# AM 121: Intro to Optimization Models and Methods Fall 2017

Save this PDF as:

Size: px
Start display at page:

Download "AM 121: Intro to Optimization Models and Methods Fall 2017" ## Transcription

1 AM 121: Intro to Optimization Models and Methods Fall 2017 Lecture 10: Dual Simplex Yiling Chen SEAS Lesson Plan Interpret primal simplex in terms of pivots on the corresponding dual tableau Dictionaries Define dual simplex and interpret it in terms of pivots on the primal tableau Applications Jensen & Bard: 3.9 1

2 Motivation: Dual Simplex Useful for solving integer programs via branch-and-bound search. Find a solution after a new constraint is added, or an existing constraint is modified. A new Phase 1 Phase 2 method (no need for artificial variables!) Naming convention m+n max j=m+1 c j x j m+n s.t. j=m+1 a ij x j b i x j 0 i = 1, 2,, m j = m+1,, m+n primal vars x m+1,,x m+n ; slack vars x 1,,x m m min i=1 b i y i m s.t. i=1 a ij y i c j y i 0 j = m+1,, m+n i = 1,, m dual vars y 1,,y m ; surplus vars y m+1,,y m+n 2

3 Example max 4x 3 + x 4 + 3x 5 s.t. x 3 + 4x 4 1 3x 3 x 4 + x 5 3 x 3, x 4, x 5 0 Dual: min y 1 + 3y 2 s.t. y 1 + 3y 2 4 4y 1 y 2 1 y 2 3 y 1, y 2 0 equivalent maximization problem max -y 1-3y 2 s.t. -y 1-3y y 1 +y 2-1 -y 2-3 y 1, y 2 0 Tableau (Review) A tableau must have isolated variables but does not need to have non-negative right hand side values. If the RHS is negative, the tableau is not (primal) feasible. 3

4 Primal <> Dual RHS non-basic Initial primal tableau: z 4x 3 x 4 3x 5 = 0 x 1 + x 3 + 4x 4 = 1 x 2 + 3x 3 x 4 + x 5 = 3 B={1,2} B ={3,4,5} feasible Initial dual tableau: z +y 1 +3y 2 = 0 -y 1-3y 2 + y 3 = -4-4y 1 + y 2 + y 4 = -1 -y 2 + y 5 = -3 B={3,4,5} B ={1,2} infeasible basic z x 1 x 2 basic z y 3 y 4 y 5 x 3 x 4 x Primal dictionary y 1 y Dual dictionary RHS (negated columns) non-basic (negated columns) Primal <> Dual Initial primal tableau: z 4x 3 x 4 3x 5 = 0 x 1 + x 3 + 4x 4 = 1 x 2 + 3x 3 x 4 + x 5 = 3 B={1,2} B ={3,4,5} Feasible; Neg shadow costs Initial dual tableau: z +y 1 +3y 2 = 0 -y 1-3y 2 + y 3 = -4-4y 1 + y 2 + y 4 = -1 -y 2 + y 5 = -3 B={3,4,5} B ={1,2} Infeasible; Non-neg shadow costs Primal dictionary B={1,2} B={3,4,5} Dual dictionary -ve transpose of primal; B (dual) = B (primal) 4

5 A Dictionary Definition. The dictionary corresponding to a tableau with basis B is v -c T B (where B is the set of non-basic) b -A B with rows of A B and b ordered by indices of basic vars and columns of c T B ordered by indices of non-basic vars Suppose tableau was reordered as: z x 4 4x 3 3x 5 = 0 x 2 x 4 + 3x 3 + x 5 = 3 x 1 + 4x 4 + x 3 = 1 Corresponding dictionary is still: B={1,2} A dictionary and basis B completely defines a tableau! A simple primal-dual correspondence Primal dictionary Basic B, non-basic B negated transpose çè Dual dictionary Basic B, non-basic B So: (1) primal tableau has corresponding dual tableau (2) can track progress of simplex in the dual! (3) primal obj value = - dual obj value at each step (negated because we converted min f(y) into max f(y)) 5

6 Primal pivot (track in dual) z 4x 3 x 4 3x 5 = 0 x 1 + x 3 + 4x 4 = 1 x 2 + 3x 3 x 4 + x 5 = 3 B={1,2} B ={3,4,5} pivot (2,5); 2 out 5 in pivot z +3x 2 +5x 3 4x 4 = 9 x 1 +x 3 + 4x 4 = 1 x 2 +3x 3 x 4 +x 5 = 3 B={1,5} B ={2,3,4} z +y 1 +3y 2 = 0 -y 1-3y 2 +y 3 = -4-4y 1 + y 2 +y 4 = -1 - y 2 +y 5 = -3 B={3,4,5} B ={1,2} pivot (5,2); 5 out 2 in pivot z +y 1 +3y 5 = -9 -y 1 +y 3-3y 5 = 5-4y 1 +y 4 +y 5 = -4 y 2 -y 5 = 3 B={2,3,4} B ={1,5} still dual infeasible New dictionaries z +3x 2 +5x 3-4x 4 = 9 x 1 +x 3 +4x 4 = 1 x 2 +3x 3 x 4 +x 5 = 3 B={1,5} B ={2,3,4} z x 1 x 5 x 2 x 3 x Primal dictionary z +y 1 +3y 5 = -9 -y 1 +y 3-3y 5 = 5-4y 1 +y 4 +y 5 = -4 y 2 -y 5 = 3 B={2,3,4} B ={1,5} z y 2 y 3 y 4 y 1 y Dual dictionary (negated transpose) 6

