Different Approaches for Solving Location Problems

8/10/2019 Different Approaches for Solving Location Problems

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Different Approaches for Solving Facility Location Problems1

Single Facility Location Problems using Centre of Gravity Approach

Given

Set of source points Set of demand points Volumes to be moved from sources Volumes to be moved to demand locations Transportation costs Objective: Minimize total transportation cost

Center-of-Gravi ty App roach

Locations of supply and demand points Volume of flow from/to facility to supply or demand points Transportation costs (rates) at each demand or supply point Problem: Find the location of the new facility in order to minimize total

transportation costs

Parameters

TC: total transportation costVi: volume at point iRi: transportation rate at point idi: distance to point i from the facility to be located

Problem: Min TC = Vi Ri diCoordinates of new facility = (

____

,YX )Coordinates of existing facilities = (Xi,Yi)

X-Equation Y-Equation

where

1Contents of this note are adapted fr om Business Logistics Management by Ronald H . Bal lou and few othersources and edited by Dr .T.A.S.Vij ayaraghavan, XLRI Jamshedpur pur ely for classroom teaching Th is

teaching note is stri ctly for private circulation only.

=

i

iii

i

iiii

dRV

dXRV

X/)(

/)(__

=

i

iii

i

iiii

dRV

dYRV

Y/)(

/)(__

22 )()( YYXXKd iii ++==


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Example

0 4 80

2

4

6

8(2000, $0.05) (3000, $0.05)

(2500, $0.075)

(1000, $0.075)

(1500, $0.075)

i X Y V R d cost

1 3 8 2000 0.050 35.52 35522 8 2 3000 0.050 42.63 6395

3 2 5 2500 0.075 31.65 59354 6 4 1000 0.075 14.48 10865 8 8 1500 0.075 40.02 4503

Total Transportation Cost 21471

d= 10 ( 3-5.16)2 + (8-5.18)2 = 35.52

i VR VRX VRY d VR/d VRX/d VRY/d

1 100 300 800 35.32 2.815 8.446 22.5232 150 1200 300 42.63 3.519 28.149 7.037

3 187.50 375 937.50 31.65 5.924 11.848 29.6214 75 450 300 14.48 5.180 31.077 20.7185 112.50 900 900 40.02 2.811 22.489 22.489

20.249 102.009 102.388


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= 102.009/20.249 = 5.038 = 102.388/20.249 = 5.057

Iteration X-coord Y-coord Total cost0 5.160 5.180 21471.001 5.038 5.057 21431.222 4.990 5.031 21427.113 4.966 5.032 21426.144 4.951 5.037 21425.695 4.940 5.042 21425.446 4.932 5.046 21425.307 4.927 5.049 21425.238 4.922 5.051 21425.199 4.919 5.053 21425.1610 4.917 5.054 21425.1511 4.915 5.055 21425.14.

,,100 4.910 5.058 21425.14

Exact Solution

Centre ofGravity

__

X

__

X__

Y


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Mixed-Integer Linear Programming

Mathematicians have labored for many years to develop efficient solution proceduresthat have a broad enough problem description to be of practical value in dealing withthe large, complex location problem frequently encountered in logistics networkdesign and yet provide a mathematically optimum solution. They have experimentedwith the use of sophisticated management science techniques, either to enrich theanalysis or to provide improved methods for solving this difficult problem optimally.These methods are goal programming, tree search methods and dynamicprogramming among others. Perhaps the most promising of this class is the mixed-integer linear programming approach. It is the most popular methodology used incommercial location models. The primary benefit associated with the mixed-integerlinear programming approach-a benefit not always offered by other methods-is itsability to handle fixed costs in an optimal way. The advantages of linear programmingin dealing with the allocations of demand throughout the network, which is at theheart of such an approach, are well known. Although optimization is quite appealing,it does exact its price. Unless special characteristics of a particular problem areexploited, computer running times can be long and memory requirements substantial.Warehouse location problems are presented in many variations. Researchers who

have applied the integer programming approach have described one suchwarehouse location problem as follows:

There are several commodities produced at several plants with knownproduction capacities. There is a known demand for each commodity ateach of the number of customer zones. This demand is satisfied byshipping via warehouses, with each customer zone being assignedexclusively to a single warehouse. There are lower as well as upper limitson the allowable total annual throughput on each warehouse. Thepossible locations for the warehouses are given, but the particular sites tobe used are to be selected so as to result in the least total distributioncost. The warehouse costs are expressed as fixed charges (imposed forthe sites actually used) plus a linear variable charge. Transportation costs

are taken as linear. Thus; the problem is to determine which warehouselocations to use, what size warehouse to have at each selected location,what customer zones should be served by each warehouse, and whatpattern of transportation flows there should be for all commodities. This isto be done so as to meet the given demands at minimum total distributioncost, subject to plant capacity and warehouse configuration of thedistribution system.

