1. Facility Location Decisions
Classifying location decisions
Driving force (critical factor - traffic, labor rates, emergency facilities, obnoxious facilities)
Number of facilities
Discrete vs. continuous choices
Data aggregation
Time Horizon

2. Facility Location
Rent Curve - The rent of land is a decreasing function of the distance to the market
Weight gaining vs. weight losing industries
Weight losing should locate close to raw materials
Weight gaining should locate close to market
Tapered (concave) transportation costs
The derivative of total transportation cost is non-increasing with the distance to the market (holds for inbound and outbound costs)
Optimal solution will always locate either at raw materials or at market (extreme point solution)

3. Single Facility Location Model
This model assumes a known set, I, of source and demand points, each with known demand volumes, Vi, and transportation rates, Ri.
The objective is to locate the facility at the point that minimizes total transportation cost, TC:
Let di denote the distance from the facility to demand point i.
Min
subject to:
The decision variables are the coordinates of the facility
Xi, Yi denote the coordinates of demand point i.

4. Single Facility Location Model
Differentiating TC w.r.t. and setting the result equal to zero gives the ‘center of gravity’:

5. Single Facility Location Model
Note however, that are given in terms of di, which is a function of
An algorithm that will converge to the optimal
This continuous problem is often called the Weber problem
These problems are restrictive because they assume continuity of location and straight-line distances
Also, only variable distance related costs are considered

6. General Facility Location Model
The general facility location problem considers the simultaneous location of a number of facilities
Notation:
I - Set of customers, indexed by i.
J - Set of facilities, indexed by j.
di - demand of customer i.
cij - cost of transporting a unit from facility j to customer i.
Fj - fixed cost of creating facility j.
xij - variable for flow from facility j to customer i.
Yj - binary variable that equals 1 if we create facility j, 0 otherwise
sj - capacity of facility j.

7. Uncapacitated Facility Location Model Formulation

8. Capacitated Facility Location Model Formulation

9. Location Decisions and Risk Pooling
Suppose we must serve n independent markets with a single product, and each market has average demand per period of D and standard deviation  (we neglect lead times for simplicity)
Assume we have a service level policy to ensure that the probability of not stocking out in each period equals 
Suppose we serve each market using a single inventory stocking location.
The standard deviation of demand as seen by the single stocking location in each period equals
If demand is normally distributed, the safety stock required at the single location equals z

10. Location Decisions and Risk Pooling
Suppose, at the other extreme, we place an inventory stocking location in each of the n markets
Each stocking location will need to hold z to meet the service level requirement
The system-wide safety stock equals zn > z
This example illustrates the risk-pooling effects of location decisions
The more stocking locations we have, the more duplication in safety stock we have
The single location, however, will incur the maximum transportation costs, while n locations should minimize the transportation costs

11. Location Decisions and Risk Pooling
The safety stock costs and transportation costs are at odds with each other
We need to strike a balance between the two
Models for this decision are currently limited (Prof. Shen has worked on a model that addresses this tradeoff)
However, this simple analysis can provide strong insights
If inventory costs dominate transportation costs (as in expensive computing chips), we are driven to have less stocking points; if transport costs dominate (as in coal), then we are driven to have more stocking locations
One thing not included in the analysis is delivery lead time and its impact on service levels – obviously more locations closer to markets can respond much more quickly to customers

12. Supply Chain Design Model
The objective of this model is to determine the warehouse and plant configuration that minimizes total costs for production and distribution of multiple products.
Based on Geoffrion and Graves, 1974, “Multicommodity distribution system design by Benders decomposition,” Management Science, v20, n5. (see Tech. Suppl., Ch. 13)
Notation:
i - index for commodities
j - index for plants
k - index for warehouses
l - index for customer zones

13. Supply Chain Design Model
Notation (continued):
Sij - production capacity for commodity i at plant j.
Dil - demand for commodity i in customer zone l.
- min and max total throughput for warehouse k.
fk - fixed part of annual costs for owning and operating warehouse k.
vk - variable unit cost of throughput for warehouse k.
Cijkl - average unit cost of producing, handling, and shipping commodity i from plant j through warehouse k to customer zone l.
Xijkl - amount of commodity i flowing from plant j through warehouse k to customer zone l.
ykl - binary variable = 1 if warehouse k serves customer zone l, 0 otherwise
zk - binary variable = 1 if warehouse k is open, 0 otherwise.

14. Supply Chain Design Model Formulation

15. Network Planning
Network planning refers to assessing or reassessing the configuration of facilities, commodities, and flows currently used to satisfy demand
Network planning data checklist:
List of all products
Customer, stocking point, and source point locations
Demand by customer location
Transportation rates
Transit times, order transmittal times, and order fill rates
Warehouse rates and costs
Purchasing/production costs
Shipment sizes by product
Inventory levels by location, by product, control methods
Order patterns by frequency, size, season, content
Order processing costs and where they are incurred

16. Network Planning Data Checklist (continued):
Capital cost
Customer service goals
Available equipment and facilities and their capacities
Current distribution patterns (flows)
Note that many of these are decision variables
Accumulating these data usually results in improvements by uncovering anomalies
We must decide our network design strategy:
Specify minimum service levels
Specify shortage costs and minimize cost
Levels of acceptable aggregation of demand
Optimization vs. heuristic methods
Which areas require the most accuracy and attention?