Manufacturing companies and some service organizations often find it necessary to maintain proximity to their markets and also to input sources. For manufacturing companies, the input sources may be raw materials, power, water, and so on. For service organizations, the input source may be a skilled labor pool for example, companies such as Silicon Graphics specializing in computer software and hardware design.
The allocation problem is then to find the quantity of raw material each supply source should be supplying to each plant, as well as the quantity of finished goods each plant:
should be supplying to each customer. For the single product case, this problem may be set up as a transportation model and hence may be solved rather easily (Das and Heragu 1988).
2.7.1. Two-Stage Transportation Model
We consider an allocation model that has two stages of distribution. We formulate a linear programming (LP) model for this problem and show how a corresponding transportation tableau may be set up. The ideas are subsequently illustrated in a numeric example.
Consider this notation:
Si capacity of supply source i, where i = 1. 2,..., p Pj capacity of plant j, where; = 1, 2, ..., q
Dk demand at customer k, where k= 1, 2, ..., r
cij cost of transporting one unit from supply source i to plant j djk cost of transporting one unit from plant j to customer k
xij number of units of raw material shipped from supply source i to plant j yjk number of units of product shipped from plant j to customer k
Suppliers Plants (Warehouses) Customers
S1 P1 D1
S2 P2 D2
S3 P3 D3
. . .
. . .
. . .
Sp Pq Dr
p.q + q.r
This is the LP model:
xij yjk
Minimize
∑∑ ∑∑
shipments. Constraint (2) ensures that the raw material shipped out from each supply source does not exceed its capacity limits. Constraint (3) ensures that the raw material shipment received from all the supply sources at each plant does not exceed its capacity limits. Constraint (4) requires that the total amount of finished products shipped from the plants to each customer be sufficient to cover the demand.Constraint (5) is a material balance equation ensuring that all the raw material that comes into each plant is shipped out as finished product to customers.
Notice that we are implicitly assuming that a unit of finished product requires one unit of raw material. If this is not the case, we can adjust the model easily, as discussed in Das and Heragu (1988).
For the above model to be transformed into an equivalent transportation model, either the plants or the raw material supply sources (but not both) must have limited capacity. (Otherwise, the problem cannot be set up as a transportation model and hence we cannot use the well-known transportation algorithm.
The problem may be formulated in the above model, however, and solved via the simplex algorithm.) Depending on whether supply sources or plants have limited capacities and whether supply exceeds demand, these four cases arise:
1. Supply source capacity is unlimited, plant capacity is limited, and total plant capacity is greater than total demand.
2. Supply source capacity is unlimited, plant capacity is limited, and total demand exceeds total plant capacity.
3. Plant capacity is unlimited, supply source capacity is limited, and total supply source capacity exceeds total demand.
4. Plant capacity is unlimited, supply source capacity is limited, and total demand exceeds total supply source capacity.
Example:
Two-stage distribution problem: RIFIN Company has recently developed a new method of manufacturing a type of chemical. The method involves refining a certain raw material that can be obtained from four overseas suppliers, A, B, C, and D, who have access to the four ports at Vancouver, Boston, Miami, and San Francisco, respectively. RIFIN wants to determine the location for plants that will refine the material. Once refined, the chemical will be transported via trucks to five outlets located in Dallas, Phoenix, Portland, Montreal, and Orlando.
After an initial study, the choice of location for RIFIN's refineries has been narrowed down to Denver, Atlanta, and Pittsburgh. Assume that one unit of the raw material is required to make one unit of the chemical. The amount of raw material that can be obtained from suppliers A, B, C, and D and the amount of chemical required at the five outlets are given in the following table (a). The cost of transporting the raw material from each port to each potential refinery and the cost of trucking the chemical to outlets are provided in tables (b) and (c), respectively. Determine the locations of RIFIN's refining plants, the capacities at these plants, and the distribution pattern for the raw material and processed chemical.
(a) Supply and demand for four sources and five outlets
(b) Inland raw material transportation cost
(c) Chemical trucking cost
Solution:
Above figure is a pictorial representation of the RIFIN problem. We can reasonably assume that there is no practical limit on the capacity of the refineries at any of the three locations, Atlanta, Denver, and Pittsburgh, because the refineries have not been built yet. This assumption allows us to use the two-stage transportation method.
The transportation problem may be solved to yield the solution (with a total cost of
$65.400) in the following figure, which indicates that refineries should be built at all three locations.