Location TeknikIndustri
Facilities Facilities Location
Lecture Notes #2
Stefanus Eko Wiratno 2007
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Location Problems and Models
A classification scheme for location models can used to help identify those problems Location problems and models may be classified in a number of ways (Daskin, 1995) :
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. Demands and candidate facility locations Type of network location models Number of Facilities to Locate Distance Metrics Nature of inputs about time Nature of inputs about uncertainty Homogenous product and demand Sector problems Objective Elasticity demand Capacity of facilities The allocation of demand to facilities Level of facilities Locating desirable facilities
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Taxonomy of Location Problems and Models
Demands and candidate facility locations Planar location models; demands occur anywhere on a plane Network location models; demands and travel between demand sites and facilities are assumed to occur only on a network or graph composed of nodes and links Discrete location models; allow for the use of arbitrary distance between nodes. As such, the structure of the underlying network is lost
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Taxonomy of Location Problems and Models
Type of network location models Tree problems; a network in which there is at most one path from any node to any other node General graph problem; consists of a connected general network and a complete graph
Number of Facilities to Locate Single facility location problem Multiple facilities location problem
(exogenous: number of facilities is parameter, endogenous: number of facilities is decision variable)
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Taxonomy of Location Problems and Models
Distance Metrics Manhattan or rightangle distance metric
d ( xi , y i ) ; ( x j , y j ) = xi x j + y i y j
[
]
Euclidean or straightline distance metric
d ( xi , yi ) ; ( x j , y j ) =
Ip distance metric
[
]
[( x x ) + ( y y ) ]
2 2 i j i j
d ( xi , yi ) ; ( x j , y j ) = xi x j
[
] [(
) +( y y )
p i j
p
]
1
p
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Taxonomy of Location Problems and Models
Nature of inputs about time Static location problem; the inputs do not depend on time Dynamic location problem; the inputs (e.g., demand, costs, etc) depend on time Nature of inputs about uncertainty Deterministic models; the inputs to models may be certain Probabilistic models; the inputs to models may be uncertain
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Taxonomy of Location Problems and Models
Homogenous product and demand Singleproduct models; products or services are homogenous and identical demand Multipleproduct models; products are distinguished by having different origins and destinations Sector problems Private sector location problems; the investment costs and benefit are typically measured in monetary units Public sector location problems; many non monetary cost and benefits must also be considered
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Taxonomy of Location Problems and Models
Objective Singleobjective problems and models Multipleobjective problems and model Elasticity demand Elastic demand; demand is given and independent of the level of service Inelastic demand; demand depends on the level of service, facility location, types and sizes of facilities
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Taxonomy of Location Problems and Models
Capacity of facilities Capacitated facilities; having unlimited capacity Uncapacitated facilities; models impose explicit capacity limits on facilities The allocation of demand to facilities Nearest facility allocation model; demands are assigned to the nearest facility provided that facility has the capacity to serve the demand General demand allocation model; problems may result in the need to split the demand at a site between several facilities
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Taxonomy of Location Problems and Models
Level of facilities Hierarchical level models; the facilities have hierarchy for serve demands Singlelevel models; the demands are served by single facility Locating desirable facilities Desirable facilities; the closer the facilities are to the people or goods being served Undesirable facilities; most people want facilities located as far away as possible
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Location Model Typology
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Current, J., Daskin, M. and D. Schilling, 2001, Discrete Network Location Models, forthcoming as chapter 3 in Facility Location Theory: Applications and Methods, Z. Drezner and H. Hamacher eds.
Location Models
Analytic Models
Closed form models Very strong assumptions Calculus
Continuous Location
Discrete Location Continuous Facility Location Weber Problem; Geom. Prob.
Network Models
Underlying Network Demands/Facilities on Network Graph Theory; Polynomial Time Alg.
Discrete Location
No underlying network IP Formulations Heuristic/IP solution techniques
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Analytic Models
1 2A A Total cost = + fN c $/item/dem 3 N demands fixed
cost avg. dist 2/3
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c 2 N = A 6f TC 1.1447 A f ( c )
* * 13
23
Fixed cost/distance balance, variable dependencies Daganzo et al.