In descriptive language, this problem can be expressed in the following manner:

Find the number, size, and locations of warehouses in a logistics networkthat will minimize the fixed and linear variable costs of moving all productsthrough the selected network subject to the following:

I. The available supply of the plants cannot be exceeded for eachproduct.II. The demand for all products must be met.III. The throughput of each warehouse cannot exceed its capacity.IV. A minimum throughput of a warehouse must be achieved before it

can be opened.


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V. All products for the same customer must be served from the samewarehouse.

The problem can be solved using general integer linear programming computersoft- ware packages. Historically, such practical problems were not solved, evenwith the most powerful computers; however, researchers now apply suchtechniques as decomposing a multi-product problem into as many sub problemsas there are products, eliminating parts of the problem irrelevant to the solution,and approximating data relationships in forms that complement the solutionapproach in order to achieve acceptable computer running times and memoryrequirements. Today, researchers are claiming to be able to substantially extendthe number of echelons in the network that can be modeled, include multiple timeperiods in the model, and cautiously handle nonlinear cost functions.

Another location method that utilizes mixed-integer programming is the p-medianapproach. It is less complicated but less robust than the previous formulation.Demand and supply points are located by means of coordinate points. Facilitiesare restricted to be among these demand or supply points. The costs affectinglocation are variable transportation rates expressed in units as $/cwt./mi. and the

annual fixed costs associated with the candidate facilities. The number offacilities to be located is specified before solution.

Considering a small multi-product problem and a standard integer programmingsoftware code, solving a location problem by integer programming can be illustrated.Suppose we have the problem as shown in Figure. There are twp products that aredemanded by three customers, but a customer can be served out of only onewarehouse. There is a choice between two warehouses. Warehouse 1 has ahandling cost of $2/cwt. of throughput; a fixed cost of $100,000 per year if held open;and a capacity of 110,000 cwt. per year. Warehouse 2 has a handling cost of $l/cwt.,a fixed cost of $500,000, and an unlimited capacity. There is no minimum volume tokeep a warehouse open. Two plants can be used to serve the warehouses. Theplants may produce either product, but the production costs per cwt. differ for each

product. Plant 1 has a product capacity constraint (60,000 cwt. for product 1 and50,000 cwt. for product 2). Plant 2 has no capacity constraint for either product. Ourtask is to find which warehouse(s) should be used, how customer demand should beassigned to them, and which warehouses and their throughput should be assigned tothe plants. The problem formulation is shown at then end of this section. The problemis solved using the MIPROG module in LOGWARE. The solution is to open onlywarehouse 2 and to serve it from plant 2. The cost summary is

Category Cost

Production $1,020,000Transportation 1,220,000Warehouse handling 310,000

Warehouse fixed cost 500,000

Total $3,050,000


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Figure:1- A Small Multi-product Warehouse Location Problem for Mixed IntegerLinear Programming


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Formulation of the Mixed Integer Linear Programming Problem for the Exampleshown in Figure -1

Def in i t ion of var iables

Formulation:

Objective Function


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Constra ints:


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P-Median Approach

Environment plus incinerates toxic chemicals used in various manufacturingprocesses. These chemicals are moved from 12 market areas around the country toits incinerators for disposal. The company provides the transportation, due to thespecial equipment and handling procedures required. Transportation services are

contracted at a cost of $1.30 per mile, and the trucks are fully loaded at 300 cwt.Trips are out and back from an incinerator. Therefore, the effective transport rate is($1.30/mi. 2)/300cwt.=$0.0867/cwt./mi. The market locations, annual processingvolume, and annual fixed operation costs, regardless of throughput volume, are asfollows

