Using the Simplex Method in Mixed Integer Linear Programming


 Horace West
 1 years ago
 Views:
Transcription
1 Integer Using the Simplex Method in Mixed Integer UTFSM Nancy, 17 december 2015 Using the Simplex Method in Mixed Integer
2 Outline Mathematical Programming Integer 1 Mathematical Programming Optimisation Problems Solving Techniques 2 3 Integer Using the Simplex Method in Mixed Integer
3 Integer Optimisation Problems Solving Techniques 1 Mathematical Programming Optimisation Problems Solving Techniques 2 3 Integer Using the Simplex Method in Mixed Integer
4 Integer Optimisation Problems Solving Techniques 1 Mathematical Programming Optimisation Problems Solving Techniques 2 3 Integer Using the Simplex Method in Mixed Integer
5 Integer Optimisation Problems Solving Techniques Constraint Satisfaction Problems A Constraint Satisfaction Problem (CSP) consists of: Variables: The unknown, what we can modify Parameters: The known, what we cannot modify Constraints: Relationships among variables and parameters Using the Simplex Method in Mixed Integer
6 Integer Optimisation Problems Solving Techniques NQueens Problem Using the Simplex Method in Mixed Integer
7 Integer Optimisation Problems Solving Techniques Map Colouring Problem Using the Simplex Method in Mixed Integer
8 Integer Optimisation Problems Solving Techniques Constraint Satisfaction Problems Given a CSP we can define: Solution: An assignment of values to variables Feasible solution: A solution satisfying all constraints A CSP can be Satisfiable: If there exists at least one feasible solution Unsatisfiable: If there does not exist a feasible solution Solving a CSP means to search for one or all feasible solutions, or to prove that the CSP is unsatisfiable. Using the Simplex Method in Mixed Integer
9 Integer Optimisation Problems Solving Techniques Constraint Satisfaction Optimisation Problems A Constraint Satisfaction Optimisation Problem (CSOP) consists of: CSP: Variables, parameters, and constraints Objective function(s): Function(s) to be optimised satisfying all constraints Using the Simplex Method in Mixed Integer
10 Integer Optimisation Problems Solving Techniques Knapsack Problem Using the Simplex Method in Mixed Integer
11 Integer Optimisation Problems Solving Techniques Transportation Problems Using the Simplex Method in Mixed Integer
12 Integer Optimisation Problems Solving Techniques Traveling Salesperson Problem (TSP) Using the Simplex Method in Mixed Integer
13 Integer Optimisation Problems Solving Techniques Vehicle Routing Problems (VRP) Using the Simplex Method in Mixed Integer
14 Integer Optimisation Problems Solving Techniques Scheduling Problem Using the Simplex Method in Mixed Integer
15 Integer Optimisation Problems Solving Techniques Given a CSOP we can define: Optimal solution: A feasible solution better than or equal to all other feasible solutions Depending on the set of constraints, a CSOP can be Feasible: If there exists at least one feasible solution Unfeasible: If there does not exist a feasible solution Depending on the set of constraints and the objective function, a feasible CSOP can have Unbounded Feasible Solutions: Feasible solutions can grow indefinitely Unique Optimal Solution: Just one optimal solution Multiple Optimal Solutions: Several optimal solutions Solving a CSOP means to search for optimal solution(s) or to prove that the CSOP is unfeasible. Using the Simplex Method in Mixed Integer
16 Integer Optimisation Problems Solving Techniques 1 Mathematical Programming Optimisation Problems Solving Techniques 2 3 Integer Using the Simplex Method in Mixed Integer
17 Integer Solving Techniques Optimisation Problems Solving Techniques Both CSPs and CSOPs can be tackled using two approaches: Global or Complete Techniques: Search the global optimal solution, the best one among all feasible solutions Local or Incomplete Techniques: Search for a local optimal solution, the best one among the considered feasible solutions Using the Simplex Method in Mixed Integer
18 Integer 1 Mathematical Programming Optimisation Problems Solving Techniques 2 3 Integer Using the Simplex Method in Mixed Integer
19 Integer Linear: All mathematical functions are linear Programming: In the sense of planning activities Industrial Applications: Blending, transportation, etc. Origin: George B. Dantzig, 1947 Using the Simplex Method in Mixed Integer
20 Integer 1 Mathematical Programming Optimisation Problems Solving Techniques 2 3 Integer Using the Simplex Method in Mixed Integer
21 Mathematical Programming Integer 1 Assign variables x 1 and x 2 to axes x and y of plane, respectively. 2 Identify the feasible area: Applying nonnegativity constraints, select positive side of both axes. Draw functional constraints. 3 Identify the feasible solution in the feasible area that optimise the value of the objective function. Using the Simplex Method in Mixed Integer
22 Example Mathematical Programming Integer Considering the following problem: Max z = 11x x 2 Subject to 1x 1 + 1x x 1 + 7x x 1 + 5x x 1 + 1x 2 30 x 1, x 2 0 Using the Simplex Method in Mixed Integer
23 Integer Variable representation x 1 and x 2 are assigned to axes x and y of a plane, respectively. Each point (x 1, x 2 ) represents a solution. The solution where x 1 = 0 x 2 = 0 is called the origin. Using the Simplex Method in Mixed Integer
24 Integer Set of feasible solutions Nonnegativity constraints, x 1 0 y x 2 0, impose (x 1, x 2 ) will be in the positive side of both axes. Using the Simplex Method in Mixed Integer
25 Integer Set of feasible solutions Constraint 1x 1 + 1x 2 17 allows solutions (x 1, x 2 ) under line 1x 1 + 1x 2 = 170: Doing the same for all constraints: Using the Simplex Method in Mixed Integer
