Location Planning and Analysis

The need for location decisions:

The nature of location decisions:

Location decision making procedure:

  1. Decide on the criteria to use for evaluating location alternatives.
  2. Identify important factors.
  3. Develop location alternatives:
    1. Identify a general region for a location.
    2. Identify a small number of community alternatives.
    3. Identify site alternatives among the community alternatives.
  4. Evaluate the alternatives and make a decision.

New firms typically locate in a certain area simply because the owner lives there. Large established companies, particularly those that already operate in more than one location, tend to take a more formal approach.

Factors that affect location decisions:

Multiple plant manufacturing strategies:

  1. Product plant strategy. Entire products or product lines are produced in separated plants. Each plant supplies to entire market. The strategy often results in economics of scale.
  2. Market area plant strategy. Plants are designed to serve a particular geographic segment of a market. Individual plant produce most products. This strategy significantly saves on shipping costs.
  3. Process plant strategy. Different plants concentrate on different aspects of a process. This strategy is best suited for products that have numerous components. Coordination of production throughout the system becomes a major issue and requires a highly informed and centralized administration to achieve effective operation.

Service and retail locations typically place traffic volume and convenience high on the list of important factors. Available public transportation and parking facility are often a consideration.

Factors relating to foreign locations:

The future will see a trend toward a smaller and automated factories, microfactories, with a narrow product focus located close to markets.

Evaluating Location Alternatives

Locational Cost-Profit-Volume Analysis

Graphical assumption:

  1. Fixed costs are constant for the range of probable output.
  2. Variable costs are linear for the range of probable output.
  3. The required level of output can be closely estimated.
  4. Only one product is involved.

Graphical Procedure:

  1. Determine the fixed and variable costs associated with each location alternative.
  2. Plot the total-cost lines for all location alternatives on the same graph.

where

FC= fixed cost

VC= variable cost per unit

Q= quantity or volume of output

  1. Determine which location will have the lowest total cost for the expected level of output. Alternatively, determine which location will have the highest profit.

For a profit analysis, compute

,

where

R= revenue per unit.

When a problem involves shipment of goods from multiple sending points to multiple receiving points, and a new location (sending or receiving point) is to be added to the system, the company should undertake a separate analysis of transportation. In such instances, the transportation model of linear programming is very helpful.

Factor Rating

Factor rating establishes a composite value for each alternative that summarizes all related factors.

The following procedure is used to develop a factor rating:

  1. Determine which factors are relevant.
  2. Assign a weight to each factor that indicates its relative importance compared with all other factors. Typically, weights sum to 1.00.
  3. Decide on a common scale for all factors.
  4. Score each location alternative.
  5. Multiply the factor weight by the score for each factor, and sum the results for each location alternative.
  6. Choose the alternative that has the highest composite score.

In some cases, managers may prefer to establish minimum thresholds for composite scores. If an alternative fails to meet that minimum, they can reject it without further consideration.

The Center of Gravity

The center of gravity method treats distribution cost as a linear function of the distance and the quantity shipped.

The quantity to be shipped to each destination is assumed to be fixed. An acceptable variation is that quantities are allowed to be different, as long as their relative amounts remain the same.

The coordination of the center of gravity is defined as

,

where

= quantity to be shipped to destination i,

= x coordinate of destination i, and

= y coordinate of destination i.

If the quantities to be shipped to every location are equal, you have

,

where

n= number of destinations.

  1. Map showing destinations.

  1. Add coordinates.

  1. Center of gravity.

Transportation Model

The transportation problem involves finding the lowest-cost plan for distributing stocks of goods or supplies from multiple origins to multiple destinations that demand the goods.

The following figure shows the nature of a transportation problem in real life.

The information needed consists of the following:

  1. A list of origins and one’s capacity or supply quantity per period.
  2. A list of the destinations and each one’s demand per period.
  3. The unit cost of shipping items from each origin to each destination.

The information is arranged into the following transportation table.

Assumptions are:

  1. The items shipped are homogeneous.
  2. Shipping cost per unit is the same regardless of the number of units shipped.
  3. There is only one route or mode of transportation being used between each origin and each destination.

Major steps in solving the transportation using the table are:

  1. Obtaining an initial solution.
  2. Testing the solution for optimality.
  3. Improving sub-optimal solutions.

Obtaining An Initial Solution --- The Intuitive Lowest-Cost Approach

The procedure involves these steps:

  1. Identify the cell with the lowest cost.
  2. Allocate as many units as possible to that cell, and cross out the row or column (or both) that is exhausted by this.
  3. Find the cells with the next lowest cost from among the feasible cells.
  4. Repeat steps (2) and (3) until all units have been allocated.

Testing For Optimality --- Stepping Stone

In the stepping-stone method, cell evaluation proceeds by borrowing one unit from a full cell and using it to assess the impact of shifting units into the empty cell. Helpful rules for obtaining evaluation paths are

  1. Start by placing a + sign in the (empty) cell you wish to evaluate.
  2. Move horizontally (or vertically) to a completed cell (a cell that has units assigned to it). It is OK to pass through an empty cell or a completed cell without stopping. Choose a cell that will permit your next move to another completed cell. Assign a minus (-) sign to the cell.
  3. Change direction and move to another completed cell. Again, choose one that will permit your next move. Assign a plus sign (+) to the cell.
  4. Continue this process of moving to a completed cell and alternating + and – signs until you can complete a closed path back to the original cell. Make only horizontal and vertical moves.
  5. You may find it helpful to keep track of cells that have been evaluated by placing the cell evaluation value in the appropriate cell with a circle around it.

