Linear Programming Lecture Notes - Penn State Personal Web Server

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finishing toys and 120 hours per week making toys. The company wishes to maximize the profit it makes by choosing how much of each toy to produce. We can represent the profit maximization problem of the company as a linear programming problem. Let x1 be the number of planes the company will produce and let x2 be the number of boats the company will produce. The profit for each plane is $10 − $3 = $7 per plane and the profit for each boat is $8 − $2 = $6 per boat. Thus the total profit the company will make is: (2.5) z(x1, x2) = 7x1 + 6x2 The company can spend no more than 120 hours per week making toys and since a plane takes 3 hours to make and a boat takes 1 hour to make we have: (2.6) 3x1 + x2 ≤ 120 Likewise, the company can spend no more than 160 hours per week finishing toys and since it takes 1 hour to finish a plane and 2 hour to finish a boat we have: (2.7) x1 + 2x2 ≤ 160 Finally, we know that x1 ≤ 35, since the company will make no more than 35 planes per week. Thus the complete linear programming problem is given as: ⎧ max z(x1, x2) = 7x1 + 6x2 s.t. 3x1 + x2 ≤ 120 ⎪⎨ x1 + 2x2 ≤ 160 (2.8) x1 ≤ 35 ⎪⎩ x1 ≥ 0 x2 ≥ 0 Exercise 10. A chemical manufacturer produces three chemicals: A, B and C. These chemical are produced by two processes: 1 and 2. Running process 1 for 1 hour costs $4 and yields 3 units of chemical A, 1 unit of chemical B and 1 unit of chemical C. Running process 2 for 1 hour costs $1 and produces 1 units of chemical A, and 1 unit of chemical B (but none of Chemical C). To meet customer demand, at least 10 units of chemical A, 5 units of chemical B and 3 units of chemical C must be produced daily. Assume that the chemical manufacturer wants to minimize the cost of production. Develop a linear programming problem describing the constraints and objectives of the chemical manufacturer. [Hint: Let x1 be the amount of time Process 1 is executed and let x2 be amount of time Process 2 is executed. Use the coefficients above to express the cost of running Process 1 for x1 time and Process 2 for x2 time. Do the same to compute the amount of chemicals A, B, and C that are produced.] 1. Modeling Assumptions in <strong>Linear</strong> <strong>Programming</strong> Inspecting Example 2.3 (or the more general Problem 2.1) we can see there are several assumptions that must be satisfied when using a linear programming model. We enumerate these below: 14
Proportionality Assumption: A problem can be phrased as a linear program only if the contribution to the objective function and the left-hand-side of each constraint by each decision variable (x1, . . . , xn) is proportional to the value of the decision variable. Additivity Assumption: A problem can be phrased as a linear programming problem only if the contribution to the objective function and the left-hand-side of each constraint by any decision variable xi (i = 1, . . . , n) is completely independent of any other decision variable xj (j = i) and additive. Divisibility Assumption: A problem can be phrased as a linear programming problem only if the quantities represented by each decision variable are infinitely divisible (i.e., fractional answers make sense). Certainty Assumption: A problem can be phrased as a linear programming problem only if the coefficients in the objective function and constraints are known with certainty. The first two assumptions simply assert (in English) that both the objective function and functions on the left-hand-side of the (in)equalities in the constraints are linear functions of the variables x1, . . . , xn. The third assumption asserts that a valid optimal answer could contain fractional values for decision variables. It’s important to understand how this assumption comes into play– even in the toy making example. Many quantities can be divided into non-integer values (ounces, pounds etc.) but many other quantities cannot be divided. For instance, can we really expect that it’s reasonable to make 1/2 a plane in the toy making example? When values must be constrained to true integer values, the linear programming problem is called an integer programming problem. These problems are outside the scope of this course, but there is a vast literature dealing with them [PS98, WN99]. For many problems, particularly when the values of the decision variables may become large, a fractional optimal answer could be obtained and then rounded to the nearest integer to obtain a reasonable answer. For example, if our toy problem were re-written so that the optimal answer was to make 1045.3 planes, then we could round down to 1045. The final assumption asserts that the coefficients (e.g., profit per plane or boat) is known with absolute certainty. In traditional linear programming, there is no lack of knowledge about the make up of the objective function, the coefficients in the left-hand-side of the constraints or the bounds on the right-hand-sides of the constraints. There is a literature on stochastic programming [KW94, BTN02] that relaxes some of these assumptions, but this too is outside the scope of the course. Exercise 11. In a short sentence or two, discuss whether the problem given in Example 2.3 meets all of the assumptions of a scenario that can be modeled by a linear programming problem. Do the same for Exercise 10. [Hint: Can you make 2/3 of a toy? Can you run a process for 1/3 of an hour?] Exercise 12. Suppose the costs are not known with certainty but instead a probability distribution for each value of ci (i = 1, . . . , n) is known. Suggest a way of constructing a linear program from the probability distributions. [Hint: Suppose I tell you that I’ll give you a uniformly random amount of money between $1 and $2. How much money do you expect to receive? Use the same reasoning to answer the question.] 15
