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A heuristic method for the vehicle routing problem with backhauls and inventory

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Abstract

The purpose of this paper is to determine the route of the vehicle routing problem with backhauls (VRPB), delivering new items and picking up the reused items or wastes, and resolve the inventory control decision problem simultaneously since the regular VRPB does not. Both the vehicle routing decision for delivery and pickup, and the inventory control decision affect each other and must be considered together. Hence, a mathematical model of vehicle routing problem with backhauls and inventory (VRPBI) is proposed. Since finding the optimal solution(s) for VRPBI is a NP-hard problem, this paper proposes a heuristic method, variable neighborhood tabu search (VNTS), adopting six neighborhood searching approaches to obtain the optimal solution. Moreover, this paper compares the proposed heuristic method with two other existing heuristic methods. The experimental results indicate that the proposed method is better than the two other methods in terms of average logistic cost (transportation cost and inventory cost).

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Abbreviations

N:

Number of customers

K:

Number of vehicles (or routes)

b :

Vehicle capacity

MaxSup :

Vehicle service capacity

c :

Cost of dispatching vehicles

cm :

Traveling cost/unit distance

A d :

Delivery ordering cost/each order

A r :

Pick-up ordering cost/each order

h d :

Delivery holding cost/unit time/unit product

h r :

Pick-up holding cost/unit time/unit product

hs :

Delivery shortage cost/each time

T p :

Pick-up shortage cost/each time

g :

Index of customers (1 ≤ g  ≤ N)

h :

Index of customers (1 ≤ h ≤ N)

i :

Index of customers (1 ≤ i  ≤ N)

k :

Index of vehicles or routes (1 ≤ k ≤ K)

V k :

Customer set for route k (1 ≤ k ≤ K)

Dis kgh :

Total distance for route k

Q dkgh :

Number of units (delivery order quantity) serviceable for route k during each production run

Q rkgh :

Number of units (pick-up order quantity) recoverable for route k during each production run

UL kgh :

Average delivery demand for route k during lead time

pl kgh :

Average pick-up demand for route k during lead time

D kgh :

Total delivery demand for route k

P kgh :

Total pick-up demand for route k

D h :

Delivery demand for customer h

P h :

Pick-up demand for customer h

d h :

Delivery order quantity during each production run for customer h

p h :

Pick-up order quantity during each production run for customer h

R kgh :

Order-up-to level for delivery replenishment of route k

PR kgh :

Order-up-to level for pick-up replenishment of route k

f k :

The ratio of pick-up and delivery demands (or pick-up and delivery order quantity) for route k \({f_k =\frac{\sum\nolimits_g \sum\nolimits_h {P_{kgh}}} {\sum\nolimits_g \sum\nolimits_h D_{kgh}} = \frac{\sum\nolimits_g \sum\nolimits_h Q_{rkgh}} {\sum\nolimits_g \sum\nolimits_h Q_{dkgh}} }\)

B(R kgh ):

Expected delivery shortage number for route k during each production run

B(PR kgh ):

Expected pick-up shortage number for route k during each production run

f L (x):

Probability density function for customer’s delivery demand, x, of each route during lead time L

f L (y):

Probability density function for customer’s pick-up demand, y, of each route during lead time L

x kgh :

1, if point g immediately precedes point h on route k; 0, otherwise

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Correspondence to Shu-Chu Liu.

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Liu, SC., Chung, CH. A heuristic method for the vehicle routing problem with backhauls and inventory. J Intell Manuf 20, 29–42 (2009). https://doi.org/10.1007/s10845-008-0101-9

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