item. The requirement should be satisfied without delay, and is delivered from the warehouse by a fleet of homogenous vehicles with limited capacity V. Denote f as the maximum frequency with which each retailer is visited and f 6 1, this constraint may, for example, be due to limited material handling capacity at the retailers. The stock in the warehouse is delivered from its supplier. No inventory capacity constraint is imposed on the warehouse or on the retailers. Whenever a vehicle is sent out from the warehouse to replenish inventory to a set of retailers S N, it incurs a fixed cost c (independent of the specific route driven) plus a variable cost proportional to the total distance traveled by the vehicle (we assume here that the unit variable cost of the vehicle is 1). Inventory holding costs are incurred at a constant rate h per unit of time, and per unit stock for all retailers, while this cost at the warehouse is denoted as h0. Let h 0 = h h0 > 0 (since the unit holding cost at the retailer normally exceeds that at the warehouse, the assumption is a natural one). In addition, a fixed ordering cost k0 is incurred each time inventory is replenished at the warehouse. The objective is for the warehouse to determine inventory policies and routing strategies for the distribution system such that each retailer can meet its demand, and so that the long-run average costs of the whole system (ordering, delivery and inventory) are minimized. Due to the complicated nature of the problem, an optimal strategy has not yet been proposed for it. Even when considering a problem with individual and uncoordinated deliveries, the optimal policy structure can be very complex making these solutions unattractive, even if their computation were tractable (Roundy, 1985). Thus the problem is usually solved by the authors restricting themselves to finding an optimal solution under a specify strategy. 3. The literature The problem under study is addressed by Anily and Federgruen (1993). These authors restrict their analyses to a class of replenishment strategies w with the following properties: a replenishment strategy in w specifies a collection of regions, each of which includes some retailers. If a retailer belongs to several regions, a specific fraction of its sales is assigned to each of these. Each time one of the outlets in a given region receives a

788


delivery, this comes from a vehicle which visits all other outlets in the region as well. Under a strategy in w, all regions are controlled independently of each other. Thus, if a retailer belongs to two regions, it is treated as two separate sub-outlets, each responsible for a specific fraction of the sales. It is therefore possible that a delivery is made to one sub-outlet at a period during which the other sub-outlet continues to have stock, thus more inventory costs may be incurred. The policy proposed by Anily and Federgruen (1993) can be deemed as one of the fixed partition (FP) policies discussed in Anily and Federgruen (1990a,b) and Bramel and Simichi-Levi (1995). In this paper, a FP policy which is derived mainly from Anily and Federgruen (1993) is also presented. However, in the proposed policy, each retailer belongs to only one region, allowing for easy integration of distribution, marketing and customer service functions. In addition, as the optimal region partition is found by employing a tabu search algorithm, the problem dealt with in this study is more general in nature than that studied by Anily and Federgruen (1993). The constraints on the demand rate, as well as the usual assumption that the retailers’ locations are at an i.i.d. random distance from the warehouse, can be relaxed. According to the proposed FP policy, if the retailer set N is partitioned into region set v = {1, 2, . . . , L}, the remaining problem is reduced to identifying the optimal inventory replenishment strategies in a classical one-warehouse L-retailer system in which each region l 2 v plays the role of a single ‘‘retailer’’ P with demand rate ml ¼ j2l Dj and a procurement cost, which is the sum of the fixed and variable transportation costs. Even in this kind of situation, the problem is compounded by determining a replenishment strategy for the warehouse that is ‘‘optimally’’ integrated with that of each region. No method is known for computing an optimal strategy. However, Roundy (1985) has shown that, in the absence of constraints on regional replenishment frequencies and without the consideration of the variable transportation cost to the retailers, a simple POT policy exists whose cost is guaranteed to be within 6% of optimality. Under a POT policy, the warehouse (region l) replenishes its inventory every T p0 ðT pl ; l ¼ 1; 2; . . . ; LÞ time units when its inventory reaches to zero; and ðT p0 ; T p1 ; . . . ; T pL Þ are POT multiples of a base planning period TB. Here, we adopt the POT strategy for a given partition in which the constraints on regional replenishment frequencies and the variable transportation costs to the retailers are accounted for, and try to find POT replenishment intervals by rounding off the replenishment intervals ðT 0 ; T 1 ; . . . ; T L Þ for which the corresponding cost is the lower bound of the cost of the POT replenishment policy proposed. Several other related problems involving inventory and routing have been addressed in the literature. A few authors account for problems similar to the one under study, but without considering central stock at the warehouse. These include the work of Anily and Federgruen (1990a), Viswanathan and Mathur (1997) and Anily and Bramel (2004). Chan et al. (1998) and Chan and Simchi-Levi (1998) construct an inventory control and vehicle routing strategy under the assumption that the number of the retailers is very large, such that the warehouse can receive fully loaded trucks from the out-supply but never hold inventory. Fixed ordering cost is not considered in the above four works. 4. A lower bound for the cost of any feasible strategy In this section, a lower bound of the long-run average cost of any feasible strategy for the considered distribution system is drawn. It can be seen from the prior sections that, for any feasible strategy R, average costs during the interval [0, t) consist of four parts: denote c1 and c2 as the average of the ordering cost and the holding cost at the warehouse respectively, c3 as the average of the holding cost at the retailers, and c4 as the average of the transportation cost incurred by the vehicles traveling from the warehouse to the retailers. Without loss of generality, we set the initial inventory as well as the final inventory of the warehouse during the interval [0, t) to zero, and do this for all the retailers. We first relax the restriction on the capacity of the vehicles and deduce the minimum of the sum of c1, c2 and c3 under such assumption, which can be taken as a lower bound of c1 + c2 + c3. The calculation of a lower bound of c4 follows that given by Chan et al. (1998). Thus the sum of these two lower bounds can be deemed as a lower bound for the average cost of any feasible strategy.


