15 Quantitative Methods of Deciding Facility Location
Vikas Singla
15.1 OBJECTIVES
This chapter would help students to understand following quantitative methods of facility location:
- Cost-Profit-Volume Analysis method
- Factor Rating Method
- Centre of Gravity Method
- Transportation Model
15.2 INTRODUCTION
Last chapter emphasized on importance of location decision as it had long term financial impact on company’s’ profitability. It was also discussed that a location decision depended on various factors. These factors vary in importance from service to manufacturing industry. A firm can have various selection alternatives but study of these factors help a location decision maker to decide on the most relevant location. The deciding factors are governed by strategic objective of the firm. A profit oriented firm would be dictated by different factors as compared to a low cost producer. Similarly, influencing factors also depends on nature of the firm i.e. whether it is service oriented or a manufacturing concern. These factors were discussed in previous chapter. This chapter would discuss quantitative methods to evaluate these factors. Precisely, the emphasis of this chapter would be to discuss quantitative ways of identifying factors and evaluating importance of each factor in location decision making taking into consideration objective and nature of the firm. Following methods have been discussed:
- Cost-Profit-Volume Analysis
- Factor Rating Method
- Centre of Gravity Method
- Transportation Model
15.3 COST-PROFIT-VOLUME ANALYSIS
A firm could decide on a particular location out of multiple alternatives if that location produces:
- at lowest cost
- gives maximum profit
This method provides calculations to achieve any of the above stated objectives if a firm fulfills following assumptions:
- Fixed costs are constant for a particular range of output at a particular location.
- Variable costs are linear for selected range of output i.e. variable cost does not change if output varies over a particular range.
- All the selected location alternatives produce same and only one item.
For cost analysis, total cost for each location can be computed by using following formula:
Total cost (TC) = Fixed cost (FC) + variable cost (V) * Quantity or volume of output (Q) ………(1)
For profit analysis, total profit can be computed by using following formula:
Profit (P) = Quantity or volume of output (Q) * (Revenue – variable cost) – Fixed cost
P =Q * (R-V) – FC ………(2)
Example 15.2.1: Fixed and variable costs for four potential locations are given below. If expected output at selected locations is estimated to be 10,000 units which location would provide the lowest cost?
| Alternative locations | Fixed Cost (Rs.) | Variable cost per unit (Rs.) |
| A | 250,000 | 11 |
| B | 100,000 | 30 |
| C | 150,000 | 20 |
| D | 200,000 | 35 |
Solution: By using formula in equation (1) total cost for each location is:
| TC (A) | = 250,000 + 11 * 10,000 | = 360,000 |
| TC (B) | = 100,000 + 30 * 10,000 | = 400,000 |
| TC (C) | = 150,000 + 20 * 10,000 | = 350,000 |
| TC (D) | = 200,000 + 35 * 10,000 | = 550,000 |
So, location C should be selected for manufacturing same product as this location would incur minimum cost out of given alternatives.
Example 15.2.2: Following data shows fixed and variable cost for a particular product which can be manufactured at two locations. The demand for this product was estimated to be constant at 8800 units annually. Also, it was assumed that per unit of the product sells at Rs.6. Determine which location would yield highest profit.
| Alternative locations | Fixed Cost (Rs.) | Variable cost (Rs.) |
| A | 8000 | 44,000 |
| B | 9400 | 35,200 |
Solution: Revenue from selling of each unit is given as Rs.6 per unit.
Annual demand is given as constant = 8800 units
| So, revenue for location A | = 6 * 8800 | = 52,800 |
| And, revenue for location B | = 6 * 8800 | = 52,800 |
By using formula in equation (2), profit for each location would be:
Profit (A) = 8800 * (52,800 – 44,000) – 8000
= Rs. 800
Profit (B) = 8800 * (52,800 – 35,200) – 9400
= RS. 8,200
Hence, location B should be selected as it gives more profit than location A.
15.4 FACTOR RATING METHOD
Factor rating method of location analysis is perhaps the most widely used method of evaluating alternative locations for a firm. This method is a combination of both qualitative and quantitative methods for identifying the most feasible location. It involves following steps:
Step 1: Firstly, by applying qualitative methods which involves taking views from experts and decision makers factors or dimensions are identified which are important for evaluating a firms’ location. These factors vary from firm to firm. For instance, factors influencing a retail firm might be totally different than factors influencing a banks’ location. Thus, views and opinions regarding dimensions influencing a firms’ location are taken from experts making it the most important aspect of factor rating method.