26 Integer Set of feasible solutions The set of feasible solutions is called feasible area. Using the Simplex Method in Mixed Integer
27 Optimising Mathematical Programming Integer Assigning arbitrary values to the objective function 11x x 2 : 11x x 2 = 100: there exist some feasible solutions 11x x 2 = 135: there exist some feasible solutions 11x x 2 = 170: there does not exist any feasible solution Using the Simplex Method in Mixed Integer
28 Optimising Mathematical Programming Integer Different values of the objective function generate different parallel lines, the higher the value, the farther the line from the origin. Feasible solution(s) on the farthest line correspond to the optimal solution(s). Optimal solution is in the intersection of constraints c3 and c4: 3x 1 + 5x 2 = 48 3x 1 + 1x 2 = 30 solving the system of two equations we get: (x 1, x 2 ) = (17/2, 9/2); z = 11 17/ /2 = 313/2 Using the Simplex Method in Mixed Integer
29 Integer Properties of feasible solutions at vertices 1 Location of optimal solutions: If there exists an optimal solution, it has to be a feasible solution at a vertex. If there exist multiple optimal solutions, at least two of them have to be feasible solutions at adjacent vertices. 2 There exists a finite number of feasible solutions at the vertices. 3 If a feasible solution at a vertex is equal or better than all the feasible solutions at adjacent vertices, then it is equal or better than all the feasible solutions at the vertices, that is, it is optimal. Using the Simplex Method in Mixed Integer
30 Standard form Mathematical Programming Integer Subject to Min z = 1x 1 + 1x 2 1x 1 + 2x x 1 + 1x x 1 2x 2 = 10 x 1, x 2 0 The standard form of this problem is the following: Subject to Min z = 1x 1 + 1x 2 + 0s 1 + 0s 2 1x 1 + 2x 2 + 1s 1 = 30 2x 1 + 1x 2 1s 2 = 28 1x 1 2x 2 = 10 x 1, x 2, s 1, s 2 0 Using the Simplex Method in Mixed Integer
31 Integer 1 Mathematical Programming Optimisation Problems Solving Techniques 2 3 Integer Using the Simplex Method in Mixed Integer
32 Integer Iterative Optimisation Algorithm Structure Initialisation procedure Iterative process Stopping criterion Optimisation Algorithm Structure Search for an initial solution Iteratively search for a better solution Optimality test Using the Simplex Method in Mixed Integer
33 Integer Overview of the Simplex Method 1 Initial step: Start in a feasible basic solution at a vertex. 2 Iterative step: Move to a better feasible basic solution at an adjacent vertex 3 Optimality test: A feasible basic solution at a vertex is optimal when it is equal or better than feasible basic solutions at all adjacent vertices. Using the Simplex Method in Mixed Integer
34 Integer Creating the initial simplex tableau Given the following linear programming problem in standard form: Subject to Max z = 11x x 2 + 0s 1 + 0s 2 + 0s 3 + 0s 4 1x 1 + 1x 2 + 1s 1 = 17 3x 1 + 7x 2 + 1s 2 = 63 3x 1 + 5x 2 + 1s 3 = 48 3x 1 + 1x 2 + 1s 4 = 30 x 1, x 2, s 1, s 2, s 3, s 4 0 Let: c j : coefficient for variable j in the objective function a ij : coefficient for variable j in constraint i b i : righthand side value of constraint i Using the Simplex Method in Mixed Integer
35 Integer A simplex tableau is the following: c 1... c n a a 1n b a m1... a mn b m Row c Matrix A Column b Using the Simplex Method in Mixed Integer
36 Integer Creating the initial simplex tableau Subject to Max z = 11x x 2 + 0s 1 + 0s 2 + 0s 3 + 0s 4 1x 1 + 1x 2 + 1s 1 = 17 3x 1 + 7x 2 + 1s 2 = 63 3x 1 + 5x 2 + 1s 3 = 48 3x 1 + 1x 2 + 1s 4 = 30 x 1, x 2, s 1, s 2, s 3, s 4 0 x 1 x 2 s 1 s 2 s 3 s b i Using the Simplex Method in Mixed Integer
37 Integer To read the solutions we add two columns: Basis: Current basic variables c j : Coefficients in the objective function of current basic variables x 1 x 2 s 1 s 2 s 3 s 4 Basis c j b i s s s s As all nobasic variables are assigned value zero, the value of basic variables is obtained from column b i at the corresponding row: (x 1, x 2, s 1, s 2, s 3, s 4 ) = (0, 0, 17, 63, 48, 30) The product of the column c j and the column b i turns in the value of the objective function (indicated under column b i ) Using the Simplex Method in Mixed Integer
38 Graphically Mathematical Programming Integer Using the Simplex Method in Mixed Integer
39 Integer Evaluating adjacent feasible basic solutions How about the quality of this basic solution with respect to adjacent basic solutions? Changing the value of a variable: Impacts directly the objective function through its coefficient in the objective function (c j ) Impacts indirectly the objective function through variations in other variables We need to take into account both impacts Using the Simplex Method in Mixed Integer
40 Integer Evaluating adjacent feasible basic solutions The direct impact in the objective function is easily computed by c j, but for the decrease and the net change we will add two rows into the tableau: z j : Decrease in the objective function when increasing variable j by one. c j z j : Net change in the objective function when increasing variable j by one. Using the Simplex Method in Mixed Integer
41 Integer Computing z j z 1 = = 0 z 2 = = 0 z 3 = = 0 z 4 = = 0 z 5 = = 0 z 6 = = 0 Adding this information into the tableau... x 1 x 2 s 1 s 2 s 3 s 4 Basis c j b i s s s s z j Entering x 1 or x 2 does not decrease the obj func Using the Simplex Method in Mixed Integer
