Linear Programming

Quantitative technique for Managerial DecisionCourse code: MAS 501 Linear ProgrammingBy NagendraBahadurAmatyaHead of Department

IntroductionLinear programming is a widely used mathematical modeling technique to determine the optimum allocation of scarce resources among competing demands. Resources typically include raw materials, manpower, machinery, time, money and space. The technique is very powerful and found especially useful because of its application to many different types of real business problems in areas like finance, production, sales and distribution, personnel, marketing and many more areas of management. As its name implies, the linear programming model consists of linear objectives and linear constraints, which means that the variables in a model have a proportionate relationship. For example, an increase in manpower resource will result in an increase in work output.

ESSENTIALS OF LINEAR PROGRAMMING MODELFor a given problem situation, there are certain essential conditions that need to be solved by using linear programming.1. Limited resources : limited number of labour, material equipment and finance2. Objective : refers to the aim to optimize (maximize the profits or minimize the costs).3. Linearity : increase in labour input will have a proportionate increase in output.4. Homogeneity : the products, workers' efficiency, and machines are assumed to be identical.5. Divisibility : it is assumed that resources and products can be divided into fractions. (in case the fractions are not possible, like production of one-third of a computer, a modification of linear programming called integer programming can be used).

PROPERTIES OF LINEAR PROGRAMMING MODELThe following properties form the linear programming model:Relationship among decision variables must be linear in nature.A model must have an objective function.Resource constraints are essential.A model must have a non-negativity constraint.

FORMULATION OF LINEAR PROGRAMMINGFormulation of linear programming is the representation of problem situation in a mathematical form. It involves well defined decision variables, with an objective function and set of constraints.Objective function:The objective of the problem is identified and converted into a suitable objective function. The objective function represents the aim or goal of the system (i.e., decision variables) which has to be determined from the problem. Generally, the objective in most cases will be either to maximize resources or profits or, to minimize the cost or time.Constraints:When the availability of resources are in surplus, there will be no problem in makingdecisions. But in real life, organizations normally have scarce resources within which the job has to be performed in the most effective way. Therefore, problem situations are within confined limits in which the optimal solution to the problem must be found.Non-negativity constraintNegative values of physical quantities are impossible, like producing negative number of chairs, tables, etc., so it is necessary to include the element of non-negativity as a constraint

GENERAL LINEAR PROGRAMMING MODELA general representation of LP model is given as follows:Maximize or Minimize, Z = p1 x1 + p2 x2 ………………pnxnSubject to constraints,w11 x1 + w12 x2 + ………………w1nxn ≤ or = or ≥ w1 ……………(i)w21 x1 + w22 x2 ………………w2n xn ≤ or = or ≥ w2 …………… (ii). . . .. . . .. . . .wm1 x1 + wm2 x2 +………………wmnxn ≤ or = ≥ wm …………(iii)Non-negativity constraint,xi ≥ o (where i = 1,2,3 …..n)

Example A company manufactures two types of boxes, corrugated and ordinary cartons. The boxes undergo two major processes: cutting and pinning operations. The profits per unit are Rs. 6 and Rs. 4 respectively. Each corrugated box requires 2 minutes for cutting and 3 minutes for pinning operation, whereas each carton box requires 2 minutes for cutting and 1 minute for pinning. The available operating time is 120 minutes and 60 minutes for cutting and pinning machines. The manager has to determine the optimum quantities to be manufacture the two boxes to maximize the profits.

Solution Decision variables completely describe the decisions to be made (in this case, by Manager). Manager must decide how many corrugated and ordinary cartons should be manufactured each week. With this in mind, he has to define: xl be the number of corrugated boxes to be manufactured.x2 be the number of carton boxes to be manufactured Objective functionis the function of the decision variables that the decision maker wants to maximize (revenue or profit) or minimize (costs). Manager can concentrate on maximizing the total weekly profit (z). Here profit equals to (weekly revenues) – (raw material purchase cost) – (other variable costs). Hence Manager’s objective function is: Maximize z = 6X2 + 4x2 Constraints show the restrictions on the values of the decision variables. Without constraints manager could make a large profit by choosing decision variables to be very large. Here there are three constraints: Available machine-hours for each machine Time consumed by each product Sign restrictions are added if the decision variables can only assume nonnegative values (Manager can not use negative negative number machine and time never negative number )

