104 Production and Operation Management

Unit 7 Facility Location

UNIT 7: FACILITY LOCATION

UNIT STRUCTURE7.1 Learning Objectives

7.2 Introduction

7.3 Factors influencing plant location

7.4 Break Even Analysis

7.5 Single facility location problems

7.6 Multifacility location problems

7.7 Minimax location problems

7.8 Gravity location problem

7.9 Euclidean-distance location problem

7.10 Covering problem

7.11 Warehouse location problem

7.12 Let Us Sum Up

7.13 Further Reading

7.14 Answers to Check Your Progress

7.15 Model Questions

7.1 LEARNING OBJECTIVESAfter going through this unit, you must be able to:

• explain the factors influencing plant location

• explain the concept of Break Even Analysis

• describe single facility location problems

• explain multifacility location problems

• describe minimax location problems

• explain gravity location problem

• describe euclidean-distance location problem

• explain covering problem

• describe warehouse location problem

7.2 INTRODUCTIONIn this unit, we are going to explain the factors influencing plant

location, the concept of Break Even Analysis, single facility location problems,

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multifacility location problems, minimax location problems, gravity location

problem, Euclidean-distance location problem, covering problem, and

describe warehouse location problem.

7.3 FACTORS INFLUENCING PLANT LOCATION

Factors influencing location decision of any production plant are:

Locational Choice for the First Time: For organisations/ companies

setting up their production/ operations facilities for the first time, the

steps are as follows:

a) Identification of a suitable region

b) Identification of a suitable locality

c) Identification of a suitable site- The factors influencing selection of

the actual site for setting up facilities for an organisation for the first

time are as follows:

General Factors:

(i) Proximity to customers (markets)

(ii) Proximity to raw material

(iii) Availability of good transportation facilities

(iv) Availability of good communication facilities

(v) Availability of power supply

(vi) Availability of basic amenities

(vii) Presence of good government policies: This applies to favourable

taxation policies, promotional facilities for industrial activities,

subsidies etc.

(viii) Considerations for environment and community

(ix) Proximity to contractors

(x) Availability of cheap land

(xi) Lesser constructions costs

(xii) Availability of cheap, skilled and efficient labour

(xiii) Availability of land/ facilities for setting residential complexes,

schools, hospitals, clubs etc.

(xiv) Availability of waste disposal facilities

(xv) Reasonable fuel cost

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Additional General Factors for locating foreign operations facilities:(xvi) Existence of favourable trade barriers(xvii) Presence of large customer base in the foreign country(xviii) Presence of operational facilities of competitors in the foreign country(xix) Presence of lower costs in the foreign country (due to low cost of

labour, material, transportation financing etc.)(xx) Presence of favourable incentives for trade in the foreign country-

These include industrial infrastructure, insurance, tax exemptions,interest free/ subsidized loans etc.

(xxi) Presence of opportunities for exploiting firm specific advantages(xxii) Existence of economies of scale(xxiii) Synergy in the form of past learning experiences and benefits while

setting up facilities in other countries(xxiv) Power and prestige of any firm may increase with the presence of

operations and facilities in a foreign country(xxv) Harsh regulatory conditions in the home country may force many

firms to have operations in another country with favourable regulatoryconditions.

Specific Factors:

(i) Economic stability and concern of the country towards outsideinvestments

(ii) Cultural factors, languages and cultural differences in the country(iii) Wage rate, policy, duties etc.

Locational Choice for an already established organisation withone or more facilities existing: For organisations/ companies withone or more facilities existing, the factors considered for locationaldecisions are applied in case of the following:

(i) Plants manufacturing distinct product or product lines(ii) Manufacturing plants each supplying to a specific market area(iii) Manufacturing plants divided according to the product/ product

line being manufactured; and these special- product plants locate

in various market areas

(iv) Plants divided on the basis of processes or stages in

manufacturing

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(v) Plants emphasizing flexibility in adapting to constantly changing

market needs

7.4 BREAK-EVEN ANALYSIS IN FACILITY LOCATIONDECISIONS

The basic steps for break-even analysis in facility location decisions

in case of Production and Operations Management are:

(i) Determination of variable costs

(ii) Determination of fixed costs

(iii) Calculating total cost (sum of fixed costs and variable costs)

(iv) Plotting the above total cost lines (sum of fixed costs and variable

costs) for all alternatives in a single graph

(v) Identifying the approximate ranges for sites with lowest total cost

Solving algebraically for the break-even points over the relevant ranges

Example 7.1: In the case of a product DRE567, there are two possible

locations for setting up a production plant in a region of a particular country.

