The assignment problem is a special type of linear programming problem where assignees are being assigned to perform tasks or Jobs. For example, the assignees might be employees who need to be given work assignments. Assigning people to jobs is a common application of the assignment problem. However, the assignees need not be people. They could be machines, or vehicles, or plants, or even time slots to be assigned tasks.

To fit the definition of an assignment problem, these kinds of applications need to be formulated in a way that satisfies the following assumptions:

(i) The number of assignees and the number of tasks are the same (Balanced).

(ii) Each assignee is to be assigned to exactly one task.

(iii) Each task is to be performed by exactly one assignee.

(iv) There is a cost associated with assignee i performing task j

(v) The objective is to determine how all assignments should be made to minimize the total cost.

The general assignment model with n assignees and n tasks is presented as follows:

The assignment model is actually a special case of the transportation model in which the assignees represent the sources, and the tasks represent the destinations. In effect, the assignment model can be solved directly as a regular transportation model. Nevertheless, the fact that all supply and demand amounts equal 1 has led to the development of a simple solution algorithm called the Hungarain method. Although the new method appears totally unrelated to the transportation model, the algorithm is actually rooted in the simplex method, just as the transportation model is.

Mathematical Formulation of the problem

Let = 0, if the task not assigned to assignee

= 1, if the task assigned to assignee

The objective of the model is to determine the unknowns that minimize the overall cost:

Minimize

Subject to the constraints

for all i and j

The Assignment Algorithm – Hungarian method

The steps of the assignment algorithm are as follows:

Step 1: For the original cost matrix, identify each row’s minimum, and subtract it from all the entries of the row.

Step 2: For the matrix resulting from step 1, identify each column’s minimum, and subtract it from all the entries of the column.

Step 3: Assign the zeroes:

(a) Examine the rows of the current matrix successively until a row with exactly one unmarked zero is found. Circle this zero, indicating that an assignment will be made there. Cross out all other zeroes lying in the column of above encircled zero. The crossed cells will not be considered for any future assignment. Continue this manner until all the rows have been taken care of.

(b) Similarly for column.

Step 4: Check for Optimality: Repeat step 3 successively till one of the following occurs.

(a) There is no row and no column without assignment. In such a case, the current assignment is optimal.

(b) There may be some row or column without assignment. In this case, the current solution is not optimal. Proceed to next step.

Step5: Draw minimum number of lines crossing all zeroes as follows: If the number of lines equal to the order of the matrix, then the current solution is optimal, otherwise it is not optimal. Proceed to the next step.

Step6: Examine the elements that do not have a line through them. Select the smallest of these elements and subtract the same from all the elements that do not have a line through them, and add this element to every element that lies in the intersection of the two lines. Repeat this until an optimal assignment is reached.

Example 3.4:

Joe’s three children – John, Karen and Terri – want to earn some money to take care of personal expenses during a school trip to the local zoo. Joe has chosen three tasks for his children: mowing the lawn, painting the garage, and washing the family cars. To avoid anticipated sibling competition, he asked them to summit (secret) bids for what they feel was a fair pay for each of the three tasks. The understanding then was that all three children would abide their father’s decision as to who gets which tasks. The following table summarizes the bids received.

	Mow	Paint	Wash
John	15	10	9
Karen	9	15	10
Terri	10	12	8

Based on this information, how should Joe assign the tasks?

Solution:

Let and be row i and column j minimum costs as defined in step 1 and 2 respectively.

	Mow	Paint	Wash	Row minimum
John	15	10	9
Karen	9	15	10
Terri	10	12	8

	Mow	Paint	Wash
John	6	1	0
Karen	0	6	1
Terri	2	4	0
Column minimum

	Mow	Paint	Wash
John	6	0	0
Karen	0	5	1
Terri	2	3	0

The cells with underscored zero entries provide the optimum solution. The total cost is 9 + 10 + 8 = 27. This amount also will always equal .

Unbalanced Assignment Problem

When the cost matrix of an assignment problem is not a square matrix (# of assignees is not equal to the # of tasks), the assignment is called an unbalanced assignment problem. In such problems, dummy rows or columns are added in the matrix so as to complete it to form a square matrix.

For a function f defined in some open interval a (but not necessarily at a itself), we say , if given any (tiny) number > 0, there is another number d > 0 such that 0 < |x – a| < d guarantees that
|f(x) – L| < .

Definition of a Limit

Let f be a function defined on an open interval containing c (except possibly at c) and let L be a real number. The statement

  = L means that for each >0, there exists a >0 such that

if 0 < <

then < .

Limits can be found Numerically and Algebraically

While almost all limits can be found graphically, as we have been discussing, it is not always practical or necessary if the function is defined algebraically.

