Location decisions are strategic decisions. The reasons for location decisions Growth –Expand existing facilities –Add new facilities Production Cost Depletion of Resources Nature of Location Decisions

Multiple Plant Strategies Community Factors Regional Factors Factors Affecting Plant location Site Factors

1-Location of raw material Raw material oriented factories; weight of input >>> weight of output Regional; Raw Material

These types of plants tends to be closer to the raw material resources. Indeed row material or any other important input. Regional; Raw Material

2-Location of market Market oriented plants; Space required for output >>> space required for input. Car manufacturing, Appliances Regional; Market

3-Labor, water, Electricity Availability of skilled labor, productivity and wages, union practices Availability of water; Blast furnace requires a high flow of water Availability of electricity; Aluminum plant strongly depends on availability and cost of electricity, it dominates all other inputs. Regional; Labor, Water, Electricity

1-Quality of Life; Cost of living, housing, schools, health care, entertainment, church 2- Financial support; Tax regulations, low rate loans for new industrial and service plants Community

1-Land; Cost of land, development of infrastructure. 2-Transportation; Availability and cost of rail road, highways, and air transportation. 3-Environment; Environmental and legal regulations and restrictions Site

Decentralization Small is beautiful; Instead of a single huge plant in one location, several smaller plants in different locations Decentralization based on product Decentralization based on geographical area Decentralization based on process

Decentralization based on Product Each product or sub-set of products is made in one plant Each plant is specialized in a narrow sub-set of products. Lower operating costs due to specialization.

Decentralization Based on Geographical Area Each plant is responsible for a geographical region, Specially for heavy or large products. Lower transportation costs.

Decentralization Based on Process Car industry is an example. Different plants for engine, transmission, body stamping, radiator. Specialization in a process results in lower costs and higher quality. Since volume is also high, they also take advantage of economy of scale. However, coordination of production of all plants becomes an important issue and requires central planning and control

Foreign producers locating in U.S. –“Made in USA” –Currency fluctuations Just-in-time manufacturing techniques Focused factories Information highway Trends in Global Locations

Cost-volume Analysis –Determine fixed and variable costs –Plot total costs –Determine lowest total costs BEP in Location Analysis

Fixed and variable costs for four potential locations Example

Solution

Annual Output (000) $(000) A B C B Superior C Superior A Superior D Graphical Solution

Center of Gravity ; Single Facility Location Center of gravity is a method to find the optimal location of a single facility The single facility is serving a set of demand centers or It is being served by a set of supply centers The objective is to minimize the total transportation Transportation is Flow ×Distance

Examples of Single Facility Location Problem There are a set of demand centers in different locations and we want to find the optimal location for a Manufacturing Plant or a Distribution Center (DC) or a Warehouse to satisfy the demand of the demand centers or There are a set of suppliers for our manufacturing plant in different locations and we want to find the optimal location for our Plant to get its required inputs The objective is to minimize total Flow × Distance

Center of Gravity ; Single Facility Location Suppose we have a set of demand points. Suppose demand of all demand points are equal. Suppose they are located at locations X i, Y i Where is the best position for a DC to satisfy demand of these points Distances are calculated as straight line not rectilinear. There is another optimal solution for the case when distances are rectilinear.

Optimal Single Facility Location The coordinates of the optimal location of the DC is

Example We have 4 demand points. Demand of all demand points are equal. Demand points are located at the following locations

Example Where is the optimal location for the center serving theses demand points (2,2) (8,5) (5,4) (3,5)

Solution Where is the optimal location for the center serving theses demand points (2,2)(8,5)(5,4)(3,5)

Solution The optimal location for the center serving theses demand points (2,2) (8,5) (5,4) (3,5)

Center of Gravity ; Single Facility Location Suppose we have a set of demand points. Suppose they are located at locations X i, Y i Demand of demand point i is Q i. Now where is the best position for a DC to satisfy demand of these points Again; the objective is to minimize transportation.

Optimal Single Facility Location The coordinates of the optimal location of the DC is

Example Where is the optimal location for the center serving theses demand points (2,2) (8,5) (5,4) (3,5)

Solution Where is the optimal location for the center serving theses demand points 800 : (2,2)900 : (3,5) 100 : (8,5) 200 : (5,4)

Solution Where is the optimal Y location for the center serving theses demand points 800 : (2,2)900 : (3,5) 100 : (8,5) 200 : (5,4)

Solution The optimal location for the center serving theses demand points (800) (100) (200) (900)

Example Where is the optimal location for the center serving theses demand points (0,0) (6,3) (3,2) (1,3)

Solution Where is the optimal location for the center serving theses demand points 800 : (0,0)900 : (1,3) 100 : (6,3)200 : (3,2)

Solution Where is the optimal location for the center serving theses demand points 800 : (0,0)900 : (1,3) 100 : (6,3)200 : (3,2)

Solution The optimal location for the center serving theses demand points is at the same location (800) (100) (200) (900)

The Transportation Problem D (demand) D (demand) D (demand) S (supply) S (supply) S (supply)

There are 3 plants, 3 warehouses. Production of Plants 1, 2, and 3 are 300, 200, 200 respectively. Demand of warehouses 1, 2 and 3 are 250, 250, and 200 units respectively. Transportation costs for each unit of product is given below Transportation problem : Narrative representation Warehouse Plant Formulate this problem as an LP to satisfy demand at minimum transportation costs.