42 Integer Computing c j z j c1 z1 = 11 0 = 11 c 2 z 2 = 14 0 = 14 c 3 z 3 = 0 0 = 0 c 4 z 4 = 0 0 = 0 c 5 z 5 = 0 0 = 0 c 6 z 6 = 0 0 = 0 Adding this information into the tableau... x 1 x 2 s 1 s 2 s 3 s 4 Basis c j b i s s s s z j c j z j It is not optimal, entering x 1 or x 2 allows to improve the obj func Using the Simplex Method in Mixed Integer
43 Integer Criterion to enter a nonbasic variable into the basis The first criterion proposed to select the entering nonbasic variable was to choose the variable with highest c j z j, i.e, with highest unit increase in the objective function. x 1 x 2 s 1 s 2 s 3 s 4 Basis c j b i s s s s z j c j z j Of course, we would like to increase the value of x 2 as much as possible, but we have to take into account all constraints! Using the Simplex Method in Mixed Integer
44 Integer Criterion to remove a basic variable from the basis The maximum value for x 2 is giving by the first current basic variable getting value zero when x 1 is increasing. Considering column j corresponding to the nonbasic variable entering into the basis, we compute rate b i a ij (only for a ij > 0) for each row i, and we select the basic variable giving the minimum b i a ij. x 1 x 2 s 1 s 2 s 3 s 4 Basis c j b i b i s s s s z j c j z j a ij 1 = = 9 = = 30 1 Using the Simplex Method in Mixed Integer
45 Pivoting Mathematical Programming Integer x 1 x 2 s 1 s 2 s 3 s 4 Basis c j b i b i a ij s s s s z j c j z j The adjacent basic solution is obtained replacing s 2 by x 2 in the basis, and operating rows to obtain in column x 2 the following vector: Using the Simplex Method in Mixed Integer
46 Pivoting Mathematical Programming Integer Using the Gauss Elimination Method, we can Multiply a constraint by a constant k (k 0) Add constraints In our case: Multiplying the second row by 1 7 we get the new second row. Multiplying the new second row by 1 and adding it to the first row we get the new first row. Multiplying the new second row by 5 and adding it to the third row we get the new third row. Multiplying the new second row by 1 and adding it to the fourth row we get the new fourth row. Using the Simplex Method in Mixed Integer
47 Pivoting Mathematical Programming Integer x 1 x 2 s 1 s 2 s 3 s 4 Basis c j b i b i a ij s s s s z j c j z j x 1 x 2 s 1 s 2 s 3 s 4 Basis c j b i 4 1 s x s s New feasible basic solution: (x 1, x 2, s 1, s 2, s 3, s 4 ) = (0, 9, 8, 0, 3, 21); z = 126 Using the Simplex Method in Mixed Integer
48 Graphically Mathematical Programming Integer Using the Simplex Method in Mixed Integer
49 Integer Evaluating adjacent feasible basic solutions x 1 x 2 s 1 s 2 s 3 s 4 Basis c j b i 4 1 s x s s Computing z j and c j z j... x 1 x 2 s 1 s 2 s 3 s 4 Basis c j b i 4 1 s x s s z j c j z j c j z j > 0 so there exists a better adjacent basic solution and the current one is not optimal Using the Simplex Method in Mixed Integer
50 Pivoting Mathematical Programming Integer x 1 x 2 s 1 s 2 s 3 s 4 Basis c j b i b i s = x = s = s = z j c j z j x 1 x 2 s 1 s 2 s 3 s 4 Basis c j b i 1 2 s x x s New feasible basic solution: (x 1, x 2, s 1, s 2, s 3, s 4 ) = ( 7 2, 15 2 a ij, 6, 0, 0, 12); z = Using the Simplex Method in Mixed Integer
51 Graphically Mathematical Programming Integer Using the Simplex Method in Mixed Integer
52 Integer Evaluating adjacent feasible basic solutions x 1 x 2 s 1 s 2 s 3 s 4 Basis c j b i 1 2 s x x s Computing z j and c j z j... x 1 x 2 s 1 s 2 s 3 s 4 Basis c j b i 1 2 s x x s z j c j z j c j z j > 0 so there exists a better adjacent basic solution and the current one is not optimal Using the Simplex Method in Mixed Integer
53 Pivoting Mathematical Programming Integer x 1 x 2 s 1 s 2 s 3 s 4 Basis c j b i b i s x a ij = x s z j c j z j x 1 x 2 s 1 s 2 s 3 s 4 Basis c j b i s x x s New feasible basic solution: (x 1, x 2, s 1, s 2, s 3, s 4 ) = ( 17 2, = 15 2 = 6, 4, 6, 0, 0); z = Using the Simplex Method in Mixed Integer
54 Graphically Mathematical Programming Integer Using the Simplex Method in Mixed Integer
55 Integer Evaluating adjacent feasible basic solutions x 1 x 2 s 1 s 2 s 3 s 4 Basis c j b i 1 1 s x x s Computing z j and c j z j... x 1 x 2 s 1 s 2 s 3 s 4 Basis c j b i 1 1 s x x s z j c j z j All values in row c j z j are zero or negative so the Stopping Criterion is verified! Optimal Solution: (x, xcarlos, scastro, s, s Using, s the ) = Simplex ( 17 Method, 9, 4, in Mixed 6, 0, Integer 0)
56 Integer 1 Mathematical Programming Optimisation Problems Solving Techniques 2 3 Integer Using the Simplex Method in Mixed Integer
57 Integer Integer Why? Many practical problems need variables to take integer values How to solve it? Exhaustive enumeration is only useful when dealing with few combinations of values In general, we deal with a combinatorial problem Solving techniques are based on divide and conquer approach Using the Simplex Method in Mixed Integer
58 Integer 1 Mathematical Programming Optimisation Problems Solving Techniques 2 3 Integer Using the Simplex Method in Mixed Integer
59 Integer Solve linear programming relaxations to bound the objective function Create branches by adding constraints that eliminate noninteger values Using the Simplex Method in Mixed Integer
60 Example Mathematical Programming Integer Considering the following Integer Programming problem: Max z = 11x x 2 Subject to 1x 1 + 1x x 1 + 7x x 1 + 5x x 1 + 1x 2 30 x 1, x 2 0 x 1, x 2 Z Using the Simplex Method in Mixed Integer
61 Integer Solving the Relaxation Max z = 11x x 2 Subject to 1x 1 + 1x x 1 + 7x x 1 + 5x x 1 + 1x 2 30 x 1, x 2 0 x = (8.5, 4.5) z = Using the Simplex Method in Mixed Integer
62 Graphically Mathematical Programming Integer Using the Simplex Method in Mixed Integer
63 Enumeration tree Mathematical Programming Integer Using the Simplex Method in Mixed Integer
