LOCATION PLANNING
Plant or warehouse location is a long term planning issue that requires a careful analysis of
several factors that affect
location decisions. These factors are listed below.
Regional Factors
Raw Materials
Markets
Labor
Taxes
Climate (for some industries)
Other factors
Community Related Factors
Service Facilities
Community Attitude
Community Size
Utilities
Environmental Regulations
Taxes
Others
Site Related Factors
Land
Transportation
Legal restrictions
In choosing a proper location for a plant, first regional factors are analyzed and a suitable
region is chosen. Then, a suitable community within the chosen region is selected; the final step is
narrowing down on the specific site within that community where the plant will be built.
Some of the Techniques Used in Location Planning
Factor Rating
Locational Break-even Analysis
Grid Method
Linear Integer Programming
Factor Rating
We talked about the various factors that affect location decisions. These factors can be
weighed and scored to yield results that are helpful in choosing a proper location.
Example:
Suppose we are evaluating two locations. All the factors and conditions are the same except
the factors specified below. Choose the best location if only the following factors are relevant.
SCORES (OUT OF 100)
FACTOR
WEIGHTED SCORES
WEIGHT
SITE 1
SITE 2
SITE 1
SITE 2
Nearness to raw materials
0.3
100
50
30
15
Labor Cost
0.1
80
90
8
9
Electricity Cost
0.4
70
80
28
32
Land Cost
0.2
40
80
8
16
74
72
1.0
On the basis of factor rating analysis we choose Site 1 because it has the highest weighted
score.
Locational Break-even Analysis
This technique compares the total cost curves of the potential sites and chooses the site with
the minimum total cost curve. Suppose the following graph depicts the total cost (TC) curves for
two potential sites.
Cost
Total Cost Curve for Site A
Total Cost Curve for Site B
5000
Quantity
According to these graphs, the two alternatives break even when the production quantity is
5,000. If the plant's production capacity is going to be less than 5,000 we choose Site A as the TC
curve for Site A is under that of site B for quantities below 5,000. If the plant capacity is going to
be more than 5,000 we choose Site B.
Grid Method
A central location is determined by superimposing a grid on the relevant geographical
region on the map, measuring the distances of the demand centers, multiplying these distances by
the corresponding weights (proportions or percentages) of demands, and then summing the
weighted distances.
Example:
Market
Demand
Proportional
Demand
(p)
Horizontal
Distance
(x)
Vertical
Distance
(y)
Weighted
Horizontal
Distance (px)
Weighted
Vertical
Distance (py)
A
80
80/200=0.40
8
9
8(0.40)=3.2
9(0.40)=3.6
B
40
40/200=0.20
6
6
6(0.20)=1.2
6(0.20)=1.2
C
20
20/200=0.10
5
9
5(0.10)=0.5
9(0.10)=0.9
D
20
20/200=0.10
3
8
3(0.10)=0.3
8(0.10)=0.8
E
10
10/200=0.05
4
5
4(0.05)=0.2
5(0.05)=0.25
F
30
30/200=0.15
8
6
8(0.15)=1.2
6(0.15)=0.9
200
1.0
6.6
7.65
Proportional demands are computed by dividing the demand of each market by the total demand.
For example, the proportional demand for Market A is given by 80 / 200 = 0.4. The sum of the
weighted horizontal distances add up to 6.6 and the sum of the vertical distances add up to 7.65 =
7.7 approximately; hence, the center of demand has the coordinates of (6.6 , 7.7). This is the point
where the plant or warehouse should be located to be able to serve the markets in question centrally.
The coordinates of center of demand can be marked on the actual map to indicate the actual location
to be chosen.
Linear Integer Programming
This technique uses a linear integer programming model that deals with the
decisions that must be made about the trade-offs between transportation costs and costs for
locating or operating facilities. The decisions concern the selection of facilities to locate or
operate and the quantity to ship from each location to any demand center.
The following example in table form below shows a situation where there are three
warehouses with supplies of 3,000, 4,000, and 3,000 and three markets with demands of
1,000, 2,000, and 3,000. The numbers in the small inner squares show the unit
transportation costs. For example, the cost of shipping one unit from W1 (Warehouse 1) to
M1 (Market 1) is $5. The basic question is: "Which warehouses should be operated and
how much should be shipped from each warehouse to each market in order to minimize the
total transportation and warehouse operation costs?"
M1
M2
5
M3
6
Supply
7
W1
3000
1
2
2
W2
4000
8
9
9
W3
Demand
3000
1000
2000
3000
Preparing a new table by placing the variables in the empty cells of the previous table will facilitate
the linear programming formulation; the new table follows.
M1
M2
5
W1
x11
6
x12
1
W2
x21
Demand
1000
2
2000
4000
x23
9
x32
3000
x13
x22
x31
Supply
7
2
8
W3
M3
9
x33
3000
3000
The x variables are the transportation variables. For example, x12 = number of units to be shipped
from Warehouse 1 to Market 2.
