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# Introductory maths analysis chapter 07 official

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### Introductory maths analysis chapter 07 official

1. 1. INTRODUCTORY MATHEMATICALINTRODUCTORY MATHEMATICAL ANALYSISANALYSISFor Business, Economics, and the Life and Social Sciences ©2007 Pearson Education Asia Chapter 7Chapter 7 Linear ProgrammingLinear Programming
2. 2. ©2007 Pearson Education Asia INTRODUCTORY MATHEMATICAL ANALYSIS 0. Review of Algebra 1. Applications and More Algebra 2. Functions and Graphs 3. Lines, Parabolas, and Systems 4. Exponential and Logarithmic Functions 5. Mathematics of Finance 6. Matrix Algebra 7. Linear Programming 8. Introduction to Probability and Statistics
3. 3. ©2007 Pearson Education Asia 9. Additional Topics in Probability 10. Limits and Continuity 11. Differentiation 12. Additional Differentiation Topics 13. Curve Sketching 14. Integration 15. Methods and Applications of Integration 16. Continuous Random Variables 17. Multivariable Calculus INTRODUCTORY MATHEMATICAL ANALYSIS
4. 4. ©2007 Pearson Education Asia • To represent geometrically a linear inequality in two variables. • To state a linear programming problem and solve it geometrically. • To consider situations in which a linear programming problem exists. • To introduce the simplex method. • To solve problems using the simplex method. • To introduce use of artificial variables. • To learn alteration of objective function. • To define the dual of a linear programming problem. Chapter 7: Linear Programming Chapter ObjectivesChapter Objectives
5. 5. ©2007 Pearson Education Asia Linear Inequalities in Two Variables Linear Programming Multiple Optimum Solutions The Simplex Method Degeneracy, Unbounded Solutions, and Multiple Solutio Artificial Variables Minimization The Dual 7.1) 7.2) 7.3) 7.4) Chapter 7: Linear Programming Chapter OutlineChapter Outline 7.5) 7.6) 7.7) 7.8)
6. 6. ©2007 Pearson Education Asia Chapter 7: Linear Programming 7.1 Linear Inequalities in 2 Variables7.1 Linear Inequalities in 2 Variables • Linear inequality is written as where a, b, c = constants and not both a and b are zero. 0 or0 or0 or0 ≥++ >++ ≤++ <++ cbyax cbyax cbyax cbyax
7. 7. ©2007 Pearson Education Asia Chapter 7: Linear Programming 7.1 Linear Inequalities in 2 Variables • A solid line is included in the solution and a dashed line is not.
8. 8. ©2007 Pearson Education Asia Chapter 7: Linear Programming 7.1 Linear Inequalities in 2 Variables Example 1 – Solving a Linear Inequality Example 3 – Solving a System of Linear Inequalities Find the region defined by the inequality y ≤ 5. Solution: The region consists of the line y = 5 and with the half-plane below it. Solve the system Solution: The solution is the unshaded region.    −≥ +−≥ 2 102 xy xy
9. 9. ©2007 Pearson Education Asia Chapter 7: Linear Programming 7.2 Linear Programming7.2 Linear Programming Example 1 – Solving a Linear Programming Problem • A linear function in x and y has the form • The function to be maximized or minimized is called the objective function. byaxZ += Maximize the objective function Z = 3x + y subject to the constraints 0 0 1232 82 ≥ ≥ ≤+ ≤+ y x yx yx
10. 10. ©2007 Pearson Education Asia Chapter 7: Linear Programming 7.2 Linear Programming Example 1 – Solving a Linear Programming Problem Solution: The feasible region is nonempty and bounded. Evaluating Z at these points, we obtain The maximum value of Z occurs when x = 4 and y = 0. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 4403 11233 12043 0003 =+= =+= =+= =+= DZ CZ BZ AZ
11. 11. ©2007 Pearson Education Asia Chapter 7: Linear Programming 7.2 Linear Programming Example 3 – Unbounded Feasible Region The information for a produce is summarized as follows: If the grower wishes to minimize cost while still maintaining the nutrients required, how many bags of each brand should be bought?
12. 12. ©2007 Pearson Education Asia Chapter 7: Linear Programming 7.2 Linear Programming Example 3 – Unbounded Feasible Region Solution: Let x = number of bags of Fast Grow bought y = number of bags of Easy Grow bought To minimize the cost function Subject to the constraints There is no maximum value since the feasible region is unbounded . yxC 68 += 0 0 802 20025 16023 ≥ ≥ ≥+ ≥+ ≥+ y x yx yx yx
13. 13. ©2007 Pearson Education Asia Chapter 7: Linear Programming 7.3 Multiple Optimum Solutions7.3 Multiple Optimum Solutions Example 1 – Multiple Optimum Solutions • There may be multiple optimum solutions for an objective function. Maximize Z = 2x + 4y subject to the constraints Solution: The region is nonempty and bounded. 0, 162 84 ≥ ≤+ −≤− yx yx yx ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (max)328402 (max)324482 82402 =+= =+= =+= CZ BZ AZ
14. 14. ©2007 Pearson Education Asia Chapter 7: Linear Programming 7.4 The Simplex Method7.4 The Simplex Method Standard Linear Programming Problem • Maximize the linear function subject to the constraints where x and b are non-negative. nn xcxcxcZ +++= 2211           ≤+++ ≤+++ ≤+++ ≤+++ mnmnmm nn nn nn bxaxaxa bxaxaxa bxaxaxa bxaxaxa     2211 11212111 11212111 11212111 . .
