### Proof Optimal Solution

Consider a symmetric square matrix A and the following linear program: Min cx St Ax > c x > 0 Prove that if x* satisfies Ax* = c and x* > 0 then x* is an optimal solution to this linear program.

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Consider a symmetric square matrix A and the following linear program: Min cx St Ax > c x > 0 Prove that if x* satisfies Ax* = c and x* > 0 then x* is an optimal solution to this linear program.

These questions are a part of a Operations Research class with a section on Reliability Theory. (See attached file for full problem description with proper symbols and equations) --- Question Let N be a non-negative, integer-valued random variable, Show that P{N > 0} >= (E[N])2

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Please see the attached document for the question all properly formatted.

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Consider the linear program: Min -2x-y St x-y<2 x+y<6 x, y> 0 a) By inspection, argue that this problem cannot have an unbounded optimal solution. b) Convert this problem to simplex standard form, enumerate all the basic solutions, identify which ones are feasible, and compute their objective values. c) What

Exercise 4.25 This exercise shows that if we bring the dual problem into standard form and then apply the primal simplex method, the resulting algorithm is not identical to the dual simplex method. Consider the following standard form problem and its dual. minimize x1 + x2 maximize p

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Exercise 4.26 Let A be a given matrix. Show that exactly one of the following alternatives must hold. (a) There exists some x does not equal 0 such that Ax = 0, x > 0. (b) There exists some p such that p'A> 0'. Exercise 4.27 Let A be a given matrix. Show that the following two statements are equivalent. (a) Every vector such

This question is from linear programming. I want to use duality (it's so obvious), farkas lemma (alternative solution) and all. (See attached file for full problem description with equations) --- (a) Let . Prove that one of the following systems has a solution but not both: (b) Prove or disprove the following cla

1. AA Auto manufactures luxury cars and trucks. The company believes that its most likely customers are high-income women and men. To reach these groups, AA Auto has embarked on an ambitious TV advertising campaign and has decided to purchase 1-minute commercial spots on two types of programs: comedy shows and football games.

See attached file for full problem descriptions with complete equations. --- 1. During the next three months, Ironco faces the following demands for steel: 100 tons (month 1); 200 tons (month 2); 5 tons (month 3). During any month, a worker can produce up to 5 tons of steel. Each worker is paid $5000 per month. Workers c

Can anyone finish up this proof by continuing my preliminery work? I started but can't finish this. I know starting by adding up the point z is correct way, but just can't continue to show if and only if. (See attached file for full problem description) --- Assume , , with rank (A) = m are given. Two different basic

Exercise 3.22 Consider the following linear programming problem with a single constraint: minimize Σ i=1 --> n cixi subject to = ... i=1,...,n. (a) Derive a simple test for checking the feasibility of this problem. (b) Assuming that the optimal cost is finite, develop a simple method for obtaining an optimal soluti

Consider the following LP problem min s.t. (a) Suppose we have a very fast routine to solve the problems of the form min s.t. for arbitrary vectors . How would you decompose the problem above the take advantage of such fast subroutine? (b) Suppose we have a very fast routine to solve pr

I am working to define constraints and am having problems in doing so. I would like a detalied explanation of how this is done and how Lindo will interrprut the information. (See attached file for full problem description) --- A company needs to lease warehouse storage space for five months at the start of the year. Th

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A company needs to lease warehouse storage space for five months at the start of the year. The space requirements (in square feet) and the leasing costs of each type of lease are given in the two tables below: Month Required Space (sq. feet) Jan 15,000 Feb 10,000 Mar 20,000 Apr 5,000 May 25,000 Lease Term (months) Co

Sunco Oil Co. manufactures three types of gasoline: Gas 1, Gas 2 and Gas 3. Each type is produced by blending three type of crude oil: Crude 1, Crude 2 and Crude 3. The sales price per barrel of gasoline and the purchase price per barrel of crude oil is given in the following table: Gasoline Type Gas Selling Price Per Barrel C

In this problem I am trying to get rid of the artificial variable using the two phase method. However all of the rows either have negatives or zeros and my final answer keeps coming out to be a negative and none of the other answers plug into the constraints. The problem is Using Simplex method minimize C:

A biologist must make a nutrient for her algae. The nutrient must contain three basic elements D, E, F, and must contain at least 10kg of D, 12kg of E, and 20Kg of F. The nutrient is made from three ingredients, I, II, III. The quantity of D, E, F in one unit of each of the ingredients is given in the following chart.

Question #1 A company produces three products. The per-unit profit, labor usage, and pollution produced per unit are given in the table 1. At most, 3 million labor hours can be used to produce the three products, and government regulations require that the company produce at most 2 lb of pollution. If we let Xi = units produ

For each statement, state whether it is true or false. Be sure to justify your answer. a) Suppose you are given a linear program in Rn with mE equality constraints and mI inequality constraints. Let x be an element of the polyhedron at which n - mE inequality constraints are active. Then x must be an extreme point of the poly

Can anyone help me to prove this? I'm really stuck with geometry in Linear Programming... (See attached file for full problem description and equations) --- Assume P is a polyhedron and H is a supporting hyperplane to P. Prove that is an extreme point of if and only if is an extreme point of P.

Find the complete optimal solution to this linear programming problem. Min 3X + 3Y s.t. 12X + 4Y > 48 10X + 5Y > 50 4X + 8Y > 32 X , Y > 0

Please help me to find out how I can do this (See attached file for full problem description) --- Let (see attachment) It is clear that we can rewrite (attached) as (attached) , i.e. as a system of linear inequalities. (I've done this). Show that in fact we can rewrite (attached) as a system of (attached) linear i

Question 1 Cully furniture buys 2 products for resale: big shelves (B) and medium shelves (M). Each big shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs $300 and requires 90 cubic feet of storage space. The company has $75000 to invest in shelves this week, and the warehouse has 1800

Why should decision makers who are primarily concerned with marketing or finance or production know about linear programming?

(See attached file for full problem description) --- Indicate whether the sentence or statement is true or false. _____ 1. Determining the production quantities of different products manufactured by a company based on resource constraints is a product mix linear programming problem. _____ 2. When using linear program

Explain a process more clearly. --- - Minimize... - Duality Principle... - Transposing Matrices... - Pivot Operation --- Please see the attached file for the fully formatted problems.