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Demand Weighted Distance
Dem wtd total distance =
(30)7+(20)11.2+(60)8.6+(50)4.5
=1175.5
20
60 11.2 8.6 4.5 50
Dem wtd avg dist =
1175.5/(30+20+60+50)=7.35
Not equal to avg dist of
(7+11.2+8.6+4.5)/4=7.825 30
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Continuous Models
Discrete demand sites Continuous facility location
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Minimize demand weighted avg. distance Single facility = Weber (center of gravity) problem Iterative numerical algorithms [Weiszfeld alg.; Newton Bracketing (Levin, Ben-Israel)]
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Example Weber (Gravity) problem
20 (-5,10) 13.4
(7.7,5.7)
Demand wtd total 60 (10,12) dist = 1107.8
6.7 3.0
Demand wtd avg dist = 6.92 Unweighted avg dist = 8.16
9.5 30 (0,0)
50 (9,3)
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Network Models
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Underlying network Demands on nodes (or links) Key areas Models Applications Algorithms (Goldmans majority algorithm for 1median on tree; Tamir/Hasin work) Properties (Hakimi property)
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Hakimi Property of P-Median
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There is at least one optimal solution that consists of locating only on the nodes of a network
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Discrete Location Objectives
Covering Set covering Maximal covering P-center Average distance based P-median Fixed charge Maximize Profit Other (undesirable facilities) P-dispersion Maximize minimum distance
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Coverage
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Solution to Maximal Covering Problem w/ 10 facilities Dc=300
Set covering Find min. # needed to cover all demands Max covering Cover max # DEMANDS w/ fixed # facilities P-center cover all demand nodes w/ fixed # facilities in smallest possible distance
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Classification of Location Problem
[Logistics Management]
1. Location Problems 2. Allocation Problems 3. Location Allocation Problems
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Location Problems
Determine the location of one or more new facilities in one or more of several potential sites The number of sites must at least equal the number of new facilities being located The cost of locating each new facility at the potential sites (fixed cost and operating & transportation cost) is assumed to be know.
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Location Problems
Location problems can be classified as: Single-facility problems Multifacility problems Classification of location problems is based on whether the set of possible locations for a facility is finite or infinite : Discrete space location problem Continuous space location problem
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Allocation Problems
Assume that the number and location of facilities are known a priori and attempt to determine how each customer is to be served (problem determines how much each facility is to supply to each customer center)
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Location-Allocation Problems
Problems determine not only how much each customer is to receive from each facility but also the number of facility along with their locations and capacities
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Facility Location Problems
Can be classified as (Hax and Candea, 1984) :
Single-facility location problem; deal with the optimal determination of the location of a single facility Location Problem Multi-facility location problem; deal with the simultaneous location determintaion for more than one facility Location-Allocation Problem
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Facility Location Problems
Another classification it is based on whether the set of possible locations for a facility is finite or infinite : 1. Discrete space location problem It have a finite feasible set of sites in which to locate a facility For most real-world problem, this models are more appropriate The solutions may be near optimal but feasible
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Facility Location Problems
1. Continuous space location problem; A facility can be located anywhere within the confines of a geographic area, then the number of possible locations is infinite Assuming that the transportation costs are proportional to distance The solutios may be infeasible but optimal
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Important Factors in Location Decisions (1)
Proximity to source of raw materials Cost and availability of energy and utilities Cost, availability, skill, and productivity of labor Government regulations Taxes Insurance Construction costs and land price Government and political stability Exchange rate fluctuation Export and import regulations, duties, and tariffs
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Important Factors in Location Decisions (2)
Transportation system Technical expertise Environmental regulations Support service Community services Weather Proximity to customer Business climate Competition-related factors
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Techniques for Discrete Space Location Problems
1. Qualitative Analysis 2. Quantitative Analysis 3. Hybrid Analysis
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Qualitative Analysis (1)
(Technique for Discrete Space Location Problems)
It usually using scoring method (subjective decision making tools), consists of these steps:
Step 1 : Step 2 : List all the factors that are important that have impact on the location decision Assign an appropriate weight (typically between 0 and 1) to each factor based on the relative importance of each Assign a score (typically between 0 and 100) to each location with respect to each factor identified in step 1
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Step 3 :
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Qualitative Analysis (2)
(Technique for Discrete Space Location Problems)
Step 4 :
Compute the weighted score for each factor for each location by multiplying its weight by the corresponding score Compute the sum of the weighted scores for each a location based on these scores
Step 5 :
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Qualitative Analysis (3)
(Technique for Discrete Space Location Problems)
A payroll processing company has recently won several major contracts in the midwest region of the U.S. and central Canada and wants to open a new, large facility to serve these areas. Because customer service is so importance, the company wants to be as near its customers as possible. Preliminary investigation has shown that Minneapolis, Winnipeg, and Springfield, Illinois, would be the three most desirable locations and the payroll company has to select one of these three. A subsequent thorough investigation of each location with respect to eighnt important factors has generated the raw scores and weights listed in table 2. Using the location scoring method, determine the best location for the new payroll processing facility.
