2016 International Conference on Electronic Information Technology and Intellectualization (ICEITI 2016) ISBN: 978-1-60595-364-9

**Flow Interception Location Problem of **

**Multi-phase and Multi-demand with **

**Changed Network **

### Zucai Zhou and Xi Zhang

**ABSTRACT **

This paper focus on multi-phase and multi-demand location problem, the information of demand and path is different on different time. The decision maker must close some facilities and open some new facilities to match the new demand. Based on the above problem, we refer the multi-phase bi-objective flow interception location model, the decision maker locate the facilities to match different condition, to get the objective of maximized benefit and minimized cost. We design the multi-objective evolutionary algorithm to solve the model.

**INTRODUCTION **

The location model should consider some aspects of future uncertainty. These are dynamic facility location problems [1]. Since Ballou proposed this problem, a lot of articles began to focus on the problem[2]. Chardaire addresses with the multi period, incapacitated facility location problem[3]. Melo focuses on the dynamic multi-commodity capacitated facility location[4]. The above articles focus on single type facility, and one type customer demand. There are some articles consider multi-type facilities location under certainty[5、6]. In this paper, we study the problem of multi - type flow interception facility location problem(FIFLP). We found multi-objective model, and take multi-objective evolutionary algorithm to solve the problem[7].

**MODEL FORMULATION **

This paper considers two types of demand which are served by facilities A and B respectively. If there are facilities A and B on the same point, then the customers

__________________________

whose need one type of facility for service original will be bring in passing demand for another type of facility. The costs consist of fixed cost, operational cost and closed facility cost. The question with the study is how to location to intercept maximize demand flow and to use minimize cost. Based on the question, the paper found one multi-period multi-demand bi-objective FIFLP (MMBF) model. MMBF model assumes that customers flow on the network between their respective O-D (origin and destination) pairs, and the customers choose the short path on O-D pairs.

Tis the set of periods, t is the t-period, *t**T*_{. The demand and path is fixed in one }

period, and changed in different period. The opened facility on t-1 period might be
closed on t period. *G V Et*( ,*t* *t*)is the network on t period, *Vt*is the set of vertex on t
period， *t*

*E* _{is the set of sides on t period}， *t*

*P* _{ is the set of path on t period}，*Pt* *Et*.
*t*

*p*

*V*

is the set of vertex of p-path. *t*, *t*are the unit revenue of interception unit flow for
facility A and B respectively on t period.*CAitOPen*：the fixed cost of opening a new

facility A at pointi on t period, *i**Vt*_{；}*CBitOPen*：the fixed cost of opening a new

facility B at pointi on t period, *i**Vt*_{；}*LAit*：the operational cost of facility A at

pointi on t period, *i**Vt*_{；}*LBit*：the operational cost of facility B at pointi on t

period, *i**Vt*_{；}*CCloseAit* ：the closed cost of facility A at pointi on t period, *i**Vt*；
*Close*

*Bit*

*C* _{：}_{the closed cost of facility A at pointi on t period, } *t*
*i**V* _{；}

*A*
*pt*

*f*

：the flow on

path p need be served by facility A on t period, *p**Pt*；

*B*
*pt*

*f* _{：}

the flow on path p

need be served by facility A on t period, *p**Pt*；

*t*
*ip*

*I*

：the set of path passpointi on t

period, *i**Vt*_{，} *p**Pt*_{；}Pr1*pt*：the passing demand probability of *A*
*pt*

*f* , *t*

*p**P* _{；}

2

Pr* _{pt}* ： the passing demand probability of B

*pt*

*f* , *p**Pt* _{；}

1 open facility at piont on period

0 ot hers

*A*
*it*

*A* *i* *t*

*x* ; 1 open facility at piont on period

0 ot hers

*B*
*it*

*B* *i* *t*

*x*

1 close facility at piont on period 0 ot her s

*A*
*it*

*A* *i* *t*

*y* ; 1 close facility at piont on period

0 ot her s

*B*
*it*

*B* *i* *t*

*y*

1 at last one facility A on path at period

0 ot hers

*A*
*pt*
*p* *t*

*z* ; 1 at last one facility on path at period

0
B
ot hers
*B*
*pt*
*p* *t*
*z*

1 the passing demand on path at period

0 ot hers

*pt*
*p* *t*
*u*
1 2

max [ *A* *A* *B* *B* ( *A*Pr *B*Pr ) ]
*t* *pt* *pt* *t* *pt* *pt* *t* *pt* *pt* *t* *pt* *pt* *pt*
*t*

*t T p P*

*f z* *f z* *f* *f* *u*

_{}

###

（1）1 1

max{ }

min [ *Open*( *A* *A* ) *A* *Open*( *B* *B* ) *B* *Close* *A* *Close* *B*]
*Ait* *it* *it* *Ait* *it* *Bit* *it* *it* *Bit* *it* *Ait* *it* *Bit* *it*
*t*

