Decision Aid Methodologies In Transportation Lecture 1: Polyhedra and Simplex method
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1 Decision Aid Methodologies In Transportation Lecture 1: Polyhedra and Simplex method Chen Jiang Hang Transportation and Mobility Laboratory April 15, 2013 Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 1 / 54 m
2 Outline 1 Teaching syllabuses 2 Decision aid tool and mathematical model 3 Basic polyhedron and convex theory 4 Linear programming and Simplex method hen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 2 / 54 m
3 Teaching syllabuses Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 3 / 54 m
4 Teaching syllabuses Topics that will cover Introduction to optimization 1 Polyhedron theory 2 Linear programming (Simplex method, Duality theory, Column generation) 3 Integer programming (Cutting plane, Branch and Bound) 4 Network problem 5 Approximation methods and heuristics Optimization in airline transport Optimization in maritime transport Optimization in railway transport Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 4 / 54 m
5 Teaching syllabuses Software will be used in laboratories IBM ILOG CPLEX Optimization Studio. You can download the trail version, free for 90 days. However, it has problem size limitation. The latest version is V12.5 (platform: Windows, Mac OS & Linux). Download HERE. Note that you need to create an IBM account in order to download. Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 5 / 54 m
6 Decision aid tool and mathematical model Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 6 / 54 m
7 Decision aid tool and mathematical model Do you still remember? 2006 FIFA World Cup (Germany) Quarter final match: Germany VS Argentina In the penalty shoot-out, Jens Lehmann (goalkeeper) guessed the directions correctly 4 out of 4 and saved two goals! The secret: Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 7 / 54 m
8 Decision aid tool and mathematical model Decision aid tool the note The legendary note was sold for ONE MILLION EUROs! Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 8 / 54 m
9 Decision aid tool and mathematical model The steps to build an optimization mathematical model Understand the system (problem analysis) Establish objective(s) Identify the key decision variables Spot all the boundary conditions (constraints) Model the problem Verify the model Solve the problem Evaluate the results Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 9 / 54 m
10 Decision aid tool and mathematical model The importance of system understanding Braess s paradox building a new road doesn t always lead to shorter traveling time experienced by individual driver. S A T/ T/100 B E Suppose all the drivers (4000) are selfish, before A and B are connected, half of the travelers will take SAE and the other half take SBE with 65 minutes of traveling time for each driver. After the road AB is built, all the drivers will take route SABE with 80 minutes of traveling time for each driver. hen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 10 / 54 m
11 Decision aid tool and mathematical model The importance of system understanding The case of Seikan Tunnel Seikan tunnel connects Honshu and Hokkaido in Japan and is both the longest and the deepest operational rail tunnel in the world. hen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 11 / 54 m
12 Decision aid tool and mathematical model The importance of system understanding Why built Seikan tunnel? A booming economy saw traffic levels on the JNR-operated Seikan. Ferry doubled to 4,040,000 persons/year from 1955 to 1965, and cargo levels rose 1.7 times to 6,240,000 tonnes/year. After operation in 1988 However, for passenger transport, 90% of people use air due to the speed and cost. For example, to travel between Tokyo and Sapporo by train takes more than ten hours and thirty minutes, with several transfers. By air, the journey is three hours and thirty minutes, including airport access times. Deregulation and competition in Japanese domestic air travel has brought down prices on the Tokyo-Sapporo route, making rail more expensive in comparison. hen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 12 / 54 m
13 Decision aid tool and mathematical model Establish objective(s) The objective should be clearly established for the problem. In practice, one problem with multiple objectives are very common. Usually, there are two ways to handle multi-objective problems: 1 Introduce the weights and transform the multi-objective problem to single objective problem. 2 Use the concept of Pareto optimality Y A Pareto front Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 13 / 54 m X
14 Decision aid tool and mathematical model Model the problem If possible, always try to build a linear model, i.e., the objective function and all the constraints only contain linear terms of decision variables. For example, min : x x 2 (1) s.t. x x 2 1 (2) x 1, x 2 0 (3) is not a good formulation. Instead, if we substitute x 2 1 by y 1, the resulting formulation is much better since it is a linear one. Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 14 / 54 m