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Qualitative Analysis (4)
(Technique for Discrete Space Location Problems)
Solution
Steps 1, 2, and 3 have already been completed for us. We now need to compute the weighted score for each location-factor pair (Step 4), and these weighted scores and determine the location based on these scores (Step 5).
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Qualitative Analysis (5)
(Technique for Discrete Space Location Problems)
Wt. Factors .25 .15 .15 .10 .10 .10 .08 .07 Proximity to customers Land/construction prices Wage rates Property taxes Business taxes Commercial travel Insurance costs Office services
Location Minn. Winn. Spring. 95 90 65 60 60 90 70 45 60 70 90 70 80 90 85 80 65 75 70 95 60 90 90 80
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Qualitative Analysis (6)
From the analysis in above Table, it is clear that Minneapolis would be the best location based on the subjective information.
(Technique for Discrete Space Location Problems)
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Qualitative Analysis (7)
(Technique for Discrete Space Location Problems)
Wt. Factors .25 .15 .15 .10 .10 .10 .08 .07 Proximity to customers Land/construction prices Wage rates Property taxes Business taxes Commercial travel Insurance costs Office services Sum of weighted scores
Location Minn. Winn. Spring. 23.75 22.5 16.25 9 9 13.5 10.5 6.75 9 7 9 8.5 8 9 8.5 8 6.5 7.5 5.6 7.6 4.8 6.3 6.3 5.6 78.15 76.65 72.15
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Qualitative Analysis (8)
(Technique for Discrete Space Location Problems)
Of course, as mentioned before, objective measures must be brought into consideration especially because the weighted scores for Minneapolis and Winnipeg are close.
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Quantitative Analysis
(Technique for Discrete Space Location Problems)
It is appropriate for a specific set of objectives and constraints : Minimax location model is approriate for determining the location of an emergency service facility, where the objective is to minimize the maximum distance traveled between the facility and any customer Transportation model is approriate for determining the location of facility, where the objective is to minimize total distance traveled between the facility and any customer
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Minimax Location Model (1)
The minimax location problem is given by: Min f ( X ) = max[ ( x ai + y bi ), i = 1,2, , m] In order to obtain the minimax solution, let c1 = minimum (ai+bi) c2 = maximum (ai+bi) c3 = minimum (-ai+bi) c4 = maximum (-ai+bi) c5 = maximum (c2c1, c4c3)
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Minimax Location Model (2)
Optimum solutions to the minimax location problem can be shown to be all points on the line segment connecting the point
( x , y ) = 0.5( c c , c + c
1 1 1 3 1
3
+ c5 )
and the point
( x , y ) = 0.5( c
2 2
2
c4 , c2 + c4 c5 )
The maximum distance will be equal to c5/2
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Minimax Location Model (3)
Example :
Consider the problem of locating a maintenance department in a production area. It is desirable to locate the maintenance facility as close to each machine as possible, in order to minimize machine downtime. Eight machines are to be maintained by crews from the central maintenance facility. The coordinate locations of the machine are (0,0), (4,6), (8,2), (10,4), (4,8), (2,4), (6,4), and (8,8).
