CERTIFICATE Certified that this is a bona fide report of the project entitled 'Linear Programming Problems' done by Mr VINOD under my supervision and guidance in the partial fulfillment for the internal assessment introduced in the degree semester course.
Place : Elthuruth Date :
Signature Name of Guide and Designation
ACKNOWLEDGEMENT I am deeply grateful to class teacher VINOD and other teachers in our Maths Department from whom I received comments and suggestions which have helped me to improve the collection of this project. I would like to express my appreciation to the group members who took my project and helped me in improving the earlier versions of the manuscript. I would like to place on record my sincere appreciation to vinod for his painstaking efforts in typing the manuscript in a record time. Last but not least, I acknowledge with thanks the sacrifices made by my dear friends and family on account of my involvement with the task of completing this project work.
INTRODUCTION Mathematics is the queen of science. In our daily life, planning is required on various occasions, especially when the resources are limited. Any planning is meant for attaining certain objectives. The best strategy is one that gives a maximum output from a minimum input. The objective which is in the form of output may be to get the maximum profit, minimum cost of production or minimum inventory cost with a limited input of raw material, manpower and machine capacity. Such problems are referred to as the problems of constrained optimization. Linear programming is a technique for determining an optimum schedule of interdependent activities in view of the available resources. Programming is just another word for 'planning' and refers to the process of determining a particular plan of action from amongst several alternatives. Linear programming applies to optimization models in which objective and constraint functions are strictly linear. The technique is used in a wide range of applications, including agriculture, industry, transportation, economics, health systems, behavioral and social sciences and the military. It also boasts efficient computational algorithms for problems with thousands of constraints and variables. Indeed, because of its tremendous computational efficiency, linear programming forms the backbone of the solution algorithms for other operative research models, including integer, stochastic and non-linear programming. The graphical solution provides insight into the development of the general algebraic simplex method. It also gives concrete ideas for the development and interpretation of sensitivity analysis in linear programming. Linear programming is a major innovation since World War II in the field of business decision making, particularly under conditions of certainty. The word 'linear' means the relationships handled are those represented by straight lines, i.e. the relationships are of the form y = a + bx and the word 'programming' means taking decisions systematically. Thus, linear programming is a decision making technique under given constraints on the assumption that the relationships amongst the variables representing different phenomena happen to be linear. A linear programming problem consists of three parts. First, there objective function which is to be either maximized or minimized. Second, there is a set of linear constraints which contains thee technical specifications of the problems in relation to the given resources or requirements. Third, there is a set of non negativity constraints since negative production has no physical counterpart. AIM
1. To find and know more about the importance and uses of 'linear programming'. 2. To formulate a linear programming problem and solve in simplex method and dual problem.
DATA COLLECTION Linear programming is a versatile mathematical technique in operations research and a plan of action to solve a given problem involving linearly related variables in order to achieve the laid down objective in the form of minimizing or maximizing the objective function under a given set of constraints
CHARACTERISTICS
Objectives can be expressed in a standard form viz. maximize/minimize z = f(x) where z is called the objective function. Constraints are capable of being expressed in the form of equality or inequality viz. f(x) = or ≤ or ≥ k, where k = constant and x ≥ 0.
Resources to be optimized are capable of being quantified in numerical terms.
The variables are linearly related to each other.
More than one solution exist, the objectives being to select the optimum solution.
The linear programming technique is based on simultaneous solutions of linear equations.
USES
There are many uses of L.P. It is not possible to list them all here. However L.P is very useful to find out the following:
Optimum product mix to maximize the profit.
Optimum schedule of orders to minimize the total cost.
Optimum media-mix to get maximum advertisement effect.
Optimum schedule of supplies from warehouses to minimize mi nimize transportation costs.
Optimum line balancing to have minimum idling time.
Optimum allocation of capital to obtain maximum R.O.I
Optimum allocation of jobs between machines for maximum utilization of machines. Optimum assignments of jobs between workers to have maximum labor productivity.
Optimum staffing in hotels, police stations and hospitals to maximize efficiency.
Optimum number of crew in buses and trains to have minimum operating costs.
Optimum facilities in telephone exchange exchange to have minimum break downs.
The above list is not an exhaustive one; only an illustration.
ADVANTAGES
Provide the best allocation of available resources.
meet overall objectives of the management.
Assist management to take proper decisions.
Provide clarity of thought and better appreciation of problem.
Improve objectivity of assessment of the situation.
Put across our view points more successfully by logical argument supported by scientific methods.
PRINCIPLES
Following principles are assumed in L.P.P Proportionality: There exist proportional relationships between objectives and constraints. Additivity: Total resources are equal to the sum of the resources used in individual activities. Divisibility: Solution need not be a whole number viz decision variable can be in fractional form. Certainty: Coefficients of objective function and constraints are known constants and do not change viz parameters remain unaltered. Finiteness: Activities and constraints are finite in number. Optimality: The ultimate objective is to obtain an optimum solution viz 'maximization' or 'minimization'.