7 Primal pivot (track in dual) z +3x 2 +5x 3-4x 4 = 9 x 1 +x 3 +4x 4 = 1 x 2 +3x 3 x 4 +x 5 = 3 B={1,5} B ={2,3,4} Pivot (1,4); 1 out 4 in z +y 1 +3y 5 = -9 -y 1 +y 3-3y 5 = 5-4y 1 +y 4 +y 5 = -4 y 2 -y 5 = 3 B={2,3,4} B ={1,5} Pivot (4,1); 4 out 1 in z +x 1 +3x 2 +6x 3 = 10 ¼x 1 +¼x 3 + x 4 = ¼ ¼x 1 +x 2 +3¼x 3 + x 5 = 3¼ B={4,5} B ={1,2,3} pivot z +¼y 4 + 3¼y 5 = -10 y 1 ¼y 4 ¼y 5 = 1 y 2 y 5 = 3 y 3 ¼y 4 3¼y 5 = 6 B={1,2,3} B ={4,5} pivot Primal feasible, Primal optimal. Dual feasible, Dual optimal Final tableaus; Final dictionaries z +x 1 +3x 2 +6x 3 = 10 ¼x 1 +¼x 3 + x 4 = ¼ ¼x 1 +x 2 +3¼x 3 + x 5 = 3¼ B={4,5} B ={1,2,3} z +¼y 4 + 3¼y 5 = -10 y 1 ¼y 4 ¼y 5 = 1 y 2 y 5 = 3 y 3 ¼y 4 3¼y 5 = 6 B={1,2,3} B ={4,5} z x 4 x 5 z y 1 y 2 y 3 x 1 x 2 x ¼ -¼ 0 -¼ 3¼ -¼ -1-3¼ Primal dictionary y 4 y ¼ -3¼ 1 ¼ ¼ ¼ 3¼ Dual dictionary (negated transpose) 7

8 From Initial to Final tableau Initial primal tableau Final primal tableau z 4x 3 x 4 3x 5 = 0 z +x 1 +3x 2 +6x 3 =10 x 1 + x 3 + 4x 4 = 1 ¼x 1 +¼x 3 + x 4 =¼ x 2 + 3x 3 x 4 + x 5 = 3 ¼x 1 +x 2 +3¼x 3 + x 5 =3¼ B={1,2} B ={3,4,5} B={4,5} B ={1,2,3} Initial dual tableau z +y 1 +3y 2 = 0 -y 1-3y 2 +y 3 =-4-4y 1 + y 2 +y 4 =-1 -y 2 +y 5 =-3 B={3,4,5} B ={1,2} Final dual tableau z +¼y 4 +3¼y 5 =-10 y 1 ¼y 4 ¼y 5 =1 y 2 y 5 =3 y 3 ¼y 4 3¼y 5 =6 B={1,2,3} B ={4,5} Primal Simplex: Effect On Dual Solution Maintains primal feasibility, terminates with primal optimality Non-negative RHS in primal <> non-negative reduced costs in dual Free reduced costs in primal <> free RHS values in dual (and dual infeasible for a while) Terminates with non-negative reduced costs in primal <> non-neg RHS in dual (and dual feasible) Pivot in primal, track in dual. Corresponding dual solution always has nonnegative reduced costs, but is initially infeasible. Note: pair (x,y) satisfy complementary slackness during pivots, but y is infeasible until termination. 8

9 Complementary Slackness z 4x 3 x 4 3x 5 = 0 x 1 + x 3 + 4x 4 = 1 x 2 + 3x 3 x 4 + x 5 = 3 B={1,2} B ={3,4,5} pivot (2,5) pivot z +3x 2 +5x 3 4x 4 = 9 x 1 +x 3 + 4x 4 = 1 x 2 +3x 3 x 4 +x 5 = 3 B={1,5} B ={2,3,4} z +y 1 +3y 2 = 0 -y 1-3y 2 +y 3 = -4-4y 1 + y 2 +y 4 = -1 - y 2 +y 5 = -3 B={3,4,5} B ={1,2} pivot (5,2) pivot z +y 1 +3y 5 = -9 -y 1 +y 3-3y 5 = 5-4y 1 +y 4 +y 5 = -4 y 2 -y 5 = 3 B={2,3,4} B ={1,5} It is because of CS that we get this particular dual <> primal construction Primal dictionary çè Dual dictionary Basic B, non-basic B negated transpose Basic B, non-basic B In particular, it is CS that gives us that: primal obj value = - dual obj value at each step (negated because we converted min f(y) into max f(y)) 9

10 Different kinds of pivots V1: pick non-basic x j with c j <0 to enter, and look at positive column entries to find basic var to leave V2: pick basic x j with b j <0 to exit, and look at negative row entries to find non-basic var to enter V1 V2 Primal simplex Primal tableau Dual tableau Dual simplex Dual tableau Primal tableau Dual Simplex Maintain dual feasibility, terminate with dual optimality Rather than work in dual tableau, can track in primal tableau: perform "dual pivots. 10

11 Dual Simplex Maintain dual feasibility, terminate with dual optimality Rather than work in dual tableau, can track in primal tableau: perform "dual pivots. Reduced costs Primal tableau properties Primal feasible Dual feasible Primal and Dual optimal Free Non-neg Non-neg RHS Non-neg Free Non-neg primal pivots to solve dual pivots to solve Example: Dual Simplex Primal problem max x 4 x 5 s.t. -2x 4 x 5 4-2x 4 + 4x 5-8 -x 4 + 3x 5-7 x 4, x 5 0 Dual problem min 4y 1-8y 2-7y 3 s.t. -2y 1 2y 2 y 3-1 -y 1 + 4y 2 +3y 3-1 y 1, y 2, y 3 0 max -4y 1 + 8y 2 +7y 3 s.t. 2y 1 + 2y 2 + y 3 1 y 1 4y 2 3y 3 1 y 1, y 2, y 3 0 equivalent maximization problem 11

