Chapter 4 The Simplex Algorithm Part II
|
|
- Corey Evans
- 6 years ago
- Views:
Transcription
1 Chapter 4 The Simple Algorithm Part II Based on Introduction to Mathematical Programming: Operations Research, Volume 4th edition, by Wayne L Winston and Munirpallam Venkataramanan Lewis Ntaimo L Ntaimo (c) 005 INEN40 TAMU
2 So Far Modeling LPs Solving -variable LPs graphically Simple method - Ma LPs Simple method - Min LPs Method : Multiply objective function by - Method : Modify Step 3 of simple algorithm for Ma LPs Simple method Special cases: Alternative optimal solutions Unbounded LPs L Ntaimo (c) 005 INEN40 TAMU
3 Degeneracy and Convergence of the Simple Algorithm L Ntaimo (c) 005 INEN40 TAMU 3
4 L Ntaimo (c) 005 INEN40 TAMU 4 4 Convergence of the Simple Method 0 st Ma z = = b A c T Recall: Where, = mn m m n n a a a a a a a a a A = n = c n c c c = b m b b b Assumption: Constraint matri A has n columns and m linearly independent rows So we can form an m m square matri called the basis by setting (n - m) variables to zero (nonbasic variables or nbv s) The remaining variables are the basic variables or bv s
5 4 Convergence of the Simple Method Recall that for any LP with m constraints, two bfs are said to be adjacent if their sets of bv s have (m - )bv s in common eg bfs of two immediate simple tableaus are adjacent bfs If an LP in standard form has m constraints and n variables, then there may be a basic solution for each choice of nonbasic variables From n variables, in how many different ways can a set of (n - m) nonbasic variables (or equivalently, m basic variables) be chosen? n = m n! ( n m)! m! n m Thus an LP can have at most basic solutions! L Ntaimo (c) 005 INEN40 TAMU 5
6 4 Convergence of the Simple Method For n = 0 and m = 0 we have: 0 = 0 0! =? (0 0)! 0! 84,756 bfs! Assuming that no bfs is repeated the simple algorithm will find the optimal bfs after a finite number of iterations Vast eperience with the simple algorithm indicates that an optimal solution is found after eamining fewer than 3m bfs (eg 3(0) = 30) Compared with eamining 84,756 bfs, the simple algorithm is quite efficient! L Ntaimo (c) 005 INEN40 TAMU 6
7 4 Degeneracy and Convergence of the Simple Method Definition: An LP is degenerate if it has at least one bfs in which a basic variable is equal to zero We say that an LP is nondegenerate if all bfs are positive Theoretically, the simple algorithm (as you have learned it so far) can fail to find the optimal solution to an LP However, LPs arising from actual applications seldom ehibit this behavior L Ntaimo (c) 005 INEN40 TAMU 7
8 4 Degeneracy and Convergence of the Simple Method Consider the following relationship for a ma LP: z-value for new bfs = z-value for current bfs (value of entering variable in new bfs)*(coefficient of entering variable in row 0 of current bfs) Recall: (coefficient of entering variable in row 0) < 0 and (value of entering variable in new bfs) >= 0 Two Facts: () If (value of entering variable in new bfs) > 0), then (z-value for new bfs) > (z-value for current bfs) () If (value of entering variable in new bfs) = 0), then (z-value for new bfs) = (z-value for current bfs) L Ntaimo (c) 005 INEN40 TAMU 8
9 4 Degeneracy and Convergence of the Simple Method Fact implies that each iteration of the simple algorithm will increase z for a nondegenerate LP This means that it is impossible to encounter the same bfs twice So if we use the simple algorithm to solve a nondegenerate LP, we are guaranteed to find the optimal solution in a finite number of iterations Fact implies that the simple algorithm may fail for a degenerate LP In this case the algorithm will encounter the same bfs at least twice This occurrence is called cycling If cycling occurs then the algorithm can loop, or cycle, forever among a set of bfs and never get to the optimal solution! L Ntaimo (c) 005 INEN40 TAMU 9
10 L Ntaimo (c) 005 INEN40 TAMU 0 4 Degeneracy and Convergence of the Simple Method Eample: Solve the following LP using the simple tableau method 0, 0 6 st 5 Ma + + = z Converting to standard form: 0,,, 0 6 st 5 Ma = + = = s s s s z
11 4 Degeneracy and Convergence of the Simple Method Eample: Solve using the simple tableau method Initial Tableau Solution Row z s s RHS BV MRT z = 0 z = s = 6 6 = s = 0 0* = 0 Second Tableau Row z s s RHS BV MRT z = 0 z = s = 6 3* = = 0 None = 0 Cycling: We encounter same bfs twice! Third (Optimal) Tableau Row z s s RHS BV MRT z = z = = 3 = = 3 = 3 Optimal Solution: z =, = 3, =3 L Ntaimo (c) 005 INEN40 TAMU
12 4 Degeneracy and Convergence of the Simple Method If an LP has many degenerate bfs (or a bfs with many bv s equal to zero), then the simple algorithm is often very inefficient Eample: Solve the previous eample using the graphical method B 6 4 D + C A 4 6 z = z = = The etreme points of the feasible region are B,C and D Three sets of basic variables correspond to etreme point C BV's BFS Etreme Point, = = 3, s = s = 0 D, s = 0, s = 6, = s = 0 C, s = 6, s = -6, = s = 0 Infeasible, s = 0, s = 6, = s = 0 C, s = 6, s = 6, s = = 0 B s, s s = 6, s = 0, = = 0 C L Ntaimo (c) 005 INEN40 TAMU
13 4 Degeneracy and Convergence of the Simple Method For an LP with n decision variables to be degenerate, (n+) or more of the LP s constraints (including the sign restrictions i 0 as constraints) must be binding at an etreme point Fortunately, the simple method can be modified to ensure that cycling will never occur Apply the so called anti-cycling rules L Ntaimo (c) 005 INEN40 TAMU 3
