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LINEAR PROGRAMMING 1. What is Liner Programming?Every organisation faces the problem of allocation of resources. They Include men, machine, material, capital, information etc Most, of these decisions are made subject to constraints. For example, production from a factory is limited capacity constraints; organisations face working capital constraints and technical constraints.If it is long time till a decision is to be made then many constraints can be avoided by changing resources, i.e., add more capacities, borrow more funds, etc. However, if the available resources cannot be expanded optimal utilization of existing resources becomes very important task of organisation. Linear programming is one of the most versatile, popular and quantitative techniques. A linear programming model provides an efficient determining an optimal decision (or an optimal strategy or an optimal plan) from a large number of possible decisions. The optimal decision is one that meets a specified objective of management, subject to various constraints and restrictions.Linear programming is a powerful quantitative technique (or operational research technique) designs to solve allocation problem.The term ‘linear programming’ consists of the two words Linear & Programming. The word ‘Linear’ is used to describe the relationship between decision variables which are directly proportional. For example, doubling (or tripling) the production of a product will exactly double (or triple) the profit and required resources, then it is linear relationship. The word ‘programming’ means planning of activities in a manner that achieves some ‘optimal’ result with available resources. A programme is ‘optimal’ if it maximises or mininimises some measure or criterion of effectiveness such as profit, contribution (i.e. sales -variable cost), sales, cost. Thus, ‘Linear Programming’ indicates the planning of decision variables which are directly proportional, to achieve the ‘optimal’ result considering the limitations within which the problem is to be solved.2. Advantages of Linear Programming Helps the management to make efficient use of factors of productionProvides flexibility in analysing a variety of multi-dimensional problemsImproves the quality of decision making Helps in detecting bottlenecks in production processHelps in improving the skill and knowledge of the young executives by providing an insight and prospective in to the problem situations.Serves as a vary good tool for adjusting to meet changing situationsProvides information base from which decisions on allocation of scarce resources can be taken. 3. Assumptions of Linear Programming To solve a linear programming problem, the following assumptions are applied:Proportionality : The proportionality exists in the objective function and the constraints inequalities. If one unit of a product contributes Rs.10 towards profit, then the total contribution would be equal to 10 x ,where x is the number of units of the product. In other words, if the output is doubled, the profit would also be doubled. In effect, there is a constant returns to scale—and there is no economies of scale. Certainty : The available quantities of the resources and the consumption per unit of the products manufactured from these resources are known with certainty. All external factors are stationery and they remain so over the period. (Selling price, variable cost and resource requirement per unit of the product remain constant over the entire range of the output). Additivity: The objective function and constraint inequalities both, the total of all the activities is given by the sum total of each activity conducted separately. In other words, total units of a constrained resource used equals sum of the individual resource requirement of the products. The assumption implies that there is no interaction among the decision variables( Decision variable does not represents any by-product)Divisibility : The products can be produced in fractions and resources require in their manufacture too can be employed in fraction of a unit. 4. Limitations of Linear Programming Linear programming methods are applicable only when values of the constraints are known with certainty. However, in many real situations the value of the constraints is not known. LP problems assume linearity of objective and constraints. But, generally, in most of the business and industrial problems particularly production related problems, the relationships are non-linear. If such problems are solved by LP, the results are likely to be incorrect. Linear relationships permit variables to take non-integer (fractional) values. Often fraction of a unit can not be procured, produced and sold. Rounding off the solution to the nearest integer may not yield optimal solution.LP deals only with a single objective problem, whereas in real life situations, there may be more than one objective. LP methods are based on the assumption that constraints and inputs remain stationary (constant) over the period whereas in practice they change due to changes in internal and external conditions. LP assumes that all inputs are quantifiable which is not always true. There may be certain costs and benefits which & cannot be quantified. Goodwill, for example, can not be considered during problem formulation as it is non-quantifiable. LP solutions do not take into account the effect of lower cost due to quantity discounts, reduction in manufacturing time /cost due to external factoRs. LP is not possible to be applied (or is difficult to be applied) to problems having alternative solutions. LP methods provide solution that is optimal for the available input data. Any opportunities left out during formulation are ignored. 