Solving the multi-compartment capacitated location routing problem with pickup–delivery routes and stochastic demands

Accepted Manuscript Solving the Multi-compartment Capacitated Location Routing Problem with Pickup-Delivery Routes and Stochastic Demands Shan-Huen Huang PII: DOI: Reference:

S0360-8352(15)00221-1 http://dx.doi.org/10.1016/j.cie.2015.05.008 CAIE 4041

To appear in:

Computers & Industrial Engineering

Received Date: Revised Date: Accepted Date:

23 June 2014 5 February 2015 6 May 2015

Please cite this article as: Huang, S-H., Solving the Multi-compartment Capacitated Location Routing Problem with Pickup-Delivery Routes and Stochastic Demands, Computers & Industrial Engineering (2015), doi: http:// dx.doi.org/10.1016/j.cie.2015.05.008

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Solving the MultiMulti-compartment Capacitated Location Routing Problem with PickupPickup-Delivery Routes and Stochastic Demands Demands Dr. Shan-Huen Huang Department of Logistics Management National Kaohsiung First University of Science and Technology No. 1, University Road, Yanchao District, Kaohsiung 824, Taiwan [email protected] Tel.: +886 76011000x3218; fax: +886 76011040

1

Solving the MultiMulti-Compartment Capacitated Location Routing Problem with PickupPickup-Delivery Routes and Stochastic Demands Demands

Abstract This paper considers an advanced capacitated location routing problem in a distribution network with multiple pickup and delivery routes, and each customer placing random multi-item demands on it. The pickup and delivery services need two fleets of vehicles and will form two different sets of routes. However, the unpredictability of variation in the multi-item demands makes the routing of multi-compartment vehicles to accommodate such demands complex. To solve this multifaceted problem, a new process employing the TABU search is proposed in this research study. This proposed approach includes three stages: location selection, customer assignment, and vehicle routing. The innovative concept is to divide all customers into assignment-determined and assignment-undetermined groups in order to narrow down the search area of a solution domain so the TABU search can be more efficient and effective. Two sets of benchmarks are then generated to verify the quality of the proposed method. According to the experiment results, the proposed solution process can both resolve the problems and yield good results in a reasonable amount of computing time. The analysis of the solution process parameters is also provided. In addition, the comparisons between stochastic demand and deterministic demand cases are calculated and discussed as well. 2

Keywords: Logistics, Location Routing Problem, Multi-Compartment, Stochastic demand, TABU Search, pickup and delivery routes

1. Introduction

The Location Routing Problem (LRP) is a complex problem that incorporates both the Facility Location Problem (FLP) and the Vehicle Routing Problem (VRP). Because the FLP is a strategical level problem and VRP is an operational level problem, these two levels of decision are most often tackled separately. However, each of the two problems constitute such a large portion of company expenses, comprehensive and simultaneous consideration could result in a better plan that significantly reduces the cost of researching them independently. Various researchers have devoted themselves to this problem and proposed several different solution processes. For instance, Tuzun and Burke (1999), Prins et al. (2007), and Caballero et al. (2007) employed the TABU Search (TS) algorithm to solve the LRP—a solution that has been proven to be an effective method. Duhamel et al. (2010) and Nguyen et al. (2012) use the Greedy Randomized Adaptive Search Procedure (GRASP) to solve the problem and obtain promising results. Yu et al. (2010) and Wu et al. (2002) employed the Simulated Annealing (SA) heuristic to solve the LRP, while Ting and Chen (2013) applied the meta-heuristic Ant Colony Optimization (ACO) algorithm to resolve the problem. Other researchers, such as Belenguer et al. (2011) and Karaoglan et al. (2011), adopted the Branch-and-Cut method. Sets of benchmarks for solving the LRP have been established by Tuzun and Burke (1999) and Prins et al. (2007), among others. Subsequent researchers frequently base their solution processes on these benchmarks to verify the quality of their own work. 3

The conventional LRP is a deterministic node-routing problem. However, randomized cases are far more realistic for long-term planning; for example, when the same problem has to be repeatedly solved but the actual data varies from one instance to another. Only a few researchers, including Laporte et al. (1989) and Chan et al. (2001), have studied such complex cases. In addition, Albareda-Sambola et al. (2007) studied a randomized location-routing problem and solved it using a two-stage model. In this study, an extended form of the LRP with uncertain demands that considers pickup/delivery routes and multi-item cargo is analyzed. For a parcel delivery service provider, for example, pickup and delivery routes both originate from depots and the cost of these two services is affected by the locations of these facilities. Moreover, the depots and routes may need to deal with more than one type of cargo because of incompatibility constraints in the real world, such as those presented by refrigerated food, frozen food, and room temperature food. The problem, in dealing with multiple products that must travel in independent compartments, is referred to as a multi-compartment vehicle routing problem, which is taken into account in this research as well. This kind of problem has been emerging in recent years. Fallahi et al. (2008) apply two algorithms to solve the problem: the memetic algorithm with a post-optimization phase based on path relinking, and the TABU search method. Mendoza et al. (2010) also employed the memetic algorithm to solve the multi-compartment vehicle routing problem with randomly determined demands. Mendoza et al. (2011) further proposed three constructive heuristics to solve the multi-compartment vehicle routing problem with random demands. The three methods are the savings-based algorithm, novel look-ahead heuristic, and post-optimization procedure based on the classical 2-Opt heuristic. In addition, Muyldermans and Pang (2010) combined a localized search procedure with the Guided Local Search meta-heuristic to solve and improve the solution of the 4

multi-compartment vehicle routing problem. Ultimately, the objective of solving the Multi-Compartment Location Routing Problem with Pickup-Delivery Routes and Stochastic Demands (MLRPPDRSD) is to minimize the total cost, including depot opening costs, route costs, travel costs, recourse costs, and any penalties. This problem is categorized as a NP-hard problem since it is an extension of LRP. The MLRPPDRSD is divided into three parts and a solution process is proposed in this research study. In the pages that follow, Section 2 describes the problem, definitions and mathematics programming. Section 3 details the methodology used to solve this complex problem. The experiments and subsequent analyses are introduced in Section 4, with results and conclusions discussed in Section 5.