15 Decision aid tool and mathematical model An example for practice A shipping company plans to acquire an aircraft and is designing a customized interior to carry thermally insulated and normal products Temperature controlled products are sold in the market for a profit of CHF 7 per unit, while normal ones yield a profit of CHF 5 per unit Temperature controlled products require 2 kwh electric power and 2 m 3 space for carrying one unit Normal products requires 1 kwh power and 4 m 3 space per unit Total power and space availability are 1000 kwh and 2400 m 3 Assuming that aircraft will always fly full capacity, how many units of temperature controlled and normal products should it be designed for? Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 15 / 54 m
16 Decision aid tool and mathematical model An example for practice Objective: To maximize the profit for the shipping company Decision variables: x 1, units of temperature controlled products; x 2, units of normal products Mathematical model max : 7x 1 + 5x 2 (4) s.t. 2x 1 + x (5) 3x 1 + 4x (6) x 1, x 2 0 (7) In this formulation, (5) reflects the power availability constraint; (6) is the space constraint; and (7) makes sure the results are non-negative. Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 16 / 54 m
17 Decision aid tool and mathematical model An example for practice x A x 1 Geometric representation of the constraints (the gray part is called the feasible domain). By the theory of linear programming, we know that the optimal solution for this example is corner point A. Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 17 / 54 m
18 Basic polyhedron and convex theory Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 18 / 54 m
19 Some notations Basic polyhedron and convex theory In this lecture, matrix A has m rows and n columns (A is the transpose of A; A 1 is the inverse of of A if it is a square matrix). We use A j to denote its jth column, that is, A j = (a 1j, a 2j,, a mj ). We also use a i to denote the vector formed by the entries of the ith row, that is, a i = (a i1, a i2,, a in ). Note that all the vectors are column vector. A = A 1 A 2 A n = a 1. a m Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 19 / 54 m
20 Basic polyhedron and convex theory Polyhedra Polyhedron A polyhedron is a set that can be described in the form {x R n Ax b}, where A is an m n matrix and b is a vector in R m. We call {x R n Ax b} the general form of a polyhedron; {x R n Ax = b, x 0} the standard form of a polyhedron. Hyperplane and halfspace Let a be a nonzero vector in R n and let b be a scalar. Then the set {x R n a x = b} is called a hyperplane and the set {x R n a x b} is called a halfspace. Note that both hyperplane and halfspace are polyhedra and a hyperplane is the boundary of a corresponding halfspace. hen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 20 / 54 m
21 Convex set Basic polyhedron and convex theory Convex set A set S R n is convex if for any x, y S, and any λ [0, 1], we have λx + (1 λ)y S. Convex combination & convex hull Let x 1,, x k be vectors in R n and let λ 1,, λ k be nonnegative scalars whose sum is unity. The vector k i=1 λ ix i is said to be a convex combination of the vector x 1,, x k ; The convex hull of the vector x 1,, x k is the set of all convex combination of these vectors. hen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 21 / 54 m
22 Basic polyhedron and convex theory Extreme point of a polyhedron Extreme point Let P be a polyhedron. A vector x P is an extreme point of P if we cannot find two vectors y, z P, both different from x, and a scalar λ [0, 1], such that x = λy + (1 λ)z. Geometrically speaking, an extreme point of a polyhedron is the corner point of the polyhedron. v w u y x hen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 22 / 54 m z
23 Basic polyhedron and convex theory Basic solution and basic feasible solution associated with a polyhedron Basic solution & basic feasible solution Consider a polyhedron P defined by linear equality and inequality constraints, and let x be an element of R n. The vector x is a basic solution if: 1 All equality constraints are active; 2 Out of the constraints that are active at x, there are n of them that are linearly independent. If x is a basic solution that satisfies all of the constraints, we say that it is a basic feasible solution. For example, if P = {x 1 + x 2 = 1, x 1 0, x 2 0}, then (1, 0) and (0, 1) are basic (feasible) solutions of P. hen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 23 / 54 m
24 Basic polyhedron and convex theory Extreme point=basic feasible solution Extreme point and basic feasible solution associated with a polyhedron are equivalent. That is, if x P is an extreme point of P then x is also a basic feasible solution of P, vice versa. For a polyhedron, extreme point is the geometric interpretation of basic feasible solution and basic feasible solution is the algebraic representation of extreme point. Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 24 / 54 m
25 Basic polyhedron and convex theory Recession cone of a polyhedron Consider a nonempty polyhedron P = {x R n Ax b} and let us fix some y P. We define the recession cone at y as the set of all directions d along which we can move indefinitely away from y, without leaving the set P. That is, the recession cone of P is the set {d R n A(y + λd) b, λ 0} or more concise {d R n Ad 0} Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 25 / 54 m