*t T i* *V*

*C* *x* *x* _{} *L x* *C* *x* *x* _{} *L x* *C* *y* *C* *y*

###

S.T

（3）

（4）

（5）

（6）

（7）

（8）

（9）

（10）

The objective function（1）is to maximize the revenue. The objective function

(2) is to minimize the cost. The constraints (3)，(4) state that 1
*A*
*pt*

*z*

and 1

*B*
*pt*

*z*

only

when facility A and B on the path p at period t, others the values of
*A*
*pt*
*z*
and
*B*
*pt*
*z*
are
zeros. The constraint（5）states that*upt*=1 when both the facility A and B are on
pointi on t period. The constraints（6）、（7）state that there are close facilities
only when opened a facility on t-1 period and unopened on t period at pointi. The
constraints（8）、（9）state that facilities opened when the flows are over with the
set value at pointi.

**ALGORITHM FOR THE MODEL **

MMBF model is NP-hard problem, evolutionary algorithm is used to solve the
problem. The main steps are as follows: Step1: to computer the short path of O-D
pairs; Step2: the maximum evolutionary generation is Max-gen, the population size is
NIND, the chromosome length is Field D, *pc* and *pm* are crossing-over rate and
aberration rate; to produce initial population *P*0_{=crtbp (NIND, Field D)}*t*1,*Pt* *P*0;

Step3: to computer the function value （Objv1，Objv2）of population*Pt*, to get
Pareto front=sortfun(Objv1,Objv2)） based on function value, to operate choice

*A* *A*
*it* *pt*
*t*
*i Vp*
*x* *z*

###

,*t*

*t**T p**P*

*B* *B*
*it* *pt*
*t*
*i Vp*
*x* *z*

###

,*t*

*t**T p**P*

*A* *B*
*it* *it* *pt*
*t*

*i Vp*

*x x* *u*

###

,*t*

*t**T p**P*

1(1 )

*A* *A* *A*

*it* *it* *it*

*x* _{} *x* *y* *t**T i V*, _{p}t

1(1 )

*B* *B* *B*

*it* *it* *it*

*x* _{} *x* *y* *t**T i V*, *pt*

*A* *A* *A*
*pt* *t* *it*
*t*

*p Iip*

*f* *f x*

###

,*t*

*p*

*t**T i V*

*B* *B* *B*
*pt* *t* *it*
*t*

*p Iip*

*f* *f x*

###

,*t*

*p*

*t**T i V*

### , ,

### ,

### ,

### ,

### {0,1}

(s_chrom=choice(Chrom,F)）、cross（s_chrom=xovsp(s_chrom,0.9)）、aberration
and to get sub-population*Qt* =mut(s_chrom,0.03); Step 4: to combine the *Pt*and *Qt*,
get the new population*Rt*; Step 5：to carry out the fast no-dominated sorting for the
new population *Rt*, get the optimize Pareto-front{*F F*1, ,2 }; Step 6: to order i=1,

*t*

*P* _{, and carry out }*P _{t}*

*P*

_{t}*F*

_{i}_{, i=i+1, until }

*Pt*

*Fi*

*N*

_{; Step 7}

_{：}

_{to use fitness }

sharing and niche size to sort *Fi*, to take the *N* *Pt* solutions of big sharing degree in

*i*

*F*_{into }*P _{t}*

_{ to order}

_{t}_{ }

_{t}_{1}

_{; Step 8}

_{：}

_{to record the chromosomes in the first dominant }

frontier, to compute objective function value.

**COMPUTATIONAL EXPERIMENTS **

This paper designs two-step example, figure 1 is the first period network which is consisted of 12 notes and 15 arcs, the number on arcs is distance. Figure 2 is the second period network. Table 1 is about cost, table 2 is flow information. To order

1 4

_{，}_{2} 5_{，}_{1}2_{，}_{2} 5_{，}_{the unit is yuan. }

Figure 1. The first period network. Figure 2. The second period network.

v1 v2 v3 v4 v5 v6 v10 v7 v8 v9 v11 v12 4 4 10 10 2 2 33 44 33 55

77 9 9 6 6 9 9 6 6 8 8 7 7 8 8 7 7 66 v13 v1 v2 v3 v4 v5 v6 v10 v7 v8 v9 v11 v12 4 4 10 10 2 2 3

