1. WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 2 Introduction to Linear Programming

2. Last Class Introduction to Operations Research Examples of OR in forestry Introduction to mathematical models –Objective function, decision variables, constraints Sept 7, 2012Wood 492 - Saba Vahid2

3. Example: Custom Cabinets company Use excess capacity for 2 new products: Pine desks & Alder hutches Has three departments that are partially committed to producing existing products Wants to determine how many units of each new product can be produced each week by using the excess capacity of departments to generate the highest profits Sept 7, 2012Wood 492 - Saba Vahid3 DepartmentCapacity per unitAvailable capacity per week Pine deskAlder hutch Solid wood0.25012 Panel00.25 Finishing0.250.518 Profit per unit $40$50 Objective Decision variableConstraints

4. Maximize:x 1 = number of desks/week 40 x 1 + 50 x 2 x 2 = number of hutches/week Formulating the Linear Program (LP) What do you want to maximize or minimize? Profits What are the constraints? Available capacity Subject to: 0.25x 1 <=12(Solid Wood Capacity) 0.20x 2 <=5(Panel Capacity) 0.25x 1 +0.50x 2 <=18(Finishing Capacity) x 1 >=0 x 2 >=0 Sept 7, 20124Wood 492 - Saba Vahid Linear

5. Note that because of the inequalities there are many feasible solutions. You have to find the best one. Maximize: $40x 1 + $50x 2 Subject to: 0.25x 1 <=12(Solid Wood Capacity) 0.20x 2 <=5(Panel Capacity) 0.25x 1 +0.50x 2 <=18(Finishing Capacity) x 1 >=0 x 2 >=0 Solving the LP by trial and error Sept 7, 20125Wood 492 - Saba Vahid Try x 1 = 10, x 2 = 5 Z=$650 All constraints are satisfied Custom Cabinet LP1

6. Matrix format for LP Sept 7, 20126Wood 492 - Saba Vahid Custom Cabinet LP2

7. Sept 7, 2012Wood 492 - Saba Vahid7 Feasible Region Custom Cabinet LP2

8. Sept 7, 2012Wood 492 - Saba Vahid8 Custom Cabinet LP2 Z=1000 Z=2000 x 1 =48 0.25*48 + 0.5* x 2 =18 x 2 =12

9. Example: Whitt Window Company (prob. 3.1-7) Has three employees Makes two types of windows: wood-framed and aluminium-framed Profits per frame: $180 for wood-framed, $90 for aluminum-framed Dough makes a maximum of 6 wood frames per day Linda makes a maximum of 4 aluminium frames per day Bob forms and cuts a maximum of 48 ft 2 of glass per day Each wood-framed window uses 6 ft 2 glass Each aluminum-framed window uses 8 ft 2 glass How many windows per day to make in order to maximize profits? Sept 7, 2012Wood 492 - Saba Vahid9 Objective Decision variables Constraints Whitt Windows LP

10. Next Week Solving an LP with Excel Solver Simplex Algorithm Sept 7, 2012Wood 492 - Saba Vahid10