DEFINITION OF TERMS
a) Basic solution: There are instances where number of unknowns (p) are more than the number of linear equations (q) available. In such cases we assign zero values to all surplus unknowns. There will be (p-q) such unknowns. With these values we solve 'q' equations and get values of 'q' unknowns. Such solutions are called Basic Solutions. b) Basic variables: The variables whose value is obtained from the basic solution is called basic variables c) Non-basic variables : The variables whose value are assumed as zero in basic solutions are called non-basic variables.
d) Solution: A solution to a L.P.P is the set of values of the variables which satisfies the set of constraints for the problem. e) Feasible solution: A feasible solution to a L.P.P the set of values of variables which satisfies the set of constraints as well as the non-negative constraints constraints of the problem. f) Basic feasible solution: A feasible solution to a L.P.P in which the vectors associated with the non-zero variables are linearly independent is called basic feasible solution Note: Linearly independent: independent: variables x 1 , x2 , x3......... are said to be linearly independent if k 1 x1+k 2 x2+.........+k n xn=0, implying k 1=0, k 2=0,........ g) Optimum (optimal) solution : A feasible solution of a L.P.P is said to be the the optimum solution if it also optimizes the objective function of the problem. h) Slack variables: Linear equations are solved through equality form of equations. Normally, constraints constraints are given in the "less than or equal" (≤) form. In such cases, we add appropriate variables to make it an "equality" (=) equation. These variables added to the constraints to make it an equality equation in L.P.P is called stack variables and is often denoted by the letter 'S'. Eg: 2x1 + 3x2 ≤ 500 2x1 + 3x2 + S1 = 500, where S1 is the slack variable i) Surplus variables: Sometime, constraints are given in the "more than or equal" (≥) form. In such cases we subtract an approximate variable to make it into "equality" (=) form. Hence variables subtracted from the constraints to make it an equality equation in L.P.P is called surplus variables and often denoted by the letter 's'. Eg: 3x1 + 4x2 ≥ 100 3x1 + 4x2 - S2 = 0, where S2 is the surplus variable. j) Artificial variable : Artificial variables are fictitious variables. These are introduced to help computation and solution of equations in L.P.P. There are used when constraints are given in (≥) "greater than equal" form. As discussed, discussed, surplus variables are subtracted in such cases to convert inequality to equality form . In certain cases, even after introducing surplus variables, the simplex tableau may not contain an 'Identity matrix' or unit vector. Thus, in a L.P.P artificial variables are introduced in order to get a unit vector in the simplex tableau to get feasible solution. Normally, artificial variables are represented by the letter 'A'. k) Big-M-method: Problems where artificial variables are introduced can be solved by two methods viz. Big-M-method and Two phase method. Big-M-method is modified simplex method for solving L.P.P when high penalty cost (or profit) has been assigned to the artificial variable in the objective function. This method is applicable for minimizing and maximizing problems.
l) Two Phase method : L.P.P where artificial variables are added can be solved by two phase method. This is a modified simplex method. Here the solution takes place in two phases as follows: foll ows: Phase I - Basic Feasible solution: Here, simplex method is applied to a specially constricted L.P.P called Auxiliary L.P.P and obtain basic feasible solution. Phase II - Optimum Basic solution : From basic feasible solution, obtain optimum feasible solution. m) Simplex Tableau : This is a table prepared pr epared to show and enter the values obtained for basic variables at each stage of Iteration. This is the derived values at each stage of calculation. Optimal solution An optimal solution of a linear programming problem is the set of r eal values of the decision variables which satisfy the constraints including the non-negativity conditions, if any and at the same time optimize the objective function. A vector (x1, x2... xn) which satisfies the constraints A x ≤ or ≥ b only is called a solution. And a solution which satisfies all the constraints including the nonnegativity condition is called a feasible solution. The set of all feasible solutions is called feasible space. Integer Programming A L.P.P in which solution requires integers is called an integer programming problem. A mathematical programming in which the objective fn is quadratic but all the constraints are linear un the decision variable is called a quadratic programming. 2
2
eg: Max z = x 1 + x2 Subject to 2x1 + x2 ≤ 6 7x1 + 8x2 ≤ 28 x1, x2 ≥ 0
Graphical L P solution
The graphical procedure includes two steps : 1. Determination of the solution space that defines all feasible solutions of the model. 2. Determination of the optimum solution from among all the feasible points in the solution space. The proper definition of the decision variables is an essential first step in the development of the model. Once done, the task of constructing the objective function and the constraints is more straight strai ght forward.
