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. directed network to satisfy demand at minimal total cost and minimal total time

1. A Bi-Objective Minimum Cost-Time
Network Flow Problem
Amirhadi Zakeri
‫#"ی‬‫&ان‬'(‫آزاد‬‫ﮕﺎه‬‫ﺸ‬0
‫دا‬

2. Islamic Azad University
Tehran Central
Dr. Mehrdad Mehrbod
Amirhadi Zakeri
ID: 940224828

3. 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

4. ‫ﻫﺰﯾﻨﻪ-زﻣﺎن‬ ‫ﺣﺪاﻗﻞ‬‫ﻣﺴﺎﻟﻪ‬
(MCTNF)
Minimum Cost-Time Network Flow
Bi-Objective
directed network to satisfy demand at minimal total cost and minimal total time
‫ﺟﮫﺖدار‬ ‫ﮔﺮاف‬

5. A B
cost
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

6. MCFBOMCTF
weighted sum scalarization technique
Bi- objective One objective
‫اوزان‬ ‫مجموع‬ ‫تکنیک‬

7. MCF
MCCF
MCIF
MCF with Continuous flows
MCF with Integer flows
variables are real valued variables
variables restricted to integer values
1. Introduction
‫حقیقی‬ ‫متغیرهای‬
‫صحیح‬ ‫عدد‬ ‫مقادیر‬

8. 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
‫اهداف‬ ‫گیری‬ ‫تصمیم‬ ‫متغیرهای‬ ‫مستقیم‬ ‫وابستگی‬

9. flow conservation
capacity constraints ‫ظرفیت‬ ‫محدودیت‬
‫جریان‬ ‫حفظ‬
In addition to the flow variables, new binary variables
dependent on the flow variables are also present
+

10. G=(N,A) directed network
cost= c
fixed time = t
capacity= u
ij
ij
ij
‫هزینه‬
‫ثابت‬ ‫زمان‬
‫کمان‬ ‫ظرفیت‬
i ∈ N
(i,j) ∈ A
b =indicates its supply or demand dependingi
b =0 ⇾ node is a transshipment nodei
bi-objective model:
auxiliary binary variable y is utilized to consider time t for positive flow xij ij ij real valued variables,

11. 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
=

12. 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
≠

13. Definition 1.
if and then it is called
dominate in decision space
dominate in decision space

14. 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

15. 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)

16. 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

17. supported
non-
dominated
in the objective space

18. non-
supported
non-
dominated

19. dominated
point
its associated feasible solution is inefficient

20. 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 ‫شرطی‬ ‫محدودیت‬

21. Solving BOMCTF Problem
M is a sufficiently large positive number

22. Solving BOMCTF Problem
M is a sufficiently large positive number

23. Solving BOMCTF Problem
M is a sufficiently large positive number
is redundant
is redundant

24. 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

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

26. A B
( c , t , u )ij ij ij
Million Rials
Hour
Million tons
Capacity
Cost
Fixed time
b = the value of supply or demand (in terms of million tons)i

27. 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

28. S : producing a specific good
Others are consumers.

29. 13.5 million tons of goods through the network to satisfy demand at minimal cost and time.

30. 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

31. 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

32. END