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4.1. A lower bound of the sum of c1, c2 and c3 Let y represent the dispatch number received by the warehouse during the interval [0, t). We denote all the replenish schedules with a given y as kðyÞ, in which cost c4 are not accounted for. For any j 2 y, denote pj and Qj as the delivery time and quantity to the warehouse respectively. To get such a lower bound, we first assume that there exists only one retailer i in the considered distribution system. Qj is in fact the sum of the items required by retailer i during the interval [pj, pj+1), each of which will be stored either in the warehouse or in the retailer at any moment before it is sold to the customers. As higher inventory costs will be incurred if the required item is stored with the retailer rather than in the warehouse, to get the optimal schedule among kðyÞ, any item required should be stored in the warehouse as long as possible. So the product should be delivered to the retailer at the maximum frequency possible, which is f here, and pj is the same as when one of the deliveries is dispatched to the retailer. See Fig. 1 for reference, where t = 18, and f = 1/4 in the example given. As the delivery frequency to the retailer is constant and equal to f in the optimal replenish schedule among kðyÞ, the warehouse’s requirement can be deemed as constant and occurs every 1/f periods with quantities Dif1. According to the traditional economic ordering quantity (EOQ) model, we can calculate the optimal delivery cycle TT 0 to the warehouse for the long-run period t, where the optimal delivery times y* to the warehouse equals t=TT 0 . Consider now a distribution system in which there are N = {1, 2, . . . , n} retailers. Similar to the above analysis, it can be further deduced that for any kðyÞ, the optimal delivery frequency to each of the retailers is identical and equals f. pj is the same as when one of the deliveries is made to any retailer, and Qj is the sum of the required quantities of all the retailers during the interval [pj, pj+1), j 2 y. These conclusions imply that the deliveries should be made to all the retailers concurrently,P and the warehouse’s requirement can be deemed con1 stant and occurs every 1/f periods with quantities . Thus we can follow the traditional EOQ i2N Di f model to calculate the optimal delivery frequency TT 0 to the warehouse for a long-run period, where the optimal delivery times y* to the warehouse equals t=TT 0 . Denote TT0 as the delivery cycle to the warehouse, and cos t1 as the average of the sum of the ordering and inventory costs of the considered distribution system in a long planning period t. Then P X Di f 1 þ 0 Di ðTT 0 f 1 Þ þ 0 k0 h. ð1Þ h0 þ cos t1 ðTT 0 Þ ¼ þ i2N 2 2 TT 0 i2N To derive the right hand side of Eq. (1) and letting (cos t1(f0)) 0 = 0, then sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2k 0 0 . TT 0 ¼ P i2N Di h0 Since the optimal delivery cycle TT 0 should be an integer in the real operation, we discuss next the value of by analyzing the following situations:

TT 0

• TT 00 P 1=f , this follows the premise for setting up Eq. (1). Furthermore, if TT 00 is an integer, TT 0 ¼ TT 00 ; otherwise, TT 0 should be the integer nearest to TT 00 , which is not less than 1/f. So

0

4

Delivery to the warehouse

8

12

16

period

Delivery to the retailer

Fig. 1. Illustration of the delivery to the warehouse as well as to the retailer.

790


TT 0

¼

dTT 00 e; ðTT 00 bTT 0 cÞ P ðdTT 0 e TT 00 Þ; bTT 00 c; ðTT 00 bTT 0 cÞ < ðdTT 0 e TT 00 Þ.

• TT 00 < 1=f , as TT 00 is outside the premise for setting up Eq. (1), TT 0 ¼ 1=f . Thus, P X Di f 1 þ 0 Di ðTT 0 f 1 Þ þ 0 k0 h; h0 þ min cos t1 ¼ þ i2N 2 2 TT 0 i2N where the value of TT 0 can be gained based on the above analysis, and c1 þ c2 þ c3 P min cos t1 .

ð2Þ

4.2. A lower bound of c4 Consider now the average of the transportation cost c4. For any feasible strategy R to the original problem, let R be the vehicles’ number sent out from the warehouse during the interval [0, t) (for the sake of notational convenience, it is assumed that each vehicle is used only once during the considered interval), let Sr be the set of retailers visited by vehicle r, r = 1, 2, . . . , R, h(Sr) be the length of the optimal traveling salesman tour through the warehouse and the retailers in the set Sr, and wri be the quantities of product received by retailer i from vehicle r during the P interval [0, t). Denote by Qr the amount of product delivered by vehicle r during the interval [0, t), i.e., Qr ¼ i2N wri . It is clear that for vehicle r and any retailer i 2 Sr, h(Sr) + c P 2di + c. Hence X X wri ½hðS r Þ þ c P wri ð2d i þ cÞ. Qr ½hðS r Þ þ c ¼ i2S r

i2S r

Since Qr 6 V, hðS r Þ þ c P

X wr i ð2d i þ cÞ. V i2S r

So, c4 ¼

X Di 1X 1 X X wri ð2d i þ cÞ P ð2d i þ cÞ. ½hðS r Þ þ c P t r2R t r2R i2S r V V i2N

ð3Þ

Denote the average cost of any feasible policy R in interval [0, t) as ‘ðR; tÞ. Then it can be seen from formulations (2) and (3) that X Di ‘ðR; tÞ ¼ c1 þ c2 þ c3 þ c4 P min cos t1 þ ð2d i þ cÞ. V i2N So B ¼ min cos t1 þ

X Di ð2d i þ cÞ V i2N

ð4Þ

is a lower bound for the minimum long-run average cost for any feasible strategy R, where the calculation of min cost1 can be traced in Section 4.1. 4.3. Further analysis of the lower bound B* It can be seen that the lower bound of the average cost of any feasible strategy R is composed of two parts. One is min cost1, which is the lower boundP of c1 + c2 + c3 and delivery frequency to each of the retailers is taken as f here. Whereas the other one is i2N DV i ð2d i þ cÞ, which is the lower bound of c4 and calculated


791

by approximating firstly the distance of any route as twice of the average of the sum of the distance between the warehouse and all of the nodes in the route, assuming that the load of any vehicle isP full. * as Denote the average cost of the optimal feasible solution corresponding to any B i¼1;2;3;4 ci , now we P analyze in the following the variation of DS ¼ i¼1;2;3;4 ci B under some cases. •