These two possible locations are X and Y. The following figures are estimated

for these two locations:

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Particulars Location X Location Y

Annual Fixed Costs (Rs.) 200,000 700,000

Variable Costs per unit (Rs. per unit) 60 32

Revenues per unit (Rs. per unit) 70 72

Determine the best possible location for setting up a production plant for

product DRE567 using Break-Even Analysis for an expected demand of

10,000 units? (Adapted from Production and Operations Management by

E.E.Adam and R.J. Ebert, Prentice Hall of India Private Limited, pp.233-234.)

Solution to Example 7.1: Given, in the case of setting up a production

plant for product DRE567, there are two possible locations in a region of a

particular country, namely, X and Y.

Let, FCX and FCY indicate the annual fixed costs in relation to the above

product in locations X and Y respectively.

Let, VCX and VCY indicate the variable costs per unit in relation to the

above product in locations X and Y respectively.

Let, RVX and RVY indicate the revenues per unit in relation to the above

product in locations X and Y respectively.

Let, BEPX and BEPY indicate the Break-Even Points in relation to the

above product in locations X and Y respectively.

Therefore,

BEPX = FCX ÷ ( RVX - VCX ) = 200,000 ÷ ( 70 - 60 ) = 20,000 units

BEPY = FCY ÷ ( RVY – VCY ) = 700,000 ÷ ( 72 - 32 ) = 70,000 units

At the expected demand of 10,000 units, profits or losses for the

alternative locations X and Y are:

It is seen that location X is the most attractive for setting up a production

plant for product DRE567 in the above region of the particular country as it

would result in a profit of Rs. 440,000.

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7.5 SINGLE FACILITY LOCATION PROBLEMS

Single facility location problems are centred around location decision

for a single production/ service delivery plant, warehouse, or retail store or

customer service point. These reflect the most common problems faced

during selection of optimal locations for any facility. These problems seeks

to minimize the total transportation cost (TC) from the present location of a

facility to the point i where it is to be located.

Minimize TC= ΣViRidi ……………….…………..(i)

In equation (i),

TC is the total transportation cost (TC) from the present location of a

facility to the point i where it is to be located,

Vi is the volume at the point i,

Ri is the transportation rate at the point i,

di is the distance to the point i from the facility to be located.

di = K√ [(Xi - X–

)2 + (Yi –Y–

)2] ………………………(ii)

where Xi and Yi are the coordinate points of the source and demand

points.

and X and Y are the coordinate points of the located facility

..(iii)……………………d / R V

d / X R VX

iiii

iiiii

∑∑

=

=

∑ Vi Ri Yi / di i

……………………..(iv) ∑ Vi Ri / di i

Example 7.2: In the case of ABC distributors, there are two production

plants supplying the warehouse which, in turn, supplies three demand

centres (M1, M2 and M3). The figure below shows the arrangement of the

above plants and market points. It is desired to determine the location of a

single warehouse that will minimise the total transportation costs. Each

plant and demand centre location is expressed as a geometric coordinate

point. Product A is supplied from plant P1 and product B from plant P2.

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These products are reshipped to the markets. Coordinate points, Volumes

and transportation rates are provided in the table given below. (Adapted

from University of Washington (2013). “Facility Location Decisions”.

Retrieved from http://courses. washington.edu/cee587/Readings/

Chapter%2013%20Facility%20Location%20Decisions.pdf.pp.557-558).

Table: Arrangement of the plants and market points Point

Product Total Volume (Vi)

Transportation Rate (Ri)

Coordinates i Marking Xi Yi 1 P1 A 2,000 0.050 3 8 2 P2 B 3,000 0.050 8 2 3 M1 A and B 2,500 0.075 2 5 4 M2 A and B 1,000 0.075 6 4 5 M3 A and B 1,500 0.075 8 8

Solution to Example 7.2: Given, in the case of ABC distributors, there are

two production plants supplying the warehouse which, in turn, supplies three

demand centres. It is desired to determine the location of a single warehouse

that will minimise the total transportation costs. Each plant and demand

centre location is expressed as a geometric coordinate point. Product A is

supplied from plant P1 and product B from plant P2. These products are

reshipped to the markets.

Let, M1, M2 and M3 indicate the three demand centres.

Let, “ indicate the location of the warehouse with coordinates (X– and Y– along

horizontal and vertical axes).