64 Branching Mathematical Programming Integer Branching on x 1 : Using the Simplex Method in Mixed Integer
65 Integer Solving the Relaxation Max z = 11x x 2 Subject to 1x 1 + 1x x 1 + 7x x 1 + 5x x 1 + 1x 2 30 x 1 8 x 1, x 2 0 x = (8, 4.8) z = Max z = 11x x 2 Subject to 1x 1 + 1x x 1 + 7x x 1 + 5x x 1 + 1x 2 30 x 1 9 x 1, x 2 0 x = (9, 3) z = 141 Using the Simplex Method in Mixed Integer
66 Enumeration tree Mathematical Programming Integer Using the Simplex Method in Mixed Integer
67 Branching Mathematical Programming Integer Branching on x 2 : Using the Simplex Method in Mixed Integer
68 Integer Solving the Relaxation Max z = 11x x 2 Subject to 1x 1 + 1x x 1 + 7x x 1 + 5x x 1 + 1x 2 30 x 1 8 & x 2 4 x 1, x 2 0 x = (8, 4) z = 144 Max z = 11x x 2 Subject to 1x 1 + 1x x 1 + 7x x 1 + 5x x 1 + 1x 2 30 x 1 8 & x 2 5 x 1, x 2 0 x = (7.67, 5) z = Using the Simplex Method in Mixed Integer
69 Enumeration tree Mathematical Programming Integer Using the Simplex Method in Mixed Integer
70 Branching Mathematical Programming Integer Branching again on x 1 : Using the Simplex Method in Mixed Integer
71 Integer Solving the Relaxation Max z = 11x x 2 Subject to 1x 1 + 1x x 1 + 7x x 1 + 5x x 1 + 1x 2 30 x 1 8 & x 2 5 & x 1 7 x 1, x 2 0 x = (7, 5.4) z = Max z = 11x x 2 Subject to 1x 1 + 1x x 1 + 7x x 1 + 5x x 1 + 1x 2 30 x 1 8 & x 2 5 & x 1 8 x 1, x 2 0 Unfeasible Using the Simplex Method in Mixed Integer
72 Enumeration tree Mathematical Programming Integer Using the Simplex Method in Mixed Integer
73 Branching Mathematical Programming Integer Branching again on x 2 : Using the Simplex Method in Mixed Integer
74 Integer Solving the Relaxation Max z = 11x x 2 Subject to 1x 1 + 1x x 1 + 7x x 1 + 5x x 1 + 1x 2 30 x 1 8 & x 2 5 & x 1 7 & x 2 5 x 1, x 2 0 x = (7, 5) z = 147 Max z = 11x x 2 Subject to 1x 1 + 1x x 1 + 7x x 1 + 5x x 1 + 1x 2 30 x 1 8 & x 2 5 & x 1 7 & x 2 6 x 1, x 2 0 x = (6, 6) z = 150 Using the Simplex Method in Mixed Integer
75 Enumeration tree Mathematical Programming Integer Using the Simplex Method in Mixed Integer
76 Integer 1 Mathematical Programming Optimisation Problems Solving Techniques 2 3 Integer Using the Simplex Method in Mixed Integer
77 Integer Solve linear programming relaxations to bound the objective function Create branches by adding constraints that fix variables Using the Simplex Method in Mixed Integer
78 Example Mathematical Programming Integer Max z = 9x 1 + 5x 2 + 6x 3 + 4x 4 Subject to 6x 1 + 3x 2 + 5x 3 + 2x 4 10 x 3 + x 4 1 x 1 + x 3 0 x 2 + x 4 0 x i {0, 1} i = 1,..., 4 Using the Simplex Method in Mixed Integer
79 Integer Constraints allowing values A constraint allowing values for variables: x i {0, 1} is equivalent to the following constraints: x i 0 x i 1 x i Z This correspond to an IP problem with upper bounds for all variables. Using the Simplex Method in Mixed Integer
80 Integer Solving the Relaxation Max z = 9x 1 + 5x 2 + 6x 3 + 4x 4 Subject to 6x 1 + 3x 2 + 5x 3 + 2x 4 10 x 3 + x 4 1 x 1 + x 3 0 x 2 + x 4 0 x i 1 i = 1,..., 4 x i 0 i = 1,..., 4 x = ( 5 6, 1, 0, 1); z = Using the Simplex Method in Mixed Integer
81 Enumeration tree Mathematical Programming Integer Using the Simplex Method in Mixed Integer
82 Branching Mathematical Programming Integer Remember that x i {0, 1} is equivalent to the following constraints: x i 0 x i 1 x i Z When branching we will add constraints x i 0 and x i 1: x i 0 x i 0 x i 1 x i 1 x i 0 x i 1 x i Z x i Z x i = 0 x i = 1 Using the Simplex Method in Mixed Integer
83 Branching Mathematical Programming Integer Subproblem 1 (original problem x 1 = 0): Max z = 9x 1 + 5x 2 + 6x 3 + 4x 4 Subject to 6x 1 + 3x 2 + 5x 3 + 2x 4 10 x 3 + x 4 1 x 1 + x 3 0 x 2 + x 4 0 x 1 = 0 x i {0, 1} i = 1,..., 4 Subproblem 2 (original problem x 1 = 1): Max z = 9x 1 + 5x 2 + 6x 3 + 4x 4 Subject to 6x 1 + 3x 2 + 5x 3 + 2x 4 10 x 3 + x 4 1 x 1 + x 3 0 x 2 + x 4 0 x 1 = 1 x i {0, 1} i = 1,..., 4 Using the Simplex Method in Mixed Integer
84 Integer Solving the Relaxation Subproblem 1: Max z = 5x 2 + 6x 3 + 4x 4 Subject to 3x 2 + 5x 3 + 2x 4 10 x 3 + x 4 1 x 3 0 x 2 + x 4 0 x i 1 i = 2,..., 4 x i 0 i = 2,..., 4 x = (0, 1, 0, 1); z = 9 Subproblem 2: Max z = 9 + 5x 2 + 6x 3 + 4x 4 Subject to 3x 2 + 5x 3 + 2x 4 4 x 3 + x 4 1 x 3 1 x 2 + x 4 0 x i 1 i = 2,..., 4 x i 0 i = 2,..., 4 x = (1, 4 5, 0, 4 5 ); z = Using the Simplex Method in Mixed Integer
85 Enumeration tree Mathematical Programming Integer Using the Simplex Method in Mixed Integer
86 Branching Mathematical Programming Integer Subproblem 3 (subproblem 2 x 2 = 0): Subject to Max z = 9 + 5x 2 + 6x 3 + 4x 4 3x 2 + 5x 3 + 2x 4 4 x 3 + x 4 1 x 3 1 x 2 + x 4 0 x 2 = 0 x i {0, 1} i = 2,..., 4 Subproblem 4 (subproblem 2 x 2 = 1): Max z = 9 + 5x 2 + 6x 3 + 4x 4 Subject to 3x 2 + 5x 3 + 2x 4 4 x 3 + x 4 1 x 3 1 x 2 + x 4 0 x 2 = 1 x i {0, 1} i = 2,..., 4 Using the Simplex Method in Mixed Integer
87 Integer Solving the Relaxation Subproblem 3: Max z = 9 + 6x 3 + 4x 4 Subject to 5x3 + 2x 4 4 x 3 + x 4 1 x 3 1 x 4 0 x i 1 i = 3, 4 x i 0 i = 3, 4 x = (1, 0, 4 5, 0); z = Subproblem 4: Max z = x 3 + 4x 4 Subject to 5x3 + 2x 4 1 x 3 + x 4 1 x 3 1 x 4 1 x i 1 i = 3, 4 x i 0 i = 3, 4 x = (1, 1, 0, 1 2 ); z = 16 Using the Simplex Method in Mixed Integer
88 Enumeration tree Mathematical Programming Integer Using the Simplex Method in Mixed Integer
89 Branching Mathematical Programming Integer Subproblem 5 (subproblem 4 x 4 = 0): Max z = x 3 + 4x 4 Subject to 5x 3 + 2x 4 1 x 3 + x 4 1 x 3 1 x 4 1 x 4 = 0 x i {0, 1} i = 3, 4 Subproblem 6 (subproblem 4 x 4 = 1): Max z = x 3 + 4x 4 Subject to 5x 3 + 2x 4 1 x 3 + x 4 1 x 3 1 x 4 1 x 4 = 1 x i {0, 1} i = 3, 4 Using the Simplex Method in Mixed Integer