P

i¼1;2;3 ci

¼ min cos t1 . As all the retailers in the optimal feasible solution are delivered as frequency f, they are more easier to be clustered P to minimize the transportation P cost due to the minimum delivered quantities possible, and Dc4 ¼ c4 i2N DV i ð2d i þ cÞ as well as DS ¼ i¼1;2;3;4 ci B is larger than P zero if at least two retailers are delivered along one route, by tracing the computational procedure of P i2N DV i ð2d i þ cÞ. Furthermore, the more the difference of V and f1 Æ Di, "i 2 N is, the larger DS ¼ i¼1;2;3;4 ci B would be. Such DS* also increases along with the increment of the unit variable transportation cost to the retailers. In addition, as the vehicles cannot always be fully loaded under the real operation, DS* should be a nondecreasing function of c. P • c4 ¼ i2N DV i ð2d i þ cÞ. As all the retailers are direct shipped by loaded, each of them is P the vehicles Pwith fully delivered as the minimal frequency possible. And D c ¼ c min cos t1 as well as i¼1;2;3 i i¼1;2;3 i P DS ¼ i¼1;2;3;4 ci B is larger than zero if at least one retailer is delivered with the frequency lower than f, by tracing the computational procedure of min cos t1. Furthermore, the more the difference of V and P f1 Æ Di, "i 2 N is, the larger DS ¼ i¼1;2;3;4 ci B would be. And DS* also increases along with the increment of the unit inventory cost in the retailers. • None of P the above two delivery are adopted in the optimal feasible solution. Both P modes P Dc4 ¼ c4 i2N DV i ð2d i þ cÞ andPD i¼1;2;3 ci ¼ i¼1;2;3 ci min cos t1 are larger than zero in the case that $i 2 N, V/Di > f1, and DS ¼ i¼1;2;3;4 ci B increases along with the increment of the difference of V and f1 Æ Di, "i 2 N. DS* also increases along with the increment of the unit inventory cost in the retailers as well as the unit variable transportation cost and the fixed cost to them. It can be drawn out from the above analysis that, as the two parts of B* are calculated based on the relaxation of another one, DS* = 0 if and only if V/Di = f1, "i 2 N. And the following conclusion can be drawn: P Conclusion 4.1. DS ¼ i¼1;2;3;4 ci B is larger than zero if $i 2 N, V/Di > f1, and P (1) DS* increases along with the increment of nV i2N Di f 1 . (2) DS* is a non-decreasing function of the unit inventory cost in the retailers, and also a non-decreasing function of the unit variable transportation cost to them. (3) DS* is a non-decreasing function of the fixed transportation to the retailers. P The above conclusion reveals to a certain extent the relation of B* and i¼1;2;3;4 ci , and are helpful for the analysis of the computational results and further the evaluation of the strategy as well as the algorithm proposed, which are given in the later sections. 5. Power-of-two replenishment policy In this section, we analyze the way of finding the POT replenishment interval for each of the regions partitioned. As a route can only be constructed among all the retailers belonging to a given region, we use ‘‘region’’ and ‘‘route’’ interchangeably in the following sections. Assume v = {1, . . . , L} is a given decomposition set of the retailers, and any retailer i can only belong to one of the subsets of v. T p ¼ fT p0 ; T p1 . . . ; T pL g is a corresponding set of the POT replenishment intervals, where T p0 is the replenishment interval of the warehouse. Based on Anily and Federgruen (1993), we can express the average cost of the system under the given decomposition and replenishment policy as follows: C v ðT p Þ ¼ K 0 =T p0 þ

L X l¼1

DT p0 ðT pl ; hl ; ml Þ;

792


where DT p0 ðT pl ; hl ; ml Þ includes the transportation costs which are incurred in region l 2 v, the carrying costs for the inventories stored in region l and the part of the warehouse inventory which is destined to be shipped to the region. According to Roundy (1985), 1 DT p0 ðT pl ; hl ; ml Þ ¼ ðhl þ cÞ=T pl þ ml ½h0 T pl þ h0 maxðT p0 ; T pl Þ. ð5Þ 2 For any given fixed region l 2 v, we can regard (hl + c0) as the ordering cost, ml as the demand rate, then let s0 ðhl ; ml Þ ¼ ½2ðhl þ cÞ=ðml ðh0 þ h0 ÞÞ 0

sðhl ; ml Þ ¼ ½2ðhl þ cÞ=ðml h Þ

1=2

;

1=2

represent respectively the optimal replenishment interval of region l gained by using the EOQ formula, where unit stock cost is regarded as (h 0 + h0) for the first formulation and h 0 for the second formulation. The value of the unit stock cost for any region l relies on the relationship between the replenishment interval of the warehouse and that of the region. If the former one is smaller than the later one, the unit stock cost is (h 0 + h0), otherwise the unit stock cost is h 0 . It is clear that the average cost of the system gained according to the above replenishment intervals is a lower bound of Cv(Tp). Denote this average cost as Uv, that is def

U v ¼ inffC v ðT p Þ; T p0 > 0 and f 1 6 T pl 6 V =ml ; l ¼ 1; . . . ; Lg.

ð6Þ

In the following, we analyze how to calculate Uv and the corresponding replenishing intervals ðT 0 ; T 1 ; . . . ; T L Þ. For v and any given T0, let U v ðT 0 Þ ¼ K 0 =T 0 þ

L X

fT 0 ðhl ; ml Þ;

ð7Þ

DT p0 ðT pl ; hl ; ml Þ;

ð8Þ

l¼1

where fT 0 ðhl ; ml Þ ¼ then

inf p

f 1 6T l 6V =ml

U v ¼ inf U v ðT 0 Þ; T 0 > 0 .

ð9Þ

Next, for any given region l 2 {1, . . . , L}, we analyze the variation of fT 0 ðhl ; ml Þ along with T0 (for the sake of notational convenience, we omit the sub-symbol l in the following expression). Denote the optimal replenishment interval of region l under a given T0 as T*(T0). When the constraints related to the maximum delivery frequency and the vehicle capacity are relaxed, we have 8 0 0 > : s; T 0 > s. On the other hand, when such constraints are taken into account, we can characterize the relationship of (s 0 , s, f, V/m) as the following: 1. s0 < s < f 1 < V =m;

8. s0 < f 1 < s ¼ V =m;

2. s0 < s ¼ f 1 < V =m;

9. s0 < f 1 < V =m < s;

3. s0 < f 1 < s < V =m;

10.

s0 ¼ f 1 < V =m < s;

4. s0 ¼ f 1 < s ¼ V =m;

11.

f 1 ¼ s0 < s < V =m;

5. f 1 < s0 < V =m < s;

12.

f 1 < s0 < s < V =m;

6. f 1 < V =m < s0 < s;

13.

f 1 < s0 < s ¼ V =m;

7. f 1 < V =m ¼ s0 < s;

14.

f 1 ¼ V =m.


