S.S. Heragu Industrial Engineering Department University of Louisville

S.S. Heragu Industrial Engineering Department University of Louisville
Facilities Design S.S. Heragu Industrial Engineering Department University of Louisville

Chapter 11 Basic Models for the Location Problem

Outline 11.1 Introduction 11.2 Important Factors in Location Decisions
11.3 Techniques for Discrete Space Location Problems Qualitative Analysis Quantitative Analysis Hybrid Analysis

Outline Cont... 11.4 Techniques for Continuous Space Location Problems
Median Method Contour Line Method Gravity Method Weiszfeld Method 11.5 Facility Location Case Study 11.6 Summary 11.7 Review Questions and Exercises 11.8 References

McDonald’s QSCV Philosophy
11,000 restaurants (7,000 in USA, remaining in 50 countries) 700 seat McDonald’s in Pushkin Square, Moscow $60 million food plant combining a bakery, lettuce plant, meat plant, chicken plant, fish plant and a distribution center, each owned and operated independently at same location

McDonald’s cont... Food taste must be the same at any McDonald, yet food must be secured locally Strong logistical chain, with no weak links between Close monitoring for logistical performance 300 in Australia Central distribution since 1974 with the help of F.J. Walker Foods in Sydney Then distribution centers opened in several cities

McDonald’s cont... 2000 ingredients, from 48 food plants, shipment of 200 finished products from suppliers to DC’s, 6 million cases of food and paper products plus 500 operating items to restaurants across Australia Delivery of frozen, dry and chilled foods twice a week to each of the 300 restaurants 98% of the time within 15 minutes of promised delivery time, 99.8% within 2 days of order placement No stockouts, but less inventory

Entities in a Supply Chain
Supplier Manufacturing Plant Raw Material(s) Assembly Plant Central Distribution Center(s) Regional Distribution Center(s) Retail Outlets

Introduction Design and Operation of a Supply chain Warehousing
Distribution Channels Freight Transportation Freight Consolidation Transportation Modes

Introduction Logistics management can be defined as the management of transportation and distribution of goods. facility location transportation goods handling and storage.

Introduction Cont... Some of the objectives in facility location decisions: (1) It must first be close as possible to raw material sources and customers; (2) Skilled labor must be readily available in the vicinity of a facility’s location; (3) Taxes, property insurance, construction and land prices must not be too “high;” (4) Utilities must be readily available at a “reasonable” price;

Introduction Cont... (5) Local , state and other government regulations must be conducive to business; and (6) Business climate must be favorable and the community must have adequate support services and facilities such as schools, hospitals and libraries, which are important to employees and their families.

Introduction Cont... Logistics management problems can be classified as: (1) location problems; (2) allocation problems; and (3) location-allocation problems.

List of Factors Affecting Location Decisions
Proximity to raw materials sources Cost and availability of energy/utilities Cost, availability, skill and productivity of labor Government regulations at the federal, state, country and local levels Taxes at the federal, state, county and local levels Insurance Construction costs, land price

List of Factors Affecting Location Decisions Cont...
Government and political stability Exchange rate fluctuation Export, import regulations, duties, and tariffs Transportation system Technical expertise Environmental regulations at the federal, state, county and local levels Support services

List of Factors Affecting Location Decisions Cont...
Community services, i.e. schools, hospitals, recreation, etc. Weather Proximity to customers Business climate Competition-related factors

11.2 Important Factors in Location Decisions
International National State-wide Community-wide

Qualitative Analysis Step 1: List all the factors that are important, i.e. have an impact on the location decision. Step 2: Assign appropriate weights (typically between 0 and 1) to each factor based on the relative importance of each. Step 3: Assign a score (typically between 0 and 100) for each location with respect to each factor identified in Step 1.

Qualitative Analysis Step 4: Compute the weighted score for each factor for each location by multiplying its weight with the corresponding score (which were assigned Steps 2 and 3, respectively) Step 5: Compute the sum of the weighted scores for each location and choose a location based on these scores.

Example 1: 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. Since customer service is of utmost importance, the company wants to be as near it’s “customers” as possible. Preliminary investigation has shown that Minneapolis, Winnipeg, and Springfield, Ill., would be the three most desirable locations and the payroll company has to select one of these three.

Example 1: Cont... A subsequent thorough investigation of each location with respect to eight 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.

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).

Table 11.2. Factors and Weights for Three Locations
Wt. Factors Location Minn.Winn.Spring. .25 Proximity to customers .15 Land/construction prices .15 Wage rates .10 Property taxes .10 Business taxes .10 Commercial travel

Table 11.2. Cont... Wt. Factors Location Minn. Winn. Spring.
.08 Insurance costs .07 Office services Click here

Solution: Cont... From the analysis in Table 3, it is clear that Minneapolis would be the best location based on the subjective information.