12 Initial Tableau max x 4 x 5 s.t. -2x 4 x 5 4-2x 4 + 4x 5-8 -x 4 + 3x 5-7 x 4, x 5 0 max -4y 1 + 8y 2 +7y 3 s.t. 2y 1 + 2y 2 + y 3 1 y 1 4y 2 3y 3 1 y 1, y 2, y 3 0 z + x 4 + x 5 = 0 x 1-2x 4 x 5 = 4 x 2-2x 4 + 4x 5 = -8 x 3 x 4 + 3x 5 = -7 z +4y 1-8y 2-7y 3 =0 2y 1 +2y 2 + y 3 +y 4 =1 y 1 4y 2 3y 3 +y 5 =1 Dual Pivot (track in primal) z +x 4 + x 5 = 0 x 1-2x 4 x 5 = 4 x 2-2x 4 + 4x 5 = -8 x 3 x 4 + 3x 5 = -7 B={1,2,3} pivot (2,4) (2 out, 4 in) z +4y 1-8y 2-7y 3 =0 2y 1 +2y 2 + y 3 +y 4 =1 y 1 4y 2 3y 3 B={4,5} +y 5 =1 pivot (4,2) (4 out, 2 in) pivot z +½x 2 +3x 5 = -4 x 1 x 2 5x 5 = 12 -½x 2 +x 4 2x 5 = 4 -½ x 2 + x 3 + x 5 = -3 B={1,3,4} z+12y 1-3y 3 +4y 4 =+4 y 1 +y 2 +½y 3 +½y 4 =½ 5y 1 y 3 + 2y 4 +y 5 =3 B={2,5} 12

13 Dual Pivot (track in primal) z +½x 2 +3x 5 =-4 x 1 x 2 5x 5 =12 -½x 2 +x 4 2x 5 =4 -½ x 2 +x 3 + x 5 =-3 B={1,3,4} pivot (3,2) z +12y 1-3y 3 +4y 4 =+4 y 1 +y 2 +½y 3 +½y 4 =½ 5y 1 - y 3 +2y 4 +y 5 =3 B={2,5} pivot (2,3) pivot z + x 3 + 4x 5 =-7 x 1-2x 3-7x 5 =18 - x 3 +x 4-3x 5 =7 x 2-2x 3-2x 5 =6 B={1,2,4} z +18y 1 +6y 2 +7y 4 =+7 2y 1 +2y 2 +y 3 +y 4 =1 7y 1 +2y 2 +3y 4 +y 5 =4 B={3,5} OPTIMAL! Upside Down Pivoting Dual pivots in primal tableau. Maintain dual feasibility, terminate with dual optimality (nonnegative RHS values in primal tableau). Primal infeasible for a while, but non-negative (primal) reduced costs. Terminate with feasible primal solution. Choose a variable with a ve RHS to exit, and do ratio test on strictly negative entries in row with negated reduced cost as numerator. 13

14 Dual simplex Assume have a dual feasible tableau (c B 0) Step 1. Pick a basic variable x r to leave with a strictly negative RHS b r < 0. If no such variable then optimal and STOP. Step 2. Pick a nonbasic variable x k to enter by considering row r and non-basic variables with strictly negative coefficients. Ratio test: k should satisfy -c k /a rk = min {-c j /a rj : j B, a rj <0}. If all a rj on j B nonnegative, then dual unbounded and primal infeasible and STOP. Step 3. Pivot on (r,k) and go to step 1. In corresponding dual tableau: basic variable y r enters and nonbasic variable y k leaves. Example: A dual simplex pivot z +½x 3 + ½x 4 = +1½ x 1 +x 3 = 2 x 2 ½x 3 +½x 4 = - ½ z +x 2 + x 4 = 1 x 1 + 2x 2 + x 4 = 1-2 x 2 + x 3 x 4 = 1 (Pivot on row with ve RHS, picking nonbasic variable to enter with strictly negative a rk and minimal ratio -c k / a rk ) OPTIMAL! 14

15 Example: Dual unbounded Suppose get to a (primal) tableau with row x 1 + 2x 2 + 2x 3 = -5 Dual unbounded, since coefficients on x 2 and x 3 are non-negative. Conclude that primal is infeasible (from weak duality theorem). Making use of the Dual Simplex 1. Find a solution after a new constraint is added (useful for sensitivity analysis, and solving IPs via branch-and-bound search) 2. Find a new solution after a change in a RHS coeff that is outside the allowable range 3. A new Phase 1 Phase 2 method (no need for artificial variables!) 15

16 1. Adding a New Constraint Recall the furniture example max z= 60x x x 3 s.t. 8x 1 + 6x 2 + x 3 + x 4 = 48 (lumber) 4x 1 + 2x x 3 + x 5 = 20 (finishing) 2x x x 3 + x 6 = 8 (carpentry) x 1,, x 6 0 x 1 desks; x 2 tables; x 3 chairs B={4,3,1}. x * =(2,0,8,24,0,0), z=280 Three kinds of new constraints: (i) Current optimal solution still feasible; e.g., add constraint x 1 + x 2 + x Easy: just check. (ii) Current basis becomes infeasible (iii) LP becomes infeasible 1a. New constraint makes optimal basis infeasible. Add: x 2 1. Can just add directly to optimal tableau: z +5x 2 +10x x 6 = 280-2x 2 + x 4 + 2x 5 8x 6 = 24-2x 2 + x 3 + 2x 5 4x 6 = 8 x x 2 0.5x x 6 = 2 -x 2 + x 7 = -1 (primal) B={1,3,4,7}. Dual feasible. Dual pivot in primal (7 out, 2 in). And so pivot (7,2): z +10x 4 +10x 5 + 5x 7 = 275 x 4 +2x 5-8x 6 2x 7 = 26 x 3 +2x 5-4x 6 2x 7 = 10 x 1 ½x 5 +1½x 6 +1¼x 7 = ¾ x 2 x 7 = 1 Find an optimal solution: x * =(¾, 1, 10, 26, 0, 0); z=

17 Remark Could add x 2 1 directly to the initial tableau: x 2 x 7 + x 8 = 1, with artificial variable x 8, and re-solve with primal simplex. By adding the constraint to the final tableau and using dual simplex we just need a single pivot. Large computational gain! 1b. New constraint makes problem infeasible. Consider: x 1 + x 2 12 z +5x x 5 +10x 6 = 280-2x 2 + x 4 + 2x 5 8x 6 = 24-2x 2 + x 3 + 2x 5 4x 6 = 8 x x 2 0.5x x 6 = 2 -x 1 x 2 + x 7 = -12 x 1 no longer isolated. Adopt: (5 ):=(4)+(5) 0.25x 2 0.5x x 6 + x 7 = -10 (5 ) (Primal) B={4,3,1,7}. Dual feasible. Dual pivot in primal: (7 out, 5 in). Get tableau on next slide. (4) (5) 17