90 Integer Solving the Relaxation Subproblem 5: Subject to Max z = x 3 5x 3 1 x 3 1 x 3 1 x 3 0 x = (1, 1, 1 5, 0); z = Subproblem 6: Subject to Max z = x 3 5x 3 1 x 3 0 x 3 1 x 3 1 x 3 0 Unfeasible Using the Simplex Method in Mixed Integer
91 Enumeration tree Mathematical Programming Integer Using the Simplex Method in Mixed Integer
92 Branching Mathematical Programming Integer Subproblem 7 (subproblem 5 x 3 = 0): Max z = x 3 Subject to 5x 3 1 x 3 1 x 3 1 x 3 0 x 3 = 0 x 3 {0, 1} Subproblem 8 (subproblem 5 x 3 = 1): Max z = x 3 Subject to 5x 3 1 x 3 1 x 3 1 x 3 0 x 3 = 1 x 3 {0, 1} Using the Simplex Method in Mixed Integer
93 Integer Solving the Relaxation Subproblem 7: Variable enumeration: x = (1, 1, 0, 0) Feasible solution z = 14 Subproblem 8: Variable enumeration: x = (1, 1, 1, 0) Unfeasible solution Using the Simplex Method in Mixed Integer
94 Enumeration tree Mathematical Programming Integer Using the Simplex Method in Mixed Integer
95 Integer Solving Mixed Integer Programming Problems Mixed Integer Programming Problems involve real, integer, and binary variables: Real variables: Simplex Method Integer variables: Simplex Method + constraints and Binary variables: Simplex Method + constraints = Using the Simplex Method in Mixed Integer
Standard Form of a Linear Programming Problem
494 CHAPTER 9 LINEAR PROGRAMMING 9. THE SIMPLEX METHOD: MAXIMIZATION For linear programming problems involving two variables, the graphical solution method introduced in Section 9. is convenient. However,
More informationChapter 6. Linear Programming: The Simplex Method. Introduction to the Big M Method. Section 4 Maximization and Minimization with Problem Constraints
Chapter 6 Linear Programming: The Simplex Method Introduction to the Big M Method In this section, we will present a generalized version of the simplex method that t will solve both maximization i and
More information3. Evaluate the objective function at each vertex. Put the vertices into a table: Vertex P=3x+2y (0, 0) 0 min (0, 5) 10 (15, 0) 45 (12, 2) 40 Max
SOLUTION OF LINEAR PROGRAMMING PROBLEMS THEOREM 1 If a linear programming problem has a solution, then it must occur at a vertex, or corner point, of the feasible set, S, associated with the problem. Furthermore,
More informationIEOR 4404 Homework #2 Intro OR: Deterministic Models February 14, 2011 Prof. Jay Sethuraman Page 1 of 5. Homework #2
IEOR 4404 Homework # Intro OR: Deterministic Models February 14, 011 Prof. Jay Sethuraman Page 1 of 5 Homework #.1 (a) What is the optimal solution of this problem? Let us consider that x 1, x and x 3
More information1 Solving LPs: The Simplex Algorithm of George Dantzig
Solving LPs: The Simplex Algorithm of George Dantzig. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example.
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 informationSimplex method summary
Simplex method summary Problem: optimize a linear objective, subject to linear constraints 1. Step 1: Convert to standard form: variables on righthand side, positive constant on left slack variables for
More information7.4 Linear Programming: The Simplex Method
7.4 Linear Programming: The Simplex Method For linear programming problems with more than two variables, the graphical method is usually impossible, so the simplex method is used. Because the simplex method
More information5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1
5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1 General Integer Linear Program: (ILP) min c T x Ax b x 0 integer Assumption: A, b integer The integrality condition
More informationDefinition of a Linear Program
Definition of a Linear Program Definition: A function f(x 1, x,..., x n ) of x 1, x,..., x n is a linear function if and only if for some set of constants c 1, c,..., c n, f(x 1, x,..., x n ) = c 1 x 1
More informationLinear Programming. March 14, 2014
Linear Programming March 1, 01 Parts of this introduction to linear programming were adapted from Chapter 9 of Introduction to Algorithms, Second Edition, by Cormen, Leiserson, Rivest and Stein [1]. 1
More informationPractical Guide to the Simplex Method of Linear Programming
Practical Guide to the Simplex Method of Linear Programming Marcel Oliver Revised: April, 0 The basic steps of the simplex algorithm Step : Write the linear programming problem in standard form Linear
More informationLinear Programming I
Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins
More informationOPRE 6201 : 2. Simplex Method
OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2
More informationLinear Programming for Optimization. Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc.
1. Introduction Linear Programming for Optimization Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc. 1.1 Definition Linear programming is the name of a branch of applied mathematics that
More informationSpecial Situations in the Simplex Algorithm
Special Situations in the Simplex Algorithm Degeneracy Consider the linear program: Maximize 2x 1 +x 2 Subject to: 4x 1 +3x 2 12 (1) 4x 1 +x 2 8 (2) 4x 1 +2x 2 8 (3) x 1, x 2 0. We will first apply the
More information24. The Branch and Bound Method
24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NPcomplete. Then one can conclude according to the present state of science that no
More informationLinear Programming. Widget Factory Example. Linear Programming: Standard Form. Widget Factory Example: Continued.
Linear Programming Widget Factory Example Learning Goals. Introduce Linear Programming Problems. Widget Example, Graphical Solution. Basic Theory:, Vertices, Existence of Solutions. Equivalent formulations.