793

Based on the above classification, we can further show the variation of fT 0 ðh; mÞ along with T0 as follows: a. Cases 1, 2 and 14. (s 0 < s 6 f1 < V/m or f1 = V/m) ( fT 0 ðh; mÞ ¼

.fT0 0

ðh þ cÞf þ 12 mðh þ h0 Þf 1 ;

T 0 6 f 1 ; ðT ðT 0 Þ ¼ f 1 Þ

ðh þ cÞf þ 12 mh0 f 1 þ 12 mh0 T 0 ;

T 0 > f 1 ; ðT ðT 0 Þ ¼ f 1 Þ f

1

b. Cases 3, 4, 8 and 11. (s 0 6 f

< s 6 V/m)

8 ðh þ cÞf þ 12 mðh0 þ h0 Þf 1 ; T 0 6 f 1 ; ðT ðT 0 Þ ¼ f 1 Þ > > < 0 fT 0 ðh; mÞ ¼ ðh þ cÞ=T 0 þ 12 mðh þ h0 ÞT 0 ; f 1 < T 0 6 s; ðT ðT 0 Þ ¼ T 0 Þ > > : 1=2 ½2ðh þ cÞmh0 þ 12 mh0 T 0 ; T 0 > s; ðT ðT 0 Þ ¼ sÞ 1

c. Case 5. (f

.fT0

f

τ

–1

T0

0

< s < V/m < s)

8 ½2ðh þ cÞmðh0 þ h0 Þ1=2 ; T 0 < s0 ; ðT ðT 0 Þ ¼ s0 Þ > > < fT 0 ðh; mÞ ¼ ðh þ cÞ=T 0 þ 12 mðh0 þ h0 ÞT 0 ; s0 < T 0 6 V =m; ðT ðT 0 Þ ¼ T 0 Þ > > : mðh þ cÞ=V þ 12 h0 V þ 12 mh0 T 0 ; T 0 > V =m; ðT ðT 0 Þ ¼ V =mÞ d. Cases 6 and 7. (f ( fT 0 ðh; mÞ ¼

T0

–1

1

.fT

0

τ ' V/m

T0

0

< V/m 6 s < s) .fT0

mðh þ cÞ=V þ 12 V ðh0 þ hÞ;

T 0 6 V =m; ðT ðT 0 Þ ¼ V =mÞ

mðh þ cÞ=V þ 12 h0 V þ 12 mh0 T 0 ;

T 0 > V =m; ðT ðT 0 Þ ¼ V =mÞ V/m

T0

–1

V/m

e. Cases 9 and 10. (s 0 6 f1 < V/m < s) 8 ðh þ cÞf þ 12 mðh0 þ h0 Þf 1 ; T 0 6 f 1 ; ðT ðT 0 Þ ¼ f 1 Þ > > < fT 0 ðh; mÞ ¼ ðh þ cÞ=T 0 þ 12 mðh0 þ h0 ÞT 0 ; f 1 < T 0 6 V =m; ðT ðT 0 Þ ¼ T 0 Þ > > : mðh þ cÞ=V þ 12 h0 V þ 12 mh0 T 0 ; T 0 > V =m; ðT ðT 0 Þ ¼ V =mÞ f. Cases 12 and 13. (f

1

.fT0

f

T0

6 s 0 < s < V/m)

8 ½2ðh þ cÞmðh0 þ h0 Þ1=2 ; T 0 6 s0 ; ðT ðT 0 Þ ¼ s0 Þ > > > < fT 0 ðh; mÞ ¼ ðh þ cÞ=T 0 þ 12 mðh0 þ h0 ÞT 0 ; s0 < T 0 6 s; ðT ðT 0 Þ ¼ T 0 Þ > > > : ½2ðh þ cÞmh0 1=2 þ 12 mh0 T 0 ; T 0 > s; ðT ðT 0 Þ ¼ sÞ

.fT0

τ'

τ

T0

It can be seen from the above formulations as well as from the figures that fT 0 ðh; mÞ is a non-decreasing function. It is differentiable at points s 0 and s, whereas it may fail to be differentiable at points f1 and V/m. We call the point at which function fT 0 ðh; mÞ changes the ‘‘breakpoint’’. Thus the following conclusions can be derived:

794


Conclusion 5.1. Function (7) is convex and differentiable over the whole interval T0 > 0, except possibly for the points {f1, V/ml, l = 1, . . . , L}. Conclusion 5.2. Function (7) can be expressed as follows: U v ðT 0 Þ ¼ aðT 0 Þ=T 0 þ bðT 0 Þ þ cðT 0 Þ T 0 ;

ð10Þ

where a(T0), b(T0), c(T0) are piecewise constant functions changing values only at the breakpoints, which are included in: fs0l : l ¼ 1; . . . ; Lg [ fsl : l ¼ 1; . . . ; Lg [ ff 1 ; V =ml ; l ¼ 1; . . . ; Lg.

ð11Þ

Conclusion 5.3. T 0 2 fT 00 ; f 1 ; V =ml ; l ¼ 1; . . . ; Lg, where T 0 is the replenishing interval of the warehouse minimizing Formulation (10), and T 00 satisfies:

dU v ðT 00 Þ dT 00

¼ 0.

According to the above analysis, for any given decomposition region set v = {1, . . . , L}, we can follow the procedures given below to find the replenishment intervals ðT 0 ; T 1 ; . . . ; T L Þ. Step 1. For any region l 2 v, label it as one of the classes {a, b, c, d, e, f} according to the non-decreasing sequence of the points ðs0l ; sl ; f 1 ; V =ml Þ. Insert this region’s breakpoints into a list s. If all regions have been considered, sort all the breakpoints in list s to form a non-decreasing sequence. Label each breakpoint with the class Pas well as the region it belongs to. Step 2. Let aðT 0 Þ ¼ k 0 ; cðT 0 Þ ¼ l2v ðml h0 Þ=2, consider the right-most point v1 in list s. If v21 6 aðT 0 Þ=cðT 0 Þ, go to step 5; else go to step 3. Step 3. Change the value of a(T0) and c(T0) according to following formulas. If there are still any unconsidered breakpoints in list s, denote v1 as v2, the left point of v2 as v1, go to step 4; if there are no unconsidered points in list s, go to step 6. (1) v1 = sl, l = 1, . . . , L; aðT 0 Þ ¼ aðT 0 Þ þ ðhl þ cÞ;

1 cðT 0 Þ ¼ cðT 0 Þ þ ml h0 ; 2

(2) v1 = V/ml, l = 1, . . . , L; aðT 0 Þ ¼

aðT 0 Þ þ ðhl þ cÞ; aðT 0 Þ;

l 2 fc; eg; l 2 fdg;

( cðT 0 Þ ¼

cðT 0 Þ þ 12 ml h0 ; cðT 0 Þ

1 mh; 2 l 0

l 2 fc; eg; l 2 fdg;

(3) v1 ¼ s0l ; l ¼ 1; . . . ; L; aðT 0 Þ ¼ aðT 0 Þ ðhl þ cÞ;

1 cðT 0 Þ ¼ cðT 0 Þ ml ðh0 þ h0 Þ; 2

(4) v1 = f1, l = 1, . . . , L; aðT 0 Þ ¼

aðT 0 Þ; aðT 0 Þ ðhl þ cÞ;

l 2 fag; l 2 fb; eg;

( cðT 0 Þ ¼

cðT 0 Þ 12 ml h0 ; cðT 0 Þ

1 m ðh0 2 l

l 2 fag; þ h0 Þ; l 2 fb; eg.