18 1b. New constraint makes problem infeasible. z +10x 2 +40x 6 +20x 7 = 80 -x 2 +x 4-2x 6 + 4x 7 = -16 -x 2 + x 3 + 2x 6 + 4x 7 = -32 x 1 + x 2 x 7 = 12 -½x 2 + x 5 3x 6 2x 7 = 12 -½x 2 + x 5 3x 6 2x 7 = 20 (Primal) B={4,3,1,5} Dual pivot in primal: 3 out, 2 in Get tableau on next slide 1b. New constraint makes problem infeasible. z +10x 3 +60x 6 +60x 7 = x 3 +x 4-4x 6 = -16 x 2 x 3-2x 6-4x 7 = 32 x 1 +x 3 +2x 6 +3x 7 = -20 (Primal) B={4,2,1,5} -½x 3 +x 5 4x 6 4x 7 = 36 Dual pivot in primal: 1 out. But ratio test fails, no variable to enter. Dual unbounded, and primal infeasible (by weak duality). 18

19 Making use of the Dual Simplex 1. Find a solution after a new constraint is added (useful for sensitivity analysis, and solving IPs via branch-and-bound search) 2. Find a new solution after a change in a RHS coeff that is outside the allowable range 3. A new Phase 1 Phase 2 method (no need for artificial variables!) 2. Changing a RHS Recall BFS for B={1,3,4} in furniture problem remains feasible for 16 b Suppose b 2 :=30. Let s consider what happens at same basis. z=y T b=( ) 48 = b = A B -1 b = = Construct tableau. Next slide. 19

20 z +5x 2 +10x x 6 = 380-2x 2 + x 4 + 2x 5 8x 6 = 44-2x 2 + x 3 + 2x 5 4x 6 = 28 x x 2 0.5x x 6 = -3 B={1,3,4}. Primal infeasible basis. Do a dual pivot in primal (1 out, 5 in.) z +20x 1 +30x 2 +40x 6 = 320 4x 1 + 3x 2 +x 4-2x 6 = 32 4x 1 + 3x 2 +x 3 +2x 6 = 16-2x 1 2½x 2 +x 5 3x 6 = 6 B={3,4,5} OPTIMAL! Solution x * =(0,0,16,32,6,0). Only make chairs! Making use of the Dual Simplex 1. Find a solution after a new constraint is added (useful for sensitivity analysis, and solving IPs via branch-and-bound search) 2. Find a new solution after a change in a RHS coeff that is outside the allowable range 3. A new Phase 1 Phase 2 method (no need for artificial variables!) 20

21 3. New Phase 1 - Phase 2 method (a) b 0: use primal simplex (no phase 1) (b) c 0, some b i < 0: use dual simplex (no phase 1) (c) Some b i < 0 and some c j < 0 E.g., suppose max x 1-2x 2. First solve max x 1-2x 2. This has z+x 1 +2x 2 first row, and so dual feasible. Use dual simplex to find a primal feasible solution. Second, modify tableau to introduce correct objective. Use primal simplex. ALTERNATIVE: Replace RHS b with b 0, use primal simplex to find a dual feasible solution, then bring back correct RHS and use dual simplex. Summary: Dual Simplex Dual simplex: simplex method applied to the dual problem. Maintains a dual feasible basis, terminates with an optimal dual basis. We track the dual simplex on primal tableau. Upside down pivots : first which basic var exits (-ve RHS), then which non-basic enters? Flexible approach: can work in primal tableau and move seamlessly from primal to dual pivots. Useful when constraints change, for phase1- phase 2, and we ll use for solving IPs! 21

### Section Notes 5. Review of Linear Programming. Applied Math / Engineering Sciences 121. Week of October 15, 2017 Section Notes 5 Review of Linear Programming Applied Math / Engineering Sciences 121 Week of October 15, 2017 The following list of topics is an overview of the material that was covered in the lectures

### Linear Programming. Linear programming provides methods for allocating limited resources among competing activities in an optimal way. University of Southern California Viterbi School of Engineering Daniel J. Epstein Department of Industrial and Systems Engineering ISE 330: Introduction to Operations Research - Deterministic Models Fall

### Read: H&L chapters 1-6 Viterbi School of Engineering Daniel J. Epstein Department of Industrial and Systems Engineering ISE 330: Introduction to Operations Research Fall 2006 (Oct 16): Midterm Review http://www-scf.usc.edu/~ise330

### Section Notes 4. Duality, Sensitivity, and the Dual Simplex Algorithm. Applied Math / Engineering Sciences 121. Week of October 8, 2018 Section Notes 4 Duality, Sensitivity, and the Dual Simplex Algorithm Applied Math / Engineering Sciences 121 Week of October 8, 2018 Goals for the week understand the relationship between primal and dual

### 5. DUAL LP, SOLUTION INTERPRETATION, AND POST-OPTIMALITY 5. DUAL LP, SOLUTION INTERPRETATION, AND POST-OPTIMALITY 5.1 DUALITY Associated with every linear programming problem (the primal) is another linear programming problem called its dual. If the primal involves

### Advanced Operations Research Techniques IE316. Quiz 2 Review. Dr. Ted Ralphs Advanced Operations Research Techniques IE316 Quiz 2 Review Dr. Ted Ralphs IE316 Quiz 2 Review 1 Reading for The Quiz Material covered in detail in lecture Bertsimas 4.1-4.5, 4.8, 5.1-5.5, 6.1-6.3 Material

### Some Advanced Topics in Linear Programming Some Advanced Topics in Linear Programming Matthew J. Saltzman July 2, 995 Connections with Algebra and Geometry In this section, we will explore how some of the ideas in linear programming, duality theory,

### Introduction. Linear because it requires linear functions. Programming as synonymous of planning. LINEAR PROGRAMMING Introduction Development of linear programming was among the most important scientific advances of mid-20th cent. Most common type of applications: allocate limited resources to competing

### Linear Programming. Course review MS-E2140. v. 1.1 Linear Programming MS-E2140 Course review v. 1.1 Course structure Modeling techniques Linear programming theory and the Simplex method Duality theory Dual Simplex algorithm and sensitivity analysis Integer

