Lecture 12: Transportation Introduction AGEC 352 Spring 2011 – March 7 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.  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.)

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?

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)

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.

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?

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.

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.

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.

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

Transportation Problem Source 1 Source 2 Destination 1 Destination 2 Destination 3 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.

Matrix Formulation Dest. 1Dest. 2Dest. 3 Source 1X(1,1)X(1,2)X(1,3) Source 2X(2,1)X(2,2)X(2,3) Dest. 1Dest. 2Dest. 3 Source 1P(1,1)P(1,2)P(1,3) Source 2P(2,1)P(2,2)P(2,3) Activities Matrix Objective Coefficient Matrix

Costs and Objective Values Warehouse 1 (12 $ SP) Warehouse 2 (14 $ SP) Warehouse 3 (15$ SP) Plant 1Cost = 8 Profit = 4 Cost = 10 Profit = 4 Cost = 12 Profit = 3 Plant 2Cost = 7 Profit = 5 Cost = 9 Profit = 5 Cost = 11 Profit = 4