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Points

Product Total

Volume (Vi)

Transportation Rate (Ri)

Coordinates Vi .Ri Vi .Ri .Xi Vi .Ri .Yi i Marking Xi Yi

1 P1 A 2,000 0.050 3 8 100.00 300.00 800.00 2 P2 B 3,000 0.050 8 2 150.00 1200.00 300.00 3 M1 A and B 2,500 0.075 2 5 187.50 375.00 937.50 4 M2 A and B 1,000 0.075 6 4 75.00 450.00 300.00 5 M3 A and B 1,500 0.075 8 8 112.50 900.00 900.00

∑Vi .Ri = 625.00

∑Vi .Ri .Xi = 3,225.00

∑Vi .Ri .Yi = 3,237.50

Therefore, coordinates of the warehouse at location D are =

d / R V

d / X R VX

iiii

iiiii

∑∑

=

= 3,225.00 ÷ 625.00 = 5.16

d / R V

d / Y R VY

iiii

iiiii

∑∑

= = 3,237.50 ÷ 625.00 = 5.18

Table: Calculation of Total Transportation Costs Point Total

Volume (Vi)

Transportation Rate (Ri)

Coordinates

(Xi - )(Yi - )di =

10√{(Xi - )2 - (Yi -)2}

Vi .Ri .di i Marking Xi Yi

1 P1 2,000 0.050 3 8 -2.16 2.82 35.36 3,536.00 2 P2 3,000 0.050 8 2 2.84 -3.18 42.64 6,396.00 3 M1 2,500 0.075 2 5 -3.16 -0.18 31.65 5,934.38 4 M2 1,000 0.075 6 4 0.84 -1.18 14.48 1,086.00 5 M3 1,500 0.075 8 8 2.84 2.82 40.02 4,502.25

∑Vi .Ri .di = 21,454.63

Therefore, Total Transportation Costs associated with the warehouse

located in coordinates (X– =5.16, Y– =5.18) is Rs.21,454.63.

CHECK YOUR PROGRESSQ 1: Which of the following are not the basic

steps for break-even analysis in facility location

decisions in case of Production and Operations

Management?

(a) Determination of variable costs

(b) Determination of fixed costs

(c) Calculating total cost (d) None of these

(sum of fixed costs and variable costs)

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Q 2: Which of the following is a factor influencing location decision of

any production plant?

(a) Proximity to raw material (b) Proximity to film stars

(c) Proximity to gymnasium (d) None of these

Q 3: Which of the following location problem are centered around

location decision for a single production/ service delivery plant?

(a) Method study (b) Multi Facility Location

Problem

(c) Single Facility Location Problem(d) None of these

7.6 MULTIFACILITY LOCATION PROBLEMS

The multifacility location problem (MFLP) aims to locate two or more

facilities simultaneously. MFLP may also be used to locate additional facilities

at a place wherein atleast one facility already exists (Iyigun and Ben-Israel,

2012).

The multifacility location problem (MFLP) aims to locate K facilities,

and assign customers to these facilities, so as to minimize the sum of

weighted distances

Minimize Σ K ΣK wi d (xi, ck) ………………..(i)

c1,c2,...,cK k=1xi”Xk

In equation (i), {ck : k Î 1:K} are the centres (facility locations), and Xk

is the cluster of customers assigned to the kth facility (Iyigun and Ben-Israel,

2012).

The MFLP may be divided into the following two problems (Iyigun

and Ben-Israel, 2012):

(a) The assignment problem: given K centres (ck), and assigning

each of the N points (xi) to the nearest centre, and

(b) The location problem: given the K clusters Xk formed in (a),

and finding the centre of each cluster.

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7.7 MINIMAX LOCATION PROBLEMS

The minimax facility location problem seeks a location which

minimizes the maximum distance to the sites, where the distance from

one point to the sites is the distance from the point to its nearest site. Minimax

location problems have received considerable attention in the literature as

models for locating facilities that are to provide emergency or convenient

service to a set of existing facilities (Dearing, 1977). In most of these

problems there is given a set of existing facilities whose locations are

represented as points in some space, and new facility locations are also to

be specified as points in that space. A distance function is chosen to

represent the travel distance between the new and existing facility locations.

The minimax objective is to locate the new facilities so that the maximum

distance, or a function of distance, between the new and existing facility

locations is minimized.

The maxmin facility location problem, on the other hand, aims at

identifying a suitable location for a facility in such a manner that this location

maximizes the minimum distance to the sites.

7.8 GRAVITY LOCATION PROBLEM

Gravity Location Problem aid in the selection of a suitable location

for any production plant. Here, the coordinates for the optimal location are

chosen as an average of the coordinates of the various neighbouring places

which are weighted according to the number of customers expected from

each such neighbouring place.

