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Discrete Optimization

Discrete Optimization is a branch of optimization which embodies a significant area of combinatorics that deals with discrete values, such as integers. There are two main branches of Discrete Optimization:

1. Combinatorial Optimization, which refers to problems which deal with combinatorial structures such as graphs and,

2. Integer Programming, which refers to problems where mathematical optimization only deals with integers.

However, although these subjects may be considered different branches of Discrete Optimization, they are in fact not completely isolated from each other as problems under Combinatorial Optimization can fall under Integer Programming and vice versa. However, with the growth in this area of Mathematics, both branches are often used in conjunction to optimize, in other words, to find efficient methods of constructing good solutions as well as measuring the quality of these particular solutions.

The applicability of this method extends into almost every facet of society, from scheduling planes to coordinating the production of steel to designing pharmaceutical drugs. The methodology itself of finding good solutions to these everyday problems includes a variety of mathematical techniques such as constructing tree-growing procedures, integer lattices as well as the analysis of algorithms. Thus, understanding Discrete Optimization is crucial for the study of Discrete Math as well as other disciplines which examines optimization.

Enumerations, Combinations, and Permutations

Enumeration Example Suppose ABC University has 3 different math courses, 4 different business courses, and 2 different sociology courses. Tell me the number of ways a student can choose one of EACH kind of course. Then tell me the number of ways a student can choose JUST one of the course. Enumeration - Handshakes Consider

Crushing Force

Shown in the attached file is a crushing device. A 200lb force is applied to the end of the handle as shown. The slider block at pt "e" is frictionless. Determine the crushing force P. (Assume all member self-weights are negligible).

Strong Induction and Gift Card Totals

Suppose that a store offers gift certificates in denominations of 25 dollars and 40 dollars. Determine the possible total amounts (below $160) you can form using these gift certificates. Prove your answer using strong induction.

Mathematic Custom Help

18. Of 100 clock radios with digital tuners and/or CD players sold recently in a department store, 70 had digital tuners and 90 had CD players. How many radios had both digital tuners and CD players? (Is the answer 70?) If not please give the answer and explain. 22. Of 50 employees of a store located in downtown Boston,

Unlevered cost of capital

The Green Paddle has a cost of equity of 13.73 percent and a pre-tax cost of debt of 7.6 percent. the debt-equity ratio is 0.65 and the tax rate is 32 percent. What is green paddle's unlevered cost of capital?

Auto Insurance Percentage of Coverage

Please help with the following problem. An auto insurance company classifies its customers in three categories: poor, satisfactory, and preferred. Each year, 30% of those in the poor category are moved to satisfactory and 5% of those in the satisfactory category are moved to preferred. Also, 5% of those in the satisfactory c

Discrete Optimization: Number of Seats Example

A theater charges $8 for main floor seats and $5 for balcony seats. If all seats are sold the ticket income is $4200. At one show 25% of the main floor seats and 40% of the balcony seats were sold and ticket income was $1200 how many seats are on the main floor and how many are in the balcony?

Buying a computer directly from manufacturer for $3,067.

If you buy a computer directly from the manufacturer for $3,067 and agree to repay it in 60 equal installments at 1.99% interest per month on the unpaid balance, how much are your monthly payments? How much total interest will be paid? Your monthly payment is _____. The total interest paid is ________?

Coefficient Of Friction

ABCD is a cross-section of a uniform rectangular block of mass 20kg. AB is 0.75m and BC is 1m. The block rests with A on rough horizontal ground with AB at 20 degrees to the horizontal. It is held in place by a horizontal force P N applied at C. Given that the block is on the point of slipping, find the value of P and the coeffi

Mechanic Principle Of Energy

A 9kg mass on a slope which is at an angle of 15 degrees is attached via a an inextensible string over a fixed, frictionless pulley to a 5kg mass. The 9kg mass is initially moving up the slope at a rate of 1.5 m/s. How far has the mass traveled when it comes to a stop and what will its velocity be when it has then slid back down

Measurement and Map Scales Conversion.

Please see the attached file for the fully formatted problems. Find the perimeter of the triangle in mm. Using the given map, find the distances between the cities Measure distances to the nearest sixteenth of an inch.

Determining Cholesterol given Standard Deviation and Mean

According to her doctor, Mrs. Brown's cholesterol level is higher than only 5% of the females aged 50 and over. The cholesterol levels among females aged 50 and over are approximately normally distributed with a mean of 230 mg/dL and a standard deviation of 20 mg/dL. What is Mrs. Brown's cholesterol level? Carry your intermediat

Distribution of 15 vehicle models

There are 15 different vehicle models available at a certain dealership. Oddly, each family living on Maple Street bought one of these vehicles. There are just enough families on Maple Street so you can be absolutely sure that 6 families all have the same model. How many families are there on Maple Street?

Discrete Optimization: GPA Increase Example

Students at ACC must earn 90 credits to obtain an Associate's degree. Three students find that they all have a GPA of 3.3 even though they do not have the same number of credits. The students hope to increase thier GPAs to a 3.8 by the time they have earned 90 credits. Once student has earned 50 credits, another has earned 40,

Discussion Regarding Sampling

Question: How long does it take to travel to work Monday through until Friday for 10 days? Monday = 30 + 35 /2 = 65/2 = 32.5 or 32 ½ or 0.10485 Tuesday = 35 + 25 /2 = 60/2 = 30 or 0.09675 Wednesday = 30 + 30 = 60/2 = 30 or 0.09675 Thursday = 25 + 30 /2 = 55/2 = 27.5 or 27 ½ or 0.0887 Friday = 35 + 35 = 70 /2 = 35 or

Truth Tables, Implications, Contrapositives and Converses

(a) Use truth tables to prove that an implication is always equivalent to its contrapositive. Site an example where this is so. (b) Use truth tables to prove that an implication may not be equivalent to its converse. Site an example where this is so.

Zeno's Paradox

Suppose that a man wants to cross to the far wall of a room that is 20ft across. First he crosses half of the distance to reach the 10ft mark. Next he crosses halfway across the remaining 10ft to arrive at the 5ft mark. Dividing the distance in half again he crosses to the 2.5ft mark and continues to cross the room in this way d

Population explosion in a rabbit farm

You own a rabbit far. Every week each pair of rabbits has two baby rabbits. However, the baby rabbits first reproduce when they are two weeks old. Initially you have a pair of newborn rabbits. How many pairs you have after n weeks?

12 students 3 ways

12 students are split 5 in room A, 4 in room B, and 3 in room C. How many different ways can this happen.