15. 15. ©2007 Pearson Education Asia Chapter 7: Linear Programming 7.4 The Simplex Method Example 1 – The Simplex Method Maximize subject to and x1, x2 ≥ 0. Solution: The maximum value of Z is 95.      ≤+− ≤+ ≤+ 123 352 20 21 21 21 xx xx xx 21 45 xxZ +=               −− − 0 12 35 20 100045 000013 001012 000111 3 2 1 32121 Z s s s RZsssxxB               − − − 95 52 15 5 101300 014500 001101 001210 3 1 2 32121 Z s x x RZsssxxB
16. 16. ©2007 Pearson Education Asia Chapter 7: Linear Programming 7.5 Degeneracy, Unbounded and Multiple Solutions7.5 Degeneracy, Unbounded and Multiple Solutions Degeneracy • A basic feasible solution (BFS) is degenerate if one of the basic variables is 0. Unbounded Solutions • An unbounded solution occurs when the objective function has no maximum value. Multiple Optimum Solutions • Multiple (optimum) solutions occur when there are different BFS.
17. 17. ©2007 Pearson Education Asia Chapter 7: Linear Programming 7.5 Degeneracy, Unbounded and Multiple Solutions Example 1 – Unbounded Solution Maximize subject to Solution: Since the 1st two rows of the x1-column are negative, no quotients exist. Hence, it has an unbounded solution. 321 4 xxxZ −+=      ≥ ≤++− ≤−+− 0,, 1263 30265 321 321 321 xxx xxx xxx           0 12 30 1001-4-1- 010631- 0012-65- Z s s 2 1 32121 RZsssxxB           16 4 6 1090- 0021- 02-114-03- 3 4 3 7 3 1 3 1 2 1 32121 Z x s RZsssxxB
18. 18. ©2007 Pearson Education Asia Chapter 7: Linear Programming 7.6 Artificial Variables7.6 Artificial Variables Example 1 – Artificial Variables • Use artificial variables to handle maximization problems that are not of standard form. Maximize subject to Solution: Add an artificial variable t, Consider an artificial objective equation, 212 xxZ +=        ≥ ≥+− ≤+ ≤+ 0, 2 202 12 21 21 21 21 xx xx xx xx 2 202 12 321 221 121 =+−+− =++ =++ tsxx sxx sxx 02 21 =++−− WMtxx
19. 19. ©2007 Pearson Education Asia Construct a Simplex Table, All indicators are nonnegative. The maximum value of Z is 17.             −−−+− −− MMMMW t s s RWtsssxxB 2100012 20110011 200001021 120000111 2 1 3212`             − − 171000 70010 10100 50001 2 1 2 3 2 1 2 1 2 1 2 3 2 1 2 1 2 1 3212` W t s s RZsssxxB Chapter 7: Linear Programming 7.6 Artificial Variables Example 1 – Artificial Variables
20. 20. ©2007 Pearson Education Asia Use the simplex method to maximize subject to Solution: Construct a Simplex Table, The feasible region is empty, hence, no solution exists.           −++− −−− MMMMW s t RWtssxxB 1101021 1001011 1011102 2 1 12121           −−−+− −− MMMMW s t RWtssxxB 210012 1001011 2010111 2 1 12121 Chapter 7: Linear Programming 7.6 Artificial Variables Example 3 – An Empty Feasible Region 2132 xxZ =      ≥ ≤+ ≥+− 0, 1 2 21 21 21 xx xx xx
21. 21. ©2007 Pearson Education Asia Chapter 7: Linear Programming 7.7 Minimization7.7 Minimization Example 1 – Minimization • To minimize a function is to maximize the negative of the function. Minimize subject to Solution: Construct a Simplex Table, The minimum value of Z is −(−4) = 4. 21 2xxZ +=      ≥ ≥+− ≥+− 0, 2 12 21 21 21 xx xx xx           −−+ − −− MMMMMW t t RWttssxxB 31002231 20101011 10010112 2 1 212121           −+− −−− −− 4122003 10111101 20101011 2 1 212121 MMW t t RWttssxxB
22. 22. ©2007 Pearson Education Asia Chapter 7: Linear Programming 7.8 The Dual7.8 The Dual Example 1 – Finding the Dual of a Maximization Problem Find the dual of the following: Maximize subject to where Solution: The dual is minimize subject to where 321 243 xxxZ ++=    ≤++ ≤+ 1022 102 321 21 xxx xx 21 1010 yyW +=      −≥− ≥+− ≥− 34 1 23 21 21 21 xx xx xx 0,, 321 ≥xxx 0, 21 ≥yy
23. 23. ©2007 Pearson Education Asia Chapter 7: Linear Programming 7.8 The Dual Example 3 – Applying the Simplex Method to the Dual Use the dual and the simplex method to maximize subject to where Solution: The dual is minimize subject to The final Simplex Table is The minimum value of W is 11/2 . 21 24 yyW += 0,, 321 ≥xxx3214 xxxZ −−=    ≤++ ≤−+ 2 43 321 321 xxx xxx     −≥+− −≥+ ≥+ 1 1 43 21 21 21 yy yy yy             − −− −− − − 2 11 2 1 2 3 4 5 4 1 4 1 2 5 2 1 2 1 4 1 4 3 4 1 1 2 2 32121 1000 0001 0100 0010 W y s y RWsssyyB