Transportation problem I : decision variables x 11 x x 13 x 21 x 31 x 22 x 32 x 23 x 33

Transportation problem I : decision variables x 11 = Volume of product sent from P1 to W1 x 12 = Volume of product sent from P1 to W2 x 13 = Volume of product sent from P1 to W3 x 21 = Volume of product sent from P2 to W1 x 22 = Volume of product sent from P2 to W2 x 23 = Volume of product sent from P2 to W3 x 31 = Volume of product sent from P3 to W1 x 32 = Volume of product sent from P3 to W2 x 33 = Volume of product sent from P3 to W3 We want to minimize Z = 16 x x x x x x x x x 33

Transportation problem I : supply and demand constraints x 11 + x 12 + x 13 = 300 x 21 + x 22 + x 23 =200 x 31 + x 32 + x 33 = 200 x 11 + x 21 + x 31 = 250 x 12 + x 22 + x 32 = 250 x 13 + x 23 + x 33 = 200 x 11, x 12, x 13, x 21, x 22, x 23, x 31, x 32, x 33  0

Origins We have a set of ORIGINs Origin Definition: A source of material - A set of Manufacturing Plants - A set of Suppliers - A set of Warehouses - A set of Distribution Centers (DC) In general we refer to them as Origins m 1 2 i s1s2sisms1s2sism There are m origins i=1,2, ………., m Each origin i has a supply of s i

Destinations We have a set of DESTINATIONs Destination Definition: A location with a demand for material - A set of Markets - A set of Retailers - A set of Warehouses - A set of Manufacturing plants In general we refer to them as Destinations n 1 2 j d1d2didnd1d2didn There are n destinations j=1,2, ………., n Each origin j has a supply of d j

Total supply is equal to total demand. There is only one route between each pair of origin and destination Items to be shipped are all the same for each and all units sent from origin i to destination j there is a shipping cost of C ij per unit Transportation Model Assumptions

C ij : cost of sending one unit of product from origin i to destination j m 1 2 i n 1 2 j C1nC1n C 12 C 11 C 2n C22C22 C 21

X ij : Units of product sent from origin i to destination j m 1 2 i n 1 2 j X 1n X 12 X 11 X 2n X 22 X 21

The Problem m 1 2 i n 1 2 j The problem is to determine how much material is sent from each origin to each destination, such that all demand is satisfied at the minimum transportation cost

The Objective Function m 1 2 i n 1 2 j If we send X ij units from origin i to destination j, its cost is C ij X ij We want to minimize

Transportation problem I : decision variables x 11 x x 13 x 21 x 31 x 22 x 32 x 23 x 33

Transportation problem I : supply and demand constraints x 11 + x 12 + x 13 =200 +x 21 + x 22 + x 23 =200 +x 31 + x 32 + x 33 =200 x 11 + x 21 + x 31 =150 x 12 + x 22 + x 32 =250 x 13 + x 23 + x 33 = 200

Transportation Problem is a special case of LP models. Each variable x ij appears only in rows i and m+j. Furthermore, The coefficients of all variables are equal to 1 in all constraints. Based on these properties, special algorithms have been developed. They solve the transportation problem much faster than general LP Algorithms. They only apply addition and subtraction If all supply and demand values are integer, then the optimal values for the decision variable will also come out integer. In other words, we use linear programming based algorithms to solve an instance of integer programming problems. Transportation Problem Solution Algorithms

Supply Demand Supply Demand Data for the Transportation Model Quantity demanded at each destination Quantity supplied from each origin Cost between origin and destination

$600 $400 $300 $200 WaxdaleBramptonSeaford Min.Milw.Chicago $700 $900 $100 $700 $800 Supply Locations Demand Locations Data for the Transportation Model

Our Task Our main task is to formulate the problem. By problem formulation we mean to prepare a tabular representation for this problem. Then we can simply pass our formulation ( tabular representation) to EXCEL. EXCEL will return the optimal solution. What do we mean by formulation?

Supply D -3D -2D -1 O -1 O -2 O -3 Demand

Excel

Assignment; Solve it using excel We have 3 factories and 4 warehouses. Production of factories are 100, 200, 150 respectively. Demand of warehouses are 80, 90, 120, 160 respectively. Transportation cost for each unit of material from each origin to each destination is given below. Destination Origin Formulate this problem as a transportation problem

Excel : Data