FORMATION OF MATHEMATICAL MODEL OF L.P.P
There are three forms :
General form of L.P.P
Canonical form of L.P.P
Standard form of L.P.P
These are written in 'statement form' or in 'matrix' form as explained in subsequent paragraphs. General form of L.P.P
a) Statement form: This is given as follows "Find the values of x 1, x2... xn which optimize z = c 1x1 + c2x2 + ... + c nxn subject to : a11x1 + a12x2 + ... + a 1nxn ≤ (or = or ≥) b1 a21x1 + a22x2 + ... + a 2nxn ≤ (or = or ≥) b2 am1x1 + am2x2 + ... + a mnxn ≤ (or = or ≥) bm x1, x2,... xn ≥ 0 where all the coefficients (c j, aij, bi) are constants and x j's are variables. (i = 1,2,... m) (j = 1,2,... n ) " b) Matrix form of general L.P.P. This is stated as follows "Find the values of x 1, x2, ... xn to maximize: z = c 1x1 + c2x2 + ... +cnxn n Let z be a linear function on a R defined by i. z = c1x1 + c2x2 + ... cnxn where c j are constants. Let a ij be m*n matrix and let { b 1, b2, ... bm } be set of constraints such that ii. a11x1 + a12x2 + ... + a 1nxn ≤ (or = or ≥) b 1 a21x1 + a22x2 + ... + a 2nxn ≤ (or = or ≥) b 2 am1x1 + am2x2 + ... + a mnxn ≤ (or = or ≥) b m And let j = 1,2,... n iii. x j ≥ 0 The problem of determining an n-tuple (x 1, x 2,... xn) which make z a minimum or a maximum is i s called 'General linear programming problem'. Canonical Form of L.P.P
a) Statement form: This form is given as follows: "Maximize z=c1x1 + c2x2 +....... + cnxn subject to constraints a i1x1 + ai2x2 +........+ ainxn ≤ bi ;
(i= 1,2,....,m) x1x2......xn ≥ 0 "
Characteristics of canonical form : 1. Objective function is of the "maximization" type. Note: minimization minimization of function function f(x) is equivalent equivalent to maximization maximization of function {-f(x)} . . . Minimize f(x) = Maximize {-f(x)} 2. All constraints are of the type "less than or equal to" viz "≤" except the non-negative restrictions. Note: An inequality of more than (≥) can be replaced by less than tha n (≤) type by multiplying both sides by -1 and vice versa. eg: 2x1 + 3x2 ≥ 100 can be written as -2x1-3x2 ≤ -100 3. All variables are non-negative viz x j ≥ 0
b) Canonical form of L.P.P with matrix notations: " Maximize Z = CX , subject to the constraints AX ≤ b X≥0 Where X= (x1 ,x2 ,.......,xn); C= (c1 ,c2 ,......,cn) T b = (b1 ,b2,..... ,bm) ; A= (aij) where i= 1, 2,..., m
j= 1, 2,...., n "
The Standard Form of L.P.P
a)
Statement form " Maximize Z= c1x1 + c2x2 +....+ cnxn Subject to the constraints a i1x1 + ai2x2 +.....+ a inxn = bi (i= 1, 2,...., m) x1x2....xn ≥ 0 " Characteristics of Standard form 1. Objective function is of maximization type. 2. All constraints are expressed in the function of equality form except the restrictions. 3. All variables are non- negative. constraints given in the form of "less than or equal" (≤) can be converted to the equality form by adding "slack" variables. Similarly, those given in "more than or equal" (≥) form can be converted to t he he equality form by subtracting "surplus" variables.
Note:
b) Standard form of L.P.P in matrix notations: " Maximize Z = CX subject to the constraints AX =b b ≥0 and X ≥0 where X = (x1, x2, ..... xn) ; C = (c1, c2, ...., cn ) T b = (b1, b2,....., bm) ; A= (aij) i = 1, 2, ....., m ; j = 1, 2, ....., n " Note: coefficients of slack and surplus variables in objective function are always assumed to be zero.
BREIF HISTORICAL SKETCH Programming problems first rise in economics, where the optimal allocation of resources has long long been of interest to economists. economists. More specifically, however, programming problems seem to be a direct outgrowth of the work done by b y a number of individuals in the 1930's. One outstanding theoretical model developed then was Von Newmann's linear model of an expanding economy, which was part of the efforts of a number of Austrian and German economists and mathematicians who were studying the generalization of wairasian equilibrium models of an economy. A more practical approach was made by Leontief, who developed input-output models of the economy. His work was concerned with determining how much various industries would have to produce to meet a specified bill of consumer demands. Input-output models did not actually involve any optimization; instead they required the solution of a system of simultaneous linear equations. During World War II, a group under the direction of Marshall K. Wood worked on allocating problems for the United States Air Force. Generalization of Leontief type models were developed to allocate resources in such a way as to maximize or minimize some linear objective function. George B. Dantzig was a member of the Air Force group ; he formulated the general linear programming problem and devised the simplex method of solution in 1947. His work was not generally available until 1951, when the Cowles-commission Monograph was published. After 1951, progress in the theoretical development and in practical applications of linear programming was rapid. Important theoretical contributions were made by David Gale, H. W. Kuhn and A. W. Tucker, who had a major share in developing the theory of duality in linear programming. A. Charnce, who also did some important theoretical work, and W.W. Cooper took the lead in encouraging industrial application of linear programming. Economic Interpretation of Duality
1. At the optimum, objective value in the primal
=
objective value in the dual
2. At any Iteration of the primal, Objective equation coefficient of the variable x j
=
left side of corresponding dual constraint
_
right side of corresponding corresponding dual constraint
These two results lead to important economic interpretations of duality and the dual variables. To present these interpretations formally, the general primal and dual problems are presented below Primal
Maximize Z =
c jx j
subject to aij x j = bi , i= 1, 2, ...., m x j ≥ 0 , j= 1, 2, ....., n
Dual Minimize Z= biyi
subject to aijyi ≥ c j , yi ≥ 0 ,
j= 1, 2,...., n i= 1, 2, ...., m
We can think of the primal model in this manner. the coefficient c j represents the profit per unit output of activity j. The available amount of resources i, b i, is allocated at the rate of a ij units of resource i per unit output of activity j. Problem of the linear- programming type had been formulated and solved before the pioneering work of Dantzig. In 1941, Hitchcock formulated and solved transportation problems which was independently solved by the Koopmans in 1947. In 1942, Kantorovich (Russian) also formulated the transportation problem, but did not solve it. The economist Stigler worked out a minimum-cost diet in 1945. Although this problem can be formulated as a linear programming problem, Stigler did not use this technique. It was not until Dantzig's work, however, that the general linear programming problem was formulated as such, and a method devised for solving it.
