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# Linear programming

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### Linear programming

1. 1. Linear Programming Umesh Hodeghatta Rao XIMB
2. 2. Linear Equation• X+Y=4• X–Y=2Solve X and Y• 2X = 6• X=3• Y=1
3. 3. What is Linear Programming?• Linear programming is widely used mathematical technique designed to help in the planning and allocation of resources.• Resources – Machinery, time, material, labour, and money• Why Linear? – All equations and inequalities should have Linear relationship – Programming is iterative process to derive optimal solution
4. 4. Linear Programming• A Linear Programming model seeks to maximize or minimize a linear function – Cost or profit• Subject to a set of linear constraints.• The linear model consists of the following components: – An objective function. – A set of decision variables. – A set of constraints.
5. 5. Application of Linear Programming• There are well-known applications in: – Manufacturing – Marketing – Finance (investment) – Advertising – Agriculture
6. 6. Typical Objective Function• Optimal allocation of inventory to minimize production cost• Creating job schedule or personnel schedule to meet business needs• Allocating funds in investment portofolio to maximize returns
7. 7. Lab Session
8. 8. Example - 1• Symphony manufactures two toy doll models: – Ray – Zappy• Resources are limited to – 1000 pounds of special plastic. – 40 hours of production time per week.
9. 9. • Marketing requirement – Total production cannot exceed 700 dozens. – Number of dozens of Rays cannot exceed number of dozens of Zappys by more than 350.• Technological input – Rays requires 2 pounds of plastic and 3 minutes of labor per dozen – Zappys requires 1 pound of plastic and 4 minutes of labor per dozen
10. 10. • The current production plan calls for: – Producing as much as possible of the more profitable product, Ray (\$8 profit per dozen). – Use resources left over to produce Zappys (\$5 profit per dozen), while remaining within the marketing guidelines
11. 11. • The current production plan calls for: – Producing as much as possible of the more profitable product, Ray (\$8 profit per dozen). – Use resources left over to produce Zappys (\$5 profit per dozen), while remaining within the marketing guidelinesThe current production plan consists of: 8(450) + 5(100) Rays = 450 dozen Zappy = 100 dozen Profit = \$4100 per week
12. 12. Symphony Manufacturers• Objective Function: – Weekly profit, to be maximized• Decisions variables: – X1 = Weekly production level of S Rays (in dozens) – X2 = Weekly production level of Zappys (in dozens).
13. 13. Linear EquationMax 8X1 + 5X2 (Weekly profit) subject to 2X1 + 1X2 ≤ 1000 (Plastic) 3X1 + 4X2 ≤ 2400 (Production Time) X1 + X2 ≤ 700 (Total production) X1 - X2 ≤ 350 (Mix) Xj> = 0, j = 1,2 (Nonnegativity)
14. 14. Using Excel Solver to Find an Optimal Solution and Analyze Results Rays ZappyDozens 320 360 Total LimitProfit 8 5 4360Plastic 2 1 1000 <= 1000Prod. Time 3 4 2400 <= 2400Total 1 1 680 <= 700Mix 1 -1 -40 <= 350
15. 15. Example 2• The Acme company manufactures two products – widgets and gadgets. The company earns \$2 per box for widgets and \$3 per box for gadgets. Each product assembled and packaged. It takes 9 minutes to assemble a box of widget and 15 minutes to assemble a box of gadgets. The packaging department can package a box of widgets in 11 minutes, while a box of gadgets takes 5 minutes. Maximum of 1800 hours available in both the assembly and packaging department. Acme wants to know the product combination, of both widgets and gadgets , that will maximize the profit.
16. 16. Acme problem• Maximize Z = 2X1 + 3X2• Subjected to:• 9X1 + 15 X2 <= 1800 * 60 minutes (assemble time constraints)• 11X1 + 5X2 < = 1800*60 minutes (packaging time constraints)• X1,x2 => 0
17. 17. Drug company problem• A drug company produces six different drugs at its plant. Production of each drug requires labor and raw material.• 4500 hours of labor and 1600 pouds of raw material available Drugs X1 X2 X3 X4 X5 X6 Z 1 1 1 1 1 1 Labor 6 5 4 3 2.5 1.5 Raw Material 3.2 2.6 1.5 0.8 0.7 0.3 seliing price/unit \$12.50 \$11 \$9 \$7 \$6 \$3Manufacturing cost \$6.50 \$5.70 \$3.60 \$2.80 \$2.20 \$1.20 Demand 960 928 1041 977 1084 1055