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Minimax Location Model (4)
i
1 2 3 4 5 6 7 8
ai
0 4 8 10 4 2 6 8
bi
0 6 2 4 8 4 4 8
ai + bi
0 10 10 14 12 6 10 16
-ai + bi
0 2 -6 -6 4 2 -2 0
c1=0
c2=16
c3=-6
c4=4
c5=16
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Minimax Location Model (5)
The optimum solutions lie on the line segment connecting the point
( x , y ) = 0.5( 6, 10) = (3, 5)
1 1
and the point
( x , y ) = 0.5(12, 4) = (6, 2)
2 2
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Minimax Location Model (6)
10
8
P5
P8
6
P2
4
P6
P7
P4
x*
2 P3
2
4
6
8
10
12
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Transportation Model (1)
It is mathematical model (linear programming) that can be solved using: 1. Optimization methods Manual: Simplex Algorithm Software: QSB, LINDO, LINGO, GAMS 1. Heuristic methods Least cost assignment routine method Northwest corner rule method Vogel approximation method
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Transportation Model (2)
Minimize Total Transportation Cost Z = cij xij
i =1 j =1 m n
Subject to
x
j =1 m
n
ij
ai , i = 1,2 ,...,m (supply restriction at warehouse i ) b j , j = 1,2 ,...,n (demand requirement at market j )
x
i =1
ij
xij 0, i,j = 1,2 ,...,n (non - negativity restrictions)
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Transportation Model (3)
Parameter cij : ai : bi : cost of transporting one unit from warehouse i to customer j supply capacity at warehouse i demand at customer j
Decision Variables xij : number of units transported from
warehouse i to customer j
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Transportation Model (4)
Seers Inc. has two manufacturing plants at Albany and Little Rock supplying Canmore brand refrigerators to four distribution centers in Boston, Philadelphia, Galveston and Raleigh. Due to an increase in demand of this brand of refrigerators that is expected to last for several years into the future, Seers Inc., has decided to build another plant in Atlanta or Pittsburgh. The expected demand at the three distribution centers and the maximum capacity at the Albany and Little Rock plants are given in Table 4. Determine which of the two locations, Atlanta or Pittsburgh, is suitable for the new plant. Seers Inc., wishes to utilize all of the capacity available at its Albany and Little Rock Locations
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Transportation Model (5)
Bost. Albany Little Rock Atlanta Pittsburgh Demand 10 19 21 17 200 Phil. 15 15 11 8 100 Galv. 22 10 13 18 300 Rale. 20 9 6 12 280 Supply Capacity 250 300 No limit No limit
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Transportation Model (6)
Bost. Albany Little Rock Atlanta Demand 10 19 21 200 Phil. 15 15 11 100 Galv. 22 10 13 300 Rale. 20 9 6 280 Supply Capacity 250 300 330 880
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Transportation Model (7)
Bost. Phil. Galv. Rale. Albany Little Rock Pittsburgh Demand 10 19 17 200 15 15 8 100 22 10 18 300 20 9 12 280 Supply Capacity 250 300 330 880
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Hybrid Analysis
(Technique for Discrete Space Location Problems)
It a is method that incorporates subjective (qualitative) as well as quantitative cost and other factors 1. Brown Gibson Model (1972) 2. Buffa Sarin Model (1987) This model classifies the objective and subjective factors important to the specific location being addressed as: 1. Critical, 2. Objective, 3. Subjective
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Buffa Sarin Model (1)
After the factors are classified, they are assigned numerical values: CFij OFij SFij wj 1 if location i satisfies critical factor j 0 otherwise cost of objective factor j at location i numeric value assigned (on a scale of 0-1) to subjective factor j for location i weight assigned to subjective factor j (0 wj 1)
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Buffa Sarin Model (2)
Determine the overal critical factor measure (CFMi), objective factor measure (OFMi), and subjective factor measure (SFMi) for each location i with these equations:
CFM i = CFi1 CFi 2 ... CFip = CFij
j =1 p
i = 1,2, ... ,m
q q max OFij OFij i j =1 j =1 OFM i = q q max OFij min OFij i i j =1 j =1
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i = 1,2 , ... ,m
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Buffa Sarin Model (3)
SFM i = w j SFij
j =1 r
i = 1,2, ... , m
The location measure LMi for each location is then calculated as: LM i = CFM [ OFM i + (1 ) SFM i ] where is the weight assigned to the objective factor measure
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Buffa Sarin Model (4)
Example
Mole-Sun Brewing company is evaluating six candidate locationsMontreal, Plattsburgh, Ottawa, Albany, Rochester and Kingston, for constructing a new brewery. There are two critical, three objective and four subjective factors that management wishes to incorporate in its decision-making. These factors are summarized in Table following. The weights of the subjective factors are also provided in the table. Determine the best location if the subjective factors are to be weighted 50 percent more than the objective factors.