5. Formulation of a Linear Programming Problem Formulation of a linear programming problem means constructing a mathematical model from the given data Formulation involves following steps Study the available data to find the key decisions to be made (i.e. identify unknowns called decision variables). Assign symbols (x, y ..) to the quantities of the decision variables identified in above step Express the feasible alternatives mathematically in terms of variables. ( Feasible alternatives are the non-negative constraints which when expressed iii mathematical terms appear as x > = 0, y > = 0 )Identify the objective function quantitively and express it as a linear function of decision variables multiplied by their unit costs or profits contributions.Identify restrictions on availability (resources) or requirements (demands) and express them also as a linear equality or inequality in terms of decision variables. Express the objective function, constraints and non-negativity condition identified in the above steps in the linear programming format. Step Involved In the Formulation of LP ProblemThe steps involved in the formation of linear programming problem are as follows Step 1: Identify the Decision Variables of interest to the decision maker and exp them as x1, x2, x3 Step 2 : Ascertain the Objective of the decision maker whether he wants to minimise or to maximize. Step 3 : Ascertain the cost (in case of minimization problem) or the profit (in case of maximization problem) per unit of each of the decision variables. Step 4 : Ascertain the constraints representing the maximum availability or minimum commitment or equality and represent them as less than or equal to (< =) inequality or greater than or equal to (> =) type / inequality or equal to (=) equality respectively. Step 5 : Put non-negativity restriction as under: x1, x2 > = 0 (non-negativity restriction) Step 6 : Now formulate the LP problem as under: Maximize (or Minimize) Z = c1x1 + c2x2 Subject to constraints: -------------------------------(Maximum availability) -------------------------------(Minimum commitment) -------------------------------(Equality)x = Decision Variablesc = Constant representing per unit contributiona. = Constant representing exchange coefficients of the decision variable b. = Constant representing constraint requirement or availability.CONCEPTS : LINEAR PROGRAMMING Artificial Variable : Artificial variable is an external variable added (in addition to the surplus variable when constraints are of more than or equal to type) to obtain an initial feasible Solution to a LP problem. Artificial variables have no physical meaning and hence must be reduced to zero at optimality. This can be ensured by assigning very high cost to them. Generally, a value M is assigned to each artificial variable, where M represents a number higher than any finite number .For this reason, the method of solving the problem where artificial variables are involved is termed as The Big M Method.Basis : The set of basic variables which are not restricted to zero in the current basic solution and are listed in the column. The variables with non-zero positive values which form the basis are called basic variables and the remaining variables are called non-basic variables. Basic Variables : The set of variables that are in the solution having positive values & are listed in the product mix column. The variables that normally take non-zero values to obtain a solution. Basic Feasible Solution : A basic solution for which the values of all variables are non negative & correspond to a corner of the LP feasible region. Constraints: The constraints indicate limitations on the resources which are to be allocated among various decision variables. These resources may be production capacity, manpower, and time. A constraint represents mathematical equation of the limitations imposed by the problem characteristics. The constraints must be capable of being expressed in mathematical terms. Contribution (C1— Z1) Row : It is called index row or net evaluation row and which is used to determine whether or not the current solution is optimal. The row indicates the net profit or loss that will result unit of a variable indicated in the solution of a LP problem. Decision Variables : The decision variables refer to the economic or physical quantities which are competing with one another for sharing the given limited resources. The relationship among these variables must be linear under linear programming. The numerical values of decision variables indicate the solution of the linear programming problem For example the variables in a problem on product mix represent quantities of the products to be produced, in a media selection problem they represent advertising units of different advertising media, in a diet mix problem they represent the quantity of different foods etc. The essential requirements of the variables are:They should be inter-related in terms of consumption of