APPLICATION OF LINEAR PROGRAMMING The primary reason for using linear programming methodology is to ensure that limited resources are utilized to the fullest extent without any waste and that utilization is made in such a way that the outcomes are expected to be the best possible. Some of the examples of linear programming are: a) A production manager planning to produce various products with the given resources of raw materials, man-hours, and machine-time for each product must determine how many products and quantities of each product to produce so as to maximize the total profit. b) An investor has a limited capital to invest in a number of securities such as stocks and bonds. He can use linear programming approach to establish a portfolio of stocks and bonds so as to maximize return at a given level of risk. c) A marketing manager has at his disposal a budget for advertisement in such media as newspapers, magazines, magazines, radio and television. t elevision. The manager would like to determine the extent of media mix which would maximize the advertising effectiveness. d) A Farm has inventories of a number of items stored in warehouses located located in different parts of the country that are intended to serve various markets. Within the constraints of the demand for the products and location of markets, the company would like to determine which warehouse should ship which product and how much of it to each market so that the total t otal cost of shipment is minimized. e) Linear programming is also used in production smoothing. A manufacturer has to determine the best production plan and inventory policy for future demands which are subject to seasonal and cyclical fluctuations. The objective here is to minimize the total total production and inventory cost. f) A marketing manager wants to assign territories to be covered by salespersons. The objective is to determine the t he shortest route for each salesperson starting from his base, visiting clients in various places and then returning to the original point of departure. Linear programming can be used to determine the shortest route. g) In the area of personnel management, management, similar to the travelling salesperson problem, the problem of assigning a given number of personnel to different jobs can be solved by this technique. The objective here is to minimize the total time taken to complete all jobs. h) Another problem in the area of personnel management management is the problem of determining the equitable salaries and sales-incentive compensation. Linear programming has been used successfully in such problems.
BASIC REQUIREMENTS OF A LINEAR PROGRAMMING MODEL a) The system under consideration can be described in terms of a series of activities and outcomes. These activities (variables) must be competing with other variables for limited resources and the relationships among these variables must be linear and the variables must be quantifiable. b) The outcomes of all activities are known with certainty. c) A well defined objective function exists which can be used to evaluate diff erent diff erent outcomes. The objective function should be expressed as a linear function of the decision variables. The purpose is to optimize the objective function which may be maximization of profits or minimization of costs, and so on. d) The resources which are to be allocated among various activities must be finite and limited. These resources may be capital, production capacity, manpower, time etc. e) There must not be just a single course of action but a number of feasible courses of action open to the decision maker, one of which would give the best result. f) All variables must assume non-negative values and be continuous so that fractional value of the variables are permissible for the purpose of obtaining an optimal solution.
L.P.P- Special cases We have seen a L.P.P is given in the form Maximize Z= CX Subject to constraints AX = b; X ≥ 0 -1
And the solution obtained is X= A b Thus we see that simplex solution obtained in the Tableau which is declared as optimum basic feasible solution (O.B.F.S) contains inverse matrix A. case of unbounded solutions : when L.P.P L.P.P does not give finite values of variable X, variabl e X can take very viz x1, x2, ...∞, such solutions are called UNBOUNDED. Here variable high values without violating the conditions of the constraints.
In such cases the final Tableau shows all Ratios viz R values are negative so that no minimum ratio condition can be applied.
Establish the UNBOUND condition. Maximize
Z= 2x1 + x2
Subject to the constraints x 1-x2 ≤ 10 2x1-x2 ≤ 40 x1, x2 ≥ 0 case of more than one optimum solution:
This is the case where L.P.P gives solutions which are optimum but not UN IQUE. This means, more than one optimum solution is possible. Case of Degeneracy: A basic basic feasible solution of a L.P.P is said to degenerate degenerate if at least one of the basic variables is zero. Types of Degeneracy
a) First Iteration: Degeneracy can occur occur right in the first (initial tableau). tableau). This normally happens when the number of constraints are less than number of variables in the objective function. Problem can be overcome by trial and error method. b) Subsequent Iteration: Degeneracy can also occur occur in subsequent subsequent iteration. This is due to the fact that minimum Ratio values are the same for two rows. This will make choice difficult in selecting the 'Replacing Row'. Case of cycling:
We have seen that when degeneracy exists, replacement of vector in the BASIS does not improve objective function. Another difficulty encountered in degeneracy is that the simplex Tableau gets repeated without getting optimum solution. Such occurrences in L.P.P are called CYCLING. Fortunately, such L.P.Ps are very rare. Also, in such cases, certain techniques are developed to prevent cycling and reach optimum solution. Case of duality :
Every L.P.P is associated with another L.P.P which is called the DUAL of the original L.P.P. The original L.P.P is called the PRIMAL. If the DUAL is stated as a given problem, PRIMAL becomes its DUAL and vice-versa. The solution of one contains the solution of the other. In other words, when optimum solution of PRIMAL is known, the solution of DUAL is also obtained from the very same optimum tableau.