Example 7.3: Suppose there are three neighbouring locations: A, B and

C. It is noticed that the coordinates of location A is centered at (5,8) meaning

5 coordinate points horizontally and 3 coordinate points vertically. Similarly,

it was seen that the coordinates of locations B and C are centered at (3,2)

and (6,3) respectively. Find the coordinates for the location a new plant?

Solution to Example 7.3: Given, there are three neighbouring locations:

A, B and C. It is noticed that the coordinates of location A is centered at

(5,8) meaning 5 coordinate points horizontally and 3 coordinate points

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vertically. Similarly, it was seen that the coordinates of locations B and C

are centered at (3,2) and (6,3) respectively.

The average of the horizontal (or x-axis) coordinates is (5+3+6) ÷ 3 = 4.67.

Again, the average of the vertical (or y-axis) coordinates is (8+2+3) ÷ 3 =

4.33.

Therefore, it may be better to locate the new plant centrally at the coordinates

(4.67, 4.33). But, there may be a problem of different numbers of customers

in the locations A, B and C. Say, the number of prospective customers in

locations A, B and C are 100, 750, and 200 respectively. As such, any

planner would definitely want the new location to be set up nearer location

B than location A. In this context, such number of customers may be used

as weights and deriving the coordinates of the new location from the following

calculations:

Optimal horizontal coordinate is:

3.76 =200) + 750 + (100

200)] × (6 + 750) × (3 + 100) × [(5= x

2.76 =200) + 750 + (100

200)] × (3 + 750) × (2 + 100) × [(8= y

It may not be feasible to locate the new plant at the coordinate (3.76,

2.76) due to unavailability of land at that location. However, location decision

maker may choose a location that is feasible and near the coordinate (3.76,

2.76).

7.9 EUCLIDEAN-DISTANCE LOCATION PROBLEM

In case of Euclidean-Distance Location Problem, distance from the

present location of a facility to a new location is given by the following formula

(i). This distance is also called Straight line or Euclidean distance from the

present location of the facility (in the coordinates (x, y)) to a new location (in

the coordinates (pi, qi)):

Straight line or Euclidean distance, d(X, Ai) = √{(x - pi)2 + (y - qi)2}

..............................(i)

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7.10 COVERING PROBLEM

Covering problem in location decisions provide a good alternative

for taking decisions regarding the selection of a proper locations for setting

up any facility. In most of the covering problems, customers may avail

services provided by facilities based on the distance between them and

facilities (Farahani, Asgari, Heidari, Hosseininia, and Goh, 2012; Miheliè and

Robiè, 2004). The customer(s) can obtain service from each facility provided

its distance from them is equal or less than a predefined number called

coverage distance or coverage radius. So it is clear that the notion of

coverage is related to a satisfactory method rather than an optimally possible

one. Covering problem in location decisions may be used in applications

like determining the number and locations of the following entities:

(i) public schools, (ii) police stations,

(iii) libraries, (iv) hospitals,

(v) public buildings, (vi) post offices,

(vii) parks, (viii) military facilities,

(ix) radar installations, (x) bank branches,

(xi) shopping centers (xii) waste-disposal facilities.

7.11 WAREHOUSE LOCATION PROBLEM

Warehouse Location problem (WLP) is an important tool for selection

of a proper location for a warehouse of a company. Through WLP, a company

may open warehouses in optimal locations so as to supply goods to its

existing stores at the optimal maintenance and supply cost (Nagy, 2004). In

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fact, the cost of supply to any store from a company’s production plant

depends on the location of such warehouses. Therefore, the objectives of

WLP is to determine the optimal location of such warehouse(s) and to

identify the distinct warehouse(s) which would supply goods to different

stores so that the total cost comprising of the sum of the maintenance and

supply costs is minimized.

However, in such problems it is assumed that each such feasible

warehouse has the following features:

(i) Each such feasible warehouse has similar maintenance cost,

(ii) Each such feasible warehouse has similar capacity in terms of the

optimal number of stores to whom goods are supplied from it.

(iii) It is also assumed that each such store must get supply of goods

precisely from a single open warehouse.

CHECK YOUR PROGRESSQ 4: In which of the following location problem,

the coordinates for the optimal location are

chosen as an average of the coordinates of the

various neighbouring places which are weighted according to the

number of customers expected from each such neighbouring place?

