The need for location decisions:
The nature of location decisions:
Location decision making procedure:
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:
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.
Graphical assumption:
Graphical Procedure:
where
FC= fixed cost
VC= variable cost per unit
Q= quantity or volume of output
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 establishes a composite value for each alternative that summarizes all related factors.
The following procedure is used to develop a factor rating:
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 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.
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:
The information is arranged into the following transportation table.
Assumptions are:
Major steps in solving the transportation using the table are:
The procedure involves these steps:
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
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.
Another method is the modified distribution method (MODI) The MODI procedure consists of these steps:
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.
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.
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.
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.
Procedure:
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.