More informationLecture 3. Linear Programming. 3B1B Optimization Michaelmas 2015 A. Zisserman. Extreme solutions. Simplex method. Interior point method
Lecture 3 3B1B Optimization Michaelmas 2015 A. Zisserman Linear Programming Extreme solutions Simplex method Interior point method Integer programming and relaxation The Optimization Tree Linear Programming
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 informationINTEGER PROGRAMMING. Integer Programming. Prototype example. BIP model. BIP models
Integer Programming INTEGER PROGRAMMING In many problems the decision variables must have integer values. Example: assign people, machines, and vehicles to activities in integer quantities. If this is
More information1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where.
Introduction Linear Programming Neil Laws TT 00 A general optimization problem is of the form: choose x to maximise f(x) subject to x S where x = (x,..., x n ) T, f : R n R is the objective function, S
More informationSolving Linear Programs
Solving Linear Programs 2 In this chapter, we present a systematic procedure for solving linear programs. This procedure, called the simplex method, proceeds by moving from one feasible solution to another,
More informationReduced echelon form: Add the following conditions to conditions 1, 2, and 3 above:
Section 1.2: Row Reduction and Echelon Forms Echelon form (or row echelon form): 1. All nonzero rows are above any rows of all zeros. 2. Each leading entry (i.e. left most nonzero entry) of a row is in
More informationApproximation Algorithms
Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NPCompleteness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms
More informationSolving Integer Programming with BranchandBound Technique
Solving Integer Programming with BranchandBound Technique This is the divide and conquer method. We divide a large problem into a few smaller ones. (This is the branch part.) The conquering part is done
More informationWhat is Linear Programming?
Chapter 1 What is Linear Programming? An optimization problem usually has three essential ingredients: a variable vector x consisting of a set of unknowns to be determined, an objective function of x to
More informationNetwork Models 8.1 THE GENERAL NETWORKFLOW PROBLEM
Network Models 8 There are several kinds of linearprogramming models that exhibit a special structure that can be exploited in the construction of efficient algorithms for their solution. The motivation
More informationLinear Programming. April 12, 2005
Linear Programming April 1, 005 Parts of this were adapted from Chapter 9 of i Introduction to Algorithms (Second Edition) /i by Cormen, Leiserson, Rivest and Stein. 1 What is linear programming? The first
More informationNonlinear Programming Methods.S2 Quadratic Programming
Nonlinear Programming Methods.S2 Quadratic Programming Operations Research Models and Methods Paul A. Jensen and Jonathan F. Bard A linearly constrained optimization problem with a quadratic objective
More informationDuality in Linear Programming
Duality in Linear Programming 4 In the preceding chapter on sensitivity analysis, we saw that the shadowprice interpretation of the optimal simplex multipliers is a very useful concept. First, these shadow
More informationLinear Programming: Theory and Applications
Linear Programming: Theory and Applications Catherine Lewis May 11, 2008 1 Contents 1 Introduction to Linear Programming 3 1.1 What is a linear program?...................... 3 1.2 Assumptions.............................
More information56:171. Operations Research  Sample Homework Assignments Fall 1992 Dennis Bricker Dept. of Industrial Engineering University of Iowa.
56:171 Operations Research  Sample Homework Assignments Fall 1992 Dennis Bricker Dept. of Industrial Engineering University of Iowa Homework #1 (1.) Linear Programming Model Formulation. SunCo processes
More information1 Determinants and the Solvability of Linear Systems
1 Determinants and the Solvability of Linear Systems In the last section we learned how to use Gaussian elimination to solve linear systems of n equations in n unknowns The section completely sidestepped
More informationLECTURE 5: DUALITY AND SENSITIVITY ANALYSIS. 1. Dual linear program 2. Duality theory 3. Sensitivity analysis 4. Dual simplex method
LECTURE 5: DUALITY AND SENSITIVITY ANALYSIS 1. Dual linear program 2. Duality theory 3. Sensitivity analysis 4. Dual simplex method Introduction to dual linear program Given a constraint matrix A, right
More informationOptimization Modeling for Mining Engineers
Optimization Modeling for Mining Engineers Alexandra M. Newman Division of Economics and Business Slide 1 Colorado School of Mines Seminar Outline Linear Programming Integer Linear Programming Slide 2
More informationUsing the Simplex Method to Solve Linear Programming Maximization Problems J. Reeb and S. Leavengood
PERFORMANCE EXCELLENCE IN THE WOOD PRODUCTS INDUSTRY EM 8720E October 1998 $3.00 Using the Simplex Method to Solve Linear Programming Maximization Problems J. Reeb and S. Leavengood A key problem faced
More informationSensitivity Analysis 3.1 AN EXAMPLE FOR ANALYSIS
Sensitivity Analysis 3 We have already been introduced to sensitivity analysis in Chapter via the geometry of a simple example. We saw that the values of the decision variables and those of the slack and
More information1. (a) Multiply by negative one to make the problem into a min: 170A 170B 172A 172B 172C Antonovics Foster Groves 80 88
Econ 172A, W2001: Final Examination, Possible Answers There were 480 possible points. The first question was worth 80 points (40,30,10); the second question was worth 60 points (10 points for the first
More informationChapter 2 Solving Linear Programs
Chapter 2 Solving Linear Programs Companion slides of Applied Mathematical Programming by Bradley, Hax, and Magnanti (AddisonWesley, 1977) prepared by José Fernando Oliveira Maria Antónia Carravilla A
More informationLinear Programming Notes V Problem Transformations
Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material
More information! Solve problem to optimality. ! Solve problem in polytime. ! Solve arbitrary instances of the problem. !approximation algorithm.