Step 4. If v21 6 aðT 0 Þ=cðT 0 Þ 6 v22 , go to step 5; else go to step 3. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Step 5. Let T 0 ¼ aðT 0 Þ=cðT 0 Þ, according to the class of (a, b, c, d, e, f) which region l belongs to, calculate T l ; l ¼ 1; . . . ; L. Step 6. For each breakpoint v 2 (f1, V/ml, l = 1, . . . , L) in list s, let T0 = v, calculate Uv(T0) according to formula (10), then T 0 ¼ fT 0 j min U v ðT 0 Þg. According to the class of (a, b, c, d, e, f) which region l belongs to, calculate T l ; l ¼ 1; . . . ; L.


795

In order to find the POT replenishment intervals fT p0 ; T p1 . . . ; T pL g, it is necessary to round off Here we adopt the method given by Anily and Federgruen (1993).

ðT 0 ; T 1 ; . . . ; T L Þ.

6. Tabu search algorithm In this section, we apply a tabu search algorithm to find the optimal region partition for all the retailers. A tabu search is a meta-strategy iterative procedure used to build an extended neighborhood with particular emphasis on avoiding being caught in a local optimum. It consists of exploring the search space by moving from a solution to its best neighbor—even if this results in a deterioration of the objective function value. To avoid cycling, when an action is performed it is considered tabu for the next T iterations, where T is the tabu status length. The best admissible move is then chosen as the highest evaluation move in the neighborhood of the current solution. The aspiration criterion is a measure solely designed to override the tabu status of a move if this move leads to a solution better than the best found by the search so far. The tabu search algorithm has been widely employed for solving the vehicle routing problems (VRP). As the objective function of the VRP is to find the shortest paths for visiting all the nodes in one period, the neighborhoods of the current solution are constructed based solely on the distance between the nodes. For the problem under considered, however, various cost factors should be accounted for when the neighborhoods and other technical parameters of the tabu search algorithm are designed. 6.1. Generation of the initial solution The algorithm starts by giving an initial solution using the following steps: Step 1. Generate the initial routes by arranging each node to a single route. Step 2. Based on the algorithm given in Section 5, calculate the POT intervals for the partition regions as well as the corresponding average cost of the total distribution system under the given decomposition and replenishment policy. The initial solution generated as above is helpful for widening the searching space. 6.2. Technical parameters The following are technical parameters used in the tabu search algorithm. 6.2.1. Neighborhood structure To avoid the situation that more nodes always tend to be clustered into one route, which may result in higher transportation cost since retailers in such route need to be visited frequently, the neighborhood of the current solution in our algorithm is constructed based on the limitation of fk at the iteration level k, which is defined as the maximum delivery frequency each route can be visited. As the POT replenishment intervals are adopted in the proposed strategy, we set fk = 2fk1, and f1 ¼ maxi2N 2ri f where Vf =2Di < 2ri 6 Vf =Di . At any level k, we adopt the insertion method to construct the neighborhood of the current feasible solution. The neighborhood structure can be described as follows: Step 1. Select randomly min{n, 5L} vertices, where n is the number of the retailers and L is the number of routes in the current solution. For each selected vertex i, find its five closest neighbored vertices. Use the GENI algorithm (see Gendreau et al., 1992) to successively attempt to insert vertex i into the route that the neighbored vertex belongs to. If the number of routes into which the vertex is inserted is less than five, set up a new route with vertex i. If the total demands of the route in which vertex i is inserted are larger than Vfk, try to divide the route into two parts so that the total distance of the two routes is as low as possible. Step 2. For each neighborhood of the current solution, apply the algorithm described in Section 5 to calculate the POT intervals as well as the corresponding objective function value, that is, the average cost

796


of the whole distribution system. Denote this objective function value as q, and calculate the artificial objective function value according to the following formula: q0 ¼ q þ Dmax ðnÞ1=2 quv ;

ð12Þ

where Dmax is the largest observed variation in q between two successive iterations at level k, q is a scaling factor equal to 0.0001 in the implementation, and uv is the times vertex i has been moved at level k. The function of the artificial objective is to limit the frequency with which vertex i is moved. Step 3. If a move leads to a solution better than the best one found by the search so far, perform it. Else perform the best non-tabu move of all the neighborhoods of the current solution. 6.2.2. Tabu status and aspiration criterion Whenever a vertex i is moved from route r to route s at iteration k, it may not be reinserted into route r until iteration k + u, where u is randomly selected in some interval ½u; u. In our implementation, we use u = 5 and u ¼ 10. As is common in the tabu search processes, the algorithm uses an aspiration criterion that overrides the tabu status of a vertex whenever moving it results in a new best value of q. 6.2.3. Tenure principle The tabu search procedure ends if fk = 2f. 6.3. Overall description of the tabu search process Now we present an overall description of the tabu search processes. Step 1. Let k = 1, and f1 ¼ maxi2N 2ri f , where Vf =2Di < 2ri 6 Vf =Di , generate an initial solution according to the procedures given in Section 6.1, where f is replaced by f1 here. Go to step 2. Step 2. According to the procedures given in Section 6.2, complete the insertion procedure and modify the tabu status. If the best solution found by now has not been improved after g iterations, take the best solution found at lever k as the current solution, let k = k + 1, and go to step 3. In our algorithm, g equals 100. Step 3. Let fk = 2fk1. If fk = 2f, record the best solution ever found and terminate; else go to step 2. 7. Numerical examples A few examples are given in this section to evaluate the FPPOT policy as well as the corresponding algorithm. To design each example, we assume that for each retailer i, f 1 Di 6 V ;

8i 2 N .