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Buffa Sarin Model (5)
Location Water Supply Albany Kingston Montreal Ottawa Plattsburgh Rochester 0 1 1 1 1 1 Factors Critical Tax Incentives 1 1 1 0 1 1
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Buffa Sarin Model (6)
Location Critical Revenue Albany Kingston Montreal Ottawa Plattsburgh Rochester 185 150 170 200 140 150 Factors Objective Labor Cost 80 100 90 100 75 75 Energy Cost 10 15 13 15 8 11
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Buffa Sarin Model (7)
Location Objective Factors Subjective Community Attitude 0.3 Albany Kingston Montreal Ottawa Plattsburgh Rochester 0.5 0.6 0.4 0.5 0.9 0.7 Ease of Transportation 0.4 0.9 0.7 0.8 0.4 0.9 0.65
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Buffa Sarin Model (8)
Location Factors Subjective Labor Unionization 0.25 Albany Kingston Montreal Ottawa Plattsburgh Rochester 0.6 0.7 0.2 0.4 0.9 0.4 Support Services 0.05 0.7 0.75 0.8 0.8 0.55 0.8
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Buffa Sarin Model (9)
Location Critical Factors Objective Sum of Objective Factors -95 -35 -67 -85 -57 -64 Subjective SFMi 0.7 0.67 0.53 0.45 0.88 0.61 LMi 0 0.4 0.53 0 0.68 0.56
Albany Kingston Montreal Ottawa Plattsburgh Rochester
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Brown Gibson Model
See Tata Letak Pabrik dan Pemindahan Bahan, Wignjosoebroto, S
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Techniques for Continuous Space Location Problems
1. Median Method 2. Contour Line Method 3. Gravity Method 4. Weiszfeld Method
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Median Method (1)
(Technique for Continuous Space Location Problems)
The median method finds the median location (defined later) and assign the new facility to it. Interaction between the new facility and existing ones is known Problem is to minimize the total interaction cost between each existing facility and the new one.
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Median Method (2)
(Technique for Continuous Space Location Problems)
Notation :
ci fi xi , yj cost of transportation between existing facility i and new facility, per unit traffic flow between existing facility i and new facility coordinates of existing facility i
Model :
Minimize TC = ci f i [ xi x + yi y ]
m i =1
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Median Method (3)
(Technique for Continuous Space Location Problems)
Model can be solved using : 1. Algorithm 2. Equivalent linear-constrained model
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Median Method (4)
(Technique for Continuous Space Location Problems)
Algorithm
Step 1 : List the existing facilities in nondecreasing order of the x coordinates Step 2 : Find the jth x coordinate in the list (created in step 1) at which the cummulative weight equals or exceeds half the total weight for the first time
w <
i =1 i
j 1
m
i =1
wi and 2
w
i =1 i
j
m
i =1
wi 2
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Median Method (5)
(Technique for Continuous Space Location Problems)
Step 3 : List the existing facilities in nondecreasing order of the y coordinates Step 4 : Find the kth y coordinate in the list (created in step 3) at which the cummulative weight equals or exceeds half the total weight for the first time
wi wi < 2 and i =1 i =1
k 1
m
wi wi 2 i =1 i =1
k
m
The optimal location of the new facility is given by the jth x coordinate and the kth y coordinate in step 2 and 4, respectively
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Median Method (6)
(Technique for Continuous Space Location Problems)
Example : Two high-speed copiers are to be located on the fifth floor of an office complex that houses four departments of the Social Security Administration. The coordinates of the centroid of each department as well as the average number of trips made per day between each department and the copiers yet-to-bedetermined location are known and given in Table. Assume the travel originates and ends at the centroid of each department. Determine the optimal location the x,y coordinates for the copiers
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Median Method (7)
(Technique for Continuous Space Location Problems)
Department x Coordinate y Coordinate Number 1 2 3 4 10 10 8 12 2 10 6 5
Average Number of Daily Trips to Copiers 6 10 8 4
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Median Method (8)