Approximation Algorithms Chapter Approximation Algorithms Q Suppose I need to solve an NPhard problem What should I do? A Theory says you're unlikely to find a polytime algorithm Must sacrifice one of
More informationA Constraint Programming based Column Generation Approach to Nurse Rostering Problems
Abstract A Constraint Programming based Column Generation Approach to Nurse Rostering Problems Fang He and Rong Qu The Automated Scheduling, Optimisation and Planning (ASAP) Group School of Computer Science,
More informationChapter 11. 11.1 Load Balancing. Approximation Algorithms. Load Balancing. Load Balancing on 2 Machines. Load Balancing: Greedy Scheduling
Approximation Algorithms Chapter Approximation Algorithms Q. Suppose I need to solve an NPhard problem. What should I do? A. Theory says you're unlikely to find a polytime algorithm. Must sacrifice one
More informationTransportation Polytopes: a Twenty year Update
Transportation Polytopes: a Twenty year Update Jesús Antonio De Loera University of California, Davis Based on various papers joint with R. Hemmecke, E.Kim, F. Liu, U. Rothblum, F. Santos, S. Onn, R. Yoshida,
More informationLinear Programming. Solving LP Models Using MS Excel, 18
SUPPLEMENT TO CHAPTER SIX Linear Programming SUPPLEMENT OUTLINE Introduction, 2 Linear Programming Models, 2 Model Formulation, 4 Graphical Linear Programming, 5 Outline of Graphical Procedure, 5 Plotting
More informationChapter 6: Sensitivity Analysis
Chapter 6: Sensitivity Analysis Suppose that you have just completed a linear programming solution which will have a major impact on your company, such as determining how much to increase the overall production
More informationA Branch and Bound Algorithm for Solving the Binary Bilevel Linear Programming Problem
A Branch and Bound Algorithm for Solving the Binary Bilevel Linear Programming Problem John Karlof and Peter Hocking Mathematics and Statistics Department University of North Carolina Wilmington Wilmington,
More information2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system
1. Systems of linear equations We are interested in the solutions to systems of linear equations. A linear equation is of the form 3x 5y + 2z + w = 3. The key thing is that we don t multiply the variables
More informationThe number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 72.
ADVANCED SUBSIDIARY GCE UNIT 4736/01 MATHEMATICS Decision Mathematics 1 THURSDAY 14 JUNE 2007 Afternoon Additional Materials: Answer Booklet (8 pages) List of Formulae (MF1) Time: 1 hour 30 minutes INSTRUCTIONS
More informationDATA ANALYSIS II. Matrix Algorithms
DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where
More informationLinear Programming in Matrix Form
Linear Programming in Matrix Form Appendix B We first introduce matrix concepts in linear programming by developing a variation of the simplex method called the revised simplex method. This algorithm,
More information(a) Let x and y be the number of pounds of seed and corn that the chicken rancher must buy. Give the inequalities that x and y must satisfy.
MA 44 Practice Exam Justify your answers and show all relevant work. The exam paper will not be graded, put all your work in the blue book provided. Problem A chicken rancher concludes that his flock
More information1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form
1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and
More informationSYSTEMS OF EQUATIONS
SYSTEMS OF EQUATIONS 1. Examples of systems of equations Here are some examples of systems of equations. Each system has a number of equations and a number (not necessarily the same) of variables for which
More informationCan linear programs solve NPhard problems?
Can linear programs solve NPhard problems? p. 1/9 Can linear programs solve NPhard problems? Ronald de Wolf Linear programs Can linear programs solve NPhard problems? p. 2/9 Can linear programs solve
More informationHow to speedup hard problem resolution using GLPK?
How to speedup hard problem resolution using GLPK? Onfroy B. & Cohen N. September 27, 2010 Contents 1 Introduction 2 1.1 What is GLPK?.......................................... 2 1.2 GLPK, the best one?.......................................
More informationLinear Programming Problems
Linear Programming Problems Linear programming problems come up in many applications. In a linear programming problem, we have a function, called the objective function, which depends linearly on a number
More information. P. 4.3 Basic feasible solutions and vertices of polyhedra. x 1. x 2
4. Basic feasible solutions and vertices of polyhedra Due to the fundamental theorem of Linear Programming, to solve any LP it suffices to consider the vertices (finitely many) of the polyhedron P of the
More information! Solve problem to optimality. ! Solve problem in polytime. ! Solve arbitrary instances of the problem. #approximation algorithm.
Approximation Algorithms 11 Approximation Algorithms Q Suppose I need to solve an NPhard problem What should I do? A Theory says you're unlikely to find a polytime algorithm Must sacrifice one of three
More information7.1 Modelling the transportation problem
Chapter Transportation Problems.1 Modelling the transportation problem The transportation problem is concerned with finding the minimum cost of transporting a single commodity from a given number of sources
More informationThis exposition of linear programming
Linear Programming and the Simplex Method David Gale This exposition of linear programming and the simplex method is intended as a companion piece to the article in this issue on the life and work of George
More informationMathematical finance and linear programming (optimization)
Mathematical finance and linear programming (optimization) Geir Dahl September 15, 2009 1 Introduction The purpose of this short note is to explain how linear programming (LP) (=linear optimization) may
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More informationNoncommercial Software for MixedInteger Linear Programming
Noncommercial Software for MixedInteger Linear Programming J. T. Linderoth T. K. Ralphs December, 2004. Revised: January, 2005. Abstract We present an overview of noncommercial software tools for the
More informationMath 215 HW #1 Solutions
Math 25 HW # Solutions. Problem.2.3. Describe the intersection of the three planes u+v+w+z = 6 and u+w+z = 4 and u + w = 2 (all in fourdimensional space). Is it a line or a point or an empty set? What
More informationInternational Doctoral School Algorithmic Decision Theory: MCDA and MOO
International Doctoral School Algorithmic Decision Theory: MCDA and MOO Lecture 2: Multiobjective Linear Programming Department of Engineering Science, The University of Auckland, New Zealand Laboratoire
More informationFractions and Linear Equations
Fractions and Linear Equations Fraction Operations While you can perform operations on fractions using the calculator, for this worksheet you must perform the operations by hand. You must show all steps
More information4 UNIT FOUR: Transportation and Assignment problems
4 UNIT FOUR: Transportation and Assignment problems 4.1 Objectives By the end of this unit you will be able to: formulate special linear programming problems using the transportation model. define a balanced
More informationOperation Research. Module 1. Module 2. Unit 1. Unit 2. Unit 3. Unit 1
Operation Research Module 1 Unit 1 1.1 Origin of Operations Research 1.2 Concept and Definition of OR 1.3 Characteristics of OR 1.4 Applications of OR 1.5 Phases of OR Unit 2 2.1 Introduction to Linear
More informationCHAPTER 9. Integer Programming
CHAPTER 9 Integer Programming An integer linear program (ILP) is, by definition, a linear program with the additional constraint that all variables take integer values: (9.1) max c T x s t Ax b and x integral
More informationAn Introduction to Linear Programming
An Introduction to Linear Programming Steven J. Miller March 31, 2007 Mathematics Department Brown University 151 Thayer Street Providence, RI 02912 Abstract We describe Linear Programming, an important
More informationDegeneracy in Linear Programming
Degeneracy in Linear Programming I heard that today s tutorial is all about Ellen DeGeneres Sorry, Stan. But the topic is just as interesting. It s about degeneracy in Linear Programming. Degeneracy? Students
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationOptimal Scheduling for Dependent Details Processing Using MS Excel Solver
BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 8, No 2 Sofia 2008 Optimal Scheduling for Dependent Details Processing Using MS Excel Solver Daniela Borissova Institute of
More informationModule1. x 1000. y 800.