2. Problem Definition and Formulation

The MLRPPDRSD can be described as two sets of customers (the pickup set and the delivery set) with undetermined multi-item demands following a particular random distribution, along with one set of potential depot locations with known capacity scattered over a graph. In this study, a depot contains two compartments to accommodate pickups and deliveries, with each compartment able to store a variety of incompatible types of cargo. Each pickup and delivery vehicle contains multiple compartments for these cargoes and originates from one of the open depots to service a set of customers. Upon completing a pickup or delivery service, each vehicle returns to the same open depot. Each customer is served once, and only once, by one vehicle. The objective then is to determine potential locations for the opening of candidate depots, and a priori vehicle routes that minimize costs, including setup costs, travel costs and penalty costs. 5

The mathematics model for the problem is described as follows. Letting G = (V, E) represent a complete, weighted, and undirected network G containing a set of vertices V = (V p ,Vd ,Vo ) and a set of edges E. Three subsets, V p , Vd , and Vo ,

constitute the set of vertices V where V p represents the customers requiring pickup services, Vd represents the customers requiring delivery service, and Vo indicates all potential depot sites. Each customer i ∈V p has a random pickup demand d iu for product u that follows the Poisson distribution, and must be serviced by a single pickup vehicle; similarly, customer i ∈Vd has a random delivery demand d iu for product u will be served by a single delivery vehicle. A depot site m ∈Vo with an opening cost Om contains compartment capacities W pum and Wdum to accommodate pickup and delivery of cargo u, respectively. Set E is a collection of edges connecting each pair of nodes in V. The travel cost for each edge (i, j) is given by cij, which depends on the Euclidean distance between nodes i ∈V and j ∈ V . K would then represent the set of vehicles and two subsets K p and K d of identical vehicles of k k capacity Q pu and Qdu for product u, respectively. A fixed cost, Fp or Fd , is

incurred by a single pickup or delivery vehicle route, operating out of an open depot

m. Total travel costs associated with a route include the fixed cost and costs of traversing edges (variable costs). Before constructing the mathematics formula, an introduction of the notations used immediately follows.

ρp

:

constant for deciding number of pickup vehicles

ρd

:

constant for deciding number of delivery vehicles

V

:

set of vertices, including customers and potential depot sites, wherein V = (V p ,Vd ,Vo ) 6

Vp

:

set of customers requiring pickup services, V p = {1, 2, …, N}

Vd

:

set of customers requiring delivery services, Vd = {1, 2, …, R}

Vo

:

set of potential depot sites, Vo = {1, 2, …, M}

K

:

set of available vehicles where K = ( K d , K p )

Kp

:

set of vehicles for pickup services

Kd

:

set of vehicles for delivery services

d iu

:

Om

:

opening cost of a depot m ∈ Vo

W pum

:

pickup capacity for product u from a depot m ∈Vo

random variable for customer demand i for product u following a known probability distribution with mean µiu and standard deviation σ iu

Wdum

delivery capacity for product u from a depot m ∈ Vo

cij

:

travel cost from node i to j where i, j ∈ V

Fp

:

fixed cost for one pickup route

Fd

:

fixed cost for one delivery route

εu

:

excess handling cost for one unit of product u exceeding the capacity of a compartment at a depot k Q pu

:

capacity for product u of vehicle k and k ∈ K p

k Qdu

:

capacity for product u of vehicle k and k ∈ K d

ym

:

binary variable in which y m = 1 if depot i is opened, otherwise ym = 0 binary variable in which z jm = 1 if customer j ∈ V p ∪ Vd is assigned to depot m ∈ Vo ,

z jm

: otherwise z jm = 0

7

binary variable in which xijk = 1 if edge (i,j) is traversed from i to j in the route driven by

k ij

x

:

vehicle k ∈ K

The objective of this problem is to minimize total cost, including facility opening costs, route costs, travel costs, and any penalty costs.




Facility opening cost

A cost is incurred when a depot location is opened for business. It can be represented as

∑O

m

(1)

ym

m∈Vo

In order to synchronize a facility’s opening cost with travel costs, the facility opening cost is calibrated to the same time horizon with other costs.




Route cost

A route cost is incurred through the dispatching of a vehicle; technically, it includes the fixed costs of vehicle and labor. Since customer demands are uncertain in this problem, a priori route must be designed in advance so as to evaluate the route cost. In this study, vehicle capacity is the main consideration in deciding the number of routes to be constructed. Customer demand is assumed to comply with the Poisson distribution here. Let ψ mp ( k ,η ) = {ϕ 0 ,ϕ1 , ϕ 2 ,...ϕη ,...,ϕ n , ϕ 0 } , η = 0,1,..., n, n + 1 as well as

ψ md ( k ,η ) = {ϕ 0 ,ϕ1 , ϕ 2 ,...ϕη ,...,ϕ n , ϕ 0 } , η = 0,1,..., n, n + 1 indicate a priori pickup and delivery routes, respectively, of vehicle k for an open depot m where ϕ 0 indicates the depot m. In order to simplify the problem, parameters ρ p and ρ d are defined to 8

help decide the number of routes needed for depot m wherein exceeded capacity for a pickup/delivery

route

cannot

be

larger

than

ρ p / ρd ,

respectively.

If

{µϕpu1 , µϕpu2 ,..., µϕpun } and {µϕdu1 , µϕdu2 ,..., µϕdun } denote the mean of product u for customers’ pickup and delivery routes, the following equations have to be held: k Pr(δ npu > Q pu ) ≤ ρp

(2)

k Pr(δ ndu > Qdu ) ≤ ρd

(3) n

Where δ npu and δ ndu are random variables within which their means are

∑ µϕ

pu n

i =1 n

and

∑ µϕ

du n

.

i =1




Travel cost

The travel cost can be formulated as follows:

∑∑∑ c x

k ij ij

(4)

i∈V j∈V k∈K

Because customer demands are uncertain, recourses might possibly be needed when capacity constraints are violated. Since the number of customers visited by each vehicle has been restricted by equations (2) and (3), the probability of recourse happening at ϕη for pickup and delivery routes can be formulated as follows: k k 2cmϕη × (Pr(δ ϕpuη −1 < Q pu ) × Pr(δ ϕpuη > Q pu ))  du k du k  2cmϕη × (Pr(δ ϕη −1 < Qdu ) × Pr(δ ϕη > Qdu ))

(5)

where equation (5) complies with equations (2) and (3). The number of customers visited by one vehicle is restricted by equations (2) and (3), so it is rational to assume that the number of recourses for one vehicle route is less than or equal to 1. Thus, the expected cost of any recourse for pickup and delivery routes can be reformulated as: 9

 n pu k pu k pu k 2∑ ( cmϕη × Pr(δ ϕη −1 < Q pu ) × Pr(δ ϕη > Q pu ) × Pr(δ ϕ n−η < Q pu )) η =1  n k k k  2∑ ( cmϕ × Pr(δ ϕdu < Qdu ) × Pr(δ ϕduη > Qdu ) × Pr(δ ϕdun−η < Qdu )) η η −1  η =1

