THE SIMPLEX METHOD

STEP BY STEP GUIDE Formulate the problem (i) Pick out important information (ii) Formulate constraints (iii) Formulate objective function Introduce slack variables Form initial tableau Obtain new tableaux (i) Identify pivotal column (ii) Find θ-values (iii) Identify pivotal row (iv) Identify pivot (v) Pivot 5. Get the solution

THE PROBLEM A small factory produces two types of toys: cars and diggers. In the manufacturing process two machines are used: the moulder and the colouriser. A digger needs 2 hours on the moulder and 1 hour on the colouriser. A car needs 1 hour on the moulder and 1 hour on the colouriser. The moulder can be operated for 16 hours a day and the colouriser for 9 hours a day. Each digger gives a profit of £16 and each car gives a profit of £14. The profit needs to be maximised. How do we formulate this problem?

STEP BY STEP GUIDE Formulate the problem (i) pick out important information (ii) formulate constraints (iii) formulate objective function Introduce slack variables Form initial tableau Obtain new tableaux Get the solution

PICKING OUT IMPORTANT INFORMATION A small factory produces two types of toys: cars and diggers. In the manufacturing process two machines are used: the moulder and the colouriser. A digger needs 2 hours on the moulder and 1 hour on the colouriser. A car needs 1 hour on the moulder and 1 hour on the colouriser. The moulder can be operated for 16 hours a day and the colouriser for 9 hours a day. Each digger gives a profit of £16 and each car gives a profit of £14.

PICKING OUT IMPORTANT INFORMATION A small factory produces two types of toys: cars and diggers. In the manufacturing process two machines are used: the moulder and the colouriser. A digger needs 2 hours on the moulder and 1 hour on the colouriser. A car needs 1 hour on the moulder and 1 hour on the colouriser. The moulder can be operated for 16 hours a day and the colouriser for 9 hours a day. Each digger gives a profit of £16 and each car gives a profit of £14.

A digger needs 2 hours on the moulder and 1 hour on the colouriser A digger needs 2 hours on the moulder and 1 hour on the colouriser. A car needs 1 hour on the moulder and 1 hour on the colouriser.

PICKING OUT IMPORTANT INFORMATION A small factory produces two types of toys: cars and diggers. In the manufacturing process two machines are used: the moulder and the colouriser. The moulder can be operated for 16 hours a day and the colouriser for 9 hours a day. Each digger gives a profit of £16 and each car gives a profit of £14.

A digger needs 2 hours on the moulder and 1 hour on the colouriser A digger needs 2 hours on the moulder and 1 hour on the colouriser. A car needs 1 hour on the moulder and 1 hour on the colouriser. The moulder can be operated for 16 hours a day and the colouriser for 9 hours a day.

STEP BY STEP GUIDE Formulate the problem (i) pick out important information (ii) formulate constraints (iii) formulate objective function Introduce slack variables Form initial tableau Obtain new tableaux Get the solution

make two constraints from this information. A digger needs 2 hours on the moulder and 1 hour on the colouriser. A car needs 1 hour on the moulder and 1 hour on the colouriser. The moulder can be operated for 16 hours a day and the colouriser for 9 hours a day. Using the decision variables d = number of diggers c = number of cars make two constraints from this information.

2d + c ≤ 16 FORMING CONSTRAINT 1 THE MOULDER A digger needs 2 hours on the moulder and 1 hour on the colouriser. A car needs 1 hour on the moulder and 1 hour on the colouriser. The moulder can be operated for 16 hours a day and the colouriser for 9 hours a day. 2d + c ≤ 16

d + c ≤ 9 FORMING CONSTRAINT 2 THE COLOURISER A digger needs 2 hours on the moulder and 1 hour on the colouriser. A car needs 1 hour on the moulder and 1 hour on the colouriser. The moulder can be operated for 16 hours a day and the colouriser for 9 hours a day. d + c ≤ 9

STEP BY STEP GUIDE Formulate the problem (i) pick out important information (ii) formulate constraints (iii) formulate objective function Introduce slack variables Form initial tableau Obtain new tableaux Get the solution

PICKING OUT IMPORTANT INFORMATION A small factory produces two types of toys: cars and diggers. In the manufacturing process two machines are used: the moulder and the colouriser. Each digger gives a profit of £16 and each car gives a profit of £14.