Table 11.3. Weighted Scores for the Three Locations in Table 11.2
Weighted Score Location Minn. Winn. Spring. Proximity to customers Land/construction prices Wage rates Property taxes Business taxes

Table 11.3. Cont... Weighted Score Location Minn. Winn. Spring.
Commercial travel Insurance costs Office services

Solution: Cont... Of course, as mentioned before, objective measures must be brought into consideration especially because the weighted scores for Minneapolis and Winnipeg are close.

11.3.2 Quantitative Analysis

General Transportation Model

Parameters cij: cost of transporting one unit from warehouse i to customer j ai: supply capacity at warehouse i bi: demand at customer j Decision Variables xij: number of units transported from warehouse i to customer j

Transportation Simplex Algorithm
Step 1: Check whether the transportation problem is balanced or unbalanced. If balanced, go to step 2. Otherwise, transform the unbalanced transportation problem into a balanced one by adding a dummy plant (if the total demand exceeds the total supply) or a dummy warehouse (if the total supply exceeds the total demand) with a capacity or demand equal to the excess demand or excess supply, respectively. Transform all the > and < constraints to equalities. Step 2: Set up a transportation tableau by creating a row corresponding to each plant including the dummy plant and a column corresponding to each warehouse including the dummy warehouse. Enter the cost of transporting a unit from each plant to each warehouse (cij) in the corresponding cell (i,j). Enter 0 cost for all the cells in the dummy row or column. Enter the supply capacity of each plant at the end of the corresponding row and the demand at each warehouse at the bottom of the corresponding column. Set m and n equal to the number of rows and columns, respectively and all xij=0, i=1,2,...,m; and j=1,2,...,n. Step 3: Construct a basic feasible solution using the Northwest corner method.

Transportation Simplex Algorithm
Step 4: Set u1=0 and find vj, j=1,2,...,n and ui, i=1,2,...,n using the formula ui + vj = cij for all basic variables. Step 5: If ui + vj - cij < 0 for all nonbasic variables, then the current basic feasible solution is optimal; stop. Otherwise, go to step 6. Step 6: Select the variable xi*j* with the most positive value ui* + vj*- cij*. Construct a closed loop consisting of horizontal and vertical segments connecting the corresponding cell in row i* and column j* to other basic variables. Adjust the values of the basic variables in this closed loop so that the supply and demand constraints of each row and column are satisfied and the maximum possible value is added to the cell in row i* and column j*. The variable xi*j* is now a basic variable and the basic variable in the closed loop which now takes on a value of 0 is a nonbasic variable. Go to step 4.

Example 2: 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. The expected demand at the three distribution centers and the maximum capacity at the Albany and Little Rock plants are given in Table 4.

Table 11.4. Costs, Demand and Supply Information
Bost. Phil. Galv. Rale. Supply Capacity Albany Little Rock Atlanta No limit Demand

Table 11.5. Transportation Model with Plant at Atlanta
Bost. Phil. Galv. Rale. Supply Capacity Albany Little Rock Atlanta Demand Click here for Excel formulation Click here for LINGO formulation

Example 3 Consider Example 2. In addition to Atlanta, suppose Seers, Inc., is considering another location – Pittsburgh. 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 it’s Albany and Little Rock Locations

Table 11.10. Costs, Demand and Supply Information
Bost. Phil. Galv. Rale. Supply Capacity Albany Little Rock Atlanta Pittsburgh Demand

Table 11.12. Transportation Model with Plant at Pittsburgh
Bost. Phil. Galv. Rale. Supply Capacity Albany Little Rock Pittsburgh Demand Click here for Excel model Click here for LINDO Model Click here for LINGO Model

Min/Max Location Problem:
d11 d12 d1n d21 d22 d2n Site dm1 dm2 dmn

Hybrid Analysis Critical Objective Subjective

Hybrid Analysis Cont... CFij = 1 if location i satisfies critical factor j, 0 otherwise OFij = cost of objective factor j at location i SFij = numerical value assigned (on scale of 0-100) to subjective factor j for location i wj = weight assigned to subjective factor (0< w < 1)

Hybrid Analysis Cont...

Hybrid Analysis Cont... The location measure LMi for each location is then calculated as: LMi = CFMi [  OFMi + (1- ) SFMi ] Where  is the weight assigned to the objective factor. We then choose the location with the highest location measure LMi

Example 4: Mole-Sun Brewing company is evaluating six candidate locations-Montreal, 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 7. 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.

Table 11.13: Critical, Subjective and Objective Factor Ratings for six locations for Mole-Sun Brewing Company, Inc.