All these characteristics explored above give the following Linear Programming (LP) problem max z = 6x1 + 4x2(The Objective function) s.t. 2x1 + 3x2 ≤ 120 (cutting timeconstraint) 2x1 + x2 ≤ 60 (pinning constraint) x1, x2 ≥ 0 (Sign restrictions) A value of (x1,x2) is in the feasible region if it satisfies all the constraints and sign restrictions. This type of linear programming can be solve by two methods Graphical method Simplex algorithm method

Graphic MethodStep 1: Convert the inequality constraint as equations and find co-ordinates of the line.Step 2: Plot the lines on the graph. (Note: If the constraint is ≥ type, then the solution zone lies away from the centre. If the constraint is ≤ type, then solution zone is towards the centre.) Step 3: Obtain the feasible zone.Step 4: Find the co-ordinates of the objectives function (profit line) and plot it on the graph representing it with a dotted line.Step 5: Locate the solution point. (Note: If the given problem is maximization, zmax then locate the solution point at the far most point of the feasible zone from the origin and if minimization, Zmin then locate the solution at the shortest point of the solution zone from the origin).Step 6: Solution typei. If the solution point is a single point on the line, take the corresponding values of x1 and x2. ii. If the solution point lies at the intersection of two equations, then solve for x1 and x2 using the two equations. iii. If the solution appears as a small line, then a multiple solution exists. iv. If the solution has no confined boundary, the solution is said to be an unbound solution.

CONT…The inequality constraint of the first line is (less than or equal to) ≤ type which means thefeasible solution zone lies towards the origin. (Note: If the constraint type is ≥ then the solution zone area lies away from the origin in the opposite direction). Now the second constraints line is drawn.When the second constraint is drawn, you may notice that a portion of feasible area is cut. This indicates that while considering both the constraints, the feasible region gets reduced further. Now any point in the shaded portion will satisfy the constraint equations.the objective is to maximize the profit. The point that lies at the furthermost point of the feasible area will give the maximum profit. To locate the point, we need to plot the objective function (profit) line.

Cont..Objective function line (Profit Line)Equate the objective function for any specific profit value Z,Consider a Z-value of 60, i.e.,6x1 + 4x2 = 60Substituting x1 = 0, we get x2 = 15 andif x2 = 0, then x1 = 10 Therefore, the co-ordinates for the objective function line are (0,15), (10,0) as indicated objective function line. The objective function line contains all possible combinations of values of xl and x2.Therefore, we conclude that to maximize profit, 15 numbers of corrugated boxes and 30 numbers of carton boxes should be produced to get a maximum profit. Substituting x1 = 15 and x2= 30 in objective function, we getZmax = 6x1 + 4x2 = 6(15) + 4(30)Maximum profit : Rs. 210.00

Graphic Method on ToraSteps for shoving linear programming by graphic method using TorashoftwareStep 1 Start  Tora select linear programming 

Simplex Method In practice, most problems contain more than two variables and are consequently too large to be tackled by conventional means. Therefore, an algebraic technique is used to solve large problems using Simplex Method. This method is carried out through iterative process systematically step by step, and finally the maximum or minimum values of the objective function are attained. The simplex method solves the linear programming problem in iterations to improve the value of the objective function. The simplex approach not only yields the optimal solution but also other valuable information to perform economic and 'what if' analysis.

ADDITIONAL VARIABLES USED IN SOLVING LPP Three types of additional variables are used in simplex method such as,(a) Slack variables (S1, S2, S3..…Sn): Slack variables refer to the amount of unused resources like raw materials, labour and money.(b) Surplus variables (-S1, -S2, -S3..…-Sn): Surplus variable is the amount of resources by which the left hand side of the equation exceeds the minimum limit.(c) Artificial Variables (a1, a2, a3.. …an): Artificial variables are temporary slack variables which are used for purposes of calculation, and are removed later. The above variables are used to convert the inequalities into equality equations, as given in the Given Table below.

Procedure of simplex MethodStep 1: Formulate the LP problem.Step 2: Introduce slack /auxiliary variables.if constraint type is ≤ introduce + S if constraint type is ≥introduce – S + a and if constraint type is = introduce aStep 3: Find the initial basic solution.Step 4: Establish a simplex table and enter all variable coefficients. If the objective function is maximization, enter the opposite sign co-efficient and if minimization, enter without changing the sign.Step 5: Take the most negative coefficient in the objective function, Zj to identify the key column (the corresponding variable is the entering variable of the next iteration table).Step 6: Find the ratio between the solution value and the coefficient of the key column. Enter the values in the minimum ratio column.