resourcesThe relationships among the variables should be linear. Degeneracy: Under Simplex Method, degeneracy occurs, where there is a tie for the minimum positive replacement ratio for selecting outgoing variable. A condition that arises when there is a tie in the values used to determine which variable will enter the solution next. When the outgoing variable in a non-optimal solution is a degenerate variable, then the solution does not improve in terms of the objective function value. When the degenerate row is a key row then, theoretically, there is a possibility that we may obtain a sequence of tables (having same objective function value)that might bring us back to a table that has been considered earlier. It can lead to cycling back and forth between two non-optimal solutions when it happens, it implies that that the simplex algorithm has cycled and the system would again move along the same route and the cycle would be repeated forever.. Degenerate Solution exists when a basic variable acquires a zero value in basic solution. Mostly whenever there is a tie in the replacement ratios for determining outgoing variable, the next table would give a degenerate solutionDivisibility means that the variables are not restricted to integer values (decision variables are continuous and as such their fractional values are permitted in the solution). In a linear function, each variable appears in only one term and only to the first power. A linear function in two variables, when set to a constant, gives an equation whose graph in two dimensions produces a straight line. Dual Problem. Corresponding to a linear programming problem is another linear programming problem formulated from the parameters of the original problem. A new LP, derived from the primal, according to a set of transformation rules.Dual Simplex Method: The dual Simplex Method is very similar to the regular simplex method. The only difference lies in the criterion used for selecting a variable to enter the basis & to leave the basis. This is just reverse of what is done in simplex method. The main advantage of dual simplex method over usual simplex method is that we do not require any artificial variables in the dual simplex method. Hence a lot of labour is saved whenever this method is applicable to find out the solution.In simplex method the starting initial basic solution is feasible but not optimal while in the Dual simplex, the initial basic solution is optimal but not feasible.Differences between simplex & dual simplex method : The Dual simplex method is similar the standard simplex method except that in simplex the starting initial basic solution is Feasible but not optimal while in dual simplex it is infeasible but optimal or better than optimum. The Dual simplex method works towards Feasibility while Simplex method works towards OptimalityFeasible Decision : One that satisfies the non-negativity conditions and all the constraints. Graphically, the feasible decisions are in one-to-one correspondence with the points in the feasible region.Feasible / Initial feasible Solution : If a linear programming problem is a solution which satisfies all the constraints including the non-negativity constraints. A set of values of the decision variables which satisfies all the constraints and non-negativity restriction is called feasible solutioninfeasibility or Infeasible Problem : A linear programming problem is said to be infeasible if there is no solution that satisfies all the constraints. It represents a state of inconsistency in the set of constraints. Under the Simplex Method, the problem is said to have no feasible solution if at least one of the artificial variable remains in the final simplex table as basic variable with non-zero quantity.Iso-profit Line :In the graphical method, all points in the feasible region represent feasible decision alternatives but they are not all meeting our desired objectives. Some provide a greater profit contribution or lower total cost than others., In other words, we seek to obtain the feasible solution which optimizes. It therefore requires adding one more line to the graph which is called Iso-profit or constant profit line. A straight line representing all non-negative combinations of X1 and X2 for a particular profit level. As the name implies, all the points on an iso-profit line yield the same profit. Iso-profit Method: The objective function expression 1 ,000 x1 + 850 x 2 is also linear and can be plotted as a line on the graph. Consider the equation 1,000 x 1 + 8,50 x2 = p, where p stands for an arbitrary profit. By assigning different values to p, the equation 10,000 x1 + 8,500x2 = p will not graph one line but a family of lines depending upon the value of p and all these lines will be parallel to each other. These are called the Iso-profit Lines. For example, a profit of Rs. 85,00, 000 will be obtained for all pairs of variables x1 and x2 satisfying the equation 10,000 x1 + 8500 x2 = 85,00,000. For any other profit, the objective function line would be parallel to the line drawn. The line would be nearer to the origin for smaller profits and would be moving away from the origin for increasing values of profits, i.e., we go on pulling the Iso profit line by assigning the higher region. The extreme may be either a corner profit or a straight line. In the latter case, a side of the feasible region