Step 2: After identifying suitable factors pertaining to a particular firm, the consumers of those firms are asked to rate the importance of those factors. Each factor is being rated on a particular scale indicating its importance to the consumer. For instance, a bank consumer might be asked to rate the importance of ‘easy accessibility’ and ‘convenient hours’ for availing services of a particular bank. Similarly, other factors are also rated. The importance of these factors is generally rated on a scale of 0-1. Higher the score more important is the factor. Importantly, it should be noted that identification of factors which was done in Step 1 involves opinions of experts for that particular firm for which factors need to be identified whereas, in Step 2 importance of these factors is judged by taking opinions of individuals who are users of products/services of firm under consideration.
Step 3: In this step, consumers are asked to rate the importance of identified factors with respect to alternative locations. For instance, an individual might be asked to rate the importance of a particular factor such as ‘easy accessibility’ regarding a bank if it is located at location ‘A’ or location ‘B’. Most likely for different locations each factor would be scored differently. The bank under consideration might not be easily accessible if it is located at ‘A’ so it would be scored lowly as compared to ‘B’. Thus, regarding this factor location ‘B’ is preferable. The location alternatives are scored generally on a scale of 0-100 where higher the score higher importance is given to particular location with regard to that factor.
Step 4: Lastly, the importance assigned to each factor in step 2 is multiplied by the importance given to location alternatives with respect to that factor. Then sum of these scores is calculated. Higher score for a particular location would suggest selection of that location for establishment of that firm.
Example 15.3.1: The company wants to make decision that whether to locate an apparel store in a shopping mall or in a traditional shopping street. The following table (Table 15.3.1) indicates the selected factors for evaluating location of such store.
| Table 15.3.1 | |
| Factors | Importance assigned on a scale of 0-1 |
| Closeness with other retail stores | 0.7 |
| Volume of customers | 0.4 |
| Rental costs | 0.5 |
| Availability of space | 0.4 |
| Operating costs other than rental costs | 0.3 |
| Easy accessibility | 0.7 |
| Sufficient availability of parking space | 0.8 |
| Location of the store depends on availability of other entertainment facilities like food outlets etc. | 0.65 |
Table 15.3.1 shows identified important factors and scores assigned to each factor by an individual. After assigning weights to each factor from 0-1 consumers are then asked to rate on a scale of 0-100 the importance of two alternative locations under study, namely shopping mall and traditional shopping street. The responses have been recorded in Table 15.3.2.
| Table 15.3.2 | ||
| Traditional Shopping street (0-100) | Shopping Mall (0-100) | |
| Closeness with other retail stores | 70 | 40 |
| Volume of customers | 55 | 65 |
| Rental costs | 40 | 50 |
| Availability of space | 75 | 70 |
| Operating costs other than rental costs | 50 | 55 |
| Easy accessibility | 80 | 75 |
| Sufficient availability of parking space | 75 | 80 |
| Location of the store depends on availability of other entertainment facilities like food outlets etc. | 60 | 65 |
To analyze the location alternative multiply the weights assigned as shown in Table 15.3.1 and scores given to different locations as shown in Table 15.3.2. The results have been shown in Table 15.3.3
| Table 15.3.3 | ||
| Traditional Shopping street (0-100) | Shopping Mall (0-100) | |
| Closeness with other retail stores | 0.7*70 = 49 | 0.7*40 = 28 |
| Volume of customers | 0.4*55 = 22 | 0.4*65 = 26 |
| Rental costs | 0.5*40 = 20 | 0.5*50 = 25 |
| Availability of space | 0.4*75 = 30 | 0.4*70 = 28 |
| Operating costs other than rental costs | 0.3*50 = 15 | 0.3*55 = 16.5 |
| Easy accessibility | 0.7*80 = 56 | 0.7*75 = 50.5 |
| Sufficient availability of parking space | 0.8*75 = 60 | 0.8*80 = 64 |
| Location of the store depends on availability of other entertainment facilities like food outlets etc. | 0.65*60 = 39 | 0.65*65 = 35.75 |
| Total | 291 | 273.75 |
The results as shown in Table 15.3.3 indicates that traditional shopping street would be a better location alternative for apparel store under study.
15.5 CENTRE OF GRAVITY METHOD
Centre of Gravity method is predominantly used to evaluate location of distribution centers such as warehouses. This method assumes that manufacturing facility or service centre is providing similar or only one kind of product/service. Also, inbound and outbound transportation costs are assumed to be equal. For instance, Best Price is a famous warehouse chain for Bharti Group of companies. It does not locate itself close to customer of retail products but at the convergence of its primary market of retailers. It is important to note that customers of such warehouse are similar in nature as they retail similar kind of products. An illustration of Best Price has been discussed as under.
