Linear Programming. Formulation and Graphical Solution

Transcription

1 Linear Programming Formulation and Graphical Solution

2 A Two Variable Model Simple LP with two decision variables Two dimensional model is hardly useful in the real world systems which normally encompass hundreds or thousands variables and constraints

3 A Two Variable Model: The Problem The Reddy Mikks Company owns a small paint factory that produces both interior and exterior house paints Two basic raw materials, A and B are used to manufacture the paint

4 A Two Variable Model: The Problem The maximum availability of A = 6 tons/day The maximum availability of B = 8 tons/day

5 The Table Raw Material Tons of Raw Material per Ton of Paint Exterior Interior Maximum Availability (tons) A B 2 1 8

6 The Problem Daily demand for interior paint cannot exceed that of exterior paint more than 1 ton Daily demand for interior paint is limited to 2 tons Price of interior paint = $2000/ton Price of exterior paint = $3000/ton

7 The Problem How much interior and exterior paint should the company produce daily to maximize gross income?

8 Construction of Mathematical Model What does the model seek to determine? What are the variables (unknowns) of the problems? What constraints must be imposed on the variables to satisfy the limitations of the modeled system? What is the objective (goal) that needs to be achieved to determine the optimum (best) solution from among all the feasible values of the variables?

9 Variables X E : tons produced daily of exterior paint X I : tons produced daily of interior paint

10 Objective Function Maximize the income Maximize z = 3X E + 2X I (thousand $)

11 Constraints: Usage Vs Availability (usage of raw material by both paint) (maximum raw material availability) X E + 2X I 6 (raw material A) 2X E + X I 8 (raw material B)

12 Constraints: Demand Restrictions (excess amount of interior over exterior paint) 1 ton/day X I X E 1 (demand for interior paint) 2 ton/day X I 2

13 Constraints: Non-negativity Restrictions Implicit constraints X I 0 (interior paint) X E 0 (exterior paint)

14 Complete Model To determine the values of X E : tons produced daily of exterior paint X I : tons produced daily of interior paint Objective Function to be satisfy: Maximize z = 3X E + 2X I Subject to these constraints X E + 2X I 6 (1) 2X E + X I 8 (2) X E + X I 1 (3) X I 2 (4) X I 0 (5) X E 0 (6)

15 Linearity Proportionality: contribution of each variable in the objective function or its usage of the resource be directly proportional to the level value of the variable Additivity: objective function is the direct sum of individual contributions of the different variables

16 Graphical Solution Plot the feasible space solution that satisfies all the constraints simultaneously Replace Variable: Slack Variable: constraint variable is associated with limit on availability of a resource Surplus Variable: constraint variable is associated with minimum specification

17 Graphical Solution Draw the Graphical Solution for The Reddy Mikks Company s problem Examine the feasible corner points = extreme points Determine the Optimum Solution for the problem

18 Graphical Solution Optimum Solution: Intersection between the (parallel function of) Objective Function with (one among any of) Constraint Functions

19 Graphical Solution X I H 2 1 F G K E D Solution SpaceC Optimum Solution 4 A 0 B X E Objective Function

20 Small Furniture Factory A small furniture factory manufactures tables and chairs. It takes 2 hours to assemble a table and 30 minutes to assemble a chair Assembly is carried out by 4 workers on the basis of a single 8-hour shift/day Customer usually buy at most 4 times as many chairs as tables; meaning that the factory should produce at most 4 times as many chairs as tables

21 Small Furniture Factory The sale price of a table = $135 per unit The sale price of a chair = $50 per unit

22 Small Furniture Factory Determine the problem formulation (variable, constraints and objective function Determine the daily production mix of chairs and tables that would maximize the total daily revenue

23 Diet Problem (of goats) A farmer owns 250 goats that consume 90 lb of special feed daily The feed is a mixture of corn and soybean The dietary requirement: At most 1% calcium At least 30% protein At most 5% fiber

24 Diet Problem: The Table Type of Feed Pounds per Pound of Feedstuff Calcium Protein Fiber Cost ($/lb) Corn Soybean

25 Diet Problem Determine the daily minimum cost feed mix

26 Assignment #2 Find a 2 variable linear programming problem in a computer science realm Determine the model (Identify the variable, the constraint and the objective function) Find the optimum solution Include your comment on your work

27 The End This is the end of Chapter 2