3 44 3

3 55

TABLE 2. FLOW INFORMATION. The first period flow on path

OD Pairs *f _{A}*

*f*Pr

_{B}_{1}1Pr

_{1}2

The second period flow on path

OD Pairs *f _{A}*

*f*Pr

_{B}_{2}1 Pr

_{2}2

v1-v3 12 14 0.01 0.03 v1-v3 9 15 0.07 0.01
v1-v4 15 8 0.02 0.02 v1-v4 14 18 0.08 0.02
v1-v6 18 15 0.06 0.07 v1-v6 3 2 0.09 0.06
v7-v9 9 15 0.03 0.08 v7-v9 12 10 0.06 0.03
v2-v6 14 18 0.09 0.09 v2-v6 5 0 0.02 0.09
v2-v11 16 25 0.04 0.06 v2-v11 12 14 0.03 0.04
v2-v9 20 16 0.05 0.02 v2-v9 15 8 0.02 0.02
v2-v10 12 10 0.01 0.03 v2-v10 18 15 0.06 0.07
v3-v6 25 18 0.02 0.10 v3-v6 4 3 0.01 0.03
v3-v10 12 16 0.07 0.01 v3-v10 16 25 0.02 0.02
v4-v9 20 22 0.08 0.02 v4-v9 20 16 0.06 0.07
v4-v6 16 12 0.09 0.06 v4-v6 0 2 0.03 0.08
v5-v8 17 8 0.06 0.03 v5-v8 9 15 0.09 0.09
v5-v11 19 20 0.02 0.09 v5-v11 14 18 0.04 0.06
v8-v11 11 14 0.03 0.04 v8-v11 16 25 0.05 0.02
v8-v10 21 12 0.10 0.02 v8-v10 12 16 0.01 0.03
--- --- --- --- --- v3-v13 25 18 0.02 0.10
--- --- --- --- --- v5-v13 12 16 0.07 0.01
--- --- --- --- --- v6-v14 20 22 0.08 0.02
--- --- --- --- --- v6-v13 16 12 0.09 0.06
--- --- --- --- --- v12-_{v13 } 3 4 0.06 0.03

--- --- --- --- --- v11-_{v14 } 19 20 0.02 0.09
--- --- --- --- --- v8-v13 11 14 0.03 0.04
--- --- --- --- --- v7-v13 21 12 0.10 0.02

We solved the example by programming in MATLAB7.1, the parameter is as
followings: NIND=50， *pc* =0.8， *pm* =0.03， Max-gen=400, 1

*A*

*f* _{=90}_{，} 1

*B*

*f* _{=85}_{，}

2

*A*

*f* _{ =100}_{，} *f*_{2}*B*_{ =90}_{，}_{share}_{=1. We get 12 pareto solutions. The results are shown as }

Table 3. THE PARETO SOLUTION SET.

The first period The second period Total income

Total cost New facility New facility closed facility

A B A B A B

v1,v3,v4v8 v1,v3,v6,v8 v6,v13 v6,v1 3

v3 v8 4226.3 135.7

v1,v3,v4,v8 v1,v3,v4,v8,v12 v13 v14 v3,v4,v8 v3,v4 3507.9 70.5 v1,v3,v4,v8 v1,v3,v4,v8 v13 --- --- --- 3203.9 54.5 v1,v3,v8 v1,v3,v4,v8 v13 --- v3,v8 v3,v4 2965.3 51.5 v1,v3,v4,v8 v1,v4,v3,v8 v6,v13 v6，

v13

v3 v3 4532.4 148.3

v1,v3,v4,v8 v1,v3,v4,v8,v9 --- --- v3,v4,v8 v3,v4,v9 2763.9 38 v1,v3,v4,v8 v1,v4,v8 v13 --- v3,v4v8 v1,v3,v8 2375.9 32.2 v1,v3,v4,v8,v

10 v1,v3,v4,v6,v7,v8,v9 v13 -- v3,v4,v8,v10 v4,v7,v9 3936.0 107.6 v1,v3,v4,v8 v1,v4,v3,v6,v8,v9 v13 v13 v3,v4 v3,v4,v6,

v9 3759.4 86.7 v1,v3,v4,v8 v1,v3,v6,v8 v13,v14 v14 v4,v3, v3,v6 3906.6 101.6 v1,v3,v4,v8 v1,v4,v8 v6,v13 v6,v1

3

v3 --- 4443.2 147.3

v1,v3,v4,v8 v1,v3,v4,v8 v6,v13 v6,v1 3

v3,v1 v3 4159.6 132.3

**CONCLUSION **

This paper studies the dynamic environment location decision. Due to urban transformation and other reasons, the demand and path information might change on the network, so it should bring the facilities opened and closed. Based on the above problems, the paper establishes a multi-demand bi-objective dynamic FIFLP model. And the multi-objective evolutionary algorithm is used to solve the model. Future work should focus on uncertain demand dynamic location problems.

**ACKNOWLEDGMENT **

This study is partially sponsored by Humanities and Social Science Youth Foundation of Hu Bei Educational Committee (NO.15Q079) and Youth Scientific fund of Wuhan Institute of Technology (NO. K201324).

**REFERENCES **

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3. Chardaire P., Sutter A., Costa M.C. Solving the dynamic facility location problem [J]. Networks, 1998, 28(2): 117-124.

5. Berman O., Huang R.B. The Minisum Multipurpose Trip Location Problem on Networks. Transportation science，2007, 41 (11): 500-515.

6. A Ghaderi. Heuristic Algorithms for Solving an Integrated Dynamic Center Facility Location-Network Design Model [J]. Location-Networks & Spatial Economics, 2014, 15: 1-27.