Cont….Step 7: Take the minimum positive value available in the minimum ratio column to identify the key row. (The corresponding variable is the leaving variable of the table).Step 8: The intersection element of the key column and key row is the pivotal element.Step 9: Construct the next iteration table by eliminating the leavin variable and introducing the entering variable.Step 10: Convert the pivotal element as 1 in the next iteration table and compute the other elements in that row accordingly. This is the pivotal equation row (not key row).Step 11: Other elements in the key column must be made zero. For simplicity, form the equations as follows: Change the sign of the key column element, multiply with pivotal equation element and add the corresponding variable.

Cont….Step 12: Check the values of objective function. If there are negative values, the solution is not an optimal one; go to step 5. Else, if all the values are positive, optimality is reached. Non-negativity for objective function value is not considered. Write down the values of x1, x2,……..xi and calculate the objective function for maximization or minimization.Note:If there are no x1, x2 variables in the final iteration table, the values of x1 and x2 are zero.(ii)Neglect the sign for objective function value in the final iteration table.

ExamplePrevious the packaging product mix problem is solved using simplex method.Maximize Z = 6x1 + 4x2Subject to constraints,2x1+3x2≤120 (Cutting machine) .....................(i)2x1+ x2≤ 60 (Pinning machine) ......................(ii)where x1, x2 ≥ 0Considering the constraint for cutting machine,2x1+ 3x2 ≤ 120To convert this inequality constraint into an equation, introduce a slack variable, S3 which represents the unused resources. Introducing the slack variable, we have the equation 2x1+ 3x2 + S3 = 120Similarly for pinning machine, the equation is2x1+ x2 + S4 = 60

Example cont…. If variables x1 and x2 are equated to zero, i.e., x1 = 0 and x2 = 0, then S3 = 120 S4 = 60This is the basic solution of the system, and variables S3 and S4 are known as Basic Variables, SB while x1 and x2 known as Non-Basic Variables. If all the variables are non negative, a basic feasible solution of a linear programming problem is called a Basic Feasible Solution.

Cont….Rewriting the constraints with slack variables gives us,Zmax = 6x1 + 4x2 + 0S3 + 0S4Subject to constraints,2x1 + 3x2 + S3 = 120 ....................(i)2x1 + x2 + S4 = 60 ....................(ii)where x1, x2 ≥ 0Which can shown in following simplex table form

Cont… If the objective of the given problem is a maximization one, enter the co-efficient of the objective function Zj with opposite sign as shown in table. Take the most negative coefficient of the objective function and that is the key column Kc. In this case, it is -6. Find the ratio between the solution value and the key column coefficient and enter it in the minimum ratio column. The intersecting coefficients of the key column and key row are called the pivotal element i.e. 2. The variable corresponding to the key column is the entering element of the next iteration table and the corresponding variable of the key row is the leaving element of the next iteration table (In other words, x1 replaces S4 in the next iteration table. Given indicates the key column, key row and the pivotal element.)

Cont..In the next iteration, enter the basic variables by eliminating the leaving variable (i.e., key row) and introducing the entering variable (i.e., key column).Make the pivotal element as 1 and enter the values of other elements in that row accordingly.In this case, convert the pivotal element value 2 as 1 in the next iteration table.For this, divide the pivotal element by 2. Similarly divide the other elements in that row by 2. The equation is S4/2.This row is called as Pivotal Equation Row Pe.The other co-efficients of the key column in iteration Table 5.4 must be made as zero in the iteration Table 5.5. For this, a solver, Q, is formed for easy calculation.

Cont…Solver, Q = SB + (–Kc * Pe)The equations for the variables in the iteration number 1 of table 8 are,For S3 Q = SB + (– Kc * Pe) = S3 + (–2x Pe) = S3 – 2Pe …………………………(i)For – Z , Q = SB + (– Kc * Pe) = – Z + ((– 6) * Pe) = – Z + 6Pe …………………………(ii)Using the equations (i) and (ii) the values of S3 and –Z for the values of Table 1 are found as shown in Table 5.4

Cont….Using these equations, enter the values of basic variables SBand objective function Z. Ifall the values in the objective function are non-negative, the solution is optimal.Here, we have one negative value – 1. Repeat the steps to find the key row and pivotal equation values for the iteration 2 and check for optimality.We get New Table as below:

Cont…. The solution is, x1= 15 corrugated boxes are to be produced and x2= 30 carton boxes are to be produced to yield a Profit, Zmax= Rs. 210.00

Cont….It can be solve in different kinds of software as Tora, EXCEL, Lindo, etc.In Tora As follows,Steps for shoving linear programming by graphic method using TorashoftwareStep 1 Start  Tora select linear programming 

Linear Programming

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Linear Programming