(6)

And the recourse cost for all routes can be described as: n  k k k 2 y m xϕkη −1ϕη (cmϕη × Pr(δ ϕpuη −1 < Q pu ) × Pr(δ ϕpuη > Q pu ) × Pr(δ ϕpun−η < Q pu )) ∑∑∑   m∈Vo k η =1  n k k k  2 ∑∑∑ y m xϕk ϕ (cmϕ × Pr(δ ϕdu < Qdu ) × Pr(δ ϕduη > Qdu ) × Pr(δ ϕdun−η < Qdu )) η −1 η η η −1  m∈Vo k η =1




(7)

Penalty

There is a certain probability for violation of capacity constraints in this problem due to the inherent uncertainty associated with each demand. A penalty cost is occurred as soon as the violation of capacity constraints happens. A violation of vehicle capacity constraint results in recourse, which has been discussed. The violation of depot capacity constraint requires an extra cost ε u for product u that goes toward handling the excess capacity. Hence, the total penalty can be formulated as follows: ∞

∑∑ ε

u

e=0 u

m

∞

∑∑ ε e=0 u

(e(Pr(( ∑∑ zim d iu − W pum ) = e)))

u

(8)

i

(e(Pr(( ∑∑ zim d iu − Wdum ) = e))) m

(9)

i

Any one cargo u for depot m has a probability to exceed the depot capacity since the total demand is a random variable and any one unit of excess incurred penalty ε u . Thus, Equation (8) and (9) indicates the sum of all expected penalty for each cargo u.

According to the description of the problem, it can be modeled as a stochastic programming problem as follows:

10

∑O

y m + ∑∑∑ cij xijk + Fp

m

m∈Vo

i∈V j∈V k ∈K

∑ ∑ ∑x

k mj

+ Fd

k ∈K p m∈Vo j∈V p

∑ ∑ ∑x

k mj

+

k ∈K d m∈Vo j∈Vd

n

k k k 2 ∑∑∑ y m xϕkη −1ϕη (cmϕη × Pr(δ ϕpuη −1 < Q pu ) × Pr(δ ϕpuη > Q pu ) × Pr(δ ϕpun−η < Q pu )) +

min

m∈Vo k η =1

(10)

n

k k k 2 ∑∑∑ y m xϕkη −1ϕη (cmϕη × Pr(δ ϕduη −1 < Qdu ) × Pr(δ ϕduη > Qdu ) × Pr(δ ϕdun−η < Qdu )) + m∈Vo k η =1 ∞

∑∑ ε

∞

u

(e(Pr(( ∑∑ zim d iu − W pum ) = e))) + ∑∑ ε u (e(Pr(( ∑∑ zim d iu − Wdum ) = e)))

e=0 u

m

e=0 u

i

m

i

s.t.

∑ ∑x

k ij

=1

∀j ∈V p

(11)

k ij

=1

∀j ∈Vd

(12)

k ∈K p i∈V p ∪Vo

∑ ∑x k ∈K d i∈Vd ∪Vo

k Pr(δ npu > Q pu ) ≤ ρp

(13)

k Pr(δ ndu > Qdu ) ≤ ρd

(14)

∑x

k ij

−

j∈V p

∑x

∑x

k ji

=0

∀k ∈ K p , ∀i ∈V p ∪ Vo

(15)

k ji

=0

∀k ∈ K d , ∀i ∈Vd ∪ Vo

(16)

j∈V p

k ij

−

∑x

j∈Vd

j∈Vd

∑ ∑x

k jm

≤1

∀k ∈ K p

(17)

k jm

≤1

∀k ∈ K d

(18)

j∈V p m∈Vo

∑ ∑x j∈Vd m∈Vo

∑∑ x

k ij

≤ S −1

∀S ⊆ V p , ∀k ∈ K p

(19)

k ij

≤ S −1

∀S ⊆ Vd , ∀k ∈ K d

(20)

i∈S j∈S

∑∑ x i∈S j∈S

11

∑x

k ji

+

i∈V p

∑x i∈Vd

∑{ x}

k im

≤ 1 + z jm

∀m ∈Vo , ∀j ∈V p , ∀k ∈ K p

(21)

∑{ x}

k im

≤ 1 + z jm

∀m ∈Vo , ∀j ∈Vd , ∀k ∈ K d

(22)

i∈V p \ j

k ji

+

i∈Vd \ j

xijk ∈ {0,1}

∀i ∈ V , ∀j ∈ V , ∀k ∈ K

(23)

y m ∈ {0,1}

∀m ∈Vo

(24)

z jm ∈ {0,1}

∀m ∈Vo , ∀j ∈V p ∪ Vd

(25)

Equation (10) is the objective function that minimizes the sum of depot opening costs, routing costs, travel costs, and penalties. Constraints (11) and (12) ensure each customer is assigned to one, and only one, route, and has only one predecessor along the way. Constraints (13) and (14) are vehicle capacity constraints. Since demand is uncertain for each customer, the probability of vehicle load is assumed to not exceed a specific constant ρ p and ρ d for pickup and delivery vehicles, respectively. Constraints (15), (16), (17) and (18) ensure the continuity of each route and its termination at the depot of origin. The subtour elimination constraint is represented as constraints (19) and (20). Constraints (21) and (22) indicate that a customer has to be assigned to a depot only if a route connecting them is opened. Constraints (23), (24) and (25) specify binary decision variables used in the mathematical formulation. As stated, the MLRPPDRSD is a combination of FLP and VRP, which are both classified as NP-Hard problems. This means MLRPPDRSD is a NP-Hard problem as well. In fact, the MLRPPDRSD is much more complex because it deals with the FLP and VRP, as well as uncertain demands and pickup-delivery routes. In this study, a heuristic solution procedure is proposed to tackle the MLRPPDRSD. Details of the methods used are introduced in the next section. 12

3. Methodology

The MLRPPDRSD contains three sub-problems: depot location selection, customer assignment, and vehicle routing. After selecting depot locations and opening them for business, customers can be assigned to them, and vehicle routes can be determined. The following are proposed methods for approaching these sub-problems in this research.

