BASIC FACILITY LOCATION PROBLEMS Facility location problems deal with the question of where to place facilities & how to assign customers Location analysis is for choosing locations by discussing very basic models where there is only one type of cost to consider, that of transportation. We will assume that each facility is sufficiently large that it can serve all customers if necessary, i.e, that there is no limitation on facility’s capacity 26/04/20131lec # 27 & 28

BASIC FACILITY LOCATION PROBLEMS Basic facility location problems covers two types of facility location problem in which the only factor considered is transportation: i.e, I.Single Facility Placement Problem II.The Multiple Facility Placement Problem 26/04/20132lec # 27 & 28

SINGLE-FACILITY PLACEMENT PROBLEM In this case the minimum cost is determined simply by multiplying each customer’s demand by each location’s unit cost Facility is then placed in the location that is found to have lowest overall cost when all of the customer’s demand costs are totaled by location 26/04/20133lec # 27 & 28

SINGLE-FACILITY PLACEMENT PROBLEM ILUSTRATIVE PROBLEM.1: Suppose that a new copying m/c is to be installed in an office building complex. The machine can be setup in any one of five possible locations, & six offices will be assigned to use it. The demand from each office (i.e, expected number of trips made daily) & the cost (measured in terms of the time that a person spends in walking to & from each location) are given in table 1 on excel sheet 26/04/20134lec # 27 & 28

SINGLE-FACILITY PLACEMENT PROBLEM Solution procedure: When there is only one facility to place, the solution procedure consists of three very basic steps, i.e. 1.Calculate the total transportation cost matrix (also called the demand cost matrix). An element of the matrix is obtained by multiplying the cost of assigning a unit of demand, from a customer to a location, by the total demand from the customer. Eg: the cost of assigning customer A to location 1 is 3x15=45 units of time 26/04/20135lec # 27 & 28

SINGLE-FACILITY PLACEMENT PROBLEM Solution procedure: 2.Take the sum of each column. These values represent the total cost of assigning all customers to each location 3.Select the location with the minimum total cost. This is the location in which the machine should be placed 26/04/20136lec # 27 & 28

SINGLE-FACILITY PLACEMENT PROBLEM Application to the Illustrative Problem 1: The total transportation cost matrix for this problem, generated in step 1, is given in table 2 on excel sheet. The last row, the total, is the sum of each column. Since the total cost for location 4 is the least, hence this site is selected, & all the customers are assigned to this machine 26/04/20137lec # 27 & 28

MULTIPLE-FACILITY PLACEMENT PROBLEM Multiple-Facility problem is slightly more complicated. two familiar algorithms for such type of problems are: a.The Brute Force Method & b.The Heuristic Approach Brute force attacks the problem directly by setting up all possible combinations of locations & calculating the minimum transportation cost for each customer to be served by each location combination. The best combination of locations is that which has the overall minimum cost when the totals for all the customers at each of the combination are compared. This procedure becomes unweildly when the number of possible combinations become large 26/04/20138lec # 27 & 28

MULTIPLE-FACILITY PLACEMENT PROBLEM a.BRUTE FORCE APPROACH. Steps of Procedure: 1.Construct a demand cost table. An entry ij in the demand cost table represents the cost of allocating all the demand from source (customer)I to location j 2.Formulate all (M/K) possible combinations 3.In the first combination, determine the demand assignment by assigning demand to the facility where the cost of assignment is the minimum, it is done in the following two manner: 26/04/2013lec # 27 & 289

MULTIPLE-FACILITY PLACEMENT PROBLEM a.BRUTE FORCE APPROACH. Steps of Procedure: 3.a. consider only the columns of the demand cost matrix associated with the particular combination 3.b. Assign each demand to the facility having the smallest demand costs entry in its row for those columns selected in step.3.a 4. Calculate the total cost of the combination by taking the sum of the costs of the assignment in step 3 26/04/2013lec # 27 & 2810

MULTIPLE-FACILITY PLACEMENT PROBLEM a.BRUTE FORCE APPROACH. Steps of Procedure: 5.Repeat step 3 & 4 for all remaining combinations 6.Select combination with the overall minimum cost as the optimum combination; the associated demand assignments are optimum assignments 26/04/2013lec # 27 & 2811

MULTIPLE-FACILITY PLACEMENT PROBLEM BRUTE FORCE APPROACH Illustrative Problem.2: A Manager of automobile service stations has sufficient funds to buy two identical diagnostic machines. He currently operates 5 service stations within the city, & he has identified five areas from which demands will occur. Transportation time from the centre of gravity of each area cluster to the each location as given in table.3 (on excel sheet). Also shown is the demand from each customer (area) for such a machine. The requirement is to place two new machines in locations that will minimize the total transportation time, since the manager considers this to be the measure of efficiency 26/04/2013lec # 27 & 2812

MULTIPLE-FACILITY PLACEMENT PROBLEM BRUTE FORCE APPROACH Illustrative Problem.2: Solution: There are (M/K) ways of selecting K facilities in M available locations (M/K) = M!/K! (M-K)! Here M = 5, K=2 There fore (M/K) = 5!/2!3!= 10 Possible ways of placing theses facilities If all combinations are examined & demand is assigned to each such that we obtain the minimum cost for the combination, then the overall minimum can be found By application of the steps of Brute force the further calculation is given in table 4 & 5 on excel sheet Result: from table 5 combination for location 3 & 4 gives the minimum cost for new machine installation & hence same locations recommended for installation of such machines 26/04/2013lec # 27 & 2813