Best Price warehouse stores FMCG products and also electronic goods. Primarily it is a supplier to various retailers of different cities of such products. The warehouse wanted to evaluate a location for its new store catering to retailers of three major cities A, B and C. City A is the capital city and has highest per capita income with largest size of population among selected three cities. City B and City C are similar on per capita and population characteristics. Distance between city A and city B is 50 kms, between A and C is 60 kms and between B and C is 40 kms. Coordinates of each city were calculated and demand was estimated. It is important to note that products warehouse is a supplier or distributor of variety of products but they are similar in nature and supplied to similar kind of retailers. Then coordinates of new location were found by using coordinates of existing locations and demands of each centre. By using this method of coordinates relevant location for new store were found which was found to be closest to city A. thus, this method was found to be quite relevant for finding locations of distributors supplying similar nature of products. Lastly, after finding coordinates of new location other information such as cost of transporting and demand should be considered to finalize the location alternative.
Numerical illustration has been discussed below.
Following steps are used in Centre of Gravity method for evaluating location alternatives:
Step 1: Locate on a coordinate system for existing locations.
Step 2: Calculate volumes being shipped or demand of each location.
Step 3: Use the following formula to find coordinates for new location
| X-coordinate | = | ∑dix * Vi / ∑Vi | |
| Y-coordinate | = | ∑diy * Vi / ∑Vi | |
| where | dix | = | X-coordinate of the ith location |
| diy | = | Y coordinate of the ith location | |
| Vi | = | volume of goods moved to or from the ith location |
Example 15.4.1: For following data find the coordinates of new location
| Destination | X | Y | Quantity |
| Plant A | 150 | 75 | 6000 |
| Plant B | 100 | 300 | 8200 |
| Plant C | 275 | 380 | 7000 |
| Solution: | ||
| X-coordinate of new location | = | (150*6000 + 100*8200 + 275*7000) / (6000 + 8200 +7000) |
| = | (900,000 + 820,000 + 1925,000) / (21200) | |
| = | 3645,000 / 212,00 | |
| = | 171.9 | |
| Y-coordinate of new location | = | (75*6000 + 300*8200 + 380*7000) / (6000 + 8200 +7000) |
| = | (450,000 + 2460,000 + 2660,000) / (21200) | |
| = | 5570,000 / 21200 | |
| = | 262.73 |
These X and Y coordinates can be used to locate new establishment and then further analysis regarding cost of this new location with respect to existing locations can be carried out.
15.6 TRANSPORTATION MODEL
The transportation model involves determination of minimum total cost for distributing a particular type of product manufactured at multiple factories or sources to multiple destinations or demand centers. For instance, a state having multiple thermal power plants might source coal used as raw material for producing electricity from multiple destinations. Now, it becomes important for state to formulate a model which can determine amount of coal to be sourced from which source in such a manner that total transportation cost is minimum. Thus, a transportation model helps in determining:
Allocation of units from multiple supply centers to various demand centers. It could be possible that a supply centre is used to supply raw material to more than one demand centre. Also, it is not necessary that a demand centre obtains all the raw material from only one source as each supplier has different capacity and each demand centre would have different demand. Thus, the model helps in finding out number of units each supply centre would provide to each demand centre to fulfill its total demand.
Total transportation cost which should be minimum. As each supply centre can supply varying amount of product to different demand centers thus, there can be ‘n’ number of ways in which material can be sourced and can be supplied. This model by taking into consideration transportation cost of each unit to be supplied from every supply centre to every demand centre and number of units allocated for transportation from a particular supply centre to a particular demand centre calculates most optimal transportation cost.
15.5.1 Assumptions of Transportation model:
- There should be known and finite number of supply centres
- There should be known and finite number of demand centres.
- All the supply centres manufactures or provide identical product.
- The demand of each destination and capacity of each resource centre should be known.
- The unit costs of transportation from each origin to each destination should be known.
15.5.2 Application of transportation model in Location decisions
The transportation model can be used to evaluate feasibility of various locating alternatives by comparing each alternative in terms of their transportation cost. For instance in the example discussed, warehouse A has a demand of 80 units which can be fulfilled by any of three supply centers. So, model would allocate 80 units to be supplied from each supply centre and then see its impact on total transportation cost. Finally, that model would be selected which would minimize total transportation cost.
| Fig. 15.5.1 | |
|
 |
To understand number of scenarios in which demand of each demand centre can be fulfilled from various supply centers following figure (Fig. 15.5.1) is quite illustrative. In fig. 15.5.1 we have illustrated by taking three supply centers and three demand centers. In this case, each demand centre can be supplied by any of the three supply centre. So determination of how many units to be allocated from each supply centre to different demand centers becomes quite difficult if done manually. Usage of computer software is recommended when number of supply and demand centers increases.