14 4 Degeneracy and Convergence of the Simple Method Bland s Rules for Anti-Cycling: Smallest Subscript Pivoting Rule (Assume that slack and ecess variables are numbered n+, n+, ) Step Choose as the entering variable (in a ma problem) the variable with a negative coefficient in row 0 that has the smallest subscript Step If there is a tie in the minimum ratio test, then break the tie by choosing the winner of the ratio test so that the variable leaving the basis has the smallest subscript L Ntaimo (c) 005 INEN40 TAMU 4
15 4 Degeneracy and Convergence of the Simple Method Additional homework problems have been posted on the course website on degeneracy and convergence of the simple method L Ntaimo (c) 005 INEN40 TAMU 5
16 The Big M Method L Ntaimo (c) 005 INEN40 TAMU 6
17 4 The Big M Method Recall that the simple method algorithm requires a starting bfs In all the LPs we have considered so far, we have found starting bfs by using the slack variables as our basic variables If an LP has or = constraints, however, a starting bfs may not be readily apparent In such a case, we need another method to solve the problem L Ntaimo (c) 005 INEN40 TAMU 7
18 4 The Big M Method Consider the following problem: Bevco manufactures an orange-flavored soft drink called Oranj by combining orange soda and orange juice Each orange soda contains 05 oz of sugar and mg of vitamin C Each ounce of orange juice contains 05 oz of sugar and 3 mg of vitamin C It costs Bevco to produce an ounce of orange soda and 3 to produce an ounce of orange juice Bevco s marketing department has decided that each 0- oz bottle of Oranj must contain at least 0 mg of vitamin C and at most 4 oz of sugar Use linear programming to determine how Bevco can meet the marketing department s requirements at minimum cost L Ntaimo (c) 005 INEN40 TAMU 8
19 4 The Big M Method Decision variables: = number of ounces of orange soda in a bottle of Oranj = number of ounces of orange juice in a bottle of Oranj LP: Min z = + 3 st (sugar constraint) = 0, 0 (vitamin C constraint) (0 oz in bottle of Oranj) L Ntaimo (c) 005 INEN40 TAMU 9
20 4 The Big M Method The LP in standard: Min z = + 3 st s = e = 0 + = 0,, s, e 0 The LP in standard form has z and s which could be used for BVs but row would violate sign restrictions and row 3 has no readily apparent basic variable L Ntaimo (c) 005 INEN40 TAMU 0
21 4 The Big M Method In order to use the simple method, a bfs is needed The idea of the Big M method is to create artificial variables to provide an initial bfs and then eliminate them from the final solution The variables will be labeled a i according to the row i in which they are used as follows Row 0: z Row : s = 4 Row : + 3 -e + a = 0 Row 3: + + a 3 = 0 L Ntaimo (c) 005 INEN40 TAMU
22 4 The Big M Method In the optimal solution, all artificial variables must be set equal to zero To accomplish this, in a min LP, a term Ma i is added to the objective function for each artificial variable a i For a ma LP, the term Ma i is added to the objective function for each a i M represents some very large number The modified Bevco LP in standard form then becomes: Min z = Ma + Ma 3 st s = e + a = a 3 = 0,, s, e, a, a 3 0 Modifying the objective function this way makes it etremely costly for an artificial variable to be positive The optimal solution should force a = a 3 =0 L Ntaimo (c) 005 INEN40 TAMU
23 4 The Big M Method The rows of the simple tableau would then be: Row 0: z Ma -Ma 3 = 0 Row : s = 4 Row : + 3 -e + a = 0 Row 3: + + a 3 = 0 But observe that to have canonical form we need to eliminate a and a 3 from row 0! Simply set new Row 0 to Row 0 + M(Row ) + M(Row 3) z Ma -Ma 3 = 0 M( e + a = 0) + M( + + a 3 = 0) z + (M-) +(4M-3) Me = 30M L Ntaimo (c) 005 INEN40 TAMU 3
24 4 The Big M Method Description of the Big M Method Modify the constraints so that the rhs of each constraint is nonnegative Identify each constraint that is now an = or constraint Convert each inequality constraint to standard form (add a slack variable for constraints, add an ecess variable for constraints) 3 For each or = constraint, add artificial variables Add sign restriction a i 0 4 Let M denote a very large positive number Add (for each artificial variable) Ma i to min problem objective functions or -Ma i to ma problem objective functions 5 Since each artificial variable will be in the starting basis, all artificial variables must be eliminated from row 0 before beginning the simple Remembering M represents a very large number, solve the transformed problem by the simple L Ntaimo (c) 005 INEN40 TAMU 4
25 4 The Big M Method If all artificial variables in the optimal solution equal zero, the solution is optimal If any artificial variables are positive in the optimal solution, the problem is infeasible L Ntaimo (c) 005 INEN40 TAMU 5
26 4 The Big M Method The Bevco eample continued (Using Method for Min LP): Details Iteration z s e a a3 rhs ratio ero 0 00 M - 4M -3 -M 30M Row 0 + M(Row ) + M(Row 3) ero z s e a a3 rhs ero 0 00 M - 4M -3 -M 30M Row / ero z s e a a3 rhs ero 0 00 (M-3)/3 (M-3)/3 (3-4M)/3 (60+0M)/3 Row 0 - (4M-3)*(Row ) ero 3 z s e a a3 rhs ero 0 00 (M-3)/3 (M-3)/3 (3-4M)/3 (60+0M)/ Row - 05*(Row ) ero 4 z s e a a3 rhs ero 0 00 (M-3)/3 (M-3)/3 (3-4M)/3 (60+0M)/ Row 3 - Row L Ntaimo (c) 005 INEN40 TAMU 6
27 4 The Big M Method Iteration z s e a a3 rhs ratio 0 00 (M-3)/3 (M-3)/3 (3-4M)/3 (60+0M)/ ero z s e a a3 rhs ero 0 00 (M-3)/3 (M-3)/3 (3-4M)/3 (60+0M)/ (Row 3)*(3/) ero z s e a a3 rhs ero (-M)/ (3-M)/ 500 Row 0 + (3-M)*(Row 3)/ ero 3 z s e a a3 rhs ero (-M)/ (3-M)/ Row - (5/)*Row 3) ero 4 z s e a a3 rhs ero (-M)/ (3-M)/ Row -(/3)*Row Optimal Solution: z = 5, = 5, =5 L Ntaimo (c) 005 INEN40 TAMU 7
28 4 The Big M Method The Bevco eample continued (Using Method for Min LP): Initial Tableau Row z s e a a3 RHS BV MRT 0 M- 4M-3 0 -M M z = 30M s = a = 0 667* a3 = 0 0 Second Tableau Row z s e a a3 RHS BV MRT 0 (M-3)/3 0 0 (M-3)/3 (3-4M)/3 0 (60+0M)/3 z = (60+0M)/3 0 5/ 0 / -/ 0 7/3 s = 7/ /3 0 -/3 /3 0 0/3 = 0/ /3 0 0 /3 -/3 0/3 a3 = 0/3 5* Third (Optimal) Tableau Row z s e a a3 RHS BV MRT / (-M)/ (3-M)/ 5 z = /8 /8-5/8 /4 s = / / / -/ 5 = / -/ 3/ 5 = 5 Optimal Solution: z = 5, = 5, =5 L Ntaimo (c) 005 INEN40 TAMU 8