Module1 1 Welcome to the first module of the course. It is indeed an exciting event to share with you the subject that has lot to offer both from theoretical side and practical aspects. To begin with,
More informationZeros of a Polynomial Function
Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we
More informationDantzigWolfe bound and DantzigWolfe cookbook
DantzigWolfe bound and DantzigWolfe cookbook thst@man.dtu.dk DTUManagement Technical University of Denmark 1 Outline LP strength of the DantzigWolfe The exercise from last week... The DantzigWolfe
More information56:171 Operations Research Midterm Exam Solutions Fall 2001
56:171 Operations Research Midterm Exam Solutions Fall 2001 True/False: Indicate by "+" or "o" whether each statement is "true" or "false", respectively: o_ 1. If a primal LP constraint is slack at the
More information11. APPROXIMATION ALGORITHMS
11. APPROXIMATION ALGORITHMS load balancing center selection pricing method: vertex cover LP rounding: vertex cover generalized load balancing knapsack problem Lecture slides by Kevin Wayne Copyright 2005
More informationApplied Algorithm Design Lecture 5
Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design
More informationLecture 1: Systems of Linear Equations
MTH Elementary Matrix Algebra Professor Chao Huang Department of Mathematics and Statistics Wright State University Lecture 1 Systems of Linear Equations ² Systems of two linear equations with two variables
More informationResearch Paper Business Analytics. Applications for the Vehicle Routing Problem. Jelmer Blok
Research Paper Business Analytics Applications for the Vehicle Routing Problem Jelmer Blok Applications for the Vehicle Routing Problem Jelmer Blok Research Paper Vrije Universiteit Amsterdam Faculteit
More informationDecision Mathematics 1 TUESDAY 22 JANUARY 2008
ADVANCED SUBSIDIARY GCE 4736/01 MATHEMATICS Decision Mathematics 1 TUESDAY 22 JANUARY 2008 Additional materials: Answer Booklet (8 pages) Graph paper Insert for Questions 3 and 4 List of Formulae (MF1)
More informationCHAPTER 11: BASIC LINEAR PROGRAMMING CONCEPTS
Linear programming is a mathematical technique for finding optimal solutions to problems that can be expressed using linear equations and inequalities. If a realworld problem can be represented accurately
More informationEquilibrium computation: Part 1
Equilibrium computation: Part 1 Nicola Gatti 1 Troels Bjerre Sorensen 2 1 Politecnico di Milano, Italy 2 Duke University, USA Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium
More informationQuestion 2: How do you solve a matrix equation using the matrix inverse?
Question : How do you solve a matrix equation using the matrix inverse? In the previous question, we wrote systems of equations as a matrix equation AX B. In this format, the matrix A contains the coefficients
More informationOptimization in R n Introduction
Optimization in R n Introduction Rudi Pendavingh Eindhoven Technical University Optimization in R n, lecture Rudi Pendavingh (TUE) Optimization in R n Introduction ORN / 4 Some optimization problems designing
More informationLinear Programming Supplement E
Linear Programming Supplement E Linear Programming Linear programming: A technique that is useful for allocating scarce resources among competing demands. Objective function: An expression in linear programming
More information7 Gaussian Elimination and LU Factorization
7 Gaussian Elimination and LU Factorization In this final section on matrix factorization methods for solving Ax = b we want to take a closer look at Gaussian elimination (probably the best known method
More informationLecture 3: Linear Programming Relaxations and Rounding
Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can
More informationIn this paper we present a branchandcut algorithm for
SOLVING A TRUCK DISPATCHING SCHEDULING PROBLEM USING BRANCHANDCUT ROBERT E. BIXBY Rice University, Houston, Texas EVA K. LEE Georgia Institute of Technology, Atlanta, Georgia (Received September 1994;
More informationA network flow algorithm for reconstructing. binary images from discrete Xrays
A network flow algorithm for reconstructing binary images from discrete Xrays Kees Joost Batenburg Leiden University and CWI, The Netherlands kbatenbu@math.leidenuniv.nl Abstract We present a new algorithm
More informationMATH2210 Notebook 1 Fall Semester 2016/2017. 1 MATH2210 Notebook 1 3. 1.1 Solving Systems of Linear Equations... 3
MATH0 Notebook Fall Semester 06/07 prepared by Professor Jenny Baglivo c Copyright 009 07 by Jenny A. Baglivo. All Rights Reserved. Contents MATH0 Notebook 3. Solving Systems of Linear Equations........................
More informationNAG Fortran Library Routine Document E02GCF.1
NAG Fortran Library Routine Document Note: before using this routine, please read the Users Note for your implementation to check the interpretation of bold italicised terms and other implementationdependent
More information0.1 Linear Programming
0.1 Linear Programming 0.1.1 Objectives By the end of this unit you will be able to: formulate simple linear programming problems in terms of an objective function to be maximized or minimized subject
More informationSolving Systems of Linear Equations. Substitution
Solving Systems of Linear Equations There are two basic methods we will use to solve systems of linear equations: Substitution Elimination We will describe each for a system of two equations in two unknowns,
More informationLECTURE: INTRO TO LINEAR PROGRAMMING AND THE SIMPLEX METHOD, KEVIN ROSS MARCH 31, 2005
LECTURE: INTRO TO LINEAR PROGRAMMING AND THE SIMPLEX METHOD, KEVIN ROSS MARCH 31, 2005 DAVID L. BERNICK dbernick@soe.ucsc.edu 1. Overview Typical Linear Programming problems Standard form and converting
More informationDecision Mathematics D1. Advanced/Advanced Subsidiary. Friday 12 January 2007 Morning Time: 1 hour 30 minutes. D1 answer book
Paper Reference(s) 6689/01 Edexcel GCE Decision Mathematics D1 Advanced/Advanced Subsidiary Friday 12 January 2007 Morning Time: 1 hour 30 minutes Materials required for examination Nil Items included
More informationLU Factorization Method to Solve Linear Programming Problem
Website: wwwijetaecom (ISSN 22502459 ISO 9001:2008 Certified Journal Volume 4 Issue 4 April 2014) LU Factorization Method to Solve Linear Programming Problem S M Chinchole 1 A P Bhadane 2 12 Assistant
More information