If we add the x variables row wise we get the supply constraints and
if we add them column wise we get the demand constraints for the linear program of the transportation
problem as follows.
Linear Program For The Transportation Problem
Min 5x11 +6x12 +7x13 +x21 +2x22 +2x23 +8x31 +9x32 +9x33
st
x11 +x12 +x13  3000
x21 +x22 +x23  4000
x31 +x32 +x33  3000
SUPPLY CONSTRAINTS
x11 +x21 +x31 = 1000
x12 +x22 +x32 = 2000
x13 +x23 +x33 = 3000
DEMAND CONSTRAINTS
All variables  0
NON-NEGATIVITY CONSTRAINTS
Solution to this transportation problem provides the answers to how much we should ship from each
warehouse to each market but does not answer which warehouses should be operated. This linear
programming model can be extended to provide the optimal answers to both which warehouses should be
operated and how much to ship from each warehouse to each market by incorporating the binary location
variables (yi) as shown below. If we are talking about nonexisting warehouses then the question "Which
warehouse(s) should be operated?" would be "At which site(s) should we locate or build the warehouse(s)?"
Suppose that the fixed warehouse operating costs are as follows:
Warehouse
Location
Fixed Warehouse
Operating Costs
W1
$ 1000
W2
$ 3000
y2 = 1 or 0
W3
$ 500
y3 = 1 or 0
y1 = 1 or 0
We associate "y" variables with each warehouse; y1 corresponds to W1, y2 corresponds to W2 and
y3 corresponds to W3. The y variables are binary, i.e., either 1 or 0 type of variables. If y = 1, it means
"locate or operate the warehouse" and if it is 0 it means "don't locate or operate the warehouse". Now, the
linear program for the transportation problem can be extended to include location problem as well as
illustrated below:
Linear Integer Program For The Location Problem
Min 5x11 + 6x12 + 7x13 + x21 + 2x22 + 2x23 + 8x31 + 9x32 + 9x33 + 1000y1 + 3,000y2 + 500y3
st
x11 + x12 + x13  3,000y1  x11 + x12 + x13 - 3,000y1  0
x21 + x22 + x23  4,000y2  x21 + x22 + x23 - 4,000y2  0
x31 + x32 + x33  3,000y3  x31 + x32 + x33 - 3,000y3  0
x11 + x21 + x31 = 1,000
x12 + x22 + x32 = 2,000
x13 + x23 + x33 = 3,000
xij  0
i = 1,2,3 and j = 1,2,3 yi = 1 or 0, i = 1,2,3
The solution to this problem can be obtained by using a very user friendly software
like LINDO (Linear Interactive Discrete Optimizer). If LINDO is used, the model for the
location problem would be typed virtually exactly as it appears followed by a "GO"
command to get the answers; it is as simple as this.
The summary of the LINDO solution for this problem is:
Min Z = $ 23000
y1 = 1
y2 = 1
y3 = 0
x11 =0
x21 =1000
x31 =0
x12 =2000
x22 =0
x32 =0
x13 =0
x23 =3000
x33 =0
The values of the location variables at optimality indicate that we should operate
warehouses at Site 1 and Site 2 and we should not operate a warehouse at Site 3. In case of
nonexisting warehouses y1 = 1, y2 = 1, and y3 = 0 values would indicate that we should
locate or build a warehouse at Site 1 and Site 2, but we should not locate or build a
warehouse at Site 3.
On the other hand, as mentioned earlier, the x variables are the transportation
variables showing how much to ship from each warehouse to each demand center. For
example, x11 = 0 shows that we would not ship anything from Warehouse 1 to Market 1 and
x12 = 2,000 shows that we would ship 2,000 units from Warehouse 1
to Market 2, etc.
Min Z = 23,000 is the value of the objective function that represents the minimum
possible total transportation and warehouse operating costs.
SUPPLEMENT
This supplement provides a summary of transportation problem in simplified, easy
to understand, and easy to use format.
Transportation Algorithm (for minimization)
1.
Get the initial solution (by Northwest Corner Method, for example)
2.
Compute the raw variables, u, and column variables, v, on the basis of the
occupied cells (initially set u1 = 0) where ui + vj = cij.
3.
Compute the circled values, eij, on the basis of empty cells where eij = cij - ui
- vj.
4.
Put a (+) sign into the cell with the most negative circled value and find a
feasible path by alternating the signs ((+) and (-)) and returning to the point
where you started, i.e. you have to complete the loop. You can go in
horizontal or vertical directions, but not diagonally. You can skip over and
into the occupied cells. On the other hand, you can ship over empty cells,
but you can NEVER skip into an empty cell. Make sure that there is an
equal number of (+) and (-) signs in each row and each column on the
feasible path so that they cancel row wise and column wise.
5.
Look at the cells with (-) sign on the feasible path and choose that cell with
the minimum quantity. Add this minimum quantity to the cells with (+)
signs and subtract it from the cells with (-) sign.
6.
Repeat the procedure until all the circled values (eij) are non-negative at
which point we reach optimality.