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

2. Last Week Solving LPs with the Excel Solver LP Matrix format Sept 17, 2012Wood 492 - Saba Vahid2

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=3x 1 +2x 2, when x 1 is increased, its contribution to the Obj. is always increased three-fold (3x 1 ). –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. Sept 17, 2012Wood 492 - Saba Vahid3

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. x 1 +12x 2 <=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=cx 1 +dx 2 + x 1.x 2 ) Sept 17, 2012Wood 492 - Saba Vahid4 Extra, nonlinear term

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. Sept 17, 2012Wood 492 - Saba Vahid5

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 Sept 17, 2012Wood 492 - Saba Vahid6

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) Sept 17, 2012Wood 492 - Saba Vahid7

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. Sept 17, 2012Wood 492 - Saba Vahid8

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 Sept 17, 2012Wood 492 - Saba Vahid9 Cut-Fill Example

10. Next Class More formulation examples Preview of the lab Sept 17, 2012Wood 492 - Saba Vahid10