Page 1: Linear Programming: Penn State Math
Page 4 and 5: 8. Caratheodory Characterization Th
Page 7 and 8: List of Figures 1.1 Goat pen with u
Page 9 and 10: 4.6 Convex Direction: Clearly every
Page 11: Preface Stop! Stop right now! This
Page 14 and 15: x Goat Pen Figure 1.1. Goat pen wit
Page 16 and 17: We have formulated the general maxi
Page 18 and 19: Figure 1.3. Contour Plot of z = x 2
Page 20 and 21: Figure 1.5. A Level Curve Plot with
Page 22 and 23: and thus our vector is v = (x1 −
Page 24 and 25: Figure 1.7. Gradients of the Bindin
Page 28 and 29: 2. Graphically Solving Linear Progr
Page 30 and 31: S r x 0 Br( x0) Figure 2.2. A Bound
Page 32 and 33: S Every point on this line is an al
Page 34: 6. Problems with Unbounded Feasible
Page 37: to “prove” your example works.
Page 40 and 41: Definition 3.6 (Matrix Transpose).
Page 42 and 43: 3. Matrices and Linear Programming
Page 44 and 45: Remark 3.16. We can deal with const
Page 46 and 47: Theorem 3.20. Each elementary row o
Page 48 and 49: Gauss-Jordan Elimination Computing
Page 50 and 51: and right hand side vector b = [7,
Page 52 and 53: Gauss-Jordan elimination to solve t
Page 54 and 55: which clearly has a solution for al
Page 56 and 57: which is a contradiction. Case 2: S
Page 58 and 59: Definition 3.46 (Basic Variables).
Page 60 and 61: This leads to the following linear
Page 62 and 63: This is not in a form Matlab likes,
Page 64 and 65: Example 4.5. Figure 4.1 illustrates
Page 66 and 67: Figure 4.3. A hyperplane in 3 dimen
Page 68 and 69: 4. Rays and Directions Recall the d
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Page 72 and 73: Theorem 4.31. Let P ⊆ R n be a po
Page 74 and 75: Figure 4.9. A Polyhedral Set: This
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Figure 4.10. Visualization of the s
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It is clear that P is bounded. In f
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x1 x5 x2 x λx2 +(1− λ)x3 x4 x3
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If there is some i such that c T di
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Consider now the fact xj = 0 for al
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Example 5.9. Consider the Toy Maker
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using this information, we can comp
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Figure 5.1. The Simplex Algorithm:
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Using this information, we can see
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Based on this information, we can c
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Using the rule you developed in Exe
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Exercise 52. Consider the diet prob
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Suppose we choose s4 as the leaving
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CHAPTER 6 Simplex Initialization In
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A basic feasible solution for our a
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variables. This is because we start
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To remove the artificial variables
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Remark 6.6. In the case of a minimi
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Example 6.13. Suppose we solve the
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That is, s1 = −12 and s2 = −20
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The complete problem describing McL
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* data section */ data; set DAY :=
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This leads to a vector of reduced c
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2.1. Lexicographic Minimum Ratio Te
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Proof. The initial basis Im yieds a
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for i = 1, . . . , n. Then we have:
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Revised Simplex Algorithm (1) Ident
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simplex tableau). We obtain: z1 −
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Proof. We can prove Farkas’ Lemma
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A1· Half-space cy > 0 A2· c Am·
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3. The Karush-Kuhn-Tucker Condition
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Remark 8.9. The expressions: ∗ A
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(x ∗ 1,x ∗ 2) = (16, 72) 3x1 +
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Since w1, w2 ≥ 0, we know that w
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Example 8.13. Consider the followin
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Figure 8.5. This figure illustrates
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(9.4) Proof. Rewrite Problem D as:
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This vector will be related to c in
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3. Strong Duality Lemma 9.7. Proble
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Likewise beginning with the KKT con
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egions are polyhedral sets 1 . Cert
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Figure 9.2. The simplex algorithm b
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If a selfish profit maximizer wishe
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Recall the optimal full tableau for
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For the sake of space, we will prov
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We can choose either s1 or s2 as a
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