Market Latitude Longitude Annual Volume, cwt Fixed OperatingCost, $

A 42.36 71.06 30,000 3,100,000B 41.84 87.64 240,000 2,900,000C 40.72 74.00 50,000 3,700,000D 44.93 93.20 140,000 -

E 33.81 84.63 170,000 1,400,000F 33.50 112.07 230,000 1,100,000G 39.23 76.53 120,000 -H 39.77 105.00 300,000 1,500,000I 39.14 84.51 100,000 1,700,000J 34.08 118.37 40,000 2,500,000K 35.11 89.96 90,000 -L 47.53 112.32 20,000 1,250,000

The metropolitan areas of D, G, and K will not permit the incinerators and thereforeare not considered as candidate locations. If five locations are to be used, whichshould they be?

The PMED software module in LOGWARE can help to solve this problem. Theresults show the preferred locations to minimize the cost.

NO Facility Name Volume AssignedNode Numbers

1 C 200,000 1 2 42 E 260,000 3 63 B 480,000 5 7 84 F 270,000 9 115 H 320,000 10 12

Total 1,530,000

Total cost: $24,739,040.000


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Multiple Centre of Gravity Approach

Suppose we have data for 10 markets and their corresponding transportation rates.

"M1" 2 1 3000000 .002"M2" 5 2 5000000 .0015"M3" 9 1 17000000 .002"M4" 7 4 12000000 .0013"M5" 10 5 10000000 .0012"M6" 2 5 9000000 .0015"M7" 2 7 24000000 .002"M8" 4 7 14000000 .0014"M9" 5 8 23000000 .0024"M10" 8 9 30000000 .0011

In addition, there is an annual fixed charge for each warehouse of $2,000,000. Allwarehouses have enough capacity to handle all of the market demand. The amount

of inventory in the logistics system is extimated by IT($) =6,000,000N, where N isthe number of warehouses in the network. Inventory carrying costs are 25 percentper year. Handling rates at the warehouses are all the same and, therefore, do notaffect the location outcome. How many warehouses should there be, where shouldthey be located, and which markets should be assigned to each warehouse?

Using the MULTICOG software module in LOGWARE and repeatedly solving forvarious numbers of warehouses, the following spreadsheet can be developed:

Number of Transportation Fixed Inventory TotalWarehouses Cost, $ Cost, $ Cost, $ Cost, $

1 41,409,628 2,000,000 1,500,000 44,909, 6282 25,989,764 4,000,000 2,121,320 32,111,0843 16,586,090 6,000,000 2,598,076 25,184,1664 11,368,330 8,000,000 3,000,000 22,368,3305 9,418,329 10,000,000 3,354,102 22,772,4316 8,032,399 12,000,000 3,674,235 23,706,6347 7,478,425 14,000,000 3,968,627 25,447,0528 2,260,661 16,000,000 4,242,641 22,503,3029 948,686 18,000,000 4,500,000 23,448,686

10 0 20,000,000 4,743,416 24,743,416

Four warehouses yield the best cost balance. Warehouses should be located inmarkets 3,7,9 and 10. Markets 2, 3, 4, and 5 are assigned to the warehouse at 3;markets 1, 6, and 7 are assigned to the warehouse at 9; and market 10 is assigned

to the warehouse at 10.


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Guided Linear Programming Approach

Consider the small, single-product problem shown in Figure 1. The first step is toconstruct a matrix that is formatted like the transportation problem of linear

programming. By giving it a special structure, to echelons of a logistics network canbe represented in the matrix of Figure 2. The heuristic process is guided by themanner in which the cell costs are entered into the matrix. Because the productionand transportation costs between plants and warehouses are linear, they enter theplant-warehouse cells directly. For example, the cell cost representing the flowbetween P2and W1 is the production plus transportation costs, or $4/cwt. + $4/cwt. =$8/cwt.

The cell block for warehouses and customers combines warehouse handling plustransportation plus inventory carrying plus fixed costs. Handling and transportationrates can be read directly from Figure 1. However, there are no rates for inventorycarrying and fixed costs, and they must be developed, depending on the throughputof each warehouse. Because this throughput is not known, we must assume starting

throughputs. For fixed costs, each warehouse is initially given the most favorablestatus by assuming that all demand flows through it. Thus for warehouse 1, the rateassociated with fixed costs would be annual warehouse fixed cost divided by totalcustomer demand, or $100,000/200,000 = $0.50/cwt. For warehouse 2, it is$400,000/200,000 = $2.00/cwt.