### Civil Engineering Systems Analysis Lecture XIV. Instructor: Prof. Naveen Eluru Department of Civil Engineering and Applied Mechanics Civil Engineering Systems Analysis Lecture XIV Instructor: Prof. Naveen Eluru Department of Civil Engineering and Applied Mechanics Today s Learning Objectives Dual 2 Linear Programming Dual Problem 3

### Math 414 Lecture 30. The greedy algorithm provides the initial transportation matrix. Math Lecture The greedy algorithm provides the initial transportation matrix. matrix P P Demand W ª «2 ª2 «W ª «W ª «ª «ª «Supply The circled x ij s are the initial basic variables. Erase all other values

### Marginal and Sensitivity Analyses 8.1 Marginal and Sensitivity Analyses Katta G. Murty, IOE 510, LP, U. Of Michigan, Ann Arbor, Winter 1997. Consider LP in standard form: min z = cx, subject to Ax = b, x 0 where A m n and rank m. Theorem:

### CSE 40/60236 Sam Bailey CSE 40/60236 Sam Bailey Solution: any point in the variable space (both feasible and infeasible) Cornerpoint solution: anywhere two or more constraints intersect; could be feasible or infeasible Feasible

### Optimization of Design. Lecturer:Dung-An Wang Lecture 8 Optimization of Design Lecturer:Dung-An Wang Lecture 8 Lecture outline Reading: Ch8 of text Today s lecture 2 8.1 LINEAR FUNCTIONS Cost Function Constraints 3 8.2 The standard LP problem Only equality

### DM545 Linear and Integer Programming. Lecture 2. The Simplex Method. Marco Chiarandini DM545 Linear and Integer Programming Lecture 2 The Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1. 2. 3. 4. Standard Form Basic Feasible Solutions

### Solutions for Operations Research Final Exam Solutions for Operations Research Final Exam. (a) The buffer stock is B = i a i = a + a + a + a + a + a 6 + a 7 = + + + + + + =. And the transportation tableau corresponding to the transshipment problem

### Introduction 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 - 05 Lecture - 24 Solving LPs with mixed type of constraints In the

### Civil Engineering Systems Analysis Lecture XV. Instructor: Prof. Naveen Eluru Department of Civil Engineering and Applied Mechanics Civil Engineering Systems Analysis Lecture XV Instructor: Prof. Naveen Eluru Department of Civil Engineering and Applied Mechanics Today s Learning Objectives Sensitivity Analysis Dual Simplex Method 2

### Lecture 9: Linear Programming Lecture 9: Linear Programming A common optimization problem involves finding the maximum of a linear function of N variables N Z = a i x i i= 1 (the objective function ) where the x i are all non-negative

### Linear Programming Problems Linear Programming Problems Two common formulations of linear programming (LP) problems are: min Subject to: 1,,, 1,2,,;, max Subject to: 1,,, 1,2,,;, Linear Programming Problems The standard LP problem

### Introduction to Mathematical Programming IE496. Final Review. Dr. Ted Ralphs Introduction to Mathematical Programming IE496 Final Review Dr. Ted Ralphs IE496 Final Review 1 Course Wrap-up: Chapter 2 In the introduction, we discussed the general framework of mathematical modeling

### Linear Programming Terminology Linear Programming Terminology The carpenter problem is an example of a linear program. T and B (the number of tables and bookcases to produce weekly) are decision variables. The profit function is an

### ORF 307: Lecture 14. Linear Programming: Chapter 14: Network Flows: Algorithms ORF 307: Lecture 14 Linear Programming: Chapter 14: Network Flows: Algorithms Robert J. Vanderbei April 10, 2018 Slides last edited on April 10, 2018 http://www.princeton.edu/ rvdb Agenda Primal Network

### Linear Programming. Revised Simplex Method, Duality of LP problems and Sensitivity analysis Linear Programming Revised Simple Method, Dualit of LP problems and Sensitivit analsis Introduction Revised simple method is an improvement over simple method. It is computationall more efficient and accurate.

### Linear programming and duality theory Linear programming and duality theory Complements of Operations Research Giovanni Righini Linear Programming (LP) A linear program is defined by linear constraints, a linear objective function. Its variables

### Outline. CS38 Introduction to Algorithms. Linear programming 5/21/2014. Linear programming. Lecture 15 May 20, 2014 5/2/24 Outline CS38 Introduction to Algorithms Lecture 5 May 2, 24 Linear programming simplex algorithm LP duality ellipsoid algorithm * slides from Kevin Wayne May 2, 24 CS38 Lecture 5 May 2, 24 CS38

### MATHEMATICS II: COLLECTION OF EXERCISES AND PROBLEMS MATHEMATICS II: COLLECTION OF EXERCISES AND PROBLEMS GRADO EN A.D.E. GRADO EN ECONOMÍA GRADO EN F.Y.C. ACADEMIC YEAR 2011-12 INDEX UNIT 1.- AN INTRODUCCTION TO OPTIMIZATION 2 UNIT 2.- NONLINEAR PROGRAMMING

### Tuesday, April 10. The Network Simplex Method for Solving the Minimum Cost Flow Problem . Tuesday, April The Network Simplex Method for Solving the Minimum Cost Flow Problem Quotes of the day I think that I shall never see A poem lovely as a tree. -- Joyce Kilmer Knowing trees, I understand

### INEN 420 Final Review INEN 420 Final Review Office Hours: Mon, May 2 -- 2:00-3:00 p.m. Tues, May 3 -- 12:45-2:00 p.m. (Project Report/Critiques due on Thurs, May 5 by 5:00 p.m.) Tuesday, April 28, 2005 1 Final Exam: Wednesday,

### MATLAB Solution of Linear Programming Problems MATLAB Solution of Linear Programming Problems The simplex method is included in MATLAB using linprog function. All is needed is to have the problem expressed in the terms of MATLAB definitions. Appendix

### Introduction to Operations Research - Introduction to Operations Research Peng Zhang April, 5 School of Computer Science and Technology, Shandong University, Ji nan 5, China. Email: algzhang@sdu.edu.cn. Introduction Overview of the Operations

### Assignment #3 - Solutions MATH 3300A (01) Optimization Fall 2015 Assignment #3 - Solutions MATH 33A (1) Optimization Fall 15 Section 6.1 1. Typical isoprofit line is 3x 1 +c x =z. This has slope -3/c. If slope of isoprofit line is

### Linear programming II João Carlos Lourenço Decision Support Models Linear programming II João Carlos Lourenço joao.lourenco@ist.utl.pt Academic year 2012/2013 Readings: Hillier, F.S., Lieberman, G.J., 2010. Introduction to Operations Research,

### Part 4. Decomposition Algorithms Dantzig-Wolf Decomposition Algorithm In the name of God Part 4. 4.1. Dantzig-Wolf Decomposition Algorithm Spring 2010 Instructor: Dr. Masoud Yaghini Introduction Introduction Real world linear programs having thousands of rows and columns.