ð13Þ

In fact, if constraint (13) is not satisfied for any retailer i, it may be possible to deal with the situation by decomposing f1Di into two parts, where one part is the multiple of the vehicle capacity and can be delivered by direct shipping and the other part is delivered according to the given method. (In our future work, we will address in detail the situation in which constraint (13) is relaxed.) The proposed examples are classified into two sets according to the number of the retailers. In the first example set, there are 50 retailers, whereas in the second example there are 75. All of the coordination and unit demand of the retailers in each example set are the same as those given in Christofides and Eilon (1969). In each of these two example sets, 17 problems are constructed respectively. The algorithm is coded in C++ program and run on a Pentium 4 computer with 1800 MHZ processor and 256 RAM memory. Table 1 lists the value of the parameters of the first problem in each example sets. And the values of the parameters of any problems different with the first one of each example set are shown in Tables 2 and 3, respectively, in which the best computational results after three time’s running of the code are listed also. In each of these two tables, p is the problem’s serial number; P lists the values of the parameters in p different with the first problem; Z is the objective function value of the best solution ever found, in which


797

Table 1 Value of the parameters of the first problem in each of the example sets V

f

k0

c

h0

h0

500

0.2

300

200

0.1

0.1

Table 2 Parameters and computational results for the first example set p

P

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 a b

h0 = h 0 = 0.05 h0 = h 0 = 0.01 f = 0.5 f = 0.5, h0 = h 0 = 0.05 f = 0.5, h0 = h 0 = 0.01 f=1 f = 1, h0 = h 0 = 0.05 f = 1, h0 = h 0 = 0.01 f = 0.5, h0 = 0.05 f = 0.5, h0 = 0.2 V = 200, f = 0.5, c = 80 V = 1000, f = 0.5, c = 400 c = 800, f = 0.5 c = 1600, f = 0.5 k0 = 1000, f = 0.5 k0 = 2000, f = 0.5 P DG ¼ i2N ðV Di f 1 Þ. DS = Z B*.

L

T p0

Z

B*

DGa

DSb

CPU

8 8 16 7 7 13 7 7 13 7 7 15 4 4 4 4 7

5 5 10 4 4 8 4 4 8 4 4 4 4 4 4 4 8

912.93 716.18 537.18 881.61 719.65 538.75 885.02 714.33 543.80 803.11 1046.24 977.41 872.92 1824.72 3077.19 1061.43 1203.80

831.71 637.46 470.94 677.46 574.76 459.28 638.61 555.34 455.40 613.61 766.31 786.08 641.26 1609.86 2853.06 855.16 1018.58

21 115 21 115 21 115 23 446 23 446 23 446 24 223 24 223 24 223 23 446 23 446 8446 48 446 23 446 23 446 23 446 23 446

81.22 78.72 66.24 204.15 144.89 79.47 246.41 158.99 88.40 189.50 280.88 191.33 231.66 217.08 310.45 209.67 185.22

159 197 162 190 187 167 185 170 172 220 175 177 178 180 195 183 185

Table 3 Parameters and computational results for the second example set p

P

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 a b

h0 = h 0 = 0.05 h0 = h 0 = 0.01 f = 0.5 f = 0.5, h0 = h 0 = 0.05 f = 0.5, h0 = h 0 = 0.01 f=1 f = 1, h0 = h 0 = 0.05 f = 1, h0 = h 0 = 0.01 f = 0.5, h0 = 0.05 f = 0.5, h0 = 0.2 V = 200, f = 0.5 V = 1000, f = 0.5 c = 800, f = 0.5 c = 1600, f = 0.5 k0 = 1000, f = 0.5 k0 = 2000, f = 0.5 P DG ¼ i2N ðV Di f 1 Þ. DS = Z B*.

L

T p0

Z

B*

DGa

DSb

CPU

14 14 18 6 11 23 6 11 24 6 6 18 6 6 6 6 6

5 5 10 2 4 8 2 4 8 4 2 2 4 2 2 4 4

1521.86 1178.82 910.61 1352.64 1145.18 907.31 1347.80 1145.18 905.60 1261.02 1501.34 1481.89 1391.16 2992.00 5202.64 1564.60 1814.60

1416.31 1075.31 799.01 1097.11 944.81 778.55 1028.92 910.71 771.73 1013.01 1233.51 1290.18 1032.76 2733.91 4916.31 1333.51 1551.71

30 680 30 680 30 680 34 772 34 772 34 772 36 136 36 136 36 136 34 772 34 772 12 272 72 272 34 772 34 772 34 772 34 772

105.55 103.51 111.60 255.53 200.37 128.76 318.88 234.47 133.87 248.01 267.83 191.71 358.40 258.09 286.33 231.09 262.89

257 225 306 233 321 242 247 286 255 260 264 269 294 278 282 286 291

the number of the routes is L and the warehouse ordering interval is T p0 ; CPU is the computational time (seconds) corresponding to Z. The meanings of DS, DG are illustrated in the annotation part of Table 2. In order to analyze the computational results, we denote xkpi as one of the items (parameter or computational result) of problem pi in example set k, for example, DG1pi is the value of DG of problem pi in the first example set. It can be seen from the computational results that,

798


(1) DZ 1pi < DZ 2pi for all i 6 17 except i = 11. As DG1pi < DG2pi for all i 6 17, the results of DZ 1pi < DZ 2pi follow the first item of Conclusion 4.1, considering that each cost item is all the same in the two problems compared.

Table 4 The detailed solution of some problems of the first example set p

Route

Vehicle load

Distance

4

38 9 16 50 21 34 30 49 31 26 8 29 20 35 36 3 28 22 2 18 13 41 40 19 42 6 23 7 43 24 25 14 12 5 11 32 1 48 27 46 10 39 33 45 15 44 37 17 4 47

488 328 500 480 480 500 496

92.01 69.17 115.60 101.20 91.63 73.38 109.28

4 8 4 4 4 4 4

5

38 49 50 21 29 20 35 36 3 34 30 10 39 33 45 15 37 25 13 40 41 19 42 44 23 7 26 8 31 28 22 1 32 46 6 14 18 4 17 47 27 48 43 24 12 5 9 16 2 11

496 464 492 488 456 424 500

118.34 110.67 116.16 96.70 65.88 75.96 68.87

4 4 4 4 4 4 4

6

38 9 34 50 14 18 47 4 44 37 13 25 24 17 42 19 40 41 48 26 8 27 22 20 36 35 5 49 30 1 31 28 3 32 23 7 43 6 15 45 33 10 39 2 29 21 16 46 11 12