3.1 Determining the facility location

The first step is to decide on the number of depots to open. For the traditional LRP, the number of open depots can be evaluated according to total customer demands. However, it is slightly different for the case where demand is uncertain. Generally speaking, the penalty for each exceedance of capacity increases ordinary operation costs, which makes the opening of more depots necessary. On the other hand, idle capacity increases cost so that fewer open depots seem more economical. In approaching this problem, rpu and rdu will denote the number of open depots according to pickup and delivery of product u, respectively. Additionally, ∆ pvu and ∆ dvu will indicate the sum of random variables for all pickup and delivery demands.

The following formulas can be drawn to determine the number of open depots: Pr( ∆ pvu > rpu × M ) ≤ ρ m

(26)

Pr( ∆ dvu > rdu × M ) ≤ ρ m

(27)

where ρ m is a given constant and M is the average capacity of all depots. 13

The number of open depots η pu and η du needed in this research is the minimum number to complete equations (26) and (27) for a specific item u. Assuming there is λ different products, a set with a total of 2λ numbers of open depots can be obtained. The suggested number of open depots η is the maximum number within the set and the suitable number of open depots should be tested around η . This can be represented as:

η = max{η p1 ,η d 1 ,η p 2 ,η d 2 ,...,η pλ ,η dλ }

(28)

To help in deciding the allocation of open depot locations, the concept of “seed” is proposed in this research. A seed represents a partition of customers, which implies the number of seeds is equal to the number of open depots determined. It is noted that the number of open depots can be decided according to equation (26) and equation (27). A set of seeds indicates the allocation of open depots. To decide the set of seeds, a function F (π 1 , π 2 ,...,π s ) is designed where π 1 , π 2 ,...,π s ∈ Vo are seeds and s denotes the number of seeds in the set. The value of function F is designed to be equal to the sum of distances between any two open depots in this research, and it can be represented as follows: s

s

F (π 1 , π 2 ,...,π s ) = ∑∑ π uπ v

∀u ≠ v

(29)

u =1 v =1

In this proposed method, the combination of open depots that yields the largest value of F denotes the number of seeds chosen. Basically, the largest value of function F indicates the most extensive allocation of open depots. The allocation of seeds is also the initial set of open depots in this solution procedure. Candidate depot locations other than seeds are divided into groups according to their closeness of proximity to seeds. While a non-seed depot can become an open depot in its group, it yields a new combination of open depots. In Figure 1, for example, the initial seed set is (m1, m2, m3), which is also the initial solution for determining the ultimate number of open 14

depots. If m3 is replaced by m10, the new allocation of open depots will be (m1, m2, m10). To help in finding a better combination of open depots, flexibility is added into the mechanism through use of a “wild depot.” A wild depot is a depot location nearly equidistant between two seeds. The wild depot can be a replacement for the two groups. For instance, the depot m5 in Figure 1 is located at a point almost equidistant from m1 and m2, so that it can replace the two groups, which means (m5, m2, m3) and (m1, m5, m3) are both valid combinations of an open depot combination in this solution procedure. <<< Insert Figure 1 here >>>

3.2 Determine the assignment

One customer is serviced by one open depot, which makes it a set-covering problem. Intuitively, it is better for an open depot to service the customers nearest to it. A customer who is served by a closer depot has the advantage of lower travel costs. Although the distance between a customer and depot is not a characteristic of the VRP, it is still a useful reference for this problem. Initially, a customer is assigned to the nearest seed. This assignment is called the “initial assignment” here for notation purposes. Figure 1 demonstrates the initial assignment for three partitions. Although the initial assignment is able to partition customers into s groups, the assignment is not final. The reason why the assignments will need to be adjusted is that the depot capacity constraint may be violated by the initial assignment. Therefore, assignments of customers need be altered under some circumstances in order to satisfy capacity constraints and facilitate the routing plan. With regard to the adjusting mechanism that helps in deciding assignments, customers who are located relatively near an open depot, and served by that nearest open depot, are presumed to be located 15

along an economical route. According to this intuition, customers can be divided into two clusters that are close to one depot and distant from depots. If the distance of one customer to one depot is smaller than r, which the assignment of it remain the same as it in the initial assignment, it is called assignment-determined. Otherwise, the customer is presumed to be assignment-undetermined, which implies its assignment can be changed accordingly. Figure 2 illustrates the boundaries of the assignment-determined and assignment-undetermined groups. Thus, more assignment-undetermined customers appear when the radius r is getting smaller, which will make the degree of freedom larger as well as the domain of solution. This mechanism then provides a chance to adjust the assignment conforming to depot capacity constraint. Another advantage of this mechanism is that it narrows the solution domain for the TABU Search (TS) algorithm, especially for a large problem because the moves in TS are limited with those assignments. <<< Insert Figure 2 here >>>

3.3 Vehicle routing problem

A modified TABU Search (TS) algorithm is proposed in this research in order to solve the VRP phase. TABU Search (Glover and Laguna, 1997) is a local search meta-heuristic used for mathematical optimization. A TABU list is used to avoid the search moving back to the already-visited solutions so as to enable the search to escape from current local optima and approach the other part of the solution domain. The reason to choose the TS here is that it is efficient in tackling this kind of problem (Tuzun and Burke, 1999; Prins et al., 2007; and Caballero et al., 2007). To apply the TS to this problem, an initial solution has to be generated. Since there are some assignment-undetermined customers within the band, the solution format is designed 16

as follows. <<< Insert Figure 3 here >>> Figure 3 demonstrates a virtual solution with three open depots marked as π 1 , π 2 and π 3 . There are two number sequences following the depots. The upper sequence contains the pickup customers, and the lower sequence consists of the delivery customers. The number sequence indicates the serving order of vehicles and it can be divided into several routes according to the vehicle capacity constraint as shown. As mentioned

above,

assignments

for

assignment-determined

customers

are

unchangeable. However, assignment-undetermined customers can be serviced by any depot. Hence, the method to construct an initial solution in this paper is denoted by inserting the assignment-determined customers into the number sequence belong to its assigned depot and insert the assignment-undetermined customers into arbitrary number sequences. These insertions are randomized so that all initial solutions vary. The general TS method contains a memory structure form known as the TABU list. This is a list of solutions recently considered. If a potential solution appears on this list, it cannot be revisited until it reaches an expiration point. The TABU list is classified as a short-term memory structure. For solving harder problems, an intermediate-term memory structure would be helpful. The definition of intermediate-term memory is a list of rules intended to bias a search towards promising areas of the search space. In this research, the intermediate-term memory structure is established so as to avoid exploring the undesired solution domain and increase efficiency. The maneuvers SWAP and MOVE are used here to explore the neighborhood of current solutions in the solution domain. The definition of SWAP is that two customers under the same operation (pickup or delivery) are exchanged. It is noted that an assignment-determined customer cannot be exchanged with another customer that belongs to a different depot. This means only assignment-undetermined 17