FORMING THE OBJECTIVE FUNCTION Each digger gives a profit of £16 and each car gives a profit of £14. Let Z be the total profit; formulate the objective function

FORMING THE OBJECTIVE FUNCTION Each digger gives a profit of £16 and each car gives a profit of £14. Z = 16d + 14c

THE LINEAR PROGRAMMING PROBLEM MAXIMISE Z = 16d + 14c subject to the constraints: (i) 2d + c ≤ 16 (ii) d + c ≤ 9 (iii) c ≥ 0 , d ≥ 0 VERY IMPORTANT DON’T FORGET YOUR NON – NEGATIVITY CONSTRAINTS !

STEP BY STEP GUIDE Formulate the problem Introduce slack variables Form initial tableau Obtain new tableaux Get the solution

INTRODUCING SLACK VARIABLES To change inequalities (i) and (ii) into equations we add slack variables s and t This gives: (i) 2d + c + s = 16 (ii) d + c + t = 9

THE NEW LINEAR PROGRAMMING PROBLEM MAXIMISE Z = 16d + 14c + 0s + 0t subject to the constraints: 2d + c + s + 0t = 16 d + c + 0s + t = 9 c ≥ 0 , d ≥ 0 , s ≥ 0 , t ≥ 0

STEP BY STEP GUIDE Formulate the problem Introduce slack variables Form initial tableau Obtain new tableaux Get the solution

We want to put all the information in the form of a table. This is called the initial tableau. To form the initial tableau we need to change the objective function from Z = 16d + 14c + 0s + 0t to Z – 16d – 14c – 0s – 0t = 0

FORMING THE INITIAL TABLEAU Label the table with your basic variables, s and t and with your non – basic variables, d and c. BASIC VARIABLES d c s t VALUE Z

CLICK HERE BASIC VARIABLES The variables solved for are called Non-basic variables are set to zero. CLICK HERE

FORMING THE INITIAL TABLEAU 2d + 1c + 1s + 0t = 16 1d + 1c + 0s + 1t = 9 Z – 16d – 14c – 0s – 0t = 0 BASIC VARIABLES d c s t VALUE Z 2 1 1 16 1 1 1 9 -16 -14

FORMING THE INITIAL TABLEAU Z t s VALUE c d BASIC VARIABLES 2 16 1 9 -16 -14 This is the objective row

STEP BY STEP GUIDE Formulate the problem Introduce slack variables Form initial tableau Obtain new tableaux (i) Identify pivotal column (ii) Find θ-values (iii) Identify pivotal row (iv) Identify pivot (v) Pivot Get the solution

PIVOTAL COLUMN Z t s c d 2 16 1 9 -16 -14 We now need to find where to pivot and we start by entering the basis by choosing the column with the most negative entry in the objective row. Z t s VALUE c d BASIC VARIABLES 2 16 1 9 -16 -14 This is the most negative coefficient with corresponding variable d and it’s column is called the pivotal column. d is now called the entering variable.

CLICK HERE ENTERING THE BASIS Choose the non-basic variable that is to become a basic variable. The most common rule for selecting this variable is to select the variable with the most negative entry in the objective row. CLICK HERE

STEP BY STEP GUIDE Formulate the problem Introduce slack variables Form initial tableau Obtain new tableaux (i) Identify pivotal column (ii) Find θ-values (iii) Identify pivotal row (iv) Identify pivot (v) Pivot Get the solution

FINDING θ-VALUES Z t s c d 2 1 1 16 1 1 1 9 -16 -14 You are now going to find the pivotal row and the leaving variable. You need to find θ-values. 1. Identify positive entries in the pivotal column. 2. Divide each entry in value column by the corresponding positive entry in the pivotal column. Z t s VALUE c d BASIC VARIABLES 2 1 1 16 1 1 1 9 -16 -14

