Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India LINEAR PROGRAMMING GRAPHIC METHOD 2 CHAPTER

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 2 Learning Objectives Requirements of a linear programming problem Graphical solution – maximisation case Graphical solution – minimisation case Technical issues in linear programming

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 3 Basic Requirements There must be an objective There must be alternative courses of action There must be limitation on resources It should be possible to express the objective functions and constraints as linear mathematical equations or inequalities

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 4 Crisis at Bhilpur Type A 26 Nos 2 Tons Type B 15 Nos 3 Tons Diesel 4000 l Drivers 30 Nos 100 l 200 l Bhilpur

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 5 Maximisation – Crisis at Bhilpur Type A Vehicle Type B Vehicle Total Available (a) Carrying Capacity2 Tons3 Tons- (b) Diesel Required100 Litres200 Litres4000 Litres (c) Drivers Required1130 (d) Numbers Available 2615-

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 6 Maximisation – Crisis at Bhilpur Identify the problem –To send maximum coal to Bhilpur –Coal can be sent by using a certain number of Type A vehicles or Type B vehicles or a combination of the two –Mr. Confused Singh must determine the number of Type A vehicles and the number of Type B vehicles that he should use in order to solve the problem.

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 7 Maximisation – Crisis at Bhilpur Identify the controls which affect the problem –Scarce resources are Diesel Drivers Number of Type A vehicles Number of Type B vehicles –Coal is available in plenty. Do not consider it.

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 8 Formulate the Objective Function. Let A be the number of Type A vehicles used and B be the number of Type B vehicles used A and B are called Decision Variables Total coal carried Since this is to be maximised, the objective function is Maximisation – Crisis at Bhilpur

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 9 Maximisation – Crisis at Bhilpur Formulate the constraints –Diesel - As only 4000 litres are available, the quantity consumed cannot exceed 4000 litres. Total diesel consumed by the vehicles is This must not be greater than 4000 litres, hence

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 10 Formulate the constraints –Drivers - As only 30 drivers are available, the number of vehicles used cannot exceed 30. Total number of vehicles used is This must not be greater than 30, hence Maximisation – Crisis at Bhilpur

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 11 Formulate the constraints –Type A Vehicles - As only 26 vehicles are available, the number of Type A vehicles used cannot exceed 26. Hence –Type B Vehicles - As only 15 vehicles are available, the number of Type B vehicles used cannot exceed 15. Hence Maximisation – Crisis at Bhilpur

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 12 Maximisation – Crisis at Bhilpur The model can now be written as

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 13 Maximisation – Crisis at Bhilpur Non-negativity Constraint - Since the value of A and B cannot be negative, we add one more constraint to the model

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 14 Maximisation – Crisis at Bhilpur When there are only two decision variables, the problem can be solved graphically. Let X axis represent the number of Type A vehicles and the Y axis represent the number of Type B vehicles. As A and B cannot be less than zero (non-negativity constraint), the solution must lie in the first quadrant only. Plot the constraint inequalities on the graph.

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 15 Maximisation – Crisis at Bhilpur The diesel constraint is Assume that the inequality is an equality and determine two points on the line

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 16 Maximisation – Crisis at Bhilpur Type A Vehicles Type B Vehicles

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 17 Maximisation – Crisis at Bhilpur Since the constraint is an inequality of the ‘less than’ type, all points that satisfy the inequality lie in the shaded area Type A Vehicles Type B Vehicles

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 18 Maximisation – Crisis at Bhilpur Plot the drivers constraint. If A=0 then B=30,and if B=0 then A=30. Plot (30,0) and (0,30). All points satisfying the two inequalities must lie in the shaded area. Type A Vehicles Type B Vehicles

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 19 Maximisation – Crisis at Bhilpur Plot the constraints for vehicles. And These would be lines parallel to the axes. All points satisfying the four inequalities must lie in the shaded area. Type A Vehicles Type B Vehicles

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 20 Type A Vehicles Type B Vehicles Type A Vehicles Type B Vehicles Maximisation – Crisis at Bhilpur The objective function is To find the point that satisfies this, assign any value to the objective function and plot that line. Plot this line. Move it parallel to itself, away from the origin, as we are maximising till it is just leaving the shaded area (solution area). This point gives the answer. This line is called an iso-profit line. A=20 and B=10. Type A Vehicles Type B Vehicles

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 21 Maximisation – Crisis at Bhilpur The solution to Mr. Confused Singh’s problem is to send 20 Type A vehicles and 10 Type B vehicles to Bhilpur and deliver 70 tons of coal.