customers can SWAP between distinct depots. The definition of MOVE is that one customer is shifted from its current position to another position. Only assignment-undetermined customers are allowed to move to a position outside of its current sequence. This means, assignment-determined customers can only move within the current sequence. Figure 4 is a flowchart depicting how the proposed methods solve the MLRPPDRSD. The number of open depots must be decided in the very beginning according to equation (26) and equation (27). Customers’ assignments can be determined thereafter so that the initial solution can be formed. With initial assignments representing the starting point for this examination, the TABU search is able to explore the neighborhoods by operating SWAP and MOVE maneuvers. The TABU list is kept updated in order to record the search track. If selected customers for SWAP or MOVE violate the intermediate-term memory, those customers in the operation should be reselected until they tally with the intermediate-term memory. The initial solution is recorded as the best solution in the beginning. Whenever a better solution is found, the best solution will be replaced by the better one. After a specific number of iterations, the recorded best solution is the global best solution, and the algorithm stops. Applying the same process mentioned above to all possible combinations of open depots can lead to a global best solution. <<< Insert Figure 4 here >>>

It is noted that the number of depots is obtained by equations (26) and (27). The number of depot may be exaggerated because the loosest situation is chosen according to equation (28). In order to gain the optimal number of open depot, hence, the same solving process depicted in the flowchart has to be executed with η -1. This step is repeated with one open depot lessen every time until no feasible solution is available. 18

4. Computational Study

4.1 Instances

Because there are no standard benchmarks for MLRPPDRSD, two sets of LRP benchmarks are employed and redesigned here to evaluate the proposed solution process for the MLRPPDRSD. The two sets of LRP benchmarks are proposed by Prins et al. (2004) and Tuzun and Burke (1999). They contain 30 and 36 instances, respectively. For instances proposed by Prins et al., the number of depots is set to be either 5 or 10, the number of clients is 20, 50, 100, or 200, and vehicle capacity is set to be either 70 or 150. Travel costs depend on Euclidean distances, which are the distances traveled multiplied by 100 and rounded up to the next integer. All other data, including demands, depot capacities and fixed costs, are integers as well. For the instances proposed by Tuzun and Burke, the number of depots is either 10 or 20, and vehicle capacity is set at 150. In this data set, the coordinates are real numbers and distances are not rounded. It is assumed that a total of three different types of cargo, denoted as C1, C2, and C3 for notational purposes, are demanded by each customer. The coordinates and means of demand of each customer for C1, C2, and C3 are generated randomly. Delivery customers, whose number is twice the size of pick-up customers, are also generated randomly in terms of coordinates and means of demands. Accordingly, all depots and vehicles have three independent compartments for the three types of cargo being transported. Each compartment’s capacity is randomly set as well. Two fleets with different fixed costs and travel costs are used for pick-up and delivery services. The fixed and travel costs for pick-up services are the same as those in the original 19

instances, and the fixed and travel costs for delivery services are set to be around 0.8 times the original cost. These two new sets of problems are thus named benchmark-P and benchmark-T for notation purposes.

4.2 Initial solution and parameter settings

In order to proceed with the TABU search, an initial solution should be generated in advance. The first step is to determine the initial assignment for all customers. Sequentially, both assignment-determined and assignment-undetermined customers are categorized to the corresponding open depot according to their initial assignment, just like in Figure 3. This means all customers are assigned to the nearest open depot. The customers’ positions in each sequence will be reassigned randomly to form the initial solution. Costs for the initial solution are calculated for further use. The parameters of this solution process are set as follows. The length of the TABU list is set at 7 and the number of iteration is set at 500. The rates perform SWAP and MOVE operations are both equal to 0.5. In addition, parameters ρ p and ρ d are both set at 0.75; ρ m is set at 0.1; penalties are εc1 = 150 , ε c2 = 150 , and ε c3 = 150 for benchmark-P, and penalties are ε c1 = 50 , εc2 = 50 , and ε c3 = 50 for benchmark-T.

4.3 Computational tests and analysis

Here, the effectiveness of the proposed solution process is demonstrated by testing it on benchmark-P and benchmark-T. The program is coded using the computer language JAVA, and tested on a PC with an Intel (R) Core (TM) i7 CPU (2.93GHz) and 4 GB memory. The radius r designed in this research remains 20

undetermined, and an analysis is conducted to determine the radius. Assume that D is equal to the longest distance between any two open depots, and 2 r = λD , where λ is a real number between 0 and 1. In order to determine the most appropriate radius, analysis

of

λ

is

conducted

in

correspondence

to

λ = {0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0} . Table 1 and Table 2 list output results of the analysis. Column indices #P, #D, and #O indicate the number of pickup customers, number of delivery customers, and number of open depots, respectively. In addition, column index OD denotes the open depots and column index CPU denotes the computing time in seconds. <<>> <<>> According to Table 1 and Table 2, it can be observed that most of the best solution occurs while λ =0.6~0.9. The best solutions for benchmark-P and benchmark-T, found during parameter analysis are represented in bold font in Table 1 and Table 2.

4.4 Discussion

Generally, the algorithm takes less than one minute to complete for small-scaled problems (for instance, if a total of 60 customers and 5 potential depots are being considered), and about 10 minutes for large-scaled problems (if, say, a total of 600 customers and 20 potential depots are being considered) for one open depot combination, respectively. Hence, computational time does not seem to be a critical issue for the proposed solution process. The analysis of radius is shown in Table 1 and Table 2. All customers are not confined to any open depot where λ =0. It is noted that not all the customers are assignment-determined even λ =1. The solutions gained 21

by conventional TABU search ( λ =0) can be treated as the upper bounds, it is discerned that all of the best solutions obtained are better than the upper bounds. The output results show that most of the better solution occurs while λ =0.6 ~ 0.9. This result implies that better solution obtained while most of the customers are served by the nearest depot. Theoretically, the TABU search method will find the best solution in the solution domain while λ =0 if it has enough time to explore the entire area. The possible reason is that the solution domain is too large for a TABU search to explore within a limited number of iterations. Accordingly, the proposed method that strategically confined the searching to a reasonable area is able to help it function efficiently and effectively.