USES OF DUAL CONCEPT
There are many uses of this concept. However, one of the very important use is solve difficult L.P.P. Many times, a given L.P.P is complicated having large number of constraints. In such cases, one method is to design its dual which invariably will have less number of constraints. Dual is now subjected to solution by simplex method.
Solution of a L.P.P
In general we use the following two methods for the solution if a linear programming problem. I.
II.
Geometrical (or graphical) method : If the objective function z is a function of two variables only; then the problem can be solved by the graphical method. A problem involving three variables can also be solved graphically, but with complicated procedure. Simplex method : This is the most powerful tool of the linear programming as any problem can be solved by this method. Also, this method is an algebraic procedure which progressively approaches the optimal solution. The procedure is straight forward and requires only time and patience to execute manually. Nowadays, computational methods (using computers), are available for solving such problems.
Geometrical (or Graphical) method for solving a L.P.P
If the L.P.P is two variable problem, it can be solved graphically. The steps required for solving a L.P.P by graphic method are : 1. Formulate the problem into a L.P.P 2. Each inequality in the constraint may be written as equality. 3. Draw straight lines corresponding to the equations obtained in step 2. So there will be as many straight lines as there are equations. 4. Identify the feasible region. Feasible region is the area which satisfies all constraints simultaneously. simultaneously. 5. The permissible region or feasible region is a many sided figure (a polygon). The corner points of the figure are to be located and their co-ordinates (ie. x 1 and x2 values) are to be measured. 6. Calculate the value of the objective function Z at each corner point. 7. The solution is given by the co-ordinates of that corner point which optimizes the objective function.
Dual Simplex Method
Any primal iteration z j - c j, the objective equation coefficient of x j, equals the difference between the left and right sides of the associated dual constraint. When, in the case of maximization, the primal iteration is not optimal, z j - c j < 0 for t least one variable. Only at the optimum do we have z j -c j ≥ 0 for all j. Looking at this condition from the stand point of duality, we have z j - c j =
aijyi - c j
Thus, when z j - c j < 0, aijyi < c j , which means that the dual is infeasible when wh en the primal is non-optimal. On the other hand, when z j - c j ≥ 0, aijyi ≥ c j , which means that the dual becomes feasible when the primal reaches optimality.
ALLOCATION PROBLEM A manufacturing firm has recently discontinued production of a certain product due to unfavourable market conditions resulting in considerable excess production capacity. The firm is planning to utilize this spare capacity by increasing the production of the remaining one or more of the existing three products. The currently available capacities are : Milling capacity
:
300 machine hours/day
Lathe capacity
:
225 machine hours/day
Grinder capacity
:
100 machine hours/day
The numbers of machine hours required for each of the products are: Machine type
Machine hours required Product A Product B Product C
Milling
12
3
4
Lathe
6
4
1
Grinding
3
1
2
The net profits realized from each of the three products are: Rs: 20, Rs. 9 and Rs. 10 from A, B and C respectively. The production manager desires to allocate the available capacities amongst the three products for a maximum m aximum profit. Let x1, x2 and x 3 be the quantity of products A, B and C produced within the available capacities to maximize the profit. From the above problem given details, we can formulate the problem as a linear programming problem as, Maximize, z = 20 x 1 + 9 x2 + 10 x3 Subject to
12 x1 + 3 x2 + 4 x3 6 x1 + 4 x2 + x3 3 x1 + x2 + 2 x3
x1, x2, x3
300
225 100 0
By using simplex method
By introducing slack slack variables, the problem can be restated restated as : Maximize
20x1 + 9x2 + 10x3 + 0x4 + 0 x5 + 0 x6
Subject to
12x1 + 3x2 + 4x3 + x4 = 300 6x1 + 4x2 + x3 + x5 = 225 3x1 + x2 + 2x3 +x6 = 100
Where x1, x2 and x3 are structural variable and x 4, x5, and x 6 are stock variables variables In order to simplify handling the educations in the problem, they can be placed in a tabular form, known as a tableau. The initial tableau is given below Simplex Tableau I: initial step
Ci
Basis
P1
P2
P3
P4
P5
P6
P0
0 0 0
x4 x5 x6
12 6 3
3 4 1
4 1 2
1 0 0
0 1 0
0 0 1
300 255 100
Body Matrix
c j z j = c j - z j
20 0 20
9 0 9
Qi =
25 37.5 33.3
Identity Matrix
10 0 10
0 0 0
0 0 0
0 0 0
Starting with the left hand column in the above tableau, the C1 column contains the contributions per unit for the basic variables x4, x5 and x6. The zero indicates that the contribution per unit is zero. The reasoning is that no profits are made on unused raw materials or labour. The second column, the basis, contains the variables in the solution which are used to determine the total contribution. In the initial solution, no th products are being manufactured. This is represented in the c j row for the column under P0. Since no units are manufactured, the first solution is, x1 = 0
x4 = 300
x2 = 0
x5 = 225
x3 = 0
x6 = 100
The body matrix consists of the coefficients for the real product (structural) variables in the first simplex tableau represents the coefficients of the slack variables that have been included to the t he original inequalities to make them equations.