For instance, say we are given that. If we are looking for , instead of having to graphically search for the answer, we can find both the left- and right-hand limits by using tables. By choosing x values that get closer and closer to x = 3 from both sides, we can analyze the behavior of f(x).

Table 3.1

x	2.99	2.999	2.9999	3	3.0001	3.001	3.01
f(x)	5.99	5.999	5.9999	?	6.0001	6.001	6.01

Notice that when we chose values on either side of x = 3, they were values that were very close to x = 3. It seems that as x approaches 3 from either side, the function values are approaching 6. Therefore, it seems reasonable to conclude that

Limit Rules

If a, c, and n, are real numbers, then

1) (The limit of a constant real number is that number.)

2) where p(x) is any polynomial (The limit value of a polynomial is the function value at that point.)

3)

(The limit of the product of a constant and a function equals the constant times the limit of the function.)

4)

(The limit of the sum or difference of two functions equals the sum or difference of the limits of the functions.)

5)

(The limit of the product of two functions is the product of the limits of the functions.)

6)

(The limit of a quotient is the quotient of the limits of the numerator and denominator if the limit of the denominator is not zero.)

Example 8 Evaluate

Solution

1. (Rule 2)
2. (Rule 6)

                                  (Rule 2)

When you get 0/0 we have what is called an indeterminate form and we must try other techniques to determine the limit. In this case, factor both the numerator and denominator and cancel common terms to remove the zero in the denominator. Then, apply the limit rules to the simplified expression.

              (Factor)

                                     (Cancel common terms)

                                       = 3 + 3 = 6 (Rule 2)

1. (Rule 6)

                                 (Rules 1 and 2)

This is not defined and whenever you get a result of a non-zero number over zero, there are no common factors in the numerator and denominator which can be cancelled. Therefore, there is no way to rid the denominator of its zero term, meaning that the limit does not exist.

Example 9 Evaluate

Solution

are all also known as indeterminate forms. When this form occurs when finding limits at infinity (or negative infinity) with rational functions, divide every term in the numerator and denominator by the highest power of x in the denominator to determine the limit.

Since is the highest power of x in the denominator of our function, we have

2.1 Symbols and Terminology

Set: a collection of objects

Elements (members): objects belonging to a set

There are three ways to designate a set:

1. Word description

2. Listing method

3. Set-builder notation

Ex: 1. The set of even counting numbers less than ten

2. { 2, 4, 6, 8 }

3. { x | x is an even counting number }

We use CAPITAL letters to denotes names of sets usually A, B, and C.

We use lower case letters for elements of sets.

Ex: A = { a, b, c}

Empty Set (null set): is the set containing no elements.

Notation: { } or are ways of denoting the empty set

{} is not the empty set {0} also is not the empty set

Sets of Real Numbers

Natural (counting) numbers: { 1, 2, 3, 4, 5, …}

Whole numbers: { 0, 1, 2, 3, 4, 5, …}

Integers: { …-3, -2, -1, 0, 1, 2, 3, …}

Rational numbers: any number that can be written in the form if a and b are intergers and b 0.

Ex: { 5, -4, ¾, .25, .3333…. }

Irrational numbers: number that aren’t rational. IE: numbers that can not be expressed as fractions.

Ex: { }

Cardinal Number (cardinality): the number of elements in a set; repeated elements should not be counted more than once.

Notation: n(A) = “n of A” = is the cardinal number of set A.

Finite Set: if the cardinal number of the set can be expressed as a whole number.

Ex: {1,2,3,4,a,b,c}

Infinite Set: if the cardinal number of a set can not be expressed as a whole number.

Ex: The set of counting numbers

Symbol Notation

:is and element of

:is not an element of

Ex: A={1,2,3} than 2A and 4A

Set Equality

Set A is equal to set B provided the following two conditions are met:

1. Every element in set A is an element is set B

2. Every element in set B is an element in set A

Ex: A={1,2,3} B={1,2,1,2,1,2,1,3,3,3,3,3} then A = B

2.2 Venn Diagrams and Subsets

Universal set: the set of all elements to be discussed: this is sometimes implied or directly given.

Notation: U = universal set

Venn diagram: a visual way of depicting the relationship between sets using a rectangle to denote the universal set and circles to denote sets with in the universal set.

The Compliment of a Set

For any set A within the universal set U, the compliment of A is the set of elements of U that are not elements of A

Notation: A’ = compliment of set A = {x | x U and xA}

Ex: U = { 1,2,3,4}, A={1,2}, A’ = {3,4}

Subset of a Set: Set A is a subset of set B if every element in set a is an element of set B

Notation: AB = “Set A is a subset of set B”

Ex: A={1,2,3} B={1,2,3,4,5,6} then AB

Alternative Def’n for Equality of sets

Set A = Set B if AB and BA

Proper Subset of a Set: Set A is a proper subset of set B if AB and set A is NOT equal to set B.