### Linear Optimization. Andongwisye John. November 17, Linkoping University. Andongwisye John (Linkoping University) November 17, / 25 Linear Optimization Andongwisye John Linkoping University November 17, 2016 Andongwisye John (Linkoping University) November 17, 2016 1 / 25 Overview 1 Egdes, One-Dimensional Faces, Adjacency of Extreme

### 16.410/413 Principles of Autonomy and Decision Making 16.410/413 Principles of Autonomy and Decision Making Lecture 17: The Simplex Method Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology November 10, 2010 Frazzoli (MIT)

### Duality. Primal program P: Maximize n. Dual program D: Minimize m. j=1 c jx j subject to n. j=1. i=1 b iy i subject to m. i=1 Duality Primal program P: Maximize n j=1 c jx j subject to n a ij x j b i, i = 1, 2,..., m j=1 x j 0, j = 1, 2,..., n Dual program D: Minimize m i=1 b iy i subject to m a ij x j c j, j = 1, 2,..., n i=1

### New Directions in Linear Programming New Directions in Linear Programming Robert Vanderbei November 5, 2001 INFORMS Miami Beach NOTE: This is a talk mostly on pedagogy. There will be some new results. It is not a talk on state-of-the-art

### Artificial Intelligence Artificial Intelligence Combinatorial Optimization G. Guérard Department of Nouvelles Energies Ecole Supérieur d Ingénieurs Léonard de Vinci Lecture 1 GG A.I. 1/34 Outline 1 Motivation 2 Geometric resolution

### CS 473: Algorithms. Ruta Mehta. Spring University of Illinois, Urbana-Champaign. Ruta (UIUC) CS473 1 Spring / 36 CS 473: Algorithms Ruta Mehta University of Illinois, Urbana-Champaign Spring 2018 Ruta (UIUC) CS473 1 Spring 2018 1 / 36 CS 473: Algorithms, Spring 2018 LP Duality Lecture 20 April 3, 2018 Some of the

### CSC 8301 Design & Analysis of Algorithms: Linear Programming CSC 8301 Design & Analysis of Algorithms: Linear Programming Professor Henry Carter Fall 2016 Iterative Improvement Start with a feasible solution Improve some part of the solution Repeat until the solution

### CASE STUDY. fourteen. Animating The Simplex Method. case study OVERVIEW. Application Overview and Model Development. CASE STUDY fourteen Animating The Simplex Method case study OVERVIEW CS14.1 CS14.2 CS14.3 CS14.4 CS14.5 CS14.6 CS14.7 Application Overview and Model Development Worksheets User Interface Procedures Re-solve

### An iteration of the simplex method (a pivot ) Recap, and outline of Lecture 13 Previously Developed and justified all the steps in a typical iteration ( pivot ) of the Simplex Method (see next page). Today Simplex Method Initialization Start with

### Discuss mainly the standard inequality case: max. Maximize Profit given limited resources. each constraint associated to a resource Sensitivity Analysis Discuss mainly the standard inequality case: ma s.t. n a i, z n b, i c i,, m s.t.,,, n ma Maimize Profit given limited resources each constraint associated to a resource Alternate

### Linear Programming Motivation: The Diet Problem Agenda We ve done Greedy Method Divide and Conquer Dynamic Programming Network Flows & Applications NP-completeness Now Linear Programming and the Simplex Method Hung Q. Ngo (SUNY at Buffalo) CSE 531 1

### Lesson 11: Duality in linear programming Unit 1 Lesson 11: Duality in linear programming Learning objectives: Introduction to dual programming. Formulation of Dual Problem. Introduction For every LP formulation there exists another unique linear

### COLUMN GENERATION IN LINEAR PROGRAMMING COLUMN GENERATION IN LINEAR PROGRAMMING EXAMPLE: THE CUTTING STOCK PROBLEM A certain material (e.g. lumber) is stocked in lengths of 9, 4, and 6 feet, with respective costs of \$5, \$9, and \$. An order for

### College of Computer & Information Science Fall 2007 Northeastern University 14 September 2007 College of Computer & Information Science Fall 2007 Northeastern University 14 September 2007 CS G399: Algorithmic Power Tools I Scribe: Eric Robinson Lecture Outline: Linear Programming: Vertex Definitions

### Mathematical and Algorithmic Foundations Linear Programming and Matchings Adavnced Algorithms Lectures Mathematical and Algorithmic Foundations Linear Programming and Matchings Paul G. Spirakis Department of Computer Science University of Patras and Liverpool Paul G. Spirakis

### Advanced Operations Research Techniques IE316. Quiz 1 Review. Dr. Ted Ralphs Advanced Operations Research Techniques IE316 Quiz 1 Review Dr. Ted Ralphs IE316 Quiz 1 Review 1 Reading for The Quiz Material covered in detail in lecture. 1.1, 1.4, 2.1-2.6, 3.1-3.3, 3.5 Background material

### CS 473: Algorithms. Ruta Mehta. Spring University of Illinois, Urbana-Champaign. Ruta (UIUC) CS473 1 Spring / 29 CS 473: Algorithms Ruta Mehta University of Illinois, Urbana-Champaign Spring 2018 Ruta (UIUC) CS473 1 Spring 2018 1 / 29 CS 473: Algorithms, Spring 2018 Simplex and LP Duality Lecture 19 March 29, 2018