The row c j contains coefficients of respective variables in the objective function. The last two rows of the first simplex tableau are used to determine whether or not the solution can be improved. The zero values in the x j row are the amounts by which contribution would be reduced if one unit of the variables x 1, x 2, x 3, x 4, x 5 and x6 was added to the product mix. Another way of defining the variables of the z j row is the contribution lost per unit of the variables.
Simplex Tableau II
Ci
Basis
P1
P2
P3
20 0 0
x4 x5 x6 c j z j = c j - z j
1 0 0 20 20 0
¼ 5 / 2 ¼ 9 5 4
1
/ 3 -1 1 10 20 / 3 10 / 3
P4
P6
P0
0 1 0 0 0 0
0 0 1 0 0 0
25 25 75 25 25
100 30 100
P5
P6
P0
Qi =
1
/ 12 12 -1 / 2 -1 / 4
0 / 12 12 -20 / 12 12 20
Simplex Tableau III
Ci
Basis
P1
P2
20
x4
1
0
13
7
-1
/ 10 10
0
70
-2
-1
2
0
30
11
-1
-1
/ 10 10
1
70
0 / 5 -8 / 5
0 0 0
9
x5
0
0
x6
0
1 0
20 20 0
9 9 0
c j z j = c j - z j
P3
P4
/ 30 30
/ 30 30
/ 5
/ 5
/ 10 10
10 / 15 15 74 / 15 15 76
/ 5
/ 5
0 / 15 15 -43 / 15 15 43
8
/ 4 / 4
Simplex Tableau IV
Ci
Basis
P1
P2
P3
20
x4
1
0
0
9
x5
0
1
0
0
x6
0
0
1
c j
20
9
10
z j
20
9
10
0
0
0
= c j - z j
P4
P5
P6
-2
0
-3
4
0
-2
-1
/ 11 11
1
/ 11 11 / 11 11 0
Qi =
P5
/ 33 33
/ 11 11 0
0
P0
Qi =
In tableau IV as all
, the solution has reached an optimum level. The
computational procedure comes to an end. Hence, the optimal solution is,
x1 = z
= = = =
x2 =
x3 =
=
= 698.4848 Z = 698.4848 By using Dual Problem
The dual of the above problem is, Minimize z* = 300 y 1 + 225 y2 + 100 y3 Subject to,
12 y1 + 6 y2 + 3 y3 ≥ 20 3 y1 + 4 y2 + y3 ≥ 9 4 y1 + y2 + 2 y3 ≥ 10 y1, y2, y3 ≥ 0
Introducing surplus variables and artificial variable, the above problem can be restated as follows: Minimize z* = 300 y 1 + 225 y2 + 100 y3 + 0 y4 + 0 y5 + 0 y6 + Subject to,
– y4 + y7 ≥ 20 12 y1 + 6 y2 + 3 y3 – y – y5 + y8 ≥ 9 3 y1 + 4 y2 + y3 – y – y6 + y9 ≥ 10 4 y1 + y2 + 2 y3 – y y1, y2, y3 …. Y9 ≥ 0
y7 +
y8 +
y9
Ci
Basis
P1
P2
P3
P4
P5
P6
P7
P8
P9
P0
Qi
y7
12
6
3
-1
0
0
1
0
0
20
5
y8
3
4
1
0
-1
0
0
1
0
9
y9
4
1
2
0
0
-1
0
0
1
10
5
300
225
100
0
0
0
19
11
5
/ 3
20
4
16
c j z j
300
300
z j-c j
19 -300
11 -225
6 - 100
y1
1
½
-¼
- 1 / 12 12
0
0
1
/ 12 12
0
0
y8
0
5
/ 2
¼
¼
-1
0
-¼
1
0
y9
0
-1
1
1
/ 3
0
-1
-1 / 3
0
1
c j
300
225
0
0
0
z j
300
150-
75+
-25
25-
z j-c j
0
-75-
-25+
-25
25-
0
0
y1
1
¾
0
¼
1
0
-¼
5
¼ -1 0
1
- / 6 -1 / 3
-¼ 1
19
11
-1 / 6 1
z j
300
125+
100
-
+
-25+
+
z j-c j
0
-100+
0
-
+
-25+
-
300
y1
1
0
0
-
3
/ 11
2
225
y8
0
1
0
4
-
1
100
y9
0
0
1
15
-
- 10
c j
300
225
100
z j
300
225
100
-
0
0
0
-
z j-c j
/ 6 / 3 0
1
-1 0 0
/ 6
0 0 300
0 1 100
0
y8 y9 c j
100
/ 4 -1 225
100
/ 33
/ 33 0
/ 11 / 11 / 11
0 -
0
-
-
-
1 0
/ 33
-
- 13
/ 11
-
/ 3
/ 6 / 6 / 3
10
3 / 2
/ 3
10
/ 3
10
/ 9
38
/ 33 33 -10 / 3
25-
0
7
10
/ 3
25-
-
-
-
4
-
38
4
10
148
/ 11 / 11
/ 33
/ 11
/ 33
-
Since all si ≤ 0, the optimal solution has been obtained in the above tableau. The optimal solution is y1=
, y2 =
and y3 =
Minimize z* =
=
= 698.4848
Z*
= 698.4848
=
=
Fields where Linear Programming can be used The problem for which linear programming provides a solution may be stated as : Maximize (or Minimize) some dependent variable which is a function of several independent variables, when the independent variables are subjected to various restrictions. The dependent variable is usually some economic objective such as profits, production, costs, workweeks, tonnage to be shipped etc. More profits are genera lly preferred to less profits and lower costs are preferred to higher costs. Hence, it is appropriate to represent either maximization or minimization of the dependent variables as one of the firm's objective. Linear programming is usually concerned with such objectives under given constraints with linearity assumptions. In fact, linear programming is powerful enough to take in its strides a wide range of business applications. The applications of L.P.P are numerous and are increasing every day. L.P is extensively used in solving resource allocation problems. Production planning and scheduling, transportation, sales and advertising, financial planning, portfolio analysis, corporate planning etc. are some of its most fertile application areas. More specifically, linear programming has been successfully applied in the following fields: a) Agricultural applications: Linear programming can be applied in farm management problems so far as it relates to the allocation of resources such as acreage, labor, water supply or working capital in such a way as to maximize net revenue. b) Contract awards : Evaluation of enders by recourse to linear programming guarantees that the awards are made in the cheapest way. c) Industrial applications: Applications of linear programming in business and industry are of the most diverse kind. Transportation problems concerning cost minimization can be solved using this technique. Te technique can be adopted in solving problems of production (product-mix) and inventory control as well. Thus, Linear Programming is the most widely used technique of decision making in business and industry in modern times in various fields as stated above.
