A Bi-Objective Minimum Cost-Time Network Flow Problem

Amirhadi Zakeri

گاه آزاد )'&ان #"ی دا0ش

Islamic Azad University Tehran Central

Dr. Mehrdad Mehrbod

Amirhadi Zakeri

ID: 940224828

Abstract The Minimum Cost-Time Network Flow (MCTNF) problem deals with shipping the available supply through the directed network to satisfy demand at minimal total cost and minimal total time. Shipping cost is dependent on the value of flow on the arcs; however shipping time is a fixed time of using an arc to send flow. In this paper, a new Bi-Objective Minimum Cost-Time Flow (BOMCTF) problem is formulated. The first and second objective functions consider the total shipping cost and the total shipping fixed time, respectively. We utilize the weighted sum scalarization technique to convert the proposed model to a well- known fixed charge minimum cost flow problem with single objective function. This problem is a parametric mixed integer programming which can be solved by the existence methods. A numerical example is taken to illustrate the proposed approach.

©20154ThheeAAuuththoorsr.s.PPubulbislihsehdedbybyElEselsveievrieBr.BV.VT.his is an open access article under the CC BY-NC-ND license (Shettlpe:c/t/icorenaatinvde/copmeemr-orenvs.ioerwg/ulincednesrerse/sbpyo-ncs-inbdil/i4ty.0o/)f. Academic World Research and Education Center. Selection and/ peer-review under responsibility of Academic World Research and Education Center

مساله حداقل هزینه-زمان(MCTNF)

Minimum Cost-Time Network Flow

Bi-Objective

directed network to satisfy demand at minimal total cost and minimal total timeگراف جھتدار

A Bcost

shipping cost

shipping time is a fixed time of using an arc to send flow

First total shipping cost

Second total shipping fixed time{objectives

A B( c , t , u )ij ij ij

MCF BOMCTF weighted sum scalarization technique

Bi- objective One objective

تکنیک مجموع اوزان

MCF

MCCF

MCIF

MCF with Continuous flows

MCF with Integer flows

variables are real valued variables

variables restricted to integer values

1. Introduction

متغیرهای حقیقی

مقادیر عدد صحیح

In this paper, we formulate and solve a new BOMCF problem, named Bi-Objective Minimum Cost-Time Flow

Main difference direct dependence of the decision variables of the

objectives

وابستگی مستقیم متغیرهای تصمیم گیری اهداف

flow conservation

capacity constraints محدودیت ظرفیت

حفظ جریان

In addition to the flow variables, new binary variables dependent on the flow variables are also present

+

G=(N,A) directed network

cost= c

fixed time = tcapacity= u

ij

ij

ij

هزینه

زمان ثابت

ظرفیت کمان

i ∈ N(i,j) ∈ A

b =indicates its supply or demand depending i

b =0 ⇾ node is a transshipment node i

bi-objective model:

auxiliary binary variable y is utilized to consider time t for positive flow xij ij ij real valued variables,

Multi-Objective Optimization (MOO) problems

cannot improve some objectives without sacrificing others

non-dominated points

Efficient solutions and their associated points in the objectives space

=

supported solution the optimal solution of a model with weighted sum scalarization objective function non-supported solution

S = the set of all feasible solutions of the problem

Z = the feasible set in objective space

≠

Definition 1.

if and then it is called dominate in decision space

dominate in decision space

Definition 2. a feasible solution is called efficient, or Pareto optimal, if such that dominates

If is an efficient solution

Vector non-dominated point in the objective space

The set of efficient solutions

the image of in is called the non-dominated

Definition 3. An efficient solution is a supported efficient solution, if it is an optimal solution of the following weighted sum single objective problem

for some . If is a supported efficient solution, then is named a supported non- dominated point.

∈ (0,1)

Definition 4. An efficient solution is a non-supported efficient solution, if there are no positive values and such that is an optimal solution of the model (2).

2

supported non-

dominated

in the objective space

non-supported

non-dominated

dominated point

its associated feasible solution is inefficient

Solving BOMCTF Problem

BOMCTF problem are classified into supported

non- supported

Finding all supported and non-supported efficient solutions

Major challenge

{ A supported solution is the optimal solution of the model (2)

correlation between and conditional constraint

To resolve this difficulty we replace constraint (1.e) with two auxiliary linear constraints, and reformulate the model (1)

همبستگی محدودیت شرطی

Solving BOMCTF Problem

M is a sufficiently large positive number

Solving BOMCTF Problem

M is a sufficiently large positive number

Solving BOMCTF Problem

M is a sufficiently large positive number

is redundant

is redundant

We formulate the following parametric problem to produce a set of supported efficient solutions of the above problem:

Note that the objective function helps us to ignore the constraints

A B( c , t , u )ij ij ij

A B( c , t , u )ij ij ij

Million Rials

Hour

Million tonsCapacity

Cost

Fixed time

b = the value of supply or demand (in terms of million tons) i

each city is considered as a node For each route For each city there is a b scalar which indicates the value of supply or demand

S : producing a specific good Others are consumers.

13.5 million tons of goods through the network to satisfy demand at minimal cost and time.

we apply the model (4) for the given data set in Fig 2

There are two alternative solutions for all . The first supported efficient solution (let for instance =0.05 ) is as bellow:

The related total shipping cost and fixed time is 1270.9 and 66, respectively.

∈ (0,1)

The next supported efficient solution can be obtained by solving the model (4) for (as an instance =0.5 ):

The total shipping cost and fixed time for this solution is 1242.5 and 77, respectively

As a result, two supported efficient solutions are achieved by applying the suggested approach.

proposed approach succeed in finding all supported efficient solution

It fails to determine unsupported efficient solutions

END