For instance, if a shift of one unit causes an increase of $5, total costs would be increased by $5 times the number of units shifted into the cell. Obviously, such a move would be unwise, since the objective is to decrease costs.

Evaluation path for cell 1-A.

The name stepping-stone derives from early descriptions of the method that likened the procedures to crossing a shallow pond by stepping from stone to stone.

It is important to mention that constructing paths using the stepping-stone method requires a minimum number of occupied cells. The number of occupied cells must equal the sum of the number of rows and columns minus 1 (full rank), or R+C-1, where R= number of rows and C= number of columns.

If there are too few occupied cells, the matrix is said to be degenerate.

Evaluation path for cell 2-A.

 

Evaluation path for cell 3-B.

 

Evaluation path for cell 2-D.

 

Evaluation path for cell 1-B.

 

Evaluation path for cell 1-C.

 

Evaluation result:

 

Cell

1A

2A

3B

2D

1B

1C

Cost

0

+12

-1

+11

0

-5

 

The fact that 1-A and 1-B are both zeros indicates that at least one other equivalent alternative exists. However, they are of no interest, because other cells have negative cell evaluations, indicating that the present solution is not an optimum.

Testing For Optimality --- MODI

Another method is the modified distribution method (MODI) The MODI procedure consists of these steps:

  1. Obtain index numbers for rows and columns using only completed cells. Note that there will always be at least one completed cell in each row and in each column.
    1. Begin by assigning a zero to the first row.
    2. Determine the column indices for completed cells in row 1 using the relationship: cell cost = column index + row index.
    3. Each new column value will permit the calculation of at least one new row value, and vice versa.
    4. Continue the computation until all rows and columns have index numbers.
  2. Obtain cell evaluations for empty cells using the relationship: cell evaluation = cell cost – (row index + column index).

Obtaining An Improved Solution

Evaluation path for cell 1-C and re-allocation of 10 units around the path.

Revised solution.

Evaluation of cell 1-A.

Evaluation of cell 1-B.

Evaluation of cell 2-A.

Evaluation of cell 2-D.

Evaluation of cell 3-B.

Evaluation of cell 3-C.

Special Problem --- Unequal Supply and Demand

When supply exceeds demand, this problem can be remedied by adding a dummy destination with a demand equal to difference between supply and demand. Unit shipping costs of the dummy cells are $0s.

The resulting numbers in the final solution indicate which resource(s) will hold the extra units or will have excess capacity.

A similar situation exists, when demand exceeds supply.

Note: When using the intuitive approach, if a dummy row or column exists, make assignments to dummy cell(s) last.

Special Problem --- Degeneracy

Degeneracy exists, when there are too few completed cells to allow all necessary paths to be constructed. Because degeneracy could occur in an initial solution or in subsequent solutions, it is necessary to test for degeneracy after each iteration using R+C-1.

Degeneracy can be remedied by placing a very small quantity, represented by the small , into one of the empty cells and then treating it as a completed cell. The quantity is so small that it is negligible; it will be ignored in the final solution.

Some experimentation may be needed to find the best spot for , because not every cell will enable construction of evaluation paths for the remaining cells. Moreover, avoid placing  in a minus position of a cell path that runs out to be negative because reallocation requires shifting the smallest quantity in a minus position. Since the smallest quantity is , which is essentially zero, no allocation is possible.

Computer Solutions

where

supply (rows)

demand (columns)

If supply and demand are not equal, add the appropriate dummy row or column to the table before writing the constraints.

Summary

Procedure:

  1. Make certain that supply demand are equal. If they are not, determine the difference between the two amounts. Create a dummy source or destination with a supply or demand equal to the difference so that demand and supply are equal.
  2. Develop an initial solution using the intuitive and low-cost approach.
  3. Check that the number of completed cells in equal to R+C-1. If it isn’t, the solution is degenerate and will require insertion of a minute quantity, , in one of the empty cells to make it a completed cell.
  4. Evaluate each of the empty cells. If any evaluation turns out to be negative, an improved solution is possible. Identify the cell that has the largest negative evaluation. Reallocate units around its evaluation path.
  5. Repeat steps (3) and (4) until all cells have zero or positive values. When the occurs, you have achieved the optimal solution.

For multiple location alternatives, the procedure involves working through a separate problem for each location being considered and then comparing the resulting total costs.

If other costs, such as production costs, differ among locations, these can be easily be included in the analysis, provided they can be determined on a per-unit basis.

Note that merely adding or subtracting a constant to all cost values in any row or column will not affect the optimum solution; any additional costs should only be included, if they have a varying effect within a row or column.

If profits are used in place of costs, each of the cell profits can be subtracted from the largest profit, and the remaining values (opportunity costs) can be treated in the same manner as shipping costs.

Other applications include assignment of personnel or jobs to certain departments, capacity planning, and transshipment problems.