29 4 The Big M Method Issues with the Big M Method ) It is difficult to determine how large M should be Generally chosen to be at least 00 times the largest coefficient in the objective function ) Introduction of such large numbers cause round off errors and other computational difficulties (numerical instability) Read section on Scaling of LPs on page 67 This motivates the need for yet another method! L Ntaimo (c) 005 INEN40 TAMU 9
30 4 The Big M Method Problem (0 pts) Get with your team and solve the following LP using the Big M method : Min st z = 3 +, 6 = 3 0 L Ntaimo (c) 005 INEN40 TAMU 30
31 L Ntaimo (c) 005 INEN40 TAMU 3 4 The Big M Method Solution to Problem (0 pts): LP is standard form: 0,, 3 6 st 3 Min = + = + = e e z Big M Formulation: 0,,,, 3 6 st 3 Min = + + = = a a e a a e Ma Ma z
32 4 The Big M Method Eliminate a and a from row 0 Simply set new Row 0 to Row 0 + M(Row ) + M(Row 3): z Ma -Ma = 0 M( + - e + a = 6) + M( + + a = 3) z + (3M-3) + (3M) Me = 9M L Ntaimo (c) 005 INEN40 TAMU 3
33 4 The Big M Method Using Method of the simple algorithm for min LPs: pts Initial Tableau Row z e a a RHS BV MRT 0 3M-3 3M -M 0 0 9M z = 9M a = a = 3 3/* 3 pts Second Tableau Row z e a a RHS BV MRT 0 (3M-6)/ 0 -M 0-3/M 45M z = 45M a = 45 3* = pts Third (Optimal) Tableau Row z e a a RHS BV MRT M -(4+M)/ 9 z = /3 /3 -/3 3 = = 0 Optimal Solution: z = 9, = 3, = 0 pts L Ntaimo (c) 005 INEN40 TAMU 33
34 The Two-Phase Simple Method L Ntaimo (c) 005 INEN40 TAMU 34
35 43 The Two-Phase Simple Method When an initial basic feasible solution is not available, the twophase simple method can be used as an alternative to the Big M method In this method we can add artificial variables, and then focus entirely on obtaining an initial basic feasible solution, any basic feasible solution (This is called Phase I) We can then start the simple algorithm with the basic feasible solution we have found (This is called Phase II) L Ntaimo (c) 005 INEN40 TAMU 35
36 43 The Two-Phase Simple Method Observe that if all we want is a basic feasible solution, then we can select any objective function Once we find a basic feasible solution, we can reconsider the original cost coefficients Also note that it is easy to bring cost coefficients into canonical form It is desirable to choose an objective function such that solving the Phase I LP will force the artificial variables to be zero Minimize the sum of artificial variables! L Ntaimo (c) 005 INEN40 TAMU 36
37 43 The Two-Phase Simple Method Consider solving the Bevco problem using the two-phase simple method: Decision variables: LP: = number of ounces of orange soda in a bottle of Oranj = number of ounces of orange juice in a bottle of Oranj Min z = + 3 st (sugar constraint) = 0, 0 (vitamin C constraint) (0 oz in bottle of Oranj) L Ntaimo (c) 005 INEN40 TAMU 37
38 43 The Two-Phase Simple Method The LP in standard: Min z = + 3 st s = e = 0 + = 0,, s, e 0 Note again that the LP in standard form has z and s which could be used for BVs but row would violate sign restrictions and row 3 has no readily apparent basic variable L Ntaimo (c) 005 INEN40 TAMU 38
39 43 The Two-Phase Simple Method Adding artificial variables we get the following: Min z = + 3 st s = e + a = a 3 = 0,, s, e, a, a 3 0 Then the Phase I LP is: Min w = a + a 3 st s = e + a = a 3 = 0,, s, e, a, a 3 0 L Ntaimo (c) 005 INEN40 TAMU 39
40 43 The Two-Phase Simple Method Note that row 0 for this tableau contains the basic variables a and a 3 : Row 0: w - a -a 3 = 0 Row : s = 4 Row : + 3 -e + a = 0 Row 3: + + a 3 = 0 As in the Big M method, we need to eliminate a and a 3 from row 0 to get a canonical form Simply add Row + Row 3 to Row 0: w - a -a 3 = e + a = a 3 = 0 w e = 30 L Ntaimo (c) 005 INEN40 TAMU 40
41 43 The Two-Phase Simple Method Description of the Two-Phase Simple Method Modify the constraints so that the rhs of each constraint is nonnegative Identify each constraint that is now an = or constraint Convert each inequality constraint to standard form (add a slack variable for constraints, add an ecess variable for constraints) 3 For each or = constraint, add artificial variables Add sign restriction a i 0 4 For now, ignore the original LP s objective function Instead solve an LP whose objective function is min w = (sum of all artificial variables) This is the Phase I LP Solving the Phase I LP will force the artificial variables to be zero L Ntaimo (c) 005 INEN40 TAMU 4
42 43 The Two-Phase Simple Method Description of the Two-Phase Simple Method cont Since each a i 0, solving the Phase I LP will result in one of the following three cases: Case : The optimal value of w > 0 In this case the original LP is infeasible Case : The optimal value of w = 0 and no artificial variables are in the optimal Phase I basis Drop all columns in the optimal Phase I tableau that correspond to the artificial variables In this case, combine the original objective function with the constraints from the optimal Phase I tableau This yields the Phase II LP The optimal solution to the Phase II LP is the optimal solution to the original LP Case 3: The optimal value of w = 0 and at least one artificial variable is in the optimal Phase I basis In this case, we can find the optimal solution to the original LP if at the end of Phase I we drop from the optimal Phase I tableau all nonbasic artificial variables and any variable from the original problem that has a negative coefficient in row 0 of the optimal Phase I tableau L Ntaimo (c) 005 INEN40 TAMU 4