Table 11.13 Cont... Location Albany 0 1 Kingston 1 1 Montreal 1 1
Ottawa 1 0 Plattsburgh 1 1 Rochester 1 1 Factors Critical Water Supply Tax Incentives

Table 11.13 Cont... Location Albany 185 80 10 Kingston 150 100 15
Montreal Ottawa Plattsburgh Rochester Factors Critical Objective Revenue Labor Cost Energy Cost

Table 11.13 Cont... Location 0.3 0.4 Albany 0.5 0.9 Kingston 0.6 0.7
Albany Kingston Montreal Ottawa Plattsburgh Rochester Factors Subjective Community Attitude Ease of Transportation

Table 11.13 Cont... Location 0.25 0.05 Albany 0.6 0.7
Albany Kingston Montreal Ottawa Plattsburgh Rochester Factors Subjective Labor Unionization Support Services

Table 11. 14 Location Analysis of Mole-Sun Brewing Company, Inc
Table Location Analysis of Mole-Sun Brewing Company, Inc., Using Hybrid Method

Table 11.14 Cont... Location Albany -95 0.7 0 Kingston -35 0.67 0.4
Montreal Ottawa Plattsburgh Rochester Factors Critical Objective Subjective LMi Sum of Obj. Factors SFMi

11.4 Techniques For Continuous Space Location Problems

11.4.1 Model for Rectilinear Metric Problem
Consider the following notation: fi = Traffic flow between new facility and existing facility i ci = Cost of transportation between new facility and existing facility i per unit xi, yi = Coordinate points of existing facility i

Model for Rectilinear Metric Problem (Cont)
The median location model is then to minimize: Where TC is the total distribution cost

Model for Rectilinear Metric Problem (Cont)
Since the cifi product is known for each facility, it can be thought of as a weight wi corresponding to facility i.

Median Method: Step 1: List the existing facilities in non-decreasing order of the x coordinates. Step 2: Find the jth x coordinate in the list at which the cumulative weight equals or exceeds half the total weight for the first time, i.e.,

Median Method (Cont) Step 3: List the existing facilities in non-decreasing order of the y coordinates. Step 4: Find the kth y coordinate in the list (created in Step 3) at which the cumulative weight equals or exceeds half the total weight for the first time, i.e.,

Median Method (Cont) Step 4: Cont... The optimal location of the new facility is given by the jth x coordinate and the kth y coordinate identified in Steps 2 and 4, respectively.

Notes 1. It can be shown that any other x or y coordinate will not be that of the optimal location’s coordinates 2. The algorithm determines the x and y coordinates of the facility’s optimal location separately 3. These coordinates could coincide with the x and y coordinates of two different existing facilities or possibly one existing facility

Example 5: Two high speed copiers are to be located in the fifth floor of an office complex which houses four departments of the Social Security Administration. 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-be-determined location are known and given in Table 9 below. Assume that travel originates and ends at the centroid of each department. Determine the optimal location, i.e., x, y coordinates, for the copiers.

Table 11.15 Centroid Coordinates and Average Number of Trips to Copiers

Table 11.15 Dept. Coordinates Average number of
# x y daily trips to copiers

Solution: Using the median method, we obtain the following solution:
Step 1: Dept. x coordinates in Weights Cumulative # non-decreasing order Weights

Solution: Step 2: Since the second x coordinate, namely 10, in the above list is where the cumulative weight equals half the total weight of 28/2 = 14, the optimal x coordinate is 10.

Solution: Step 3: Dept. y coordinates in Weights Cumulative
# non-decreasing order Weights

Solution: Step 4: Since the third y coordinates in the above list is where the cumulative weight exceeds half the total weight of 28/2 = 14, the optimal y coordinate is 6. Thus, the optimal coordinates of the new facility are (10, 6).

Equivalent Linear Model for the Rectilinear Distance, Single-Facility Location Problem
Parameters fi = Traffic flow between new facility and existing facility i ci = Unit transportation cost between new facility and existing facility i xi, yi = Coordinate points of existing facility i Decision Variables x, y = Optimal coordinates of the new facility TC = Total distribution cost

The median location model is then to

Since the cifi product is known for each facility, it can be thought of as a weight wi corresponding to facility i. The previous equation can now be rewritten as follows

Contour Line Method

Algorithm for Drawing Contour Lines:
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

Algorithm for Drawing Contour Lines (Cont)
 Step 3: Set i = j = 1; N0 = D0 = wi 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 i=1

Algorithm for Drawing Contour Lines:
Step 6: 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 Step 7: 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

Notes on Algorithm for Drawing Contour Lines
1. The number of vertical and horizontal lines need not be equal 2. 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

Notes on Algorithm for Drawing Contour Lines (Cont)

By noting that the Vi’s and Hj’s calculated in Step 2 of the algorithm correspond to the sum of the weights of facilities whose x, y coordinates are equal to the x, y coordinates, respectively of the ith, jth distinct lines and that we have p, q such coordinates or lines (p < m, q < m), the previous equation can be written as follows