496 496 472 488 472 496 472 464 480 488 496 472 424

64.67 43.17 57.50 81.40 93.32 58.21 93.86 63.31 79.51 74.89 98.81 69.36 34.71

8 8 8 8 8 8 8 8 8 8 8 8 8

14

27 21 23 47

6 48 8 26 31 28 3 36 35 20 2 16 11 32 46 29 22 1 7 43 24 14 25 13 41 40 19 42 44 4 18 17 37 15 45 33 39 10 30 34 50 9 49 38 5 12

500 464 500 496

141.97 78.60 171.08 134.96

2 16 2 2

15

46 16 23 47

11 32 1 22 2 20 35 36 3 28 31 26 8 48 6 27 21 29 7 43 24 14 25 13 41 40 19 42 44 4 18 17 37 15 45 33 39 10 30 34 50 9 49 38 5 12

500 464 500 496

148.19 68.40 171.08 134.96

2 16 2 2

16

5 38 49 6 43 24 27 48 7 11 2 16

486 328 490 496

158.70 76.65 148.18 139.61

2 8 2 2

17

3 36 35 20 29 2 11 23 24 43 7 26 8 48 27 31 28 22 1 32 46 12 37 15 45 33 10 49 5 41 13 25 14 6 47 17 44 42 40 19 4 18 16 50 21 34 30 39 9 38

488 472 456 500 456 492 472

99.57 100.08 70.82 81.12 81.17 101.76 105.70

4 4 8 4 4 4 4

34 21 29 20 35 36 3 28 31 26 8 22 1 32 46 23 14 25 13 41 40 19 42 4 18 50 9 30 10 39 33 45 15 44 37 17 47 12

Delivery cycle


799

Table 5 The detailed solution of some problems of the second example set p 4

Route 3 32 50 25 55 18 49 23 67 8 35 53 14 19 54 13 26 7 12 40 17 51 16 63 45 29 48 5 15 37 20 70 24 56 41 64 42 43 1 73 10 38 11 59 66 65 31 9

44 57 33 60 62 39

27 52 34 46 6 68 4 75 71 69 36 47 21 74 30 22 61 28 2 72 58

Vehicle load

Distance

Delivery cycle

492 496 500 490 498 498

138.10 103.00 99.25 129.11 159.65 150.41

4 2 2 2 2 2

5

44 3 24 18 55 50 3 26 67 46 34 4 30 74 21 61 62 73 2 7 35 53 14 19 54 8 45 29 5 15 57 13 52 27 58 10 65 31 25 9 28 22 64 42 41 43 49 23 38 11 66 59 20 37 70 60 71 69 36 47 48 16 63 56 1 33 68 51 17 12 72 39 40 6 75

500 496 500 500 500 496 496 488 488 492 500

109.64 34.12 83.73 79.21 75.00 109.03 126.11 95.95 111.25 99.22 66.24

4 4 4 4 4 4 4 4 4 4 4

6

45 29 5 73 22 64 42 50 18 55 25 75 17 26 48 47 30 13 54 14 37 36 70 20 9 40 63 23 56 43 51 32 44 34 8 4 3 24 49 16 68 2 6 35 19 53 7 52 27 57 15 65 10 58 66 59 33 41 1 11 38 4 67 74 60 71 69 21 39 31 72 12 28 61 62

440 456 448 464 488 472 472 496 488 456 496 496 440 496 464 448 488 480 488 480 496 464 496

50.32 90.33 92.18 27.23 54.30 74.94 94.76 47.59 80.96 52.41 33.25 69.50 32.32 59.45 61.92 68.39 89.92 72.88 64.14 19.46 101.11 75.97 72.16

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

500 496 484 494 496 500

165.75 121.73 52.46 124.40 146.25 105.64

2 2 4 2 2 2

492 496 500 490 498 498

138.10 103.00 99.25 129.11 159.65 150.41

14

15

33 63 16 49 24 18 25 55 50 9 39 34 46 8 14 59 19 54 13 57 15 27 17 40 32 44 3 51 68 45 29 48 5 37 20 70 60 71 69 75 2 28 61 22 43 42 64 41 56 23 6 67 7 35 53 11 66 65 38 10 58 3 32 50 25 55 18 49 23 67 8 35 53 14 19 54 13 26 7 12 40 17 51 16 63 45 29 48 5 15 37 20 70 24 56 41 64 42 43 1 73 10 38 11 59 66 65 31 9

44 57 33 60 62 39

31 72 12 26 52 4 36 47 21 74 30 1 73 62

27 52 34 46 6 68 4 75 71 69 36 47 21 74 30 22 61 28 2 72 58

4 2 2 2 2 2 (continued on next page)

800


Table 5 (continued) p

Route

Vehicle load

Distance

Delivery cycle

16

12 58 72 39 9 32 50 55 18 24 49 16 33 6 34 46 8 35 14 59 19 54 13 57 27 52 4 75 17 40 26 67 68 30 48 21 47 36 69 71 60 70 20 37 5 15 29 45 2 74 28 61 22 43 42 64 41 56 23 63 1 73 62 51 3 44 25 31 10 38 65 66 11 53 7

500 500 484 490 498 498

136.10 118.50 42.35 126.49 154.60 129.74

2 2 4 2 2 2

17

16 9 25 55 31 10 58 39 72 38 65 66 11 34 52 27 13 57 15 37 45 73 62 28 61 21 48 29 5 20 70 60 71 69 36 47 12 40 32 44 3 17 75 67 46 8 35 26 4 30 74 2 6 51 7 53 14 59 54 19 50 18 24 49 23 56 41 63 68 33 1 22 64 42 43

