1 utdallas.edu/~metin SC Design Facility Location Models

2 utdallas.edu/~metin Analytical Models for SC Design –Public sector deals with fairness and equity: minimizes the distance to the furthest customer –Location of emergency response units u Demand allocation »Distance vs. Price vs. Quality: Recall Hotelling model u Demand pattern over a geography: Discrete vs. Continuous u Distances »Euclidean vs. Rectilinear »Triangular inequality u Objective functions –Private sector deals with total costs: minimizes the sum of the distances to the customers Customer 1 Customers 2-10 Public sector locates Private sector locates

3 utdallas.edu/~metin Network Optimization Models u Allocating demand to production facilities u Locating facilities u Determining capacity – some in aggregate planning module Key Costs: Fixed facility cost Transportation cost Production cost

4 utdallas.edu/~metin A transportation network Defined by data K, D and c D2D2 m demand points/markets D4D4 D3D3 D1D1 n supply points/plants K1K1 K2K2 K3K3 c 11 c 12 c 14 c 22 c 23 c 31 c 32 c 34

5 utdallas.edu/~metin Demand Allocation Model: Transportation Problem Which market is served by which plant? Which supply sources are used by a plant? Given m demand points, j=1..m with demands D j Given n supply points, i=1..n with capacity K i Each unit of shipment from supply point i to demand point j costs c ij Send supplies from supply points to demand points x ij = Quantity shipped from plant site i to customer j Objective Constraints Decision Variables

6 utdallas.edu/~metin A transportation network Defined by data K, D, c and f m demand points D4D4 D3D3 D2D2 D1D1 n supply points f 1,K 1 f 2,K 2 f 3,K 3 c 11 c 12 c 14 c 22 c 23 c 31 c 32 c 34 Which supply point operates? y 1 =yes or no y 2 =yes or no y 3 =yes or no

7 utdallas.edu/~metin Plant Location with Multiple Sourcing Which market is served by which plant? Which supply sources are used by a plant? None of the plants are open, a cost of f i is paid to open plant i At most k plants will be opened y i = 1 if plant is located at site i, 0 otherwise x ij = Quantity shipped from plant site i to customer j

8 utdallas.edu/~metin Plant Location with Single Sourcing Each customer/market has exactly one supplier Which market is served by which plant? Which supply sources are used by a plant? None of the plants are open, a cost of f i is paid to open plant i y i = 1 if plant is located at site i, 0 otherwise x ij = 1 if market j is supplied by factory i, 0 otherwise Can a plant satisfy the demand of two or more customers with this formulation?

9 utdallas.edu/~metin Network Optimization Models vs. Gravity Location Models u Both Network Optimization Models and Gravity Location Models determine the optimal location of a new facility or facilities. u In the Network Optimization Models, the new location must be one of the discrete set of potential locations. –That is why they are more amenable to be solved by a linear program. u In the Gravity Location Models, the new location can be anywhere on a continuous line or a rectangle (coordinate system); we do not need to make a list of potential locations in advance. u In the Gravity Location Models, the distance is modeled explicitly and generally as Euclidean distance. –That is why we end up with a nonlinear objective and cannot use a linear program. –We shall develop alternative methods.

10 utdallas.edu/~metin Gravity Methods for Location Ton Mile-Center Solution Given n delivery locations, i=1..n, a i, b i : Coordinates of delivery location i d i : Distance to delivery location i F i : Annual tonnage to delivery location i Locate a warehouse at (x,y)

11 utdallas.edu/~metin Gravity Methods for Location – A Variant Change the distance Given n delivery locations, i=1..n, a i, b i : Coordinates of delivery location i d i : Distance to delivery location i F i : Annual tonnage to delivery location i Locate a warehouse at (x,y)

12 utdallas.edu/~metin Case Study: Applichem Demand Allocation

13 utdallas.edu/~metin Demand Allocation with Duties Demand Mexico Canada Frankfurt Gary Sunchem Mexico30 Canada26 Latin America 160 Europe 200 U.S.A 264 Japan Venezuela 220 Capacity

14 utdallas.edu/~metin Ven Annual Cost = $72,916,400 Mex Can Fra Sun Mex Can LatAm Eur U.S.A Jap Gar Demand Allocation with Duties without Duties Without duties, Venezuela and Canada plants are closed and Frankfurt satisfies the excess Canada, Latin America and USA demand. There is consolidation without duties. Mex Can LatAm Mex Can Ven Fra Gar Sun Eur U.S.A Jap Annual Cost = 66,328,100

15 utdallas.edu/~metin Value of Adding 0.1 M Pounds Capacity (1982) Capacity should be evaluated as an option and priced accordingly. Shadow (dual) prices from LP tells you where to invest.

16 utdallas.edu/~metin Summary u Network optimization models u Gravity location models