Limitations 1) There is no guarantee that linear programming will give integer valued equations. For instance, solution may result in producing 8.291 cars. In such a situation, the manager will examine the possibility of producing 8 as well as 9 cars and will take a decision which ensures higher profits subject to given constraints. Thus, rounding can give reasonably good solutions in many cases but in some situations we will get only a poor answer even by rounding. Then, integer programming techniques alone can handle such cases. 2) Under linear programming approach, uncertainty is not allowed. The linear programming model operates only when values for costs, constraints etc. are known but in real life such factors may be unknown. 3) The assumption of linearity is another formidable limitation of linear programming. The objective functions and the constraint functions in the L.P model are all linear. We are thus dealing with a system that has constant returns to scale. In many situations, the input-output rate for an activity varies with the activity level. The constraints in real life concerning business and industrial problems are not linearly related to the variables, in most economic situations, sooner or later, the law of diminishing marginal returns begins to operate. In this context, it can, however be stated that non-linear programming techniques are available for dealing with such situations. 4) Linear programming will fail to give a solution if management have conflicting multiple goals. In L.P model, there is only one goal which is expressed in the objective function. Eg. maximizing the value of the profit function or minimizing he cost function, one should resort to Goal programming in situations involving multiple goals. All these limitations of linear programming indicate only one thing- that linear programming cannot be made use of in all business problems. Linear programming is not a panacea for all management and industrial problems. But for those problems where it can be applied, the linear programming is considered a ver useful and powerful tool.
ANALYSIS Linear programming is a resources allocation model that seeks the best allocation of limited resources to a number of competing activities. L.P has been applied with considerable success success to a multitude of practical problems. The suitability of the graphical L.P solution is limited to variable problems. However, the graphical method reveals the important result that for solving L.P problems it is only necessary to consider the corner (or extreme) points of the solution space. This result is the key point in the development of the simplex method, which is an algebraic procedure designed to solve the general L.P problem. Sensitivity analysis should be regarded as an integral part of solving any optimization problem. It gives the L.P solutions dynamic characteristics that are absolutely necessary for making sound decisions in a constantly changing decisionmaking environment. According to Ferguson and Sargent : "Linear programming is a technique for specifying how to use limited resources or capacities of a business to obtain a particular objective such as the least cost, the highest margin or the least time, when these resources have alternative uses. It is a technique that systemizes certain conditions, the process of selecting the most desirable course of action from a number of available courses of action, thereby giving the management the information for making a more effective decision about the resources under control." Since the objective of any organization is to make the best utilization of the given resources, linear programming provides powerful technique technique for effective utilization of these given resources under certain well-defined circumstances. For instance, an industrial process consists of a number of activities relating to the capital invested and the capital required for operational activities, products to be produced and marketed, raw materials to be used, machines to be utilized, products to be stored and consumed or a combination of the above activities. Some or all of these activities are interdependent and inter-related so that there are many ways where by these resources can be allocated to various competing demands. Linear programming helps the decision maker in arranging for such combination of resources which results in optimization of objectives. Linear programming method was first formulated by a Russian mathematician Shri. L.V. Kantorovich. Today, this method is being used in solving a wide range of practical business problems. The advent of electronic computers has further increased its applications to solve many other problems in industry. It is being considered as one of the most versatile management techniques. techniques.
