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Lecture 13: Transportation Introduction

Lecture 13: Transportation Introduction. AGEC 352 Spring 2012 – March 5 R. Keeney. 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.

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Lecture 13: Transportation Introduction

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  1. Lecture 13: Transportation Introduction AGEC 352 Spring 2012 – March 5 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 many 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. • Should it be 1?

  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. 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.

  9. 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

  10. 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

  11. Matrix Formulation Activities Matrix Objective Coefficient Matrix

  12. Costs and Objective Values

  13. Lab 6 Problem *Could compare these routes or compare sources and destinations *Statistician might average costs from a source or to a destination *What should we do?

  14. Information for a Model *All of the locations are not the same, they have different capacities and requirements. Simple averaging would be incorrect…

  15. Problem Size • Transportation Problem • S = # of sources • D = # of destinations • Then • SxD = # of decision variables • S+D = # of constraints (not counting non-negativity constraints) • Problems can get big quickly…

  16. Algebraic Simplification *We use subscripts to keep track. We use s to indicate a source and d a destination. *X23 is a shipment from source 2 to destination 3

  17. Spreadsheet Setup • Three matrix approach • First • Unit cost coefficients (from the data) • Second • Decision variables (including consraints) • Third • Cost contributions (links the first two and determines the total cost)

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