26 Basic polyhedron and convex theory Recession cone of a polyhedron A polyhedron and its recession cone Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 26 / 54 m
27 Basic polyhedron and convex theory Extreme rays of a polyhedron Intuitively, extreme rays of a polyhedron are the directions associated with edges of the polyhedron that extend to infinity. Extreme ray A nonzero element x of a recession cone C R n is called an extreme ray if there are n 1 linearly independent constraints that are active at x. An extreme ray of the recession cone associated with a nonempty polyhedron P is also called an extreme ray of P. Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 27 / 54 m
28 Basic polyhedron and convex theory Why to study the extreme points & extreme rays of a polyhedron? Consider a linear programming problem min : s.t. c x Ax b Let {x 1,, x p } be the extreme points of the polyhedron P = {Ax b} and {w 1,, w q } be the set of extreme rays of P. The we have the following theorem: Optimality of extreme points If c w i 0, 1 i q, there exists an extreme point in {x 1,, x p } which is optimal; Otherwise (i.e., there exists i, c w i < 0), the optimal value is equal to. hen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 28 / 54 m
29 Basic polyhedron and convex theory Why to study the extreme points & extreme rays of a polyhedron? Besides, we have the following famous theorom: Resolution theorem Let {x 1,, x p } be the extreme points of the polyhedron P = {Ax b} and {w 1,, w q } be the set of extreme rays of P. The polyhedron P can also be expressed as p q p P = λ i x i + θ j w j λ i 0, θ j 0, λ i = 1 i=1 j=1 Therefore, a polyhedron can be expressed as (1) intersection of finite number of halfspaces (2) the sum of convex combination of extreme points and nonnegative combination (also call conic combination) of extreme rays. Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 29 / 54 m i=1
30 Linear programming and Simplex method Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 30 / 54 m
31 Linear programming and Simplex method General form to standard form How to transform an LP in general form to standard form? 1 a x b: Introduce a slack variable s 0 and a x + s = b. 2 a x b: Introduce a surpass variable p 0 and a x p = b. 3 x i is a free variable: Introduce two new variables y, z 0 and x i = y z. Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 31 / 54 m
32 Linear programming and Simplex method Why standard form? min : s.t. c x Ax = b x 0 Without loss of generality, we can assume that matrix A has m linearly independent rows and n m (in reality, this is usually the case). Question: how to obtain a basic solution quickly? A = A 1 A 2 A 3 A n A 1, A 3,, A n are linearly independent and we call them the base and their associated m variables (x 1, x 3,, x n ) basic variables. Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 32 / 54 m
33 Linear programming and Simplex method The basic idea of Simplex method? C D B? A Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 33 / 54 m
34 Linear programming and Simplex method The basic idea of Simplex method 4 K Q The objective of this card game is to increase the sum of card numbers in your hand (at most 4 cards). Which two cards you will switch? Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 34 / 54 m
35 Linear programming and Simplex method Simplex method in math form Let x be a basic feasible solution to the standard form problem, let B(1), B(2),, B(m) be the indices of the basic variables, and let B = [A B(1) A B(m) ] be the corresponding basis matrix. In particular, we have x i = 0 for every nonbasic variable, while the vector x B = (x B(1),, x B(m) ) of basic variables is given by x B = B 1 b We consider the possibility of moving away from x, to a new vector x + θd (θ > 0). Here we choose a special direction d: select a nonbasic variable x j (which is initially at zero level) and try to increase it to a positive value while keep the remaining nonbasic variables at zero. Algebraically, d j = 1, and d i = 0 for every nonbasic index i other than j. At the same time, the vector x B of basic variables changes to x B + θd B, where d B = (d B(1), d B(2),..., d B(m) ) is the vector with those components of d that correspond to the basic variables. Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 35 / 54 m
36 Linear programming and Simplex method Simplex method in math form Since x + θd should be a feasible solution, we have A(x + θd) = Ax + θad = b. That is Ad = 0. Thus, we have: 0 = Ad = n A i d i = i=1 m A B(i) d B(i) + A j = Bd B + A j i=1 Since the basis matrix B is invertible, we have d B = B 1 A j The obtained direction d is called the jth basic direction. Question: how many basic directions are there? Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 36 / 54 m
37 Linear programming and Simplex method Simplex method in math form What would be the effects on the cost function if we move along a basic direction? If d is the jth basic direction, then the rate c d of cost change along the direction d is given by c Bd B + c j, where c B = (c B(1),..., c B(m) ). This is the same as c j c BB 1 A j The above value is defined as the reduced cost c j of the variable x j. Reduced costs for basic variables?! The reduced cost for any basic variable is always 0. hen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 37 / 54 m