(Technique for Continuous Space Location Problems)
Department Number
x Coordinates in Nondecreasing Order 8 10 10 12
Weights
Cumulative Weights
3 1 2 4
8 6 10 4
8 14 24 28
Cumulative Weights = half of the total weights (28/2=14)
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Median Method (9)
(Technique for Continuous Space Location Problems)
Department Number
y Coordinates in Nondecreasing Order 2 5 6 10
Weights
Cumulative Weights
1 4 3 2
6 4 8 10
6 10 18 28
Cumulative Weights > half of the total weights (28/2=14)
Thus the optimal coordinates of the new facility are (10 , 6)
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Median Method (10)
(Technique for Continuous Space Location Problems)
Transforming the nonlinear-unconstrained model into an equivalent linear-constrained model Consider the following notation:
( xi x ) if ( xi x ) > 0 x = otherwise 0
+ i
( x xi ) if ( xi x ) 0 x = otherwise 0
i
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Median Method (11)
(Technique for Continuous Space Location Problems)
We can observe that
xi x = x + x
+ i
i
xi x = xi+ xi
A similar definition,
yi y = y + y yi y = y y
+ i
+ i
i
i
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Median Method (12)
(Technique for Continuous Space Location Problems)
Thus the transformed linear model is:
Min
Subject to
w (x
n i =1 i
+ i
+x +y +y
i
+ i
i
)
xi x = xi+ xi yi y = y y
+ i i
i = 1,2, , n i = 1,2, , n i = 1,2, , n
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xi+ , xi , yi+ , yi 0
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x , y unrestricted in sign
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Median Method (13)
(Technique for Continuous Space Location Problems)
Solve the problem using LINDO
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Median Method (14)
(Technique for Continuous Space Location Problems)
Solve the problem using LINDO
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Contour Line Method (1)
(Technique for Continuous Space Location Problems)
Step 1 : Draw a vertical line through the x coordinate and a horizontal line through the y coordinate of each facility Step 2 : Label each vertical line Vi, i=1, 2, ..., p and horizontal line Hj, j=1, 2, ..., q where Vi= the sum of weights of facilities whose x coordinates fall on vertical line i and where Hj= sum of weights of facilities whose y coordinates fall on horizontal line j
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Contour Line Method (2)
(Technique for Continuous Space Location Problems)
Step 3 : Set i = j = 1; N0 = D0 =
w i=1
m
i
Step 4 : Set Ni = Ni-1 + 2Vi and Dj = Dj-1 + 2Hj. Increment i = i + 1 and j = j + 1 Step 5 : If i < p or j < q, go to Step 4. Otherwise, set i = j = 0 and determine Sij, the slope of contour lines through the region bounded by vertical lines i and i + 1 and horizontal line j and j + 1 using the equation Sij = Ni/Dj. Increment i = i + 1 and j = j + 1
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Contour Line Method (3)
(Technique for Continuous Space Location Problems)
Step 6 :
Step 7 :
If i < p or j < q, go to Step 5. Otherwise select any point (x, y) and draw a contour line with slope Sij in the region [i, j] in which (x, y) appears so that the line touches the boundary of this line. From one of the end points of this line, draw another contour line through the adjacent region with the corresponding slope Repeat this until you get a contour line ending at point (x, y). We now have a region bounded by contour lines with (x, y) on the boundary of the region
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TeknikIndustri
Contour Line Method (4)
(Technique for Continuous Space Location Problems)
The number of vertical and horizontal lines need not be equal The Ni and Dj as computed in Steps 3 and 4 correspond to the numerator and denominator, respectively of the slope equation of any contour line through the region bounded by the vertical lines i and i + 1 and horizontal lines j and j + 1
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TeknikIndustri
Contour Line Method (5)
(Technique for Continuous Space Location Problems)
Example
Consider Example 4. Suppose that the weight of facility 2 is not 10, but 20. Applying the median method, it can be verified that the optimal location is (10, 10) - the centroid of department 2, where immovable structures exist. It is now desired to find a feasible and near-optimal location using the contour line method.