In order to investigate the effect of uncertain demand, another experiment regarding the deterministic demand is executed. In this set of problem, all the instances used above are modified to deterministic set by using the demand mean as the new customer demand. Other settings are remaining the same. The output results are listed in Table 3-a and Table 3-b. The column indexed by SC, DC, d% indicating the output of stochastic case, output of deterministic case and difference between SC and DC in percentage, respectively.

<<>> <<>>

Generally speaking, the SC will be bigger than DC because of taking the penalties into account. The percentage of difference can be less than 5 percent or as high as 76 percent. How the stochastic demand influences the total cost is related to several different conditions as follows. Because the two experiments are for stochastic and 22

deterministic cases, it is noted that this comparison is highly affected by the standard deviations. Higher standard deviation indicates higher risk to take resulting in higher cost. Consequently, the penalty for each unit affects the comparison as well. Higher penalty per unit will increase the difference. In addition, the number of open depot might be different in two cases because all the customers’ demands have to be strictly covered by open depots in deterministic cases. That means one more depot might be needed in deterministic case and that surely increase the cost. Additionally, the bandwidth in deterministic case may be different from the stochastic case. That is because some of the bandwidth is too big so that the depot capacity constraint is violated.

5. Conclusions

This research presents a new approach to solving the multi-item capacitated location routing problem with random demand and pickup-delivery routes. The MLRPPDRSD is broken down into three phases: location selection, customer assignment, and route determination. Thus, the proposed approach contains three stages: 1) determining the number of open depots; 2) determining customer assignments; and 3) using a modified TABU search method to figure out vehicle allocation and routing. The innovative concept in the proposed approach is to control customer assignments so as to confine the search method to a limited and reasonable area of the solution domain. This is accomplished by dividing customers into two groups—an assignment-determined group and an assignment-undetermined group. With a narrower search domain, the approach to solving the vehicle routing problem becomes more effective and efficient during reasonable computing time. The proposed solution approach is applied to two sets of benchmarks, benchmark-P and 23

benchmark-T. The analysis of parameter λ is conducted to determine the better settings. According to output results, the parameter λ =0.6~0.9 can yield a better solution within 500 iterations during a reasonable computing time.

References:

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Dupont, L. (2008). Branch and bound algorithm for a facility location problem with concave site dependent costs. International Jounal of Production Economics, 112, 245-254. Duhamel, C., Lacomme, P., Prins, C., Prodhon, C. (2010). A GRASP×ELS approach for the capacitated location-routing problem. Computers & Operations Research, 37, 1912-1923. Fallahi, A.E., Prins, C., Calvo, R.W. (2008). A memetic algorithm and a tabu search for the multi-compartment vehicle routing problem. Computer & Operations Research, 35, 1725-1741. Glover, F., Laguna, M. (1997). Tabu Search. Boston, MA: Kluwer Academic Publishers. Harkness, J., and ReVelle, C. (2003). Facility location with increasing production costs. European Journal of Operational Research, 145, 1-13. Karaoglan, I., Altiparmak, F., Kara, I., Dengiz, B. (2011). A branch and cut algorithm for the location-routing problem with simultaneous pickup and delivery. European Journal of Operational Research, 211, 318-332. Klose, A. (1999). A Lagrangean relax-and-cut approach for the two-stage capacitated facility location problem. European Journal of Operational Research, 126, 408-421. Klose, A., and Drexl, A. (2005). Facility location models for distribution system design. European Journal of Operational Research, 162, 4-29. Laporte, G., Louveaux, F., Mercure, H. (1989). Models and exact solutions for a class of stochastic location-routing problems. European Journal of Operational Research, 39(1), 71-78. Li, J., Chu, F., Prins, C. (2009). Lower and upper bounds for a capacitated plant location problem with multicommodity flow. Computers & Operations Research, 36, 3019-3030. 25

Lin, C.K.Y. (2009). Stochastic single-source capacitated facility location model with service level requirements. International Journal of Production Economics, 117, 439-451. Mendoza, J.E., Castanier, B., Gueret, C., Medaglia, A.L., Velasco, N. (2010). A memetic algorithm for the multi-compartment vehicle routing problem with stochastic demands. Computers & Operations Research, 37, 1886-1898. Mendoza, J.E., Castanier, B., Gueret, C., Medaglia, A.L., Velasco, N. (2011). Constructive heuristics for the multicompartment vehicle routing problem with stochastic demands. Transportation Science, 45(3), 346-363. Muyldermans, L. and Pang, G. (2010). On the benefits of co-collection: Experiments with a multi-compartment vehicle routing algorithm. European Journal of Operational Research, 206, 93-103. Nguyen, V.P., Prins, C., Prodhon, C. (2012). Solving the two-echelon location routing problem by a GRASP reinforced by a leaning process and path relinking. European Journal of Operational Research, 216, 113-126. Prins, C., Prodhon, C., Ruiz, A., Soriano, P., Calvo, R.W. (2007). Solving the capacitated

Location-Routing

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a

cooperative

Lagrangean

relaxation-granular Tabu Search heuristic. Transportation Science, 41(4), 470-483. Shiode, S., Yeh, K.Y., Hsia, H.C. (2012). Optimal location policy for three competitive facilities. Computers & Industrial Engineering, 62, 703-707. Snyder, L.V., Daskin, M.S. (2005). Reliability models for facility location: The expected failure cost case. Transportation Science, 39(3), 400-416. Ting, C.J., Chen, C.H. (2013). A multiple ant colony optimization algorithm for the capacitated location routing problem. Int. J. Production Economics, 141, 34-44. Tuzun, D., Burke, L.I. (1999). A two-phase tabu search approach to the location routing problem. European Journal of Operational Research, 116, 87-99. 26

Wu, T.H., Low, C., Bai, J.W. (2002). Heuristic solutions to multi-depot location-routing problems. Computers & Operations Research, 29, 1393-1415. Yu, V.F., Lin, S.W., Lee, W., Ting, C.J. (2010). A simulated annealing heuristic for the capacitated location routing problem. Computers & Industrial Engineering, 58, 288-299.

27

m3 m1 m7

m6 m4

m9

m5 m8

h

m2

Figure 1 the illustration of seeds determination and original assignment

m3 m1 m7

m6 m4

m9

m5 m8

r

m2

Figure 2 the illustration of assignment-determination process

Route 1

Route 2

Route 3

Route 4

Route 5

Route 6

pickup 1

2

3

4

5

6

7

π1

1

2

3

4

5

6

7

8

delivery 1 2 3 4 5 6 7 8 Route 1

1

2

3

4

5

6

1

2

3

4

5

6

π3

π2

1

2

Route 2

3

4

5

6

Route 3

assignment-determined customers assignment-undetermined customers

Figure 3 illustration of initial solution

28

Route 4

Route 5

7

Choose the open-depot combination

Decide the number of open depots

Set up the parameters including TABU list, number of iteration IT(variable it), swap rate SR(variable sr), bandwidth b, current best solution BE (current solution be).