CLICK HERE LEAVING VARIABLE The leaving variable is the variable in the pivotal row chosen to leave the basis. The entering variable is the new basic variable. CLICK HERE

STEP BY STEP GUIDE Formulate the problem Introduce slack variables Form initial tableau Obtain new tableaux (i) Identify pivotal column (ii) Find θ-values (iii) Identify pivotal row (iv) Identify pivot (v) Pivot 5. Get the solution

PIVOTAL ROW For row (i) For row (ii) The row with the smallest θ-value is called the pivotal row. Here the pivotal row is row (i)

STEP BY STEP GUIDE Formulate the problem Introduce slack variables Form initial tableau Obtain new tableaux (i) Identify pivotal column (ii) Find θ-values (iii) Identify pivotal row (iv) Identify pivot (v) Pivot 5. Get the solution

THE PIVOT The pivot! Z t s c d 2 1 1 16 1 1 1 9 -16 -14 VALUE c d BASIC VARIABLES 2 1 1 16 1 The pivotal row 1 1 9 -16 -14 The pivotal column

STEP BY STEP GUIDE Formulate the problem Introduce slack variables Form initial tableau Obtain new tableaux (i) Identify pivotal column (ii) Find θ-values (iii) Identify pivotal row (iv) Identify pivot (v) Pivot 5. Get a solution

PIVOTING Replace the leaving variable with the entering variable. 2. Divide all entries in the pivotal row by the pivot. The pivot becomes 1. 3. Add suitable multiples of the pivotal row to all other rows until all entries, apart from the pivot, in the pivotal column are zero.

Step 1 - Replace the leaving variable with the entering variable. Z t s VALUE c d BASIC VARIABLES 2 16 1 9 -16 -14 Step 2 - Divide all entries in the pivotal row by the pivot. The pivot becomes 1. BASIC VARIABLES d c s t VALUE s d 1 1/2 1/2 8 t Z

PIVOTING Z t s c d 2 16 1 9 -16 -14 t 1/2 -1/2 1 1 VALUE BASIC VARIABLES 2 16 1 9 -16 -14 Step 3 - Add suitable multiples of the pivotal row to all other rows until all entries, apart from the pivot, in the pivotal column are zero. row (ii) – ½ row (i) gives t 1/2 -1/2 1 1

PIVOTING Z t 8 1/2 1 d VALUE s c BASIC VARIABLES t 1/2 -1/2 1

PIVOTING Z t s c d 2 16 1 9 -16 -14 Z -6 8 128 VALUE BASIC VARIABLES 1 9 -16 -14 Step 3 - Add suitable multiples of the pivotal row to all other rows until all entries, apart from the pivot, in the pivotal column are zero. row (iii) + 8 row (i) gives Z -6 8 128

This is our second tableau Z t 8 1/2 1 d VALUE s c BASIC VARIABLES t 1/2 -1/2 1 Z -6 8 128 This is our second tableau

PIVOTING Follow the rules for finding a pivot on your second tableau. Pivot as before. Continue this process until there are no negative entries in the objective row. This will be your final tableau. This is called the optimal tableau.

OPTIMAL TABLEAU d c s t 1 -1 7 2 Z 12 140 BASIC VARIABLES d c s t VALUE 1 -1 7 2 Z 12 140 Note there are no negative entries in the objective row. Can you see the solution?

STEP BY STEP GUIDE Formulate the problem Introduce slack variables Form initial tableau Obtain new tableaux 5. Get the solution

OBTAINING THE SOLUTION BASIC VARIABLES d c s t VALUE 1 -1 2 12 d 7 c 2 Z 140 Remember that since s and t are now non–basic variables they are set to zero. This corresponds to the solution: s = 0, t = 0, d = 7 c = 2 Z = 140

THE SOLUTION Don’t forget to put your solution back into the context of the problem. Z = 140 d = 7 c = 2 The maximum profit is £140 To make this profit the factory should produce 7 diggers and 2 cars.