Linear Programming

• Linear Inequality


  Mathematical expression in which both sides are not equal
• Linear Inequalities


  • x > 2
  • y - 2x < 0
  • y > x
• Graphing linear inequalities


  1. Rearrange inequalities to match linear equation y = f(x)

  2. Graph the lines y = x - 2 and y = -2x + 10

  3. Draw solid line for y≤ or y≥ and dashed line for y< or y>

  4. Shade the region based on the inequalities

  
• Linear Programming


  Mathematical concept used to find the optimal solution of a linear function
• Linear Programming


  • Uses simple assumptions for optimizing the given function
  • Has huge real-world application and is used to solve various types of problems
• Linear Programming Components


  • Decision Variables: Variables to determine to achieve optimal solution
  • Objective Function: Mathematical equation representing the goal
  • Constraints: Limitations or restrictions decision variables must follow
  • Non-Negativity Restrictions: Decision variables cannot be negative in some real-world scenarios
• General Linear Programming Formula


  • Objective Function: Z = ax + by
  • Constraints: cx + dy ≥ e, px + qy ≤ r
  • Non-Negative restrictions: x ≥ 0, y ≥ 0
• Solving Linear Programming Problems

  1. Step 1: Mark the decision variables

  2. Step 2: Build the objective function and check if it needs to be minimized or maximized

  3. Step 3: Write down all the constraints

  4. Step 4: Ensure non-negative restrictions

  5. Step 5: Solve the problem using any method
• Graphical Method


  • Method to optimize two-variable linear programming
  • Plots the inequalities on the XY plane to find the feasible region and optimal solution

  
• Simplex Method


  • Iterative method to find the optimal solution by improving the objective function value
  • One of the greatest inventions of modern times due to its broad applicability in solving business-related problems
• Simplex Method Steps


  Formulate the linear programming problem

  2. Convert inequalities to equations by adding slack variables

  3. Construct the initial simplex table

  4. Identify the pivot column

  5. Identify the pivot row and element

  6. Use matrix operations to make pivot column entries zero

  7. Check for non-negative entries in bottom row, repeat from step 4 if negative

  8. The final simplex table gives the optimal solution
• Solving linear programming problems using the simplex method

  1. Add the slack variable to each inequality where ever required

  2. Construct the initial simplex table

  3. Identify the greatest negative entry in the bottom row, the column of the element with the highest negative entry is called the pivot column

  4. Divide the entries of the right-most column with the entries of the respective pivot column, excluding the entries of the bottommost row. The row containing the least entry is called the pivot row. The pivot element is obtained by the intersection of the pivot row and the pivot column

  5. Using matrix operation and with the help of the pivot element make all the entries in the pivot column to be zero

  6. Check for the non-negative entries in the bottommost row, if there are no negative entries in the bottom row, end the process else start the process again from step 4

  7. The final simplex table so obtained gives the solution to the problem
• Websites for learning linear programming


  • https://byjus.com/maths/linear-inequalities/
  • https://www.geeksforgeeks.org/linear-programming/
  • https://openstax.org/books/contemporary-mathematics/pages/5-11-linear-programming
• Simple interest


  When money is borrowed, the person borrowing the money (borrower) typically has to pay the person or entity that lent the money (the lender) more than the amount of money that was borrowed. This extra money is the interest that is to be paid.
• Interest


  The cost to borrow, the cost of the loan, or the finance cost
• Calculating simple interest and loan payoff amount

  • Principal P = $4,000, annual interest rate r = 5.5%, and number of years t = 4
  • Principal P = $14,800, annual interest rate r = 7.9%, and number of years t = 7
• Calculating simple interest and loan payoff amount for an auto repair shop loan

  • Principal P = $11,995, annual interest rate r = 14.9%, and number of years t = 2
• Calculating interest earned and future value of investments with simple interest


  • Principal P = $1,000, annual interest rate r = 2.01%, and time t = 5 years
  • Principal P = $5,000, annual interest rate r = 1.85%, and time t = 30 years
  • Principal P = $10,000, annual interest rate r = 1.25%, and time t = 18 months
  • Principal P = $7,000, annual interest rate r = 3.26%, and time t = 100 days
• Compound interest


  Accounts offering compound interest pay interest at regular intervals. After each interval, the interest is added to the original principal. Later, interest is calculated on the original principal plus the interest that has been added previously.
• Calculating future value with compound interest

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• Calculating present value with compound interest

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• Loans


  Borrowed money that the borrower agrees to pay back with interest
• Loan terms


  • Principal - total value of the loan
  • Fixed interest rate loan - has an interest rate that is fixed for the whole duration
  • Variable interest rate loan - has an interest rate that may change during the life of the loan
  • Term - life of the loan
  • Installment loan - loan with a fixed period
  • Loan amortization - process used to calculate how much of each payment will be applied to principal and how much is applied to interest
• Calculating loan payments


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• Credit cards


  Used to borrow money, usually from the bank, to pay for a purchase. The payment is a type of loan so the borrower will pay the lender back with interest. The credit card charges the lender with an annual fee, which is the fee charged for the privilege of using the card. The card also grants reward programs which are benefits granted from the usage of the card. There is also a credit limit which is the maximum amount the card can grant.
• Types of credit cards

  • Bank-issued credit cards
  • Store-issued credit cards
  • Travel/entertainment cards or charge cards
• Credit score


  A number that will show the ability of the borrower in paying off loans based on the credit report
• Factors included in credit score calculation

  • Credit Payment History: is the borrower paying on time and enough
  • The amount owed or credit utilization ratio: how much of the credit limit is reached
  • Length of credit history: age of the loan or credit card and time of being used
  • Types of credit used
  • New Credit: how often new accounts are opening
• Credit card statements


  The cardholders will receive monthly statements and they should pay the amount for at most 21 day for the minimum amount. The billing period is one month from the registration of the card. The statement will include the current balance, interest rate, the minimum payment due, and the due date. Missing the due date will result in paying late fee and a negative report to credit bureau and negatively affects the credit score.
• Credit card balance


  The previous balance, plus all expenses, minus all payments and credits, plus the interest on the card
• Included in the calculation of credit score

  • Credit Payment History
  • Amount owed or credit utilization ratio
  • Length of credit history
  • Types of credit used
  • New Credit
• Credit Payment History


  Is the borrower paying on time and enough
• Amount owed or credit utilization ratio

  How much of the credit limit is reached
• Length of credit history


  Age of the loan or credit card and time of being used
• New Credit


  How often new accounts are opening
• Credit Card Statements


  1. Cardholder receives monthly statements

  2. Pay the minimum amount due within 21 days

  3. Includes current balance, interest rate, minimum payment due, and due date

  4. Missing due date results in late fee and negative report to credit bureau
• Balance of the Credit Card

  Previous balance, plus all expenses, minus all payments and credits, plus the interest on the card
• Compute the average daily balance

  1. Add the amount on the same day

  2. The payment will be treated as a negative value

  3. Accumulate the amounts

  4. Add a column with the days before the balance changes and another column for balance times days
• Interest


  Charged for the credit card based on the average daily balance, annual interest rate, and billing cycle
• Minimum Payment


  • The smallest required amount to be paid on a credit card to avoid late fees and penalties
  • Depends on the balance of the credit card
  • For larger balances, the minimum payment will be a percentage of the balance due
  • For moderate balances, the minimum would be a specified dollar amount
  • If the balance is small, the minimum payment is the balance