152 488 496 496 464 488 500 484 476 492 500 496

38.8 104.41 89.49 79.84 82.26 102.84 53.93 46.71 56.20 95.58 119.41 97.43

8 4 4 4 4 4 4 4 4 4 4 4

(2) DZ 1pi < DZ 1pj for problems pi and pj, in which each item except f is the same and fp1i < fp1j (see problems in subsets {p1, p4, p7}, {p2, p5, p8}, and {p3, p6, p9} of the first example set). As DG1pi < DG1pj for any two of the problems under comparison, such computational results also follow the first item of Conclusion 4.1. And we can get the same conclusion for the problems in subsets {p1, p4, p7}, {p2, p5, p8}, and {p3, p6, p9} of the second example set. (3) DZ 1pi 6 DZ 1pj for problems pi and pj, in which each item is the same except that h1pi < h1pj (see problems in subsets {p1, p2, p3}, {p11, p4, p10, p5, p6}, and {p7, p8, p9} of the first example set). Such computational results follow the second item of Conclusion 4.1. And we can get the same conclusion for any two problems satisfying the same conditions in the second example set except for the pairs of {p1, p3} and {p2, p3}. (4) DZ 1p12 < DZ 1p4 < DZ 1p13 . As V 1p12 < V 1p4 < V 1p13 and

c1p

4

V 1p

4

c1

c1

¼ V p112 ¼ V p113 , where all other items are the same for p12

p13

these three problems, it can be deduced that the computational results of such three problems also follow the first item of Conclusion 4.1. And we can draw out the same conclusion for the problems in subset {p4, p12, p13} of the second example set. (5) DZ 1p4 < DZ 1p14 < DZ 1p15 . As c1p4 < c1p14 < c1p15 , where all other items are the same for these three problems, it can be deduced that the computational results of such three problems follow the third item of Conclusion 4.1. And we can draw out the same conclusion for the problems in subset {p4, p14, p15} of the second example set. (6) Z as well as B* increases if the unit cost of any items becomes larger, such conclusion suits to the computation results of all the problems of both example sets. And the replenishment and delivery schedule of the logistics system vary along with the variation of the composition of the cost items, see details of the computational results of the problems in the subsets {p4, p5, p6, p14, p15, p16, p17} of the two example sets, which are listed in Tables 4 and 5, respectively. (7) The computational time changes with the variation of the parameters. However, it can be seen from the second example set that the number of retailers is the main factor affecting the duration of computation process. It can be drawn from the above analysis that most of the computational results follow the conclusions charactering the relation of the lower bound and the optimal solution of the problem, revealing the effectiveness as well as the robustness of the policy and the algorithm. And further study should be paid on the analysis on the computational results which do not follow such conclusions, focusing mainly on the influence of the parameters to the POT policy as well as computational results of the TS algorithm, such that more effective strategy


801

can be drawn out. In addition, in-depth analysis of the relation of the proposed lower bound and the optimal solution should be done in the further study, such at a reasonable lower bound can investigated. 8. Conclusions In this study, we focus on the integration of inventory and vehicle routing schedules for a distribution system in which the warehouse is responsible for the replenishment of a single item to retailers with demands occurring at a specific constant (but retailer-dependent) rate, combining deliveries into efficient routes. This research proposes a fixed partition policy for this problem, in which the replenishment intervals of the warehouse—as well as of the retailers—are in accord with the POT principle. A lower bound of the long-run average cost of any feasible strategy for the considered distribution system is drawn. A tabu search algorithm is designed to find the optimal retailer partition region. Computational results reveal the effectiveness as well as the robustness of the policy and the algorithm. In future work, a reasonable lower bound should be investigated, and the tabu search algorithm should be modified further so as to cope with various problems, including those in which constraint (13) is relaxed. References Achabal, D.D., Mcintyre, S.H., Smith, S.A., Kalyanam, K., 2000. A decision support system for vendor managed inventory. Journal of Retailing 76 (4), 430–454. Andel, T., 1996. Manage inventory, own information. Transportation and Distribution 37 (5), 54–58. Angulo, A., Nachtmann, H., Waller, M.A., 2004. Supply chain information sharing in a vendor managed inventory partnership. Journal of Business Logistics 25, 101–125. Anily, S., Bramel, J., 2004. An asymptotic 98.5%-effective lower bound on fixed partition policies for the inventory-routing problem. Discrete Applied Mathematics 145 (1), 22–39. Anily, S., Federgruen, A., 1990a. One warehouse multiple retailers systems with vehicle routing costs. Management Science 36, 92–114. Anily, S., Federgruen, A., 1990b. A class of Euclidean routing problems with general route cost functions. Mathematics of Operations Research 15, 269–285. Anily, S., Federgruen, A., 1993. Two-echelon distribution systems with vehicle routing costs and central inventories. Operations Research 41, 37–47. Anupindi, R., Akella, R., 1993. Diversification under supply uncertainty. Management Sciences 39 (8), 944–963. Aviv, Y., Federguen, A., 1998. The operational benefits of information sharing and vendor managed inventory programs. Technical Report, Columbia University, New York. Bramel, J., Simichi-Levi, D., 1995. A location based heuristic for general routing problems. Operations Research 43, 649–660. Cetinkaya, S., Lee, C.Y., 2000. Stock replenishment and shipment scheduling for vendor-managed inventory systems. Management Science 46 (2), 217–232. Chan, L.M.A., Simchi-Levi, D., 1998. Probabilistic analyses and algorithms for three-level distribution systems. Management Science 44, 1562–1576. Chan, L.M.A., Federgruen, A., Simchi-Levi, D., 1998. Probabilistic analyses and practical algorithms for inventory-routing models. Operations Research 46 (1), 96–106. Cheung, K.L., Lee, H.L., 2002. The inventory benefit of shipment coordination and stock rebalancing in a supply chain. Management Science 48 (2), 300–306. Christofides, N., Eilon, S., 1969. An algorithm for the vehicle dispatching problem. Operational Research Quarterly 20, 309– 318. Dong, Y., Xu, K., 2002. A supply chain model of vendor managed inventory. Transportation Research, Part E 38, 75–95. Dror, M., Ball, M.O., 1987. Inventory/routing: Reduction from an annual to a short-period problem. Naval Research of Logistics Quarterly 34, 891–908. Dror, M., Levy, L., 1986. A vehicle routing improvement algorithm comparison of a ‘‘greedy’’ and a matching implementation for inventory routing. Computation and Operations Research 13, 33–45. Federgruen, A., Zipkin, P., 1984. A combined vehicle routing and inventory allocation problem. Operation Research 32, 297–373. Fumero, F., Vercellis, C., 1999. Synchronized development of production, inventory and distribution schedules. Transportation Science 33, 330–350. Gendreau, M., Hertz, A., Laporte, G., 1992. New insertion and post-optimization procedures for the traveling salesman problem. Operations Research 40 (6), 1086–1094. Kohli, R., Park, H., 1994. Coordinating buyer–seller transactions across multiple products. Management Science 40 (9), 45–50. Lee, H., Rosenblatt, M.J., 1986. A general quantity discount pricing model to increase supplier profits. Management Science 32 (9), 1177– 1185. Parker, K., 1996. Demand management and beyond. Manufacturing Systems 6, 2A–14A.

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