Generate initial solution and calculate its cost

Decide the attribution of customer assignment

Select a customer and a position for moving.

no

sr
Pickup part/delivery part

yes no MOVE allowed?

no

Select two customers for swap.

SWAP allowed?

yes move the customer to a the position

yes Swap the two customers

Evaluate the solution

yes

tabued?

aspiration? no

no

yes

Accept it as current solution be

Renew TABU list

yes

be
BE=be

no it=it+1

no

it=IT? yes Both Pickup& delivery finished?

no

yes

yes

Examine possible combinations?

no

stop

Figure 4 the flowchart of the proposed method 29

Table 1 the analysis of λ for benchmark-P #P #D #O OD CPU

λ =0

λ =0.1 λ =0.2 λ =0.3 λ =0.4 λ =0.5 λ =0.6 λ =0.7 λ =0.8 λ =0.9 λ =1.0

P1

20 40 5 2,3,5

34

215506 203647 191234 175837 158237 146550 145520 149780 150525 163937 180235

P2

20 40 5

23

127005 130081 121059 112898 110116 104459 104219 106306 114653 110360 117639

P3

20 40 5 2,4,5

25

199469 193863 194165 170137 167919 169501 155240 153131 149990 151098 157566

P4

20 40 5

4

14

94493

98191

103022

P5

50 100 5

1,4

59

478902 460263 454192 442011 424985 412519 411530 404811 410991 396904

402542

P6

50 100 5

2,4

62

312613 311316 309055 293396 285298 273611 262042 267885 249748 268665 262582

P7

50 100 5 2,3,4

60

538886 529729 468789 414054 364698 346501 337237 344045 347159 377578 417926

P8

50 100 5 1,3,4

80

350703 330200 300354 272015 250925 252589 258455 269015 278515 312284 320351

P9

50 100 5 1,4,5

82

288183 283219 253789 237855 216501 206976 201699 214871 192049 210542 197697

P10 50 100 5 3,4,5

80

437683 450077 432947 434537 432765 411591 412090 390402 391428 394926 393862

P11 50 100 5

60

411109 417165 407205 397903 395515 385697 371606 369051 369347 371709 355658

80

261914 263416 254892 255518 234377 222021 229981 232322 228342 235864 241187

3,4

1,5

P12 50 100 5 2,4,5

95414

99198

104449

93598

97106

93500

98671

97077

P13 100 200 5 2,3,5 162 1255265 1207045 1110862 1062477 947016 907242 937824 1006639 1086283 1150771 1180795 P14 100 200 5 2,3,5 162 812347 802027 751366 726176 683187 651645 639981 635150 660740 703866 754241 P15 100 200 5

2,4

113 1050437 1032185 1001805 942913 925405 910029 855564 842031 849068 893140 934094

P16 100 200 5 2,4,5 158 815843 803782 720594 655777 642656 625275 620169 624476 642012 684057 722127 P17 100 200 5

2,5

111 888088 871995 855143 804110 764706 730127 694832 685291 667315 656205 692989

P18 100 200 5 3,4,5 159 736755 746158 724461 690128 668545 652478 641312 624081 616461 612445 627741 P19 100 200 10 3,5,10 117 1159656 1147574 1118036 1018015 985792 946673 907443 894901 891666 892074 905325 P20 100 200 10

5,7

120 723276 848770 834536 814057 794695 770930 742377 736811 665952 645256 645636

P21 100 200 10 4,6,10 116 1098602 1071920 1049795 1013083 957394 928070 867283 852262 833383 823307 832938 P22 100 200 10

3,5

118 641070 638659 642038 640685 633147 630007 616485 599079 667984 691280 667871

P23 100 200 10 3,4,8 118 1035146 1004076 988810 946953 889199 839871 822999 820332 849595 861497 884367 P24 100 200 10 5,8,10 116 693231 696867 688524 690949 689383 684836 689743 689010 674213 674998 678085 P25 200 400 10 6,9,10 333 2397571 2285967 2054662 1888112 1779965 1700938 1685981 1820273 1820273 2206417 2318614 P26 200 400 10 6,9,10 343 1568493 1562424 1466597 1384802 1275076 1219374 1190093 1242160 1331985 1429997 1516873 P27 200 400 10 6,7,8 329 2364893 2318182 2212729 2039013 1866857 1818749 1824463 1850495 1970446 2100313 2244749 P28 200 400 10 2,4,7 249 1423381 1415729 1395258 1379929 1350212 1309579 1303772 1270184 1281035 1315764 1330738 P29 200 400 10 1,4 ,5 251 2008559 1962520 1912482 1817678 1737840 1663979 1625930 1612072 1596254 1677252 1730387 P30 200 400 10 2,4,6 341 1268055 1258996 1220761 1232397 1229023 1179701 1211654 1241856 1291627 1337724 1373316