43 43 The Two-Phase Simple Method The Bevco eample continued (Using Method for Min LP): Initial Phase I Tableau Row w' s e a a3 RHS BV MRT w' = s = a = 0 667* a3 = 0 0 Second Phase I Tableau Row w' s e a a3 RHS BV MRT 0 /3 0 0 /3-4/3 0 0/3 w' = 0/3 0 5/ 0 / -/ 0 7/3 s = 7/3 8/5 0 /3 0 -/3 /3 0 0/3 = 0/ /3 0 0 /3 -/3 0/3 a3 = 0/3 5* Third (Optimal) Phase I Tableau Row w' s e a a3 RHS BV MRT w' = /8 /8-5/8 /4 s = / / / -/ 5 = / -/ 3/ 5 = 5 We will now drop columns a and a 3 L Ntaimo (c) 005 INEN40 TAMU 43
44 43 The Two-Phase Simple Method We begin Phase II with the following set of equations: Row 0: z - 3 = 0 Row : s -/8e = /4 Row : -/e = 5 Row 3: + /e = 5 We need to eliminate and from row 0 to get a canonical form Add 3(Row ) + (Row 3) to Row 0: z - 3 = 0 3( -/e = 5) + ( + /e = 5) z - /e = 5 L Ntaimo (c) 005 INEN40 TAMU 44
45 43 The Two-Phase Simple Method Initial (Optimal) Phase II Tableau Row z s e RHS BV MRT / 0 z = /8 /4 s = / / 5 = / 5 = 5 In this case Phase II requires no pivots to find an optimal solution If the Phase II row 0 does not indicate an optimal tableau, then simply continue with the simple until an optimal row 0 is obtained The optimal Phase II tableau shows that the optimal solution to the Bevco problem is: Optimal Solution: z = 5, = 5, = 5 Note that we obtained the same solution using the Big M Method L Ntaimo (c) 005 INEN40 TAMU 45
46 43 The Two-Phase Simple Method For eamples of Case and Case 3 of the two-phase simple method read Eample 6 and 7 of Chapter 4 Important note: Eample 7 should be a ma problem and not a min problem! L Ntaimo (c) 005 INEN40 TAMU 46
47 43 The Two-Phase Simple Method Problem (0 pts) Get with your team and solve the following LP using the two-phase simple method : Min st z = 3 +, 6 = 3 0 L Ntaimo (c) 005 INEN40 TAMU 47
48 L Ntaimo (c) 005 INEN40 TAMU The Two-Phase Simple Method Solution to Problem (0 pts): 0, 3 6 st 3 Min = + = + = e z LP is standard form: 0, 3 6 st 3 Min = + + = z Maintaining nonegative RHS: 0, 3 6 st ' Min = + + = = a a e a a w Phase I Formulation:
49 43 The Two-Phase Simple Method Eliminate a and a from row 0 Add Row + Row 3 to Row 0: w - a -a = e + a = a = 3 w e = 9 L Ntaimo (c) 005 INEN40 TAMU 49
50 43 The Two-Phase Simple Method Using Method of the simple algorithm for min LPs: pts Initial Phase I Tableau Row w' e a a RHS BV MRT w' = a = 6 3* a = 3 3 pts Second Phase I Tableau Row w' e a a RHS BV MRT 0 0 3/ / -3/ 0 0 w' = 0 0 / -/ / 0 3 = / / -/ 0 a = 0 0* pts Third (Optimal) Phase I Tableau Row w' e a a RHS BV MRT / - 0 w' = /6 7/ -/3 3 = /3 -/6 /3 0 = 0 We will now drop columns a and a L Ntaimo (c) 005 INEN40 TAMU 50
51 43 The Two-Phase Simple Method We begin Phase II with the following set of equations: Row 0: z - 3 = 0 Row : -5/6e = 3 Row : + /3e = 0 We need to eliminate from row 0 to get a canonical form Add 3(Row ) to Row 0: z - 3 = 0 + 3( -5/6e = 3) z - 5/e = 9 L Ntaimo (c) 005 INEN40 TAMU 5
52 43 The Two-Phase Simple Method pts Initial Phase II Tableau Row z e RHS BV MRT / 0 z = /6 3 = /3 0 = 0 In this case Phase II requires no pivots to find an optimal solution If the Phase II row 0 does not indicate an optimal tableau, then simply continue with the simple until an optimal row 0 is obtained The optimal Phase II tableau shows that the optimal solution to the problem is: Optimal Solution: z = 9, = 3, = 0 pts Note that we obtained the same solution using the Big M Method L Ntaimo (c) 005 INEN40 TAMU 5
53 Summary 43 The Two-Phase Simple Method To get started with the simple method, add an artificial basis, but ensure that the artificial variables do not occur in an optimal solution Big M method: place a large cost (penalty) on each of the artificial variables Two-phase method: In phase I minimize the sum of the artificial variables At the end of phase I, we will have a basic feasible solution to the original problem Use this as a starting point for Phase II, which solves the original problem L Ntaimo (c) 005 INEN40 TAMU 53
IE 400: Principles of Engineering Management. Simplex Method Continued
IE 400: Principles of Engineering Management Simplex Method Continued 1 Agenda Simplex for min problems Alternative optimal solutions Unboundedness Degeneracy Big M method Two phase method 2 Simplex for
More informationThe Simplex Algorithm
The Simplex Algorithm How to Convert an LP to Standard Form Before the simplex algorithm can be used to solve an LP, the LP must be converted into a problem where all the constraints are equations and
More informationChapter 4 The Simplex Algorithm Part I
Chapter 4 The Simplex Algorithm Part I Based on Introduction to Mathematical Programming: Operations Research, Volume 1 4th edition, by Wayne L. Winston and Munirpallam Venkataramanan Lewis Ntaimo 1 Modeling
More informationDr. Maddah ENMG 500 Engineering Management I 10/21/07
Dr. Maddah ENMG 500 Engineering Management I 10/21/07 Computational Procedure of the Simplex Method The optimal solution of a general LP problem is obtained in the following steps: Step 1. Express the
More informationThe Big M Method. Modify the LP
The Big M Method Modify the LP 1. If any functional constraints have negative constants on the right side, multiply both sides by 1 to obtain a constraint with a positive constant. Big M Simplex: 1 The
More informationAM 121: Intro to Optimization
AM 121: Intro to Optimization Models and Methods Lecture 6: Phase I, degeneracy, smallest subscript rule. Yiling Chen SEAS Lesson Plan Phase 1 (initialization) Degeneracy and cycling Smallest subscript
More informationIE 400 Principles of Engineering Management. The Simplex Algorithm-I: Set 3
IE 4 Principles of Engineering Management The Simple Algorithm-I: Set 3 So far, we have studied how to solve two-variable LP problems graphically. However, most real life problems have more than two variables!