Figure-1: A single product Location Problem with Warehouse Fixed Costs andInventory Costs

Plant P1

Plant P2

Production= 4

Capacity=60,000

Production=4,Capacity= Unrestricted

0

5

4

2

Handling=$2/cwtFixed= $100,000Capacity=60,000 cwt

Warehouse W1

Warehouse W2

Handling=$1/cwtFixed= $400,000Capacity=Unrestricted

4

3

5

2

1

2

Customer C150000

Customer C2100000

Customer C350000

7.0)(100cos

Throughput

tcarryingInventory =

For inventory carrying costs, the per-cwt. rate depends on the number of warehousesand the demand assigned to them. Again, to give each warehouse the greatestopportunity to be selected, the assumed throughput for each warehouse is equal, or


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total customer demand divided by the number of warehouses being evaluated. Theinventory carrying rate is defined as the inventory cost in a warehouse divided by thewarehouse throughput, or ICi = K(Throughputi)

a/Throughputi. Initially for eachwarehouse, the per-cwt. inventory carrying rate is

The estimated per-unit fixed and inventory carrying rates are now entered into thewarehouse-customer cells of the matrix of Figure 2. The problem is solved in anormal manner using the TRANLP module of LOGWARE. The computational resultsare shown as the bold values in Figure 2. This now completes round one of thecomputations. Subsequent computational rounds utilize warehouse throughputs fromits previous round to improve upon the estimate of the per-unit inventory carrying andfixed costs for a warehouse. To make these estimates, we note that the throughputfor W1 is 60,000 cwt., and for W2 it is 140,000 cwt. (see Figure 2). The allocatedcosts for the warehouses will be

Figure-2: Matrix of Cell Costs and Solution Values for the First location in theExample Problem


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The cell costs in the matrix for warehouses to customers (see Figure 13-2) arerecalculated to be

C1 C2 C3

W1 11.361 10.36 12.36W2 9.432 7.72 8.72

12+4+1.67+3.69 = 11.3621+2+3.57+2.86 = 9.43

The remaining cells are unaltered. Now, solve the problem again.

The results of the second iteration solution show that all production is at plant 2 andall product is to be served from warehouse 2. That is,

C1 C2 C3

W1 0 0 0W2 50,000 100,000 50,000

Subsequent iterations repeat the second iteration solution since the allocation of

inventory and fixed costs remain unchanged. A stopping point has been reached. Tofind the solution costs, recalculate them from the actual costs in the problem. Do notuse the cell costs of Figure 2 since they contain the estimated values for warehousefixed and inventory carrying costs. Rather, compute costs as follows using the ratesfrom Figure 1:

The previous example illustrates a heuristic procedure for a single product. However,many practical location problems require that multiple products be included in the

computational procedure. With slight modification where fixed costs for a warehouseare shared among the products according to their warehouse throughput, the guidedlinear programming procedure can be extended to handle the multi-product case.


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A Warehouse-location ProblemWilliam J.Baumol and Philip Wolfe

In this case, our allocation problem is solved using the transshipment problem oflinear programming or network flow. We only need to assume that transport costs

are proportional to tonnage (and not necessarily proportional to distance). Transportcosts can vary nonlinearly (tapering) with distance. Assume we have eight demandpoints (r), two factories (f), and five potential warehouse sites (w) and we wish todetermine the number, size and location of warehouse the minimize the total ofwarehousing and transport costs.


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The warehouses can be of any size but total costs depend upon their throughput.The marginal cost of a unit through each warehouse is the derivative of the totalwarehouse costs with respect to Z.

Warehouse 1 2 3 4 5

Total Ware.

Cost 75Z1.5 80Z2

.5 75Z3.5 80Z4

.5 70Z5.5

MarginalWare.

Cost 37.5Z1-.5 40Z2

-.5 37.5Z3-.5 40Z4

-.5 35Z5.5


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PlantsWarehouses Retailers

Our initial solution assumes that there are zero cost of going through a warehouse


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Heuristics for Location of Facilities


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Different Approaches for Solving Location Problems