The simplex method shows that a corner point is essentially indentified by a basic solution of the standard form of linear programming. The optimality and feasibility conditions of the simplex method that, starting from a feasible corner point (basic solution), the simplex method will move to an adjacent corner point which has the potential to improve the value of the objective function. The maximum number of iterations (basic solutions or corner points) until the optimum is reached is limited by n!/[(n-m)!m!] in an n-variable m-equation standard L.P model. The special case of alternative optima point to the desirable adoption of one optimum solution over another, even though both may have the same optimum objective value depending, for example, on the activity "mi x" that each solution offers. An unbounded optimum or a solution space, as well as a non-existing solution, points to the possibility that some irregularities exist in the original formulation of the model. As a result, the model must be checked. The optimum tableau offers more than just the optimum values of the variables. Additionally, it gives the status and worth (shadow prices) of the different resources. The sensitivity study shows the resources can be changed within certain limits while maintaining the same activity mix in the solution. Also, the marginal profits/costs can be changed within certain ranges without changing the optimum values of the variables.
The Four Steps Involved in the Simplex Method Step 1: Select the column with the highest plus value
Evaluation of the last row in the initial tableau represents the first step in the computational procedure for the above maximization problem. The final row is the net contribution that results firm adding one unit of variable (product) to the production. The C j+ C j row reveals that the largest positive value 20. A plus value indicates that a greater contribution can be made by the firm by the including this variable (product) in the production activity. A negative value would indicate the amount at by which contribution would decrease if one unit of the variable for that column were brought into the solution. The largest positive amount in the last row (indicated by as an inclusion variable) is selected as the optimum column, since we want to maximize the total contribution. When no more positive values remain in the c j – z j row and the values are zero or negative in a maximization problem, the total contribution is all its greatest value.
Step 2 : To determine the replaced (old) row.
Once the first simplex tableau has been constructed and the variable (key column) has been selected which contributes the most per unit, the second step is to determine which variable should be replaced. In other words, inspection of the key column – mix, replacing one of indicates that the variable x 1 should be included in the product – mix, the variable x 2, x5 or x6. To determine which variable will be replaced, divide the value in the P0 column by the corresponding coefficient in the optimum column. That is, calculate Qi =
– negative quantity ( Select the row with the smallest non – negative replaced (indicated by )
) as the row to be
Having selected the key column and there placed row, we can now work on an improved solution found in tableau II. The element common to the key column and the replaced row is said to be pivotal element (indicated by ) and the elements common to the key column and the remaining row are called intersectional elements. Step 3: Computation of values for the replacing (new)row.
In the next step, the first row (elements) to determine in the second tableau is the new x1 (replacing) row for x4 (replaced) row. The elements of the x 1 row are computed by dividing each value in the replaced row(x 4) by the pivotal element. These become the values for the first row x 1 in tableau . In our illustration, we divide the elements of x 4 row by 12 to get the new elements of x 1 row.
Step 4 : Calculate new values for the remaining (rows)
The fourth and the final step in the computational procedure is to calculate all new values for the remaining row(s). The formula for calculating these new elements of row is:
( )[ ] ∑ = (New Value)
For example, in our illustration, the first element (new) in the x 5 row is = 6 - [6
1] = o
Similarly, the other elements in the x 5 row are calculated. Having completed the computation of these values, we proceed to calculate the values for z j and = c j – z j. j = 1, 2, ……, n
The formula for calculation of z j is,
Where xij are the elements of the body and the unit matrix for example, in tableau I; z j = 20
1 + 0 + 0 = 20
These steps complete the computations involved for obtain the second tableau from the first one. We continue this process, when are less than or equal to zero. If all are less than or equal to zero, then the computational procedure has come to an end and the solution (basis) of the last tableau is the optimal solution. If one of the is positive, we shall have to compute the next tableau (in the same way as before: step 1 to 4) until all are less than or equal to zero in our above example, since c j – z j (= ) th corresponding to P1 column is large positive (4), we shall then compute the third, 4 th and 5 simplex tableau in the same fashion. We shall now consider solving a minimization problem using the simplex procedure. Procedure adopted to solve a minimization problem is the same as that of – c j, instead of c j – z – z j and the test for maximization, except for the last row. Here = z j – c optimality is the same as the one used for a maximization problem.
METHODOLOGY The method of this project is by collecting data as much as possible and analyzing it. By this analysis, we can understand that importance and use of Linear Programming Problem in mathematics. It is decided to observe and collect various books in our library. Net sources, books and instruments are my teaching aids. We can conclude the project by knowing the various purposes and brief sketch of linear programming.
CONCLUSION From this project we came to a conclusion that 'Linear programming' is like a vast ocean where many methods, advantages, uses, requirements etc. can be seen. Linear programming can be done in any sectors where there is less waste and more profit. By this, the production of anything is possible through the new methods of L.P. As we had collected many datas about L.P, we came to know more about this, their uses, advantages and requirements. Also, there are many different ways to find out the most suitable L.P. Also, we formulate an example for linear programming problem and done using the two methods simplex method and dual problem. And came to a conclusion that L.P is not just a technique but a planning the process of determining a particular plan of action from amongst several alternatives. Even there are limitations, L.P is a good technique, especially in the business sectors.