More informationStandard Form An LP is in standard form when: All variables are non-negativenegative All constraints are equalities Putting an LP formulation into sta
Chapter 4 Linear Programming: The Simplex Method An Overview of the Simplex Method Standard Form Tableau Form Setting Up the Initial Simplex Tableau Improving the Solution Calculating the Next Tableau
More informationReview Solutions, Exam 2, Operations Research
Review Solutions, Exam 2, Operations Research 1. Prove the weak duality theorem: For any x feasible for the primal and y feasible for the dual, then... HINT: Consider the quantity y T Ax. SOLUTION: To
More informationSensitivity Analysis and Duality
Sensitivity Analysis and Duality Part II Duality Based on Chapter 6 Introduction to Mathematical Programming: Operations Research, Volume 1 4th edition, by Wayne L. Winston and Munirpallam Venkataramanan
More informationThe Simplex Algorithm and Goal Programming
The Simplex Algorithm and Goal Programming In Chapter 3, we saw how to solve two-variable linear programming problems graphically. Unfortunately, most real-life LPs have many variables, so a method is
More informationSimplex Algorithm Using Canonical Tableaus
41 Simplex Algorithm Using Canonical Tableaus Consider LP in standard form: Min z = cx + α subject to Ax = b where A m n has rank m and α is a constant In tableau form we record it as below Original Tableau
More informationTheory of Linear Programming
SCHOOL OF BUSINESS, ECONOMICS AND MANAGEMENT BF360 Operations Research Unit Two Theory of Linear Programming Moses Mwale e-mail: moses.mwale@ictar.ac.zm BF360 Operations Research Contents Unit 2: Theory
More informationExample. 1 Rows 1,..., m of the simplex tableau remain lexicographically positive
3.4 Anticycling Lexicographic order In this section we discuss two pivoting rules that are guaranteed to avoid cycling. These are the lexicographic rule and Bland s rule. Definition A vector u R n is lexicographically
More informationLinear Programming. Xi Chen. Department of Management Science and Engineering International Business School Beijing Foreign Studies University
Linear Programming Xi Chen Department of Management Science and Engineering International Business School Beijing Foreign Studies University Xi Chen (chenxi0109@bfsu.edu.cn) Linear Programming 1 / 148
More informationLinear Programming. Dr. Xiaosong DING
Linear Programming Dr. Xiaosong DING Department of Management Science and Engineering International Business School Beijing Foreign Studies University Dr. DING (xiaosong.ding@hotmail.com) Linear Programming
More informationECE 307 Techniques for Engineering Decisions
ECE 7 Techniques for Engineering Decisions Introduction to the Simple Algorithm George Gross Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign ECE 7 5 9 George
More information21. Solve the LP given in Exercise 19 using the big-m method discussed in Exercise 20.
Extra Problems for Chapter 3. Linear Programming Methods 20. (Big-M Method) An alternative to the two-phase method of finding an initial basic feasible solution by minimizing the sum of the artificial
More informationAM 121 Introduction to Optimization: Models and Methods Example Questions for Midterm 1
AM 121 Introduction to Optimization: Models and Methods Example Questions for Midterm 1 Prof. Yiling Chen Fall 2018 Here are some practice questions to help to prepare for the midterm. The midterm will
More informationSimplex Method for LP (II)
Simplex Method for LP (II) Xiaoxi Li Wuhan University Sept. 27, 2017 (week 4) Operations Research (Li, X.) Simplex Method for LP (II) Sept. 27, 2017 (week 4) 1 / 31 Organization of this lecture Contents:
More informationLecture 2: The Simplex method
Lecture 2 1 Linear and Combinatorial Optimization Lecture 2: The Simplex method Basic solution. The Simplex method (standardform, b>0). 1. Repetition of basic solution. 2. One step in the Simplex algorithm.
More informationVariants of Simplex Method
Variants of Simplex Method All the examples we have used in the previous chapter to illustrate simple algorithm have the following common form of constraints; i.e. a i x + a i x + + a in x n b i, i =,,,m
More informationThe Simplex Method. Lecture 5 Standard and Canonical Forms and Setting up the Tableau. Lecture 5 Slide 1. FOMGT 353 Introduction to Management Science
The Simplex Method Lecture 5 Standard and Canonical Forms and Setting up the Tableau Lecture 5 Slide 1 The Simplex Method Formulate Constrained Maximization or Minimization Problem Convert to Standard
More informationMATH 445/545 Homework 2: Due March 3rd, 2016
MATH 445/545 Homework 2: Due March 3rd, 216 Answer the following questions. Please include the question with the solution (write or type them out doing this will help you digest the problem). I do not
More informationCO350 Linear Programming Chapter 8: Degeneracy and Finite Termination
CO350 Linear Programming Chapter 8: Degeneracy and Finite Termination 27th June 2005 Chapter 8: Finite Termination 1 The perturbation method Recap max c T x (P ) s.t. Ax = b x 0 Assumption: B is a feasible
More information1 Review Session. 1.1 Lecture 2
1 Review Session Note: The following lists give an overview of the material that was covered in the lectures and sections. Your TF will go through these lists. If anything is unclear or you have questions
More informationGauss-Jordan Elimination for Solving Linear Equations Example: 1. Solve the following equations: (3)
The Simple Method Gauss-Jordan Elimination for Solving Linear Equations Eample: Gauss-Jordan Elimination Solve the following equations: + + + + = 4 = = () () () - In the first step of the procedure, we
More informationThe simplex algorithm
The simplex algorithm The simplex algorithm is the classical method for solving linear programs. Its running time is not polynomial in the worst case. It does yield insight into linear programs, however,
More information9.1 Linear Programs in canonical form
9.1 Linear Programs in canonical form LP in standard form: max (LP) s.t. where b i R, i = 1,..., m z = j c jx j j a ijx j b i i = 1,..., m x j 0 j = 1,..., n But the Simplex method works only on systems
More informationMATH 445/545 Test 1 Spring 2016
MATH 445/545 Test Spring 06 Note the problems are separated into two sections a set for all students and an additional set for those taking the course at the 545 level. Please read and follow all of these
More informationMath 273a: Optimization The Simplex method