38 Linear programming and Simplex method The importance of reduced costs Consider a basic feasible solution x associated with a basis matrix B, and let c be the corresponding vector of reduced costs. If c 0, then x is optimal. Proof: Assume that c 0 and let y be an arbitrary feasible solution, and define d = y x. Feasibility implies that Ax = Ay = b and, therefore, Ad = 0. The latter equality can be rewritten in the form Bd B + i N A id i = 0, where N is the set of indices corresponding to the nonbasic variables under the given basis. Since B is invertible, we have d B = i N B 1 A i d i and c d = c Bd B + i N c i d i = (c i c BB 1 A i )d i = i N i N Since, i N, x i = 0, y i 0. Thus, d i 0 and c i d i 0 and c d = c (y x) 0. x is optimal. hen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 38 / 54 m c i d i
39 The pivot process Linear programming and Simplex method At a basic feasible solution x and we have computed the reduced cost vector c. If c 0, we can stop since x is optimal. If c j < 0, then the jth basic direction d is a feasible and profitable direction. While moving along d, the nonbasic variable x j becomes positive and all other nonbasic variables remain at 0 (i.e., x j is brought into the basis). Since costs decrease along the direction d, it is desirable to move as far as possible (Greedy!). θ = max{θ 0 x + θd 0} If d i < 0 for some i, the largest possible value of θ is ( ) θ xi = min {i d i <0} If θ is chosen as θ, then at least one of the basic variables, say, x i will become 0 (i.e., x i leaves the basis). Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 39 / 54 m d i
40 Linear programming and Simplex method Simplex method essential steps 1 Start with a basis consisting of the basic columns A B(1),..., A B(m), and an associated basic feasible solution x. 2 Compute the reduced costs c j = c j c BB 1 A j for all nonbasic indices j. If they are all nonnegative, the current basic feasible solution is optimal, and the algorithm terminates; else, choose some j for which c j < 0. 3 Calculate θ = min {i di <0} ( xi to leave the basis. d i ) and determine the variable Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 40 / 54 m
41 Linear programming and Simplex method An example for practice max : 7x 1 + 5x 2 s.t. 2x 1 + x x 1 + 4x x 1, x 2 0 min : 7x 1 5x 2 s.t. 2x 1 + x 2 + x 3 = x 1 + 4x 2 + x 4 = 2400 x 1, x 2, x 3, x 4 0 A = [ ] c = [ ] Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 41 / 54 m
42 Linear programming and Simplex method An example for practice [ ] B 1 =,x = [0, 0, 1000, 2400],c 1 = c [0, 0]B 1 1 A = [ 7, 5, 0, 0], d 1 = [1, 0, 2, 3], θ1 = min{1000/2, 2400/3}, x 3 will be out of basis. [ ] B 2 =,x = [500, 0, 0, 900],c 2 = c [ 7, 1]B 1 2 A = [0, 1.5, 3.5, 0], d 2 = [ 0.5, 1, 0, 2.5], θ2 = min{500/0.5, 900/2.5}, x 4 will be out of basis. [ ] B 3 =,x = [320, 360, 0, 0],c 3 = c [ 7, 5]B 1 3 A = [0, 0, 2.6, 0.6], x 3 is optimal. Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 42 / 54 m
43 Linear programming and Simplex method Tabular implementation of Simplex c BB 1 b B 1 b c c BB 1 A B 1 A In more detail, c Bx B c 1 c n x B(1). B 1 A 1 B 1 A n x B(m) Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 43 / 54 m
44 Linear programming and Simplex method Tabular implementation of Simplex min 7x 1 5x 2 s.t. 2x 1 +x 2 +x 3 = x 1 +4x 2 +x 4 = 2400 x 1, x 2, x 3, x 4 0 [ 1 0 It is easy to observe that initially, we can choose B = 0 1 the tabular form: x 3 = x 4 = ]. In Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 44 / 54 m
45 Linear programming and Simplex method Tabular implementation of Simplex c 1 c 2 c 3 c x 3 = = 500 x 4 = = 800 Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 45 / 54 m
46 Linear programming and Simplex method Tabular implementation of Simplex c 1 c 2 c 3 c Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 46 / 54 m
47 Linear programming and Simplex method Tabular implementation of Simplex c 1 c 2 c 3 c Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 47 / 54 m
48 Linear programming and Simplex method Tabular implementation of Simplex c 1 c 2 c 3 c Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 48 / 54 m
49 Linear programming and Simplex method Tabular implementation of Simplex c 1 c 2 c 3 c x 1 = x 4 = Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 49 / 54 m
50 Linear programming and Simplex method Tabular implementation of Simplex c 1 c 2 c 3 c x 1 = = 1000 x 4 = = 360 Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 50 / 54 m
51 Linear programming and Simplex method Tabular implementation of Simplex c 1 c 2 c 3 c Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 51 / 54 m
52 Linear programming and Simplex method Tabular implementation of Simplex c 1 c 2 c 3 c Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 52 / 54 m
53 Linear programming and Simplex method Tabular implementation of Simplex c 1 c 2 c 3 c Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 53 / 54 m
54 Linear programming and Simplex method Tabular implementation of Simplex c 1 c 2 c 3 c x 1 = x 2 = Chen Jiang Hang (Transportation and Mobility Decision Laboratory) Aid Methodologies In TransportationLecture 1: Polyhedra and Simplex 54 / 54 m
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