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WOOD 492 MODELLING FOR DECISION SUPPORT

WOOD 492 MODELLING FOR DECISION SUPPORT. Lecture 2 Introduction to Linear Programming. Last Class. Introduction to Operations Research Examples of OR in forestry Introduction to mathematical models Objective function, decision variables, constraints. Example: Custom Cabinets company.

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WOOD 492 MODELLING FOR DECISION SUPPORT

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  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 Wood 492 - Saba Vahid

  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 Constraints Decision variable Objective Wood 492 - Saba Vahid

  4. Formulating the Linear Program (LP) • What do you want to maximize or minimize? Profits • What are the constraints? Available capacity Maximize: x1 = number of desks/week 40 x1+ 50 x2 x2= number of hutches/week Subject to: 0.25x1 <= 12 (Solid Wood Capacity) 0.20x2 <= 5 (Panel Capacity) 0.25x1 + 0.50x2 <= 18 (Finishing Capacity) x1 >= 0 x2 >= 0 Linear Wood 492 - Saba Vahid

  5. Solving the LP by trial and error Try x1= 10, x2= 5 Z=$650 All constraints are satisfied Maximize: $40x1+ $50x2 Subject to: 0.25x1 <= 12 (Solid Wood Capacity) 0.20x2 <= 5 (Panel Capacity) 0.25x1 + 0.50x2 <= 18 (Finishing Capacity) x1 >= 0 x2 >= 0 Note that because of the inequalities there are many feasible solutions. You have to find the best one. Custom Cabinet LP1 Wood 492 - Saba Vahid

  6. Matrix format for LP Custom Cabinet LP2 Wood 492 - Saba Vahid

  7. Feasible Region Custom Cabinet LP2 Wood 492 - Saba Vahid

  8. x1 =48 0.25*48 + 0.5* x2 =18 x2 =12 Z=2000 Z=1000 Custom Cabinet LP2 Wood 492 - Saba Vahid

  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 ft2 of glass per day • Each wood-framed window uses 6 ft2 glass • Each aluminum-framed window uses 8 ft2 glass • How many windows per day to make in order to maximize profits? Constraints Objective Decision variables Whitt Windows LP Wood 492 - Saba Vahid

  10. Next Week • Solving an LP with Excel Solver • Simplex Algorithm Wood 492 - Saba Vahid

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