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TeknikIndustri
Contour Line Method (6)
(Technique for Continuous Space Location Problems)
Solution
The contour line method is illustrated using Figure 1
Step 1: The vertical and horizontal lines V1, V2, V3 and H1, H2, H3, H4 are drawn as shown. In addition to these lines, we also draw line V0, V4 and H0, H5 so that the exterior regions can be identified Step 2: The weights V1, V2, V2, H1, H2, H2, H4 are calculated by adding the weights of the points that fall on the respective lines. Note that for this example, p=3, and q=4
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TeknikIndustri
Contour Line Method (7)
(Technique for Continuous Space Location Problems)
Step 3 : Since wi = 38
i =1
4
Set N0 = D0 = 38 Step 4 : Set N1 = -38 + 2(8) = -22; -26; N2 = -22 + 2(26) = 30; -18; N3 = 30 + 2(4) = 38; D1 = -38 + 2(6) = D2 = -26 + 2(4) = D3 = -18 + 2(8) = -2; D4 = -2 + 2(20) = 38;
85
(These valuesI-1504/Facilities Planning/2007/#2 bottom of each are entered at the R
TeknikIndustri
Contour Line Method (8)
(Technique for Continuous Space Location Problems)
Step 5 : Compute the slope of region S00 = -(-38/-38) = -1; S01 = -(-38/-26) = -1.46; S02 = -(-38/-18) = -2.11; S03 = -(-38/-2) = -19; S04 = -(-38/38) = 1; S10 = -(-22/-38) = -0.58; S11 = -(-22/-26) = -0.85; S12 = -(-22/-18) = -1.22; S13 = -(-22/-2) = -11; S14 = -(-22/38) = 0.58; S20 = -(30/-38) = 0.79; S21 = -(30/-26) = 1.15; S22 = -(30/-18) = 1.67; S23 = -(30/-2) = 15; S24 = -(30/38) = -0.79; S30 = -(38/-38) = 1; S31 = -(38/-26) = 1.46; S32 = -(38/-18) = 2.11;
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TeknikIndustri
Contour Line Method (9)
(Technique for Continuous Space Location Problems)
Step 5 : Compute the slope of region (cont) S33 = -(38/-2) = 19; S34 = -(38/38) = -1;
(The above slope values are shown inside each region.)
Step 6 : When we draw contour lines through point (9, 10), we get the region shown in figure 1.
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Contour Line Method (10)
(Technique for Continuous Space Location Problems)
Example:
Since the copiers cannot be placed at the (10, 10) location, we drew contour lines through another nearby point (9, 10). Locating anywhere possible within this region give us a feasible, near-optimal solution.
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Contour Line Method (11)
(Technique for Continuous Space Location Problems)
TeknikIndustri
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TeknikIndustri
Gravity Method (1)
The cost function is
Minimize TC =
ci f i ( xi x ) 2 + ( yi y ) 2
i =1
m
[
]
As before, we substitute wi = ci fi, i = 1, 2, ..., m and rewrite the objective function as
Minimize TC =
wi ( xi x ) 2 + wi ( yi y ) 2
i =1 i =1
90
m
m
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TeknikIndustri
Gravity Method (2)
Since the objective function can be shown to be convex, partially differentiating TC with respect to x and y, setting the resulting two equations to 0 and solving for x, y provides the optimal location of the new facility
m m TC = 2 wi x 2 wi xi = 0 x i =1 i =1
x = wi xi
i =1
m
w
i =1
m
i
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Gravity Method (3)
Similarly,
m m TC = 2 wi y 2 wi yi = 0 y i =1 i =1
y = wi yi
i =1
m
w
i =1
m
i
Thus, the optimal locations x and y are simply the weighted averages of the x and y coordinates of the existing facilities
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Gravity Method (4)
Example:
Consider Example 4. Suppose the distance metric to be used is squared Euclidean. Determine the optimal location of the new facility using the gravity method.