Notation: A B = “set A is a proper subset of set B”

Ex: A={1,2,3} B={1,2,3,4} then A B

Finding the number of Subsets and Proper Subsets:

The number of subsets of a given set = 2 ⁿ

if n = # of elements in a set.

The number of proper subsets of a given set = 2 ⁿ – 1

if n = # of elements in a set.

2.3 Set Operations and Cartesian Products

Intersection of Sets: the intersection of sets A and B, written “”, is the set of all elements common to both set A and set B, or = { x | xA and xB } .

Ex: If A = {1,2,3} B = {3,4,5} then = { 3 }

Union of Sets: the union of set A and set B, written “”, is the set of all elements belonging to either set A or set B or both, or = { x | xA or xB }.

Ex: If A = {1,2,3} and B = {2,3,4} then = {1,2,3,4}

Difference of Sets: the difference of set A and set B, written “A – B”, is the set of all elements belonging to set A and not to set B, or A – B = { x | xA and xB }

Ex: If A = {1,2,3,} and B = {3,4,5} then A – B = {1,2}

Ordered Pairs: in the ordered pair (a,b), a is called the first component and b is called the second component. In general (a,b)(b,a). Two ordered pairs are equal only if there first components are equal and their second components are equal, (a,b) = (c,d) if a = c and b = d.

Ordered pairs are NOT sets

Cartesian Product of Sets: the Cartesian product of sets A and B, written A x B, is

A x B = {(a,b)| aA and bB}.

Ex: Let A = {1,2,3} and B = {a,b} then A x B = {(1,a),(1,b),(2,a),(2,b),(3,a),(3,b)}

Cardinal Number of a Cartesian Product: If n(A) = a and n(B) = b, then n(AxB) = n(A) x n(B) = n(B) x n(A) = ab

De Morgan’s Laws: for any sets A and B

(A B)’ = A’ B’

(A B) ‘= A’ B’

2.4 Cardinal Numbers and Surveys

Cardinal Number Formula: for any sets A and B,

Ex: Find n(AB) if n(A) = 3, n(B) = 4, and n(AB) = 2

2.5 Infinite Sets and Their Carndinalities

One-to-One Correspondence: there will be a one-to-one correspondence between two sets A and B if each element in set A is paired with an element of set B and Each element in set B is paired with each element in set A.

Equivalent Sets: two sets are equivalent, written A~B, if you can put the two sets in a one-to-one correspondence.

Algebra of sets

Associative laws

A (B C) = (A B) C

A (B C) = (A B) C

Commutative laws

A B = B A

A B = B A

Identity laws

A = A

A U = A

Idempotent laws

A A = A

A A = A

Distributive laws

A (B C) = (A B) (A C)

A (B C) = (A B) (A C)

Complement laws

A A’ = U

A A’=

Example:

Each of the 63 first-year students studying computing at the Univ. can study a number of optional units. If 16 chose to study the accounting option, 37 chose to study the business option and 5 studied both of these options, how many took neither accounting nor business.

A ={computing students who took the accounting option}

B ={computing students who took the business option}

Then |A| = 16 |B| = 37 and |A B| = 5

|A B| = 16 + 37 – 5 = 48

63 students

- 48 who took accounting or business or both

15 took neither

              11           5              32

2.6. Power sets

The set of all subsets of a set A is called the power set of A and denoted as P(A)

For example, if A = {a,b}, P(A) = {, {a}, {b}, {a,b}}.

If S is the set {x, y, z}, then the complete list of subsets of S is as follows:

• { } (also denoted , the empty set)
• {x}
• {y}
• {z}
• {x, y}
• {x, z}
• {y, z}
• {x, y, z}

and hence the power set of S is

The power set P({1, 2, 3}) of {1, 2, 3} is equal to the set

{{1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, }.

The cardinality of the original set is 3, and the cardinality of the power set is 2³, or 8.

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Statistical methods can be used to summarize or describe a collection of data; this is called descriptive statistics. In addition, patterns in the data may be modeled in a way that accounts for randomness and uncertainty in the observations, and are then used to draw inferences about the process or population being studied; this is called inferential statistics. Descriptive, predictive, and inferential statistics comprise applied statistics. There is also a discipline called mathematical statistics, which is concerned with the theoretical basis of the subject. Moreover, there is a branch of statistics called Exact Statistics that are based on exact probability statements.

A note on terminology. The word “statistics” can either be singular or plural. In its singular form, “statistics” refers to the mathematical science discussed in this article. In its plural form, “statistics” is the plural of the word “statistic”, which is either a single data point or datum, or a quantity (such as a mean) calculated from a set of data.