30

Table 2 the analysis of µ for benchmark-T #P #D #O

OD

CPU

λ =0 λ =0.1 λ =0.2 λ =0.3 λ =0.4 λ =0.5 λ =0.6 λ =0.7 λ =0.8 λ =0.9 λ =1.0

T01 100 200 10 2,6,7

323 10282

9405

8148

6517

5937

5401

5394

5303

5282

5106

5275

T02 100 200 20

204 5492

5579

5661

5678

5518

5386

5538

5365

5611

5896

6485

296 5712

5809

5641

5450

5574

5266

5125

5556

5547

5867

6294

T04 100 200 20 2,10,16 303 5614

5599

5221

4974

5092

5263

5478

5391

5387

5537

5421

T05 100 200 10 1,2,10 292 9784

9738

8514

6930

6170

5575

5315

5164

5289

5077

5087

T06 100 200 20

4,12

206

5111

5293

5268

5224

5235

5245

4892

5021

5406

5865

6529

T07 100 200 10

2,5

201 7266

7287

6999

6837

6519

5937

5795

5690

5963

5864

6201

T08 100 200 20 12,15 207 5880

6048

5778

5825

5622

5401

5345

5285

5126

5161

5270

T09 100 200 10 1,4,5

299 5918

6015

5938

6193

6100

5595

5436

5463

5567

5844

6283

T10 100 200 20 2,7,9

301 5937

5992

6157

6142

6254

6361

6638

6899

7486

8184

9108

T11 100 200 10

6,7

209 7192

7084

7395

7368

6386

5778

5341

4770

4788

4766

4834

T12 100 200 20

3,5

207 6255

6346

5909

5665

5094

5065

4806

5222

5558

6429

8001

T13 150 300 10

7,9

341 7419

7309

7483

7381

7387

6753

6934

7200

7163

7215

7874

T14 150 300 20 15,16 323 9685

9706

9656

9280

8930

8264

8111

8225

8972

9491

10346

T15 150 300 10

326 9704

9651

9489

9486

9484

9407

9416

9638

9357

9392

9856

T16 150 300 20 4,7,16 454 9207

9108

8210

7374

7186

7375

7414

7722

8351

9193

10061

T17 150 300 10

7,9

317 7871

7887

7373

7152

6808

6735

6725

6724

7178

7341

7935

T18 150 300 20

1,11

323 11825 11668 11549 11718 11240 10876 10487

9626

9083

8707

8301

T19 150 300 10

6,8

332 8197

3,5

T03 100 200 10 4,5,7

1,5

7755

7360

7021

6748

6827

6847

6789

7039

7115

T20 150 300 20 1,6,17 455 16106 15425 11645

6862

6687

6589

6685

6557

6592

6670

6782

T21 150 300 10 4,8,9

455 8202

7874

7695

7538

7660

7566

8114

8303

8277

T22 150 300 20

325 12898 13002 12652 10680

9353

8587

8210

8197

8062

7964

8122

9,19

8033

8061

7967

T23 150 300 10 1,2,10 465 6127

6204

6002

5912

6049

6673

8057

9859

10836 11634

12016

T24 150 300 20

8033

7755

7360

7021

6748

6827

6847

6789

7039

7115

639 11205 11101 10531 10019

9799

9569

8889

9297

9222

8899

8924

T26 200 400 20 1,17,20 626 12601 12475 11834 10790 10008 10320 11227 11417 11238 11249

11495

T27 200 400 10

443 18971 19070 17638 16564 14741 13323 12161 11520 10581 10087

10132

T28 200 400 20 16,18 449 18659 17642 14878 13070 11665 11021 10771 10863 10636 10528

10455

T29 200 400 10

6,10

T25 200 400 10 4,7,9

3,8

7,9

324 8197

441 9084

9193

9094

9296

9150

9075

9391

9828

10567

11779

T30 200 400 20 15,17 436 13154 12713 11544 10359

9731

9456

9051

9192

9711

9866

8594

T31 200 400 10 1,6,9

633 8222

8110

8303

7795

7541

7843

8309

9736

11402 11778

12127

T32 200 400 20

8,14

443 8908

9174

7851

6837

6711

6697

6612

7372

8357

9852

10776

T33 200 400 10

1,9

444 16034 16013 15983 15669 15705 14964 14678 14302 13878 13712

13246

T34 200 400 20

4,5

438 10065

10050

9716

9186

9637 31

9405

9250

9478

9537

9432

9462

9579

T35 200 400 10 3,4,9

638 8510

8459

8562

8427

8741

9319

9592

9672

9636

9650

9770

T36 200 400 20 5,13,20 623 8458

8081

7682

7646

7481

8488

9983

12234 13381 14503

14656

Table 3-a the comparisons of stochastic and deterministic cases P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

SC 145520 104219 149990 93500 396904 249748 337237 250925 192049 390402

DC 137059 98830 140127 80893 331449 223913 297020 209471 176219 327757

d% 6.17% 5.45% 7.04% 15.58% 19.75% 11.54% 13.54% 19.79% 8.98% 19.11%

P11 P12 P13 P14 P15 P16 P17 P18 P19 P20

SC 355658 222021 907242 635150 842031 620169 656205 612445 891666 645256

DC 299994 211739 817963 605178 794017 588749 643158 583943 868168 629851

d% 18.56% 4.86% 10.91% 4.95% 6.05% 5.34% 2.03% 4.88% 2.71% 2.45%

P21 P22 P23 P24 P25 P26 P27 P28 P29 P30

SC 823307 599079 820332 674213 1685981 1190093 1818749 1270184 1596254 1179701

DC 808090 587944 760419 607317 1479216 1076661 1728772 1151296 1485708 1110263

Table 3-b the comparisons of stochastic and deterministic cases T01 T02 T03 T04 T05 T06 T07 T08 T09 T10 T11 T12

SC 5106 5365 5125 4974 5077 4892 5690 5126 5436 6142 4766 4806

DC 3738 3896 3802 4112 3535 3618 3798 3269 3912 3942 3767 3897

d% 36.60% 37.71% 34.80% 20.96% 43.62% 35.21% 49.82% 56.81% 38.96% 55.81% 26.52% 23.33%

T13 T14 T15 T16 T17 T18 T19 T20 T21 T22 T23 T24

SC 6753 8111 9407 7186 6724 8301 6748 6557 7538 7964 5912 6748

DC 5079 5940 6565 5640 5008 5418 4939 4542 5381 5746 5467 6028

32

d% 32.96% 36.55% 43.29% 27.41% 34.27% 53.21% 36.63% 44.36% 40.09% 38.60% 8.14% 11.94%

T25 T26 T27 T28 T29 T30 T31 T32 T33 T34 T35 T36

SC 8889 10008 10087 10455 9075 9051 7843 6612 13246 9250 8427 7646

DC 6561 7616 6820 7879 6605 7160 5790 5998 7495 6611 6616 5971

d% 35.48% 31.41% 47.90% 32.69% 37.40% 26.41% 35.46% 10.24% 76.73% 39.92% 27.37% 28.05%

d% 1.88% 1.89% 7.88% 11.02% 13.98% 10.54% 5.20% 10.33% 7.44% 6.25%

Highlights     

LRP with pickup and delivery routes considering multi-item and stochastic demands A new solving process with three stages is proposed. Design the “seed” concept to help on selecting the open depots. Group customers into assignment-determined and assignment-undetermined sets Most of the best solutions are occurred on bandwidth rate λ=0.6 ~ 0.9.

Acknowledgments

The author gratefully acknowledges the helpful comments of the editor and the anonymous reviewer who provided valuable input and comments that have contributed to improving the content and exposition of this paper. This research was supported in part by a grant from the National Science Council of the Republic of China in Taiwan under grants NSC 102-2221-E-327 -026 -MY2.

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