Math 273a: Optimization The Simplex method Instructor: Wotao Yin Department of Mathematics, UCLA Fall 2015 material taken from the textbook Chong-Zak, 4th Ed. Overview: idea and approach If a standard-form
More informationLinear Programming, Lecture 4
Linear Programming, Lecture 4 Corbett Redden October 3, 2016 Simplex Form Conventions Examples Simplex Method To run the simplex method, we start from a Linear Program (LP) in the following standard simplex
More informationYinyu Ye, MS&E, Stanford MS&E310 Lecture Note #06. The Simplex Method
The Simplex Method Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/ yyye (LY, Chapters 2.3-2.5, 3.1-3.4) 1 Geometry of Linear
More informationLesson 27 Linear Programming; The Simplex Method
Lesson Linear Programming; The Simplex Method Math 0 April 9, 006 Setup A standard linear programming problem is to maximize the quantity c x + c x +... c n x n = c T x subject to constraints a x + a x
More informationCPS 616 ITERATIVE IMPROVEMENTS 10-1
CPS 66 ITERATIVE IMPROVEMENTS 0 - APPROACH Algorithm design technique for solving optimization problems Start with a feasible solution Repeat the following step until no improvement can be found: change
More informationLinear Programming and the Simplex method
Linear Programming and the Simplex method Harald Enzinger, Michael Rath Signal Processing and Speech Communication Laboratory Jan 9, 2012 Harald Enzinger, Michael Rath Jan 9, 2012 page 1/37 Outline Introduction
More informationSpecial cases of linear programming
Special cases of linear programming Infeasible solution Multiple solution (infinitely many solution) Unbounded solution Degenerated solution Notes on the Simplex tableau 1. The intersection of any basic
More informationSimplex method(s) for solving LPs in standard form
Simplex method: outline I The Simplex Method is a family of algorithms for solving LPs in standard form (and their duals) I Goal: identify an optimal basis, as in Definition 3.3 I Versions we will consider:
More informationLinear & Integer programming
ELL 894 Performance Evaluation on Communication Networks Standard form I Lecture 5 Linear & Integer programming subject to where b is a vector of length m c T A = b (decision variables) and c are vectors
More informationOPERATIONS RESEARCH. Linear Programming Problem
OPERATIONS RESEARCH Chapter 1 Linear Programming Problem Prof. Bibhas C. Giri Department of Mathematics Jadavpur University Kolkata, India Email: bcgiri.jumath@gmail.com MODULE - 2: Simplex Method for
More informationFebruary 17, Simplex Method Continued
15.053 February 17, 2005 Simplex Method Continued 1 Today s Lecture Review of the simplex algorithm. Formalizing the approach Alternative Optimal Solutions Obtaining an initial bfs Is the simplex algorithm
More informationIE 5531: Engineering Optimization I
IE 5531: Engineering Optimization I Lecture 5: The Simplex method, continued Prof. John Gunnar Carlsson September 22, 2010 Prof. John Gunnar Carlsson IE 5531: Engineering Optimization I September 22, 2010
More informationLinear Programming. Linear Programming I. Lecture 1. Linear Programming. Linear Programming
Linear Programming Linear Programming Lecture Linear programming. Optimize a linear function subject to linear inequalities. (P) max " c j x j n j= n s. t. " a ij x j = b i # i # m j= x j 0 # j # n (P)
More informationContents. 4.5 The(Primal)SimplexMethod NumericalExamplesoftheSimplexMethod
Contents 4 The Simplex Method for Solving LPs 149 4.1 Transformations to be Carried Out On an LP Model Before Applying the Simplex Method On It... 151 4.2 Definitions of Various Types of Basic Vectors
More informationThe Graphical Method & Algebraic Technique for Solving LP s. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 1
The Graphical Method & Algebraic Technique for Solving LP s Métodos Cuantitativos M. En C. Eduardo Bustos Farías The Graphical Method for Solving LP s If LP models have only two variables, they can be
More informationMetode Kuantitatif Bisnis. Week 4 Linear Programming Simplex Method - Minimize
Metode Kuantitatif Bisnis Week 4 Linear Programming Simplex Method - Minimize Outlines Solve Linear Programming Model Using Graphic Solution Solve Linear Programming Model Using Simplex Method (Maximize)
More informationMath Models of OR: Some Definitions
Math Models of OR: Some Definitions John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 12180 USA September 2018 Mitchell Some Definitions 1 / 20 Active constraints Outline 1 Active constraints
More informationOPRE 6201 : 3. Special Cases
OPRE 6201 : 3. Special Cases 1 Initialization: The Big-M Formulation Consider the linear program: Minimize 4x 1 +x 2 3x 1 +x 2 = 3 (1) 4x 1 +3x 2 6 (2) x 1 +2x 2 3 (3) x 1, x 2 0. Notice that there are
More informationLecture 4: Algebra, Geometry, and Complexity of the Simplex Method. Reading: Sections 2.6.4, 3.5,
Lecture 4: Algebra, Geometry, and Complexity of the Simplex Method Reading: Sections 2.6.4, 3.5, 10.2 10.5 1 Summary of the Phase I/Phase II Simplex Method We write a typical simplex tableau as z x 1 x
More informationSummary of the simplex method
MVE165/MMG631,Linear and integer optimization with applications The simplex method: degeneracy; unbounded solutions; starting solutions; infeasibility; alternative optimal solutions Ann-Brith Strömberg
More informationAnn-Brith Strömberg. Lecture 4 Linear and Integer Optimization with Applications 1/10
MVE165/MMG631 Linear and Integer Optimization with Applications Lecture 4 Linear programming: degeneracy; unbounded solution; infeasibility; starting solutions Ann-Brith Strömberg 2017 03 28 Lecture 4
More informationAM 121: Intro to Optimization Models and Methods Fall 2018
AM 121: Intro to Optimization Models and Methods Fall 2018 Lecture 5: The Simplex Method Yiling Chen Harvard SEAS Lesson Plan This lecture: Moving towards an algorithm for solving LPs Tableau. Adjacent
More informationF 1 F 2 Daily Requirement Cost N N N
Chapter 5 DUALITY 5. The Dual Problems Every linear programming problem has associated with it another linear programming problem and that the two problems have such a close relationship that whenever
More informationORF 522. Linear Programming and Convex Analysis
ORF 5 Linear Programming and Convex Analysis Initial solution and particular cases Marco Cuturi Princeton ORF-5 Reminder: Tableaux At each iteration, a tableau for an LP in standard form keeps track of....................