(a) Selection (b) Gravity Location Problem

(c) Minimax Location Problem (d) None of these

Q 5: Which of the following location problem seeks a location which

minimizes the maximum distance to the sites, where the distance

from one point to the sites is the distance from the point to its

nearest site?

(a) Work sampling (b) Gravity Location Problem

(c) Minimax Location Problem (d) None of these

Q 6: Which of the following location problem aims to locate two or

more facilities simultaneously?

(a) Gravity Location Problem (b) Linear Programming

Problem

(c) Minimax Location Problem (d) Multi Facility Location Problem

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7.12 LET US SUM UP

Decisions regarding selection of a proper location for a production

facility are highly essential in production and operations management. The

important points under facility location are:

For organisations/ companies setting up their production/ operations

facilities for the first time, the steps are as follows:

a) Identification of a suitable region

b) Identification of a suitable locality

c) Identification of a suitable site.

• Break-even analysis in facility location decisions in case of Production

and Operations Management involves determination of variable costs

and fixed costs

• Single facility location problems are centered around location decision

for a single production/ service delivery plant, warehouse, or retail store

or customer service point.

• The multifacility location problem (MFLP) aims to locate two or more

facilities simultaneously.

• The minimax facility location problem seeks a location which minimizes

the maximum distance to the sites, where the distance from one point

to the sites is the distance from the point to its nearest site.

• In Gravity Location Problem, the coordinates for the optimal location

are chosen as an average of the coordinates of the various neighbouring

places which are weighted according to the number of customers

expected from each such neighbouring place.

• Covering problem in location decisions provide a good alternative for

taking decisions regarding the selection of a proper locations for setting

up any facility.

• The objectives of Warehouse Location problem (WLP) is to determine

the optimal location of such warehouse(s) and to identify the distinct

warehouse(s) which would supply goods to different stores so that the

total cost comprising of the sum of the maintenance and supply costs

is minimized.

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7.13 FURTHER READINGS

• Cem Iyigun and Adi Ben-Israel (2012), “The Multi–Facility Location Problem:

A Probabilistic Decomposition Method”, 2000 Mathematics Subject

Classification.

• Industrial Engineering and Production Management by M.S. Mahajan,

Dhanpat Rai and Company Private Limited, pp.780-784.

•Operations Management by M.S. Mahadevan, Pearson Education, pp.474-

509.

•P.M. Dearing (1977), “Minimax Location Problems With Nonlinear Costs”,

Journal of Research, Vol. 82, No. 1, pp.65-72.

•Production and Operations Management by E.E.Adam and R.J. Ebert,

Prentice Hall of India Private Limited.

Computers & Industrial Engineering, Vol. 62, pp. 368–407.

Thomas Nagy (2004), “Warehouse Location Problem”. Retrieved from

www.freehackers.org/~ tnagy/constraint/warehousemodel.pdf.

7.14 ANSWERS TO CHECK YOUR PROGRESS

Answer to Question No.1: (d) None of these

Answer to Question No.2: (a) Proximity to raw material

Answer to Question No.3: (c) Single Facility Location Problem

Answer to Question No.4: (b) Gravity Location Problem

Answer to Question No.5: (c) Minimax Location Problem

Answer to Question No.6: (d) Multi Facility Location Problem

7.15 MODEL QUESTIONS

Q. 1) Explain Warehouse Location problem (WLP).

Q. 2) Explain the factors to be considered for setting up an airport near a

city.

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Q. 3) In the case of a product, there are two possible locations for setting

up a production plant in a particular country. These two possible

locations are M and N. The following figures are estimated for these

two locations:

Determine the best possible location for setting up a production

plant for the above product using Break-Even Analysis for an

expected demand of 5,000 units?

Q. 4) Discuss Gravity Location Problem.

Q. 5) Suppose there are three neighbouring locations for setting up a

production plant: M, N and O. It is noticed that the coordinates of

location M is centred at (4,8). Similarly, it was seen that the

coordinates of locations N and O are centred at (2,3) and (5,2)

respectively. Find the coordinates for the location a new production

plant using gravity location technique?

Q. 6) In the case of XYZ company, there are two production plants

supplying the warehouse. The latter, in turn, supplies three demand

centres (D1, D2 and D3). The figure below shows the arrangement

of the above plants and market points. Determine the location of a

single warehouse that will minimise the total transportation costs.

Each plant and demand centre location is expressed as a geometric

coordinate point. Product P is supplied from plant L1 and product Q

from plant L2. These products are reshipped to the markets.

Coordinate points, Volumes and transportation rates are provided

in the table given below.

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Q. 7) Explain the concept of Minimax Location Problem.

* * *