### Department of Mathematics Oleg Burdakov of 30 October Consider the following linear programming problem (LP): Linköping University Optimization TAOP3(0) Department of Mathematics Examination Oleg Burdakov of 30 October 03 Assignment Consider the following linear programming problem (LP): max z = x + x s.t. x x

### Solving Linear Programs Using the Simplex Method (Manual) Solving Linear Programs Using the Simplex Method (Manual) GáborRétvári E-mail: retvari@tmit.bme.hu The GNU Octave Simplex Solver Implementation As part of the course material two simple GNU Octave/MATLAB

### BCN Decision and Risk Analysis. Syed M. Ahmed, Ph.D. Linear Programming Module Outline Introduction The Linear Programming Model Examples of Linear Programming Problems Developing Linear Programming Models Graphical Solution to LP Problems The Simplex Method

### Notes for Lecture 20 U.C. Berkeley CS170: Intro to CS Theory Handout N20 Professor Luca Trevisan November 13, 2001 Notes for Lecture 20 1 Duality As it turns out, the max-flow min-cut theorem is a special case of a more general

### 1 Linear programming relaxation Cornell University, Fall 2010 CS 6820: Algorithms Lecture notes: Primal-dual min-cost bipartite matching August 27 30 1 Linear programming relaxation Recall that in the bipartite minimum-cost perfect matching

### 4.1 The original problem and the optimal tableau Chapter 4 Sensitivity analysis The sensitivity analysis is performed after a given linear problem has been solved, with the aim of studying how changes to the problem affect the optimal solution In particular,

### Econ 172A - Slides from Lecture 9 1 Econ 172A - Slides from Lecture 9 Joel Sobel October 25, 2012 2 Announcements Important: Midterm seating assignments. Posted. Corrected Answers to Quiz 1 posted. Midterm on November 1, 2012. Problems

### Linear and Integer Programming :Algorithms in the Real World. Related Optimization Problems. How important is optimization? Linear and Integer Programming 15-853:Algorithms in the Real World Linear and Integer Programming I Introduction Geometric Interpretation Simplex Method Linear or Integer programming maximize z = c T x

### Improved Gomory Cuts for Primal Cutting Plane Algorithms Improved Gomory Cuts for Primal Cutting Plane Algorithms S. Dey J-P. Richard Industrial Engineering Purdue University INFORMS, 2005 Outline 1 Motivation The Basic Idea Set up the Lifting Problem How to

### Recap, and outline of Lecture 18 Recap, and outline of Lecture 18 Previously Applications of duality: Farkas lemma (example of theorems of alternative) A geometric view of duality Degeneracy and multiple solutions: a duality connection

### Mid-term Exam of Operations Research Mid-term Exam of Operations Research Economics and Management School, Wuhan University November 10, 2016 ******************************************************************************** Rules: 1. No electronic

### Advanced Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras Advanced Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras Lecture 18 All-Integer Dual Algorithm We continue the discussion on the all integer

### 4.1 Graphical solution of a linear program and standard form 4.1 Graphical solution of a linear program and standard form Consider the problem min c T x Ax b x where x = ( x1 x ) ( 16, c = 5 ), b = 4 5 9, A = 1 7 1 5 1. Solve the problem graphically and determine

### Chapter II. Linear Programming 1 Chapter II Linear Programming 1. Introduction 2. Simplex Method 3. Duality Theory 4. Optimality Conditions 5. Applications (QP & SLP) 6. Sensitivity Analysis 7. Interior Point Methods 1 INTRODUCTION

### Other Algorithms for Linear Programming Chapter 7 Other Algorithms for Linear Programming Chapter 7 Introduction The Simplex method is only a part of the arsenal of algorithms regularly used by LP practitioners. Variants of the simplex method the dual

### Homework 2: Multi-unit combinatorial auctions (due Nov. 7 before class) CPS 590.1 - Linear and integer programming Homework 2: Multi-unit combinatorial auctions (due Nov. 7 before class) Please read the rules for assignments on the course web page. Contact Vince (conitzer@cs.duke.edu)

### ISE203 Optimization 1 Linear Models. Dr. Arslan Örnek Chapter 4 Solving LP problems: The Simplex Method SIMPLEX ISE203 Optimization 1 Linear Models Dr. Arslan Örnek Chapter 4 Solving LP problems: The Simplex Method SIMPLEX Simplex method is an algebraic procedure However, its underlying concepts are geometric Understanding

### VARIANTS OF THE SIMPLEX METHOD C H A P T E R 6 VARIANTS OF THE SIMPLEX METHOD By a variant of the Simplex Method (in this chapter) we mean an algorithm consisting of a sequence of pivot steps in the primal system using alternative rules

### Practice Final Exam 2: Solutions lgorithm Design Techniques Practice Final Exam 2: Solutions 1. The Simplex lgorithm. (a) Take the LP max x 1 + 2x 2 s.t. 2x 1 + x 2 3 x 1 x 2 2 x 1, x 2 0 and write it in dictionary form. Pivot: add x

### THE simplex algorithm  has been popularly used Proceedings of the International MultiConference of Engineers and Computer Scientists 207 Vol II, IMECS 207, March 5-7, 207, Hong Kong An Improvement in the Artificial-free Technique along the Objective

### Econ 172A - Slides from Lecture 8 1 Econ 172A - Slides from Lecture 8 Joel Sobel October 23, 2012 2 Announcements Important: Midterm seating assignments. Posted tonight. Corrected Answers to Quiz 1 posted. Quiz 2 on Thursday at end of

### George B. Dantzig Mukund N. Thapa. Linear Programming. 1: Introduction. With 87 Illustrations. Springer George B. Dantzig Mukund N. Thapa Linear Programming 1: Introduction With 87 Illustrations Springer Contents FOREWORD PREFACE DEFINITION OF SYMBOLS xxi xxxiii xxxvii 1 THE LINEAR PROGRAMMING PROBLEM 1

### Introduction 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 - 03 Tabular form (Minimization) In this

### Math Models of OR: The Simplex Algorithm: Practical Considerations Math Models of OR: The Simplex Algorithm: Practical Considerations John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 12180 USA September 2018 Mitchell Simplex Algorithm: Practical Considerations

