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Linear Programming

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.

Reliabilty Theory Questions

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

Linear Programming : Formulate Decision and Solve by Computer

9-1 Horrible Harry's Horrible Harry's is a chain of 47 self service gas stations served by a small refinery and mixing plant. Each day's product requirements are met by blending feedstocks on hand at midnight. The volumes vary daily, depending on the previous day's refinery output and on bulk receipts. The entire operatio

Dual Simplex Method and Primal Simplex Method

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

Formulate a Linear Programming model

5. Bob Brown's 40th birthday party promised to be the social event of the year in Illinois. To prepare, Bob stocked up on the following liquors. Liquor On Hand (ounces) Bourbon 52 Brandy 38 Vodka 64 Dry Vermouth 24 Sweet Vermouth 36 Bob decided to mix four drinks for the party: Chaunceys, Sweet Italians, Bourbon on th

Matrix Proofs : Linear Programming, Duality, Feasibility and Optimal Solutions

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

Linear Programming : Proof using Duality and the Farkas Lemma

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

Linear Programming : Finding the Optimal Value by a Graphical Method

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.

Linear Programming to minimize costs

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

Linear Programming : Adjacency of Basic Feasible Solution and Hyperplanes

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

Feasibility of a Linear Programming Problem

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

Linear Programming : Dantzig-Wolfe and Bender Decomposition

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

Lindo Defining Constraints

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

Linear Programming : Using Lindo or Excel to Develop Constraint Equations

The following table is a list of all of the stocks that you have in your stock portfolio. The original purchase price, current price and your best guess for the "anticipated" price (one year into the future) is given below: Share Price ($) Stock # Shares Owned Purchase Current Expected In One Year 1 234 20 30 36 2 272 2

A company needs to lease warehouse storage space

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

Create and solve a linear program which maximizes Sunco's daily profits. What are the optimum decisions, i.e. the barrels of crude oil used to create the gasoline and the advertising dollars spent on stimulating the demand for gasoline?

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

Linear Programming : Using the Simplex Method to Minimize 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:

Linear Programming : Formulating Equations and Minimizing Cost

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.

Linear Programming : Scaling, Unbounded LPs and Feasible Region

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

Justify Linear Programming Statements : Equality Constraints

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

Proof in Linear Programming - Extreme Point

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.

Linear programming

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

Proof in Linear programming

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

Linear Programming (5 Problems)

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

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