Case Study 41
41
Facility Location Problem
Facility Location Problem
Problem Description
The facility location problem is a well-known problem in the areas of production and
operations management and combinatorial optimization. The problem finds an optimal
location of facilities considering facility construction costs, transportation costs, etc. This
problem is very popular because it is faced by many companies. A large number of
researchers have studied this problem and proposed solution approaches.
The purpose of this project is to build a decision support system that helps the managers
decide where to locate a facility. We give a mathematical formulation of the un-capacitated
facility location problem and describe a primal-dual algorithm. We do not give details about
why this algorithm works. To learn more about the problem and the primal-dual algorithm,
we refer the student to Francis et al. (1974) and Erlenkotter (1978).
Problem Formulation
Let I and J denote a set of facilities and customers, respectively (i = 1,…,m and j = 1,…,n),
and let fi denote the fixed cost of locating a facility at location i. Let cij be the cost of
supplying customer j from facility i. The decision variables are as follows:
xij:
1
yi  
0
denotes the fraction of customer j’s demand satisfied by facility i
if we locate a facility at location i
otherwise.
The following is a mixed-integer linear programming formulation of the un-capacitated facility
location problem.
min :  fi y i    cij xij
iI
iI jJ
Subject to :
 xij  1
iI
y i  xij  0
for all j  J ,
(1)
for all i  I, j  J ,
(2)
xij  0
for all i  I, j  J ,
(3)
y i  {0,1}
for all i  I.
(4)
The objective is to minimize the total costs. The first set of constraints shows that the
demand of customer j (for j = 1,...,J) will be satisfied. The second set of constraints shows
that if a facility is not located at a potential location i, there will be no shipments from that
location. The third set of constraints is the non-negativity constraints, and the last set of
constraints is the integrality constraints.
The following is the dual formulation of the linear programming relaxation of the uncapacitated facility location problem.
Case Study 41
Facility Location Problem
 vj
jJ
Subject to :
max :
 w ij  fi
jJ
v j  w ij  cij
w ij  0
for all i  I,
(5)
for all i  I, j  J ,
(6)
for all i  I, j  J .
(7)
vj are the dual variables for constraint set (1), and wij are the dual variables for constraint set
(2).
Dual Ascent Algorithm
Initialization:
- Let J

J

J
initially J
denote the set of customers whose vj’s are eligible to change,
.
- Set vj = c1j for all jJ.
- Calculate: s  f 
i
i
- Let
 max 0, v j  cij .
jJ


k ( j )  min k : v j  ckj for all jJ.
Step 1:
Set j = 1 and  = 0.
Step 2:
If
j  J  , go to Step 6.
Step 3:
Set


 j  min si : v j  cij  0 .
i I
Step 4:
If  j
 ck ( j ), j  v j , then set:  j  ck ( j ), j  v j ;  = 1 and k(j) = k(j) +1.
Step 5:
For each iI, such that v j
 cij decrease si by  j . Increase vj by  j .
Step 6:
If
j  n , let j = j + 1 and return to Step 2.
Step 7:
If  = 1, return to Step 1, otherwise stop.
The dual adjustment procedure is used in the case that the dual variables found from the
dual ascent algorithm do not satisfy the complementary slackness conditions.
Dual Adjustment Procedure
Case Study 41
Facility Location Problem
Step 1:
Set j = 1.
Step 2:
If I j  1 , go to Step 7.
Step 3:
If J

is empty and
i  ( j)
J '
i ( j)
is empty, go to Step 7.
Step 4:
For each iI such that v j
 cij , decrease si by v j  c j and decrease vj to c j .
Step 5:
(a) Set
J   J 
i ( j)
 J '
i ( j)
and execute the dual ascent procedure.
(b) Repeat the dual ascent procedure using all of the j’s that were not included in
J  in
Step 5 (a).
(c) Repeat the dual ascent procedure with
J  J
.
Step 6:
If vj did not go back to its original value, return to Step 2.
Step 7:
If
j  n , let j = j + 1 and return to Step 2. Otherwise, stop.
In this procedure we used the following notation:

 i  I

c 
I *j  i  I * : v j  cij
The set of tight facilities.
I j
The set of open facilities.