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 22 Maximisation – Crisis at Bhilpur The answer always lies at one of the vertices of the solution space. We can also find the solution by evaluating the value of the objective function at the vertices. The objective function is The value at the vertices is: (0,0)= 0(0,15)= 45 (10,15)= 65(20,10)= 70 (26,4)= 64(26,0)= 52

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 23 Minimisation Problem Plant 1Plant 2Monthly Require- ment Type A tyres Type B tyres Type C tyres Operating cost Rs 2500Rs 3500

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 24 Minimisation Problem Identify the problem –To meet monthly demand at minimum cost –Management must decide the number of days for which Plant 1 should be run and the number of days for which Plant 2 should be run

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 25 Minimisation Problem Let Plant 1 be run for X days and Plant 2 be run for Y days. The cost of running the plants is to be minimised. Hence, the Objective Function is

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 26 Minimisation Problem Formulate the constraints –Type A tyres – The minimum monthly requirement is 2500 tyres. In X days, 50X tyres can be produced at Plant 1 and in Y days 60Y tyres can be produced in Plant 2. –The constraint can be written as an inequality:

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 27 Minimisation Problem Similarly, constraints for Type B and Type C tyres can be written Add non-negativity constraint

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 28 Minimisation Problem The complete model can be written as

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 29 Minimisation Problem X Y Plot constraints as explained for the maximisation case. The constraint for Type A tyre is The shaded area represents the solution space. It is away form the origin as the constraint is of the ‘greater than’ type.

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 30 Plot other constraints also. The constraints for Type B and Type C tyres are The shaded area represents the solution space. Minimisation Problem X Y

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 31 Minimisation Problem X Y The objective function is To find the point that satisfies this, assign any value to the objective function and plot that line. Plot this line. Move it parallel to itself, towards the origin, as we are minimising till it is just leaving the shaded area (solution area). This point gives the answer. X=20 and Y=25.

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 32 Technical Issues No matter what the slope of the objective function, it shall leave the solution space only at one of the vertices of the polygon. The solution occurs at an extreme point of the solution space.

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 33 Technical Issues Infeasibility – No solution satisfies all the constraints. Consider the problem: x y

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 34 Technical Issues Unbounded Solution – If the solution can be made infinitely large or small without violating constraints x y

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 35 Technical Issues Redundancy – A constraint which does not affect the feasible area is called a redundant constraint Multiple optimal solutions – If the objective function line (iso-profit line) coincides with one of the sides of the solution space polygon, then all points on the line are optimal solutions.

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 36 Assumptions of Linear Programming Proportionality – The amount of each resource used and its contribution to the objective function is proportional to the value of the decision variable. If the number of units produced of an item is doubled, the resources required to produce it are also doubled and the profit contribution will also double.

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 37 Assumptions of Linear Programming Divisibility – The decision variables and the amount of resources used are assumed to have continuous values. Their values need not be integers.

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 38 Assumptions of Linear Programming Additivity – The value of the objective function and the amount of resources used is equal to the sum of the contributions of all the decision variables. E.g. If we were to produce 100 pieces of Product A we will require 100 sheets of metal. If we were to produce 100 pieces of Product A and 100 pieces of B we would still require 100 sheets. But as per additivity law we will require 200 sheets. A B

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 39 Assumptions of Linear Programming Deterministic Parameters – All coefficients, i.e. profit and cost associated with each product; amount of resources required for each product and the amount of input-output technological coefficients are known with certainty. E.g. Type A vehicle consumes only 100 litres of diesel.

Quantitative Techniques for Decision Making M.P. Gupta & R.B. Khanna © Prentice Hall India Page 40 Problem Formulation Understand the problem Collect all relevant data Identify the decision variables. (E.g. number of Type A and Type B vehicles to use) Choose a numerical measure of effectiveness. (E.g. tons of coal carried) Represent measure of effectiveness as a linear expression involving the decision variables (E.g. Max Z = 2A + 3B) Identify and represent all constraints as linear expressions involving the decision variables (E.g. 100A + 200B = 4000)