Assume P, Q are non-empty polyhedra. Let P + Q := {x + y: Prove that P + Q is a polyhedron. Prove that every extreme point of P + Q is the sum of an extreme point of P and an extreme point of Q. ---
Transportation Problem : Find Optimal Solution
Find the optimal solution for the following problem: TO FROM Chicago Atlanta supply St louis 40 63 250 Richmond 70 30 400 demand 300 350 650
Queueing Theory : Swimmers and Exponential Distribution
Queueing Theory Question 1 An average of 10 people per hour arrive (inter-arrival times are exponential) intending to swim laps at the local YMCA. Each intends to swim an average of 30 minutes. The YMCA has 3 lanes open for lap swimming. If one swimmer is in a lane, he or she swims up and down the right side of the lane. ...continues
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 ...continues
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 ...continues
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 ...continues
Stochastic Processes : M/M/s Queue, Stationary Distribution and Long-Run Proportion
Consider the M/M/s queue, with arrival rate... 5. Consider the M/M/s queue, with arrival rate λ >0 and service rate μ >0. (a) Find the condition involving λ, μ and s that is necessary for there to exist a stationary distribution. Why does this condition makes sense? (b) Find the long-run proportion of ti ...continues
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 ...continues
Reliability Theory : Minimal Cut Sets and Probability of System Life
Consider a structure in which the minimal path sets are {1, 2, 3} and {3, 4, 5} a. What are the minimal cut sets? b. If the component lifetimes are independent uniform (0,1) random variables, determine the probability that the system life will be less than ½ .
Regression and Forecasting : Using Data Regression to Predict Sales
YEAR GNP ln(gnp) 1975 1060 6.966024187 1976 1170 7.064759028 1977 1305 7.17395832 1978 1455 7.28276118 1979 1630 7.396335294 1980 1800 7.495541944 1981 2000 7.60090246 1982 2220 7.705262475 1983 2450 7.803843304 1984 2730 7.912056888 The US BNP during the years 1975-1984 is given in the above table. a. Plot x ...continues
