Presentation on theme: "Facility Location Decisions"— Presentation transcript:
1 Facility Location Decisions Classifying location decisionsDriving force (critical factor - traffic, labor rates, emergency facilities, obnoxious facilities)Number of facilitiesDiscrete vs. continuous choicesData aggregationTime Horizon
2 Facility LocationRent Curve - The rent of land is a decreasing function of the distance to the marketWeight gaining vs. weight losing industriesWeight losing should locate close to raw materialsWeight gaining should locate close to marketTapered (concave) transportation costsThe 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.Minsubject to:The decision variables are the coordinates of the facilityXi, 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 ofAn algorithm that will converge to the optimalThis continuous problem is often called the Weber problemThese problems are restrictive because they assume continuity of location and straight-line distancesAlso, 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 facilitiesNotation: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 otherwisesj - capacity of facility j.
7 Uncapacitated 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 equalsIf 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 marketsEach stocking location will need to hold z to meet the service level requirementThe system-wide safety stock equals zn > zThis example illustrates the risk-pooling effects of location decisionsThe more stocking locations we have, the more duplication in safety stock we haveThe 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 otherWe need to strike a balance between the twoModels for this decision are currently limited (Prof. Shen has worked on a model that addresses this tradeoff)However, this simple analysis can provide strong insightsIf 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 locationsOne 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 commoditiesj - index for plantsk - index for warehousesl - 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 otherwisezk - binary variable = 1 if warehouse k is open, 0 otherwise.
15 Network PlanningNetwork planning refers to assessing or reassessing the configuration of facilities, commodities, and flows currently used to satisfy demandNetwork planning data checklist:List of all productsCustomer, stocking point, and source point locationsDemand by customer locationTransportation ratesTransit times, order transmittal times, and order fill ratesWarehouse rates and costsPurchasing/production costsShipment sizes by productInventory levels by location, by product, control methodsOrder patterns by frequency, size, season, contentOrder processing costs and where they are incurred
16 Network Planning Data Checklist (continued): Capital costCustomer service goalsAvailable equipment and facilities and their capacitiesCurrent distribution patterns (flows)Note that many of these are decision variablesAccumulating these data usually results in improvements by uncovering anomaliesWe must decide our network design strategy:Specify minimum service levelsSpecify shortage costs and minimize costLevels of acceptable aggregation of demandOptimization vs. heuristic methodsWhich areas require the most accuracy and attention?