More informationThe Simplex Method. Standard form (max) z c T x = 0 such that Ax = b.
The Simplex Method Standard form (max) z c T x = 0 such that Ax = b. The Simplex Method Standard form (max) z c T x = 0 such that Ax = b. Build initial tableau. z c T 0 0 A b The Simplex Method Standard
More informationDr. S. Bourazza Math-473 Jazan University Department of Mathematics
Dr. Said Bourazza Department of Mathematics Jazan University 1 P a g e Contents: Chapter 0: Modelization 3 Chapter1: Graphical Methods 7 Chapter2: Simplex method 13 Chapter3: Duality 36 Chapter4: Transportation
More informationOptimization (168) Lecture 7-8-9
Optimization (168) Lecture 7-8-9 Jesús De Loera UC Davis, Mathematics Wednesday, April 2, 2012 1 DEGENERACY IN THE SIMPLEX METHOD 2 DEGENERACY z =2x 1 x 2 + 8x 3 x 4 =1 2x 3 x 5 =3 2x 1 + 4x 2 6x 3 x 6
More informationAM 121: Intro to Optimization Models and Methods
AM 121: Intro to Optimization Models and Methods Fall 2017 Lecture 2: Intro to LP, Linear algebra review. Yiling Chen SEAS Lecture 2: Lesson Plan What is an LP? Graphical and algebraic correspondence Problems
More informationMATH2070 Optimisation
MATH2070 Optimisation Linear Programming Semester 2, 2012 Lecturer: I.W. Guo Lecture slides courtesy of J.R. Wishart Review The standard Linear Programming (LP) Problem Graphical method of solving LP problem
More informationΩ R n is called the constraint set or feasible set. x 1
1 Chapter 5 Linear Programming (LP) General constrained optimization problem: minimize subject to f(x) x Ω Ω R n is called the constraint set or feasible set. any point x Ω is called a feasible point We
More informationIn Chapters 3 and 4 we introduced linear programming
SUPPLEMENT The Simplex Method CD3 In Chapters 3 and 4 we introduced linear programming and showed how models with two variables can be solved graphically. We relied on computer programs (WINQSB, Excel,
More informationGETTING STARTED INITIALIZATION
GETTING STARTED INITIALIZATION 1. Introduction Linear programs come in many different forms. Traditionally, one develops the theory for a few special formats. These formats are equivalent to one another
More informationLINEAR PROGRAMMING I. a refreshing example standard form fundamental questions geometry linear algebra simplex algorithm
Linear programming Linear programming. Optimize a linear function subject to linear inequalities. (P) max c j x j n j= n s. t. a ij x j = b i i m j= x j 0 j n (P) max c T x s. t. Ax = b Lecture slides
More informationmin 4x 1 5x 2 + 3x 3 s.t. x 1 + 2x 2 + x 3 = 10 x 1 x 2 6 x 1 + 3x 2 + x 3 14
The exam is three hours long and consists of 4 exercises. The exam is graded on a scale 0-25 points, and the points assigned to each question are indicated in parenthesis within the text. If necessary,
More information3 Does the Simplex Algorithm Work?
Does the Simplex Algorithm Work? In this section we carefully examine the simplex algorithm introduced in the previous chapter. Our goal is to either prove that it works, or to determine those circumstances
More informationThe Simplex Method: An Example
The Simplex Method: An Example Our first step is to introduce one more new variable, which we denote by z. The variable z is define to be equal to 4x 1 +3x 2. Doing this will allow us to have a unified
More informationSection 4.1 Solving Systems of Linear Inequalities
Section 4.1 Solving Systems of Linear Inequalities Question 1 How do you graph a linear inequality? Question 2 How do you graph a system of linear inequalities? Question 1 How do you graph a linear inequality?
More informationLecture slides by Kevin Wayne
LINEAR PROGRAMMING I a refreshing example standard form fundamental questions geometry linear algebra simplex algorithm Lecture slides by Kevin Wayne Last updated on 7/25/17 11:09 AM Linear programming
More informationPrelude to the Simplex Algorithm. The Algebraic Approach The search for extreme point solutions.
Prelude to the Simplex Algorithm The Algebraic Approach The search for extreme point solutions. 1 Linear Programming-1 x 2 12 8 (4,8) Max z = 6x 1 + 4x 2 Subj. to: x 1 + x 2
More information1. Algebraic and geometric treatments Consider an LP problem in the standard form. x 0. Solutions to the system of linear equations
The Simplex Method Most textbooks in mathematical optimization, especially linear programming, deal with the simplex method. In this note we study the simplex method. It requires basically elementary linear
More informationChap6 Duality Theory and Sensitivity Analysis
Chap6 Duality Theory and Sensitivity Analysis The rationale of duality theory Max 4x 1 + x 2 + 5x 3 + 3x 4 S.T. x 1 x 2 x 3 + 3x 4 1 5x 1 + x 2 + 3x 3 + 8x 4 55 x 1 + 2x 2 + 3x 3 5x 4 3 x 1 ~x 4 0 If we
More informationCO 602/CM 740: Fundamentals of Optimization Problem Set 4
CO 602/CM 740: Fundamentals of Optimization Problem Set 4 H. Wolkowicz Fall 2014. Handed out: Wednesday 2014-Oct-15. Due: Wednesday 2014-Oct-22 in class before lecture starts. Contents 1 Unique Optimum
More informationPart 1. The Review of Linear Programming
In the name of God Part 1. The Review of Linear Programming 1.2. Spring 2010 Instructor: Dr. Masoud Yaghini Outline Introduction Basic Feasible Solutions Key to the Algebra of the The Simplex Algorithm
More informationOPTIMISATION 3: NOTES ON THE SIMPLEX ALGORITHM
OPTIMISATION 3: NOTES ON THE SIMPLEX ALGORITHM Abstract These notes give a summary of the essential ideas and results It is not a complete account; see Winston Chapters 4, 5 and 6 The conventions and notation
More informationA = Chapter 6. Linear Programming: The Simplex Method. + 21x 3 x x 2. C = 16x 1. + x x x 1. + x 3. 16,x 2.
