Lecture 12: Transportation Introduction

1. Lecture 12: Transportation Introduction AGEC 352 Spring 2011 – March 7 R. Keeney

2. Units in the equations of a model • Setup of the fertilizer mix model and getting the right coefficients. • First step: Identify the units for the activity definitions. • Tons of stock fertilizer (F1, F2, F3, F4) • Second step: Identify the units for the right hand side of constraints. • Tons of nutrient element (Nitrogen etc.)

3. Units continued • 1: Tons of F1 • 2: Tons of N • How many tons of N are in 1 ton of F1? • The answer to that is the coefficient. • These can be changed but you have to keep everything consistent • 1: 100 tons of F1 • 2: Lbs. of N • How mant lbs. of N are in 100 tons of F1?

4. Another Example • Farm problem focused on corn growing • Corn acres planted and harvested • Bushels of corn marketed • Bushels of corn put in storage • Bushels of corn fed to hogs • Requires a constraint that converts corn acres harvested to bushels of corn • How many bushels are in an acre of corn? • Yield (bushels/acre)

5. Units and Specification • For almost every type of problem units can be an issue • One type where it is typically not is the transportation problem • General name for any problem where activities are defined by movement of products rather than their production or use.

6. Transportation coefficients • Source supply: 100 units of product • Ship no more than this amount • Destination demand: 60 units of product • Ship no less than this amount • Activity = ship from source to destination. • A unit at the source converts exactly to a unit at the destination, making the coefficient = 1. • But is this true?

7. Commodity Properties • Based on final use • Form • Products are converted from original to one or more consumable types. • Time • Products are inventoried converting them from current to future consumption possibilities. • Place • Products are moved converting them to consumption possibilities at another location.

8. Commodity Properties • Based on final use • Form • Products are converted from original to one or more consumable types. • Time • Products are inventoried converting them from current to future consumption possibilities. • Place • Products are moved converting them to consumption possibilities at another location.

9. Classes of problems • Production type model: Basic resources are converted to consumable or saleable products. • Ex. Labor and lumber to make chairs & tables. • Blending type model: Basic consumables are blended together to meet requirements. • Ex. Combine fertilizers together to make a new product with different composition.

10. Models to date have been about form, now we deal with place • Company has two plants and three warehouses (all in different locations) • Must transport the output of the plants to the warehouses • Production capacity is limited at each plant • Demand at each warehouse is limited and each warehouse location faces a different price

11. Transportation Problem Source 1 Destination 1 Source 2 Destination 2 All material must be moved from a source to a destination. Decision variables have two dimensions (from, to) = (source, dest.) Objective coefficients have two dimensions (from, to) = (s,d). Notation P(1,2) = profit per unit from shipping from S1 to D2. X(1,2) = amount moved from shipping from S1 to D2. P(1,2)*X(1,2) = total profit from shipping from S1 to D2. Summing all P*X’s gives total profit for firm. Destination 3

12. Matrix Formulation Activities Matrix Objective Coefficient Matrix

13. Costs and Objective Values