:vj
ij

J i  j : I *j  {i} ;
one with
The set of customers such that facility iI* is the only
v j  cij .
i ' ( j)  I  ;

c j  max cij : v j  cij
i I
i’(j) is the second best facility in I+ for customer j.

The next lower
cij that we might decrease vj to.
Case Study 41
Facility Location Problem
Excel Spreadsheet
1.
Build a spreadsheet that presents the fixed cost of opening a facility (fi for iI).
2.
Build a spreadsheet that presents the variable unit cost cij (for iI and jJ).
User Interface
1.
Create the welcome form.
2.
Build a data entry form. The following are suggestions to help you design this form. In
this form insert two option buttons. These option buttons allow the user to select
whether to read the data from a file or manually enter the data. Include a command
button that, when clicked on, performs these actions:
3.
4.
a.
If the user chose to read the data from a file, a text box should appear where the
user types in the name of the file.
b.
If the user chose to enter the data manually, a text box appears where the user
types in the total number of facilities (m) and the total number of customers (n).
Upon submission of this information two tables appear. The first table with
dimensions m by 1 allows the user to type in the fixed cost of opening a facility. The
second table with dimensions m by n allows the user to type in the total variable
cost of supplying each customer from different suppliers.
Build a form that allows the user to understand the facility location problem by looking at
an example. This form includes the following:
a.
A problem statement.
b.
A problem formulation.
c.
Present how the dual algorithm was implemented for solving this example.
d.
The optimal objective function value and the corresponding optimal solution.
e.
A graphical representation of the optimal allocation of customers to facilities.
Build a form that allows the user to solve the problem and view the results. In this form
include three frames. In the first frame include two option buttons and a command
button. The option buttons allow the user to select a method to solve the problem. The
two methods available are the dual ascent algorithm and the Excel solver. When the
command button is clicked on, the problem is solved using the method selected by the
user. The second frame has a number of option buttons that allow the user to open any
of the reports described below. The third frame, titled “Sensitivity Analysis,” has option
buttons that enable the user to choose a parameter for the sensitivity analysis. The user
is interested in testing the sensitivity of the optimal solution with respect to fixed cost,
variable unit costs, etc.
Design a logo for this project. Insert this logo in the forms created above. Pick a background
color and a font color for the forms created. Include the following in the forms created: record
navigation command buttons, record operations command buttons, and form operations
command buttons as needed.
Reports
1.
Report the following results:
K T
K T
K T
k 1t 1
k 1t 1
k 1t 1
min :   ckt xkt    hkt I kt    Fkt z kt
Case Study 41
Subject to :
K
 zkt  1
Facility Location Problem
a. forThe
t optimal
1,..., T , allocation of customers
(1) to facilities.
k 1
b.
xkt  I k ,t 1  I kt  d kt
xkt  Pkt z kt
xkt , I kt  0
z kt  {0,1}
c.
d.
The optimal allocation costs, consisting of fixed and variable costs.
for k  1,..., K ; t  1,..., T ,
(2)
for k  1,..., K ; t  1,..., T ,
(3)
for k  1,..., K ; t  1,..., T ,
(4)
The fixed facility location costs.
The total variable costs.
2.
Report the results from the sensitivity analysis.
3.
Give a graph representation of the optimal allocation of customers to facilities.
for k  1,..., K ; t  1,..., T .
(5)
Reference
Erlenkotter, D., “A Dual-Based Procedure for the Uncapacitated Facility Location.”
Operations Research, Vol 26(6), pg. 992-1009, 1978.
Francis, R.L., McGinnis, F.L., Jr., White, J.A., “Facility Layout and Location: An Analytical
Approach.” Prentice Hall, 2nd Ed., 1974.