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Reliability assessment of a multi-state distribution network under cost and spoilage considerations

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Abstract

The fundamental mission of a distribution network is to satisfy the customer demand by providing sufficient delivery capacity. However, the capacity of a distribution network is practically stochastic because of unexpected events, and moreover, the commodities may rot or be spoilt during delivery owing to inclement weather, traffic accidents, collisions, and so on, such that the intact commodity flow may not meet market demand. This paper focuses on the reliability of a multi-state distribution network (MSDN) with cost and spoilage characteristics, defined as the probability that the MSDN is able to distribute a sufficient quantity of goods to meet the market demand under delivery spoilage and budget limit considerations. A specific spoilage rate associated with each route is adopted to characterize the perishability of commodity flows, and the critical routes whose spoilage rate change has the biggest impact on network reliability are identified with the use of sensitivity analysis method. Apart from delivery cost, the cost involved with the disposal of spoilt goods is also incorporated into the reliability indicator. A minimal paths based algorithm is presented to calculate network reliability, together with an example to illustrate the procedure. A real fruit distribution network is accordingly discussed to demonstrate the utility of the algorithm and the managerial implication of network reliability.

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Abbreviations

MP:

Minimal path

MCV:

Minimal capacity vector

MSDN:

Multi-state distribution network

V :

Set of nodes

E :

Set of routes

(V, E):

An MSDN

m :

The number of routes in (V, E)

a i :

The ith route in (V, E)

s:

The single supplier which represents the single source node

λ :

The number of markets in (V, E)

tk :

The kth market in (V, E) which represent the sink node

π i :

The number of states owned by route ai

h ij :

The jth capacity of ai

H :

The maximal capacity vector of (V, E)

c i :

The cost per unit of used capacity through ai, i = 1, 2, …, m

C :

The cost per unit of spoilt commodity

w :

The capacity required for each unit of commodity

z k :

The number of MPs connecting the single source node s and sink node tk

MP k , j :

The MP connecting source s and sink tk

d k :

The demand of market tk

D :

Demand vector D = (d1, d2,…,dλ)

f k , j :

The normal flow through MPk,j

NF :

The normal-flow vector: \( (f_{{1,1}} , \ldots ,f_{{1,z_{1} }} ,f_{{2,1}} \ldots,f_{{2,z_{2} }} , \ldots,f_{{k,1}} , \ldots,f_{{k,z_{k} }} , \ldots,f_{{\lambda ,1}} , \ldots,f_{{\lambda ,z_{\lambda } }} ) \)

r i :

The spoilage rate of route ai

R :

Spoilage rate vector R = (r1, r2,…,rm)

r k , j :

The spoilage rate of MP MPk,j

o k , j :

The special flow through MPk,j

SF :

The special-flow vector: \( (o_{{1,1}} , \ldots ,o_{{1,z_{1} }} ,o_{{2,1}} , \ldots ,o_{{2,z_{2} }} , \ldots ,o_{{k,1}} , \ldots ,o_{{k,z_{k} }} , \ldots ,o_{{\lambda ,1}} , \ldots ,o_{{\lambda ,z_{\lambda } }} ) \)

x i :

The current capacity of ai for i = 1, 2,…, m

X :

X = (x1, x2,…,xm)

B :

Budget limit

R D , B , R :

Network reliability

(D, B, R)-MCV:

MCV that can meet the market demand vector D with the consideration of budget limit B and route spoilage pattern R

ΔH :

Sets of the feasible special-flow vectors under H

ΔX :

Sets of the feasible special-flow vectors under X

Ψ:

Set of feasible MCVs under (D, B, R)

[\( x \)]:

The smallest integer that is larger than or equal to x

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Funding

This work is jointly supported by the National Natural Science Foundation of China (Grant Nos. 71601072, 61872126), the Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant Nos. KJQN201900634, KJQN201900649), the Doctoral Project of Chongqing Federation of Social Science Circles (Grant No. 2019BS064), and the Planning Project of Human Social Science of Chongqing Municipal Education Commission (Grant Nos. 20SKGH069, 20SKGH061, 20SKGH063).

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Authors and Affiliations

  1. School of Economics and Management, Chongqing University of Posts and Telecommunications, Chongqing, 400065, China

    Yi-Feng Niu, Can He & De-Qiang Fu

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  1. Yi-Feng Niu
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  2. Can He
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  3. De-Qiang Fu
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Correspondence to Yi-Feng Niu.

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Niu, YF., He, C. & Fu, DQ. Reliability assessment of a multi-state distribution network under cost and spoilage considerations. Ann Oper Res 309, 189–208 (2022). https://doi.org/10.1007/s10479-021-04322-7

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