### Advanced Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras Advanced Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras Lecture 16 Cutting Plane Algorithm We shall continue the discussion on integer programming,

### 5.4 Pure Minimal Cost Flow Pure Minimal Cost Flow Problem. Pure Minimal Cost Flow Networks are especially convenient for modeling because of their simple nonmathematical structure that can be easily portrayed with a graph. This

### The Ascendance of the Dual Simplex Method: A Geometric View The Ascendance of the Dual Simplex Method: A Geometric View Robert Fourer 4er@ampl.com AMPL Optimization Inc. www.ampl.com +1 773-336-AMPL U.S.-Mexico Workshop on Optimization and Its Applications Huatulco

### Math 5490 Network Flows Math 590 Network Flows Lecture 7: Preflow Push Algorithm, cont. Stephen Billups University of Colorado at Denver Math 590Network Flows p./6 Preliminaries Optimization Seminar Next Thursday: Speaker: Ariela

### CSc 545 Lecture topic: The Criss-Cross method of Linear Programming CSc 545 Lecture topic: The Criss-Cross method of Linear Programming Wanda B. Boyer University of Victoria November 21, 2012 Presentation Outline 1 Outline 2 3 4 Please note: I would be extremely grateful

### Lecture 16 October 23, 2014 CS 224: Advanced Algorithms Fall 2014 Prof. Jelani Nelson Lecture 16 October 23, 2014 Scribe: Colin Lu 1 Overview In the last lecture we explored the simplex algorithm for solving linear programs. While

### 3 INTEGER LINEAR PROGRAMMING 3 INTEGER LINEAR PROGRAMMING PROBLEM DEFINITION Integer linear programming problem (ILP) of the decision variables x 1,..,x n : (ILP) subject to minimize c x j j n j= 1 a ij x j x j 0 x j integer n j=

### Heuristic Optimization Today: Linear Programming. Tobias Friedrich Chair for Algorithm Engineering Hasso Plattner Institute, Potsdam Heuristic Optimization Today: Linear Programming Chair for Algorithm Engineering Hasso Plattner Institute, Potsdam Linear programming Let s first define it formally: A linear program is an optimization

### GENERAL ASSIGNMENT PROBLEM via Branch and Price JOHN AND LEI GENERAL ASSIGNMENT PROBLEM via Branch and Price JOHN AND LEI Outline Review the column generation in Generalized Assignment Problem (GAP) GAP Examples in Branch and Price 2 Assignment Problem The assignment

### Graphs that have the feasible bases of a given linear Algorithmic Operations Research Vol.1 (2006) 46 51 Simplex Adjacency Graphs in Linear Optimization Gerard Sierksma and Gert A. Tijssen University of Groningen, Faculty of Economics, P.O. Box 800, 9700

### 11 Linear Programming 11 Linear Programming 11.1 Definition and Importance The final topic in this course is Linear Programming. We say that a problem is an instance of linear programming when it can be effectively expressed

### An introduction to pplex and the Simplex Method An introduction to pplex and the Simplex Method Joanna Bauer Marc Bezem Andreas Halle November 16, 2012 Abstract Linear programs occur frequently in various important disciplines, such as economics, management,

### 9.4 SOME CHARACTERISTICS OF INTEGER PROGRAMS A SAMPLE PROBLEM 9.4 SOME CHARACTERISTICS OF INTEGER PROGRAMS A SAMPLE PROBLEM Whereas the simplex method is effective for solving linear programs, there is no single technique for solving integer programs. Instead, a

### Solvers for Mixed Integer Programming /425 Declarative Methods - J. Eisner 1 Solvers for Mixed Integer Programming 600.325/425 Declarative Methods - J. Eisner 1 Relaxation: A general optimization technique Want: x* = argmin x f(x) subject to x S S is the feasible set Start by getting:

### Finite Math Linear Programming 1 May / 7 Linear Programming Finite Math 1 May 2017 Finite Math Linear Programming 1 May 2017 1 / 7 General Description of Linear Programming Finite Math Linear Programming 1 May 2017 2 / 7 General Description of

### Copyright 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin Introduction to the Design & Analysis of Algorithms, 2 nd ed., Ch. Iterative Improvement Algorithm design technique for solving optimization problems Start with a feasible solution Repeat the following step until no improvement can be found: change the current feasible

### Unit.9 Integer Programming Unit.9 Integer Programming Xiaoxi Li EMS & IAS, Wuhan University Dec. 22-29, 2016 (revised) Operations Research (Li, X.) Unit.9 Integer Programming Dec. 22-29, 2016 (revised) 1 / 58 Organization of this

### Chapter 1 Linear Programming. Paragraph 4 The Simplex Algorithm Chapter Linear Programming Paragraph 4 The Simplex Algorithm What we did so far By combining ideas of a specialized algorithm with a geometrical view on the problem, we developed an algorithm idea: Find

### Algorithmic Game Theory and Applications. Lecture 6: The Simplex Algorithm Algorithmic Game Theory and Applications Lecture 6: The Simplex Algorithm Kousha Etessami Recall our example 1 x + y

### Approximation Algorithms Approximation Algorithms Group Members: 1. Geng Xue (A0095628R) 2. Cai Jingli (A0095623B) 3. Xing Zhe (A0095644W) 4. Zhu Xiaolu (A0109657W) 5. Wang Zixiao (A0095670X) 6. Jiao Qing (A0095637R) 7. Zhang

### Math Introduction to Operations Research Math 300 Introduction to Operations Research Examination (50 points total) Solutions. (6 pt total) Consider the following linear programming problem: Maximize subject to and x, x, x 3 0. 3x + x + 5x 3 The Simplex Algorithm with a New rimal and Dual ivot Rule Hsin-Der CHEN 3, anos M. ARDALOS 3 and Michael A. SAUNDERS y June 14, 1993 Abstract We present a simplex-type algorithm for linear programming The Simplex Algorithm Uri Feige November 2011 1 The simplex algorithm The simplex algorithm was designed by Danzig in 1947. This write-up presents the main ideas involved. It is a slight update (mostly