WOOD 492 MODELLING FOR DECISION SUPPORT

1. WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 6 LP Assumptions

2. Last Week • Solving LPs with the Excel Solver • LP Matrix format Wood 492 - Saba Vahid

3. Assumptions of LP • For a system to be modelled with an LP, 4 assumptions must hold: Proportionality, Additivity, Divisibility, and Certainty • Proportionality: Contribution of each activity (decision variable) to the Obj. Fn. is proportional to its value (represented by its coefficient in the Obj. Fn.), • e.g. Z=3x1+2x2 , when x1 is increased, its contribution to the Obj. is always increased three-fold (3x1). • invalid assumption: e.g. manufacturing a product has a startup-cost: • If there is no products made (x=0), the total profits would be zero (Z=c.x=0), but if any products are made (x>0), the profits are not proportional to the volume of products (Z=c.x-d), where d is the start-up cost. Wood 492 - Saba Vahid

4. Assumptions of LP- Cont’d • Additivity: Every function in an LP (Obj. Fn. or the constraints) is the linear sum of individual contributions of the respective activities (decision variables) • e.g. x1+12x2 <=100, is the sum of two linear functions, each showing the level of contribution of a variable (x1 or x2) to the constraint • invalid assumption: e.g. the products are complementary • profits of the combined production is more than the sum of the individual production profits (Z=cx1+dx2+ x1.x2) Extra, nonlinear term Wood 492 - Saba Vahid

5. Assumptions of LP- Cont’d • Divisibility: Activities can be run at fractional level, i.e., decision variables can have any level (not just integer values). • e.g. x1=33.3, x2=0.01 • invalid assumption: no fractional values for decision variables allowed • e.g. assigning workers to different processes, scheduling shifts, building roads. • Certainty: Parameter values (coefficients in the functions) are known with certainty • e.g. required hours to produce each product is known with certainty. • invalid assumption: e.g.when production costs are not known with certainty • This happens commonly and therefore sensitivity analysis is an important part of any LP solution analysis. Wood 492 - Saba Vahid

6. Examples of Objective functions • Max profit • Min costs • Max utility • Max turnover • Max Return on Investment • Max Net Present Value • Min number of employees • Min redundancy • Max customer satisfaction Wood 492 - Saba Vahid

7. Examples of LP constraints • Upper & lower bounds (on raw material or products) • Productive capacity • Raw material availability • Marketing demands & limitations • Material balance (for balancing the input-output conversions within the model) • Production ratio (link between the production of two or more products) Wood 492 - Saba Vahid

8. Example: Cut-Fill areas for road building • In order to even out the road: earth should be transferred from cut areas (C1-C3) or borrow pit to Fill areas (F1-F4) or waste pit. Wood 492 - Saba Vahid

9. Example: Cut-Fill areas • What is our objective? • Minimize total earth transfer costs ($) • What are our decision variables? • How much earth (m3) to transfer from each cut area or borrow pit to each fill area or waste site • What are our constraints? • The available volume of earth in the cut areas (m3) • The required volume of earth for fill areas (m3) • Naming our variables: • C1F1: volume of earth (m3) transferred from C1 to F1 • C1W: volume of earth (m3) transferred from C1 to waste site • BF1: volume of earth (m3) transferred from the borrow pit to F1 Cut-Fill Example Wood 492 - Saba Vahid

10. Next Class • More formulation examples • Preview of the lab Wood 492 - Saba Vahid