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Gravity Method (5)
Department i 1 2 3 4 Total We conclude that : xi 10 10 8 12 yi 2 10 6 5 wi 6 10 8 4 28 wixi 60 100 64 48 wiyi 12 100 48 20
272 180
x = 272
28
= 9.7 and y = 180
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28
= 6.4
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Gravity Method (6)
If this location is not feasible, we only need to find another point which has the nearest Euclidean distance to (9.7, 6.4) and is a feasible location for the new facility and locate the copiers there. Another way, we can again draw contour lines from neighboring points to find a feasible, near-optimal location
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Weiszfeld Method (1)
The objective function for the single facility location problem with Euclidean distance can be written as:
Minimize TC = ci f i ( xi x ) 2 + ( yi y ) 2
i =1
m
As before, substituting wi=ci fi and taking the derivative of TC with respect to x and y yields
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Weiszfeld Method (2)
Although the Weiszfeld method is theoritically suboptimal, it provides x, y values that are very close to optimal For practical purposes the algorithm works very well and can be readily implemented on a spreadsheet If the optimal location is not feasible, use the contour line method to draw contour lines and then choose a suitable, feasible, near-optimal location for the new facility
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Weiszfeld Method (3)
wi [ 2( xi x )] TC 1 m = x 2 i =1 ( xi x ) 2 + ( yi y ) 2 =
i =1
m
wi xi ( xi x ) 2 + ( yi y ) 2
i =1
m
wi x ( xi x ) 2 + ( yi y ) 2
=0
x =
i =1 m
m
wi xi ( xi x ) 2 + ( yi y ) 2 wi ( xi x ) 2 + ( yi y ) 2
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i =1
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Weiszfeld Method (4)
wi [ 2( yi y )] TC 1m = y 2 i =1 ( xi x ) 2 + ( yi y ) 2 =
i =1
m
wi yi ( xi x ) 2 + ( yi y ) 2
i =1
m
wi y ( xi x ) 2 + ( yi y ) 2
=0
y=
m
wi yi
i =1 m
( xi x ) 2 + ( yi y ) 2 wi ( xi x ) 2 + ( yi y ) 2
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Weiszfeld Method (5)
Step 0 : Set iteration counter k=1
xk =
w x
i =1 m
m
ii
w
i =1
;
yk =
w y
i =1 m i
m
i
i
w
i =1
i
Step 1 : Set
x k +1 =
i =1 m
m
wi xi ( xi x ) + ( yi y ) 2 wi
k
2
k
i =1
( xi x k ) 2 + ( yi y k ) 2
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TeknikIndustri
Weiszfeld Method (6)
x
k +1
=
i =1 m
m
wi xi ( xi x k ) 2 + ( yi y k ) 2 wi ( xi x k ) 2 + ( yi y k ) 2
i =1
Step 2 : If xk+1 xk and yk+1 yk, Stop. Otherwise, set k = k + 1 and go to Step 1
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Weiszfeld Method (7)
Example:
Consider Example 5. Assuming the distance metric to be used is Euclidean, determine the optimal location of the new facility using the Weiszfeld method. Data for this problem is shown in Table below
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Weiszfeld Method (8)
Departements # 1 2 3 4 xi 10 10 8 12 yi 20 10 6 5 wi 6 20 8 4
Using the gravity method, the initial seed can be shown to be (9.8, 7.4). With this as the starting solution, we can apply Step 1 of the Weiszfeld method repeatedly until we find that two consecutive x, y values are equal.
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Weiszfeld Method (9)
Optimal location
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Facility Location Case Study (1)
TeknikIndustri
A small manufacturing company currently located in a university tech park has witnessed major growth since introducing an innovative technology into the marketplace. Its owner now wants to find a new location and build a bigger facility. In January she hired senior industrial and management engineering (IME) students at the university to investigate several potential locations and select the one that est suits her needs. The student group adopted the following five-step approach, which based on the hybrid analysis discussed earlier. (see Heragu, pp 546 551)
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Facility Location Case Study (2)
TeknikIndustri
Step 1 : determine of requirements (the students conducted interviews with the owner and facility manager to determine these company-spesific requirements for the new facility) Step 2 : classification of location factors (the requirements classified into three categories) Step 3 : data collection (this step requires the most time, but it is very important and should be done carefull) Step 4 : elimination of sites not meeting critical objectives and development of a rating chart Step 5 : site visits and site evaluation
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Selesai
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