15.5.3 An Illustration of Transportation Model:
Table 15.5.1 gives information about transportation cost of a product being transported from any of the given three supply centers to four demand centers. The demand of each demand centre and capacity of each supply centre is also given. By using transportation model number of units being transported from which supply centre to which destination needs to be determined in such a way that total transportation comes out to be minimum.
| Table 15.5.1 | |||||
| Warehouse | |||||
| Factory | A | B | C | D | Supply |
| 1 | 4 | 7 | 7 | 1 | 100 |
| 2 | 12 | 3 | 8 | 8 | 200 |
| 3 | 8 | 10 | 16 | 5 | 150 |
| Demand | 80 | 90 | 120 | 160 | Total = 450 units |
Table 15.5.1 shows:
- Three supply centres (factories) 1, 2 and 3.
- Four demand centres (warehouses) A, B, C and D.
- Transportation cost from each factory to each warehouse. For instance, the first cost cell indicates that Rs. 4 is required to transport one unit from factory ‘1’ to warehouse ‘A’.
- Supply capacity of each factory and demand of each warehouse is given. For instance, Factory 1 can supply maximum of 100 units and warehouse A has a maximum demand of 80 units.
- It is important to note that total demand of 4 warehouses and total supply of three supply centres should be same. In this case it is 450 units.
The solution to discussed problem is shown in table 15.5.2 by using computer software. The methodology of obtaining optimal transportation cost is tedious and is not discussed in this chapter.
| Table 15.5.2 | |||||
| Warehouse | |||||
| Factory | A | B | C | D | Supply |
| 1 | 4 | 7 | 7 | 1 | 100 |
| 2 | 12 | 3 | 8 | 8 | 200 |
| 3 | 8 | 10 | 16 | 5 | 150 |
| Demand | 80 | 90 | 120 | 160 | Total = 450 units |
Total transportation cost = 7*10 + 1*90 + 3*90 + 8*110 + 8*80 + 5*70
= 70 + 90 + 270 + 880 + 640 + 350
= Rs. 2300
Thus 450 units can be supplied from three factories to four warehouses by incurring a transportation cost of Rs. 2300.
15.6 SUMMARY
This chapter discusses quantitative methods of deciding an appropriate location for a firm when it faces multiple location alternatives. Factor rating method is one of the most widely used methods in location decision making strategy. It involves identifying most important factors used in evaluating a location and also the importance given to various location alternatives based on these factors. Cost-Volume-Profit analysis method is used to calculate costs and profits involved in producing similar product at various locations. The location which provides maximum profit at minimum cost is considered to be most suitable. Centre of Gravity Method is an appropriate method to find location of firms involved in distribution of similar kind of products such as supply of products from a warehouse to similar retailers. Lastly, transportation method uses transportation cost per unit from various supply centers to multiple demand centers to calculate transportation cost of various locations.
15.7 GLOSSARY
- Cost-Volume-Profit method: is a quantitative method used in deciding on an appropriate location based on minimum cost of production and maximum profit.
- Factor Rating method: identifies most important factors in evaluating multiple locations.
- Centre of Gravity Method: is used to evaluate appropriate location especially for distribution centres such as warehouses by using coordinates and volume of units required of existing locations.
- Transportation Model: allocates number of units to be transported to multiple demand centres from various supply centres in such a manner which minimizes total transportation cost.
15.8 REFERENCES/ SUGGESTED READINGS
- Chase, B.R., Shankar, R., Jacobs, F.R. and Aquilano, N.J., Operations & Supply Chain Management, 12th Edition, McGraw Hill.
- Stevenson, W.J., Operations Management, 9th Edition, Tata McGraw Hill.
- Lee J. Krajewski, Operations Management, Prentice-Hall of India, New Delhi, 8th Edition.
15.9 SHORT ANSWER QUESTIONS
1. The total demand and supply from various centres in transportation model should be equal.
(a) True (b) False
Answer: a
2. The cost cells in transportation model represent cost of transportation of one unit.
(a) True (b) False
Answer: a
3. Centre of Gravity method is used to find coordinates of distribution centres providing similar type of products.
(a) True (b) False
Answer: a
15.10 MODEL QUESTIONS
1. The owner of a ice cream chain hopes to expand its operations by opening new outlets. Three locations have been shortlisted. Each would have same labour and materials costs of Rs.1.76 per unit. Ice creams sell for Rs.2.65 each in all locations. Rent and equipment costs would be Rs.5000 per month for location A, Rs.5500 per month for location B and Rs.5800 for location C.
(a) Determine the volume necessary at each location to realize a monthly profit of Rs.10,000.
(b) If expected sales at a, B and C are 21000, 22000 and 23000 per month respectively, which location would yield the greatest profits?
2. A clothing manufacturer produces women’s clothes at four locations. The location of a central shipping point for bolts of cloth must now be determined. Weekly quantities to be shipped to each location are also given. Determine the coordinates of the location that will minimize distribution costs.
| Location | A | B | C | D |
| Coordinates (x,y) | 5,7 | 6,9 | 3,9 | 9,4 |
| Weekly Quantities | 15 | 20 | 25 | 30 |