Suppose that x is between the sth and s+1th (distinct) x coordinates or vertical lines (since we have drawn vertical lines through these coordinates in Step 1). Similarly, let y be between the tth and t+1th vertical lines. Then

Rearranging the variable and constant terms in the above equation, we get

The last four terms in the previous equation can be substituted by another constant term c and the coefficients of x can be rewritten as follows Notice that we have only added and subtracted the term

Since it is clear from Step 2 that the coefficient of x can be rewritten as Similarly, the coefficient of y is

The Ni computation in Step 4 is in fact calculation of the coefficient of x as shown above. Note that Ni=Ni-1+2Vi. Making the substitution for Ni-1, we get Ni=Ni-2+2Vi-1+2Vi Repeating the same procedure of making substitutions for Ni-2, Ni-3, ..., we get Ni=N0+2V1+2V Vi-1+2V1=

Similarly, it can be verified that

The above expression for the total cost function at x, y or in fact, any other point in the region [s, t] has the form y= mx + c, where the slope m = -Ns/Dt. This is exactly how the slopes are computed in Step 5 of the algorithm

3. The lines V0, Vp+1 and H0, Hq+1 are required for defining the “exterior” regions [0, j], [p, j], j = 1, 2, ..., p, respectively) 4. Once we have determined the slopes of all regions, the user may choose any point (x, y) other than a point which minimizes the objective function and draw a series of contour lines in order to get a region which contains points, i.e. facility locations, yielding as good or better objective function values than (x, y)

Example 6: Consider Example 5. 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.

Solution: The contour line method is illustrated using the figure below

Solution: Step 1: The vertical and horizontal lines V1, V2, V2 and H1, H2, H2, 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

Solution: Step 3: Since set N0 = D0 = -38 Step 4: Set
(These values are entered at the bottom of each column and left of each row in figure 1)

Solution: Step 5: Compute the slope of each region.
S00 = -(-38/-38) = -1; S14 = -(-22/38) = 0.58; S01 = -(-38/-26) = -1.46; S20 = -(30/-38) = 0.79; S02 = -(-38/-18) = -2.11; S21 = -(30/-26) = 1.15; S03 = -(-38/-2) = -19; S22 = -(30/-18) = 1.67; S04 = -(-38/38) = 1; S23 = -(30/-2) = 15; S10 = -(-22/-38) = -0.58; S24 = -(30/38) = -0.79; S11 = -(-22/-26) = -0.85; S30 = -(38/-38) = 1; S12 = -(-22/-18) = -1.22; S31 = -(38/-26) = 1.46; S13 = -(-22/-2) = -11; S32 = -(38/-18) = 2.11;

Solution: Step 5: Compute the slope of each region.
(The above slope values are shown inside each region.)

Solution: Step 6: When we draw contour lines through point (9, 10), we get the region shown in the previous figure. 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.

11.4.3 Single-facility Location Problem with Squared Euclidean Distances

La Quinta Motor Inns Moderately priced, oriented towards business travelers Headquartered in San Antonio Texas Site selection - an important decision Regression Model based on location characteristics classified as: Competitive, Demand Generators, Demographic, Market Awareness, and Physical

La Quinta Motor Inns (Cont)
Major Profitability Factors - Market awareness, hotel space, local population, low unemployment, accessibility to downtown office space, traffic count, college students, presence of military base, median income, competitive rates

Gravity Method: The cost function is
As before, we substitute wi = fi ci, i = 1, 2, ..., m and rewrite the objective function as

Gravity Method (Cont) 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

Gravity Method (Cont) Similarly,
Thus, the optimal locations x and y are simply the weighted averages of the x and y coordinates of the existing facilities

Example 7: Consider Example 5. Suppose the distance metric to be used is squared Euclidean. Determine the optimal location of the new facility using the gravity method.

Solution - Table 11.16 Department i xi yi wi wixi wiyi 1 10 2 6 60 12
Total

Example 6. Cont... 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

Weiszfeld Method

Weiszfeld Method: The objective function for the single facility location problem with Euclidean distance can be written as: As before, substituting wi=cifi and taking the derivative of TC with respect to x and y yields

Weiszfeld Method:

Weiszfeld Method: Step 0: Set iteration counter k = 1;

Weiszfeld Method: Step 1: Set
Step 2: If xk+1 = xk and yk+1 = yk, Stop. Otherwise, set k = k + 1 and go to Step 1

Example 8: Consider Example 6. 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 11.

Table 11.17 Coordinates and weights for 4 departments

Table 11.17: Departments # xi yi wi 1 10 2 6 2 10 10 20 3 8 6 8

Solution: 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.

Summary: Methods for Single-Facility, Continuous Space Location Problems
Rectilinear Squared Euclidean Euclidean Method Median Gravity Weiszfeld

Facility Location Case Study
See Section 11.5

S.S. Heragu Industrial Engineering Department University of Louisville

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