IE 8580 Module 2: Transportation in the Supply Chain Lecture 2.3: Transportation Network Models

Capacitated Plant Location Model Where do you place plants to serve customers? Plants in each region? Plants in a few regions? Conflicting factors Transportation and avoiding duties vs. reducing economies of scale There are fixed and variable costs IE 8580, mason@clemson.edu

Capacitated Plant Location Model Market 1 Plant 2 Market 2 … … Plant n Market m Fixed annual cost to keep a plant open Cost per unit to transport IE 8580, mason@clemson.edu

Variables and Parameters n = number of potential plants m = number of markets Dj = annual demand in market j Ki = maximum capacity in plant i fi = annualized fixed cost of plant i cij = cost to produce and ship 1 unit from factory i to market j Decision variables yi = 1 if plant i is open; 0 otherwise xij = quantity shipped from factory i to market j IE 8580, mason@clemson.edu

… … Graphically… c11, x11 K1, f1, y1 Plant 1 Market 1 D1 c12, x12 Kn, fn, yn Plant n cnm, xnm Market m Dm IE 8580, mason@clemson.edu

Mathematical Model IE 8580, mason@clemson.edu

Flow Allocation Model Brownfield design in which the locations of the plants, warehouses, and customers are known and fixed. The possible modes of transportation are fixed and the costs are deterministic How do you allocate the flow of materials to meet customer demand with minimum cost? IE 8580, mason@clemson.edu

Plants, warehouses, and customers are fixed Flow Allocation Model Plant 1 Warehouse 1 Customer 1 Plant 2 Warehouse 2 Customer 2 … … … Plant I Warehouse J Customer K Plants, warehouses, and customers are fixed Demand and capacities known IE 8580, mason@clemson.edu

Two-Stage Transportation Problem xij = flow from plant i to warehouse j yjk = flow from warehouse j to customer k cij = cost to ship one unit from i to j c’jk = cost to ship one unit from j to k Si = capacity of plant i Dk = demand of customer k IE 8580, mason@clemson.edu

Two-Stage Transportation Problem c11, x 11 c’11, y 11 D1 S1 Plant 1 Warehouse 1 Customer 1 S2 Plant 2 Warehouse 2 Customer 2 D2 … … … SI Plant I Warehouse J Customer K DK cIJ, x IJ c’JK, x JK IE 8580, mason@clemson.edu

Network Flow Model IE 8580, mason@clemson.edu

This is how you get the minimum cost we discussed last time! Single product, two plants, two warehouses, 3 customers Plant capacities: 140,000 and 60,000 Customer demands: 50,000, 100,000, and 50,000 Unit distribution costs ($) P1 P2 C1 C2 C3 W1 4 3 5 W2 2 1 IE 8580, mason@clemson.edu

It’s the same problem from before! 50 3 140 Customer 1 4 5 5 Plant 1 W/H 1 100 2 4 1 Customer 2 2 2 60 50 Plant 2 W/H 2 Customer 3 IE 8580, mason@clemson.edu 13

Mathematical Model (Long Form) IE 8580, mason@clemson.edu

Locating Plants and Warehouses Simultaneously c1hi Fi c2ie fe c3ej Supplier 1 Plant 1 Warehouse 1 Market 1 S1 K1 W1 D1 Supplier 2 Plant 2 Warehouse 2 Market 2 S2 K2 W2 D2 … … … … Supplier l Plant n Warehouse t Market m SI Kn We Dm x1hi yPi x2ie yWe x3ej IE 8580, mason@clemson.edu

Mathematical Model IE 8580, mason@clemson.edu