Chapter 6 Linear rogramming: The Simple Method Section The Dual roblem: Minimization with roblem Constraints of the Form Learning Objectives for Section 6. Dual roblem: Minimization with roblem Constraints
More informationIntroduce the idea of a nondegenerate tableau and its analogy with nondenegerate vertices.
2 JORDAN EXCHANGE REVIEW 1 Lecture Outline The following lecture covers Section 3.5 of the textbook [?] Review a labeled Jordan exchange with pivoting. Introduce the idea of a nondegenerate tableau and
More informationLinear programs Optimization Geoff Gordon Ryan Tibshirani
Linear programs 10-725 Optimization Geoff Gordon Ryan Tibshirani Review: LPs LPs: m constraints, n vars A: R m n b: R m c: R n x: R n ineq form [min or max] c T x s.t. Ax b m n std form [min or max] c
More informationSummary of the simplex method
MVE165/MMG630, The simplex method; degeneracy; unbounded solutions; infeasibility; starting solutions; duality; interpretation Ann-Brith Strömberg 2012 03 16 Summary of the simplex method Optimality condition:
More informationSystems Analysis in Construction
Systems Analysis in Construction CB312 Construction & Building Engineering Department- AASTMT by A h m e d E l h a k e e m & M o h a m e d S a i e d 3. Linear Programming Optimization Simplex Method 135
More informationIntroduction to the Simplex Algorithm Active Learning Module 3
Introduction to the Simplex Algorithm Active Learning Module 3 J. René Villalobos and Gary L. Hogg Arizona State University Paul M. Griffin Georgia Institute of Technology Background Material Almost any
More informationLecture 5 Simplex Method. September 2, 2009
Simplex Method September 2, 2009 Outline: Lecture 5 Re-cap blind search Simplex method in steps Simplex tableau Operations Research Methods 1 Determining an optimal solution by exhaustive search Lecture
More informationSensitivity Analysis
Dr. Maddah ENMG 500 /9/07 Sensitivity Analysis Changes in the RHS (b) Consider an optimal LP solution. Suppose that the original RHS (b) is changed from b 0 to b new. In the following, we study the affect
More informationWeek_4: simplex method II
Week_4: simplex method II 1 1.introduction LPs in which all the constraints are ( ) with nonnegative right-hand sides offer a convenient all-slack starting basic feasible solution. Models involving (=)
More informationTIM 206 Lecture 3: The Simplex Method
TIM 206 Lecture 3: The Simplex Method Kevin Ross. Scribe: Shane Brennan (2006) September 29, 2011 1 Basic Feasible Solutions Have equation Ax = b contain more columns (variables) than rows (constraints),
More informationAlgebraic Simplex Active Learning Module 4
Algebraic Simplex Active Learning Module 4 J. René Villalobos and Gary L. Hogg Arizona State University Paul M. Griffin Georgia Institute of Technology Time required for the module: 50 Min. Reading Most
More informationChapter 5 Linear Programming (LP)
Chapter 5 Linear Programming (LP) General constrained optimization problem: minimize f(x) subject to x R n is called the constraint set or feasible set. any point x is called a feasible point We consider
More informationOPERATIONS RESEARCH. Michał Kulej. Business Information Systems
OPERATIONS RESEARCH Michał Kulej Business Information Systems The development of the potential and academic programmes of Wrocław University of Technology Project co-financed by European Union within European
More informationIntroduction to Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras
Introduction to Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras Module - 03 Simplex Algorithm Lecture 15 Infeasibility In this class, we
More informationNotes: Deterministic Models in Operations Research
Notes: Deterministic Models in Operations Research J.C. Chrispell Department of Mathematics Indiana University of Pennsylvania Indiana, PA, 5705, USA E-mail: john.chrispell@iup.edu http://www.math.iup.edu/~jchrispe
More informationLinear Programming Redux
Linear Programming Redux Jim Bremer May 12, 2008 The purpose of these notes is to review the basics of linear programming and the simplex method in a clear, concise, and comprehensive way. The book contains
More information4.6 Linear Programming duality
4.6 Linear Programming duality To any minimization (maximization) LP we can associate a closely related maximization (minimization) LP Different spaces and objective functions but in general same optimal
More informationTHE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS. Operations Research I
LN/MATH2901/CKC/MS/2008-09 THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS Operations Research I Definition (Linear Programming) A linear programming (LP) problem is characterized by linear functions
More informationCSCI 1951-G Optimization Methods in Finance Part 01: Linear Programming
CSCI 1951-G Optimization Methods in Finance Part 01: Linear Programming January 26, 2018 1 / 38 Liability/asset cash-flow matching problem Recall the formulation of the problem: max w c 1 + p 1 e 1 = 150
More informationCSC Design and Analysis of Algorithms. LP Shader Electronics Example
CSC 80- Design and Analysis of Algorithms Lecture (LP) LP Shader Electronics Example The Shader Electronics Company produces two products:.eclipse, a portable touchscreen digital player; it takes hours
More information3. THE SIMPLEX ALGORITHM
Optimization. THE SIMPLEX ALGORITHM DPK Easter Term. Introduction We know that, if a linear programming problem has a finite optimal solution, it has an optimal solution at a basic feasible solution (b.f.s.).
More informationChapter 3, Operations Research (OR)
Chapter 3, Operations Research (OR) Kent Andersen February 7, 2007 1 Linear Programs (continued) In the last chapter, we introduced the general form of a linear program, which we denote (P) Minimize Z
More informationLinear Programming for Planning Applications
Communication Network Planning and Performance Learning Resource Linear Programming for Planning Applications Professor Richard Harris School of Electrical and Computer Systems Engineering, RMIT Linear
More informationc) Place the Coefficients from all Equations into a Simplex Tableau, labeled above with variables indicating their respective columns
BUILDING A SIMPLEX TABLEAU AND PROPER PIVOT SELECTION Maximize : 15x + 25y + 18 z s. t. 2x+ 3y+ 4z 60 4x+ 4y+ 2z 100 8x+ 5y 80 x 0, y 0, z 0 a) Build Equations out of each of the constraints above by introducing
More informationA Review of Linear Programming
A Review of Linear Programming Instructor: Farid Alizadeh IEOR 4600y Spring 2001 February 14, 2001 1 Overview In this note we review the basic properties of linear programming including the primal simplex
More informationUNIT-4 Chapter6 Linear Programming
UNIT-4 Chapter6 Linear Programming Linear Programming 6.1 Introduction Operations Research is a scientific approach to problem solving for executive management. It came into existence in England during
More information