is coincidently parallel to the Iso-profit line. If it is a corner profit, this yields the optimal solution.Inequality : A mathematical expression containing a greater-than or equal-to relation or a less-than or equal-to relation used to indicate that the total consumption of a resource must be more than or less than some limiting valueKey column : The column with the largest positive index number and it indicates which variable will enter the solution next. Key row : The row with the smallest of the replacement ratios of the constraint rows. The replacement ratios are obtained by dividing element in the solution Column (quantity column) by the corresponding element in the key column. The key row indicates the variable that will leave the basis in order to make room for entering variable (as indicated by the key column) Key element: The element at the intersection of key row and key column. It is also known as Intersecting element (the element at the intersection of other than key row and key column) Linear Relationships : Linear programming deals with problems in which the objective function as well as constraints, can be expressed as linear mathematical functions. Linear relationship has two properties: Proportionality & divisibility.Multiple Optimal Solutions : This situation occurs when there can be infinite number of solutions possible for a given problem. Under graphical method, the existence of multiple solutions is indicated by a situation under which the objective function line coincides with one of the half planes generated by a constraint. In other words, where both the objective function line and one of constraint lines have the same slope. Under Simplex Method, the existence of multiple optimal solutions is indicated by a situation under which a non- basic variable in the final simplex table showing optimal solution to a problem , has a net zero contribution. In other words, if at least one of the non-basic variable in the (C1 - Z) row of the final simplex table has a zero value, it indicates that there is more than one optimal solution.Non-negativity restrictions : Essentially, the value of decision variables must be either zero or positive. Negative values of the decision variables imply negative production (i.e. the state of dismantling or stale of destruction). Since such a stale in a real life situation is not non-existent, decision variables must assume either zero or positive values. If X & Y are the decision variables, their non-negativity restrictions, shall be expressed as under: x < 0 & Y > 0 Objective Function : An objective function represents the mathematical equation of the major goals of the system in terms of unknown called decision variables. The objective function in linear programming is of optimization type- it can be maximizing profit function or minimizing Cost function. The objective function like constraints must be capable of being expressed in mathematical terms. Optimal solution A feasible solution which optimises the objective function is called optimal solution. The optimal solution thus provides the best feasible choice of values – that choice which yields the highest (in case of maximization) or lowest (in case of minimization) value of the objective function of the decision variables. The solution which does not have multiple options is called Unique Optimal SolutionOptimal Value : The optimal value of the objective function is the value of the objective function when evaluated at the optimal solution. Primal Problem : it is the original linear programming problem. Post-optimality (Sensitivity) Analysis of a linear programming problem : It is a study of the effect of changes of the profit or resource level on the solution. The study of how sensitive an optimal solution is to the model assumptions and to data changes. Uses of post-optimality analysis :Sensitivity analysis forms an integral part of solving LP models, in fact any OR models. The analysis renders the model dynamic characteristics that allow the analyst to check changes in the optimum solution resulting from future changes in the data of the model. It deals with the changes the optimal solution when the parameters of the problem are changed. The changes in the parameter include 1.Changes in the co-efficient of the objective functions 2.Changes in the right hand side of the constraints 3.Addition of new variable 4.Deletion of existing variable 5.Changes in technological co-efficient Proportionality implies that the relationships are directly proportional (if resource availability increases by some percentage, then the output shall also increase by the same percentage or if one unit of variable consumes X hours of capacity, two units shall consume ‘2x’ hours of capacity and three units shall consume ‘3 X’ hours of capacity)Range of Optimality : The range of values over which a basic variable’s coefficient can change without causing a change in the optimal solution mix.Redundant Constraint : Redundant constraint is a constraint which does not affect the feasible region.Shadow Price : Shadow price of a resource represents the amount by which the contribution could increase if one extra unit of the resource was made available. Expressed in other words, shadow price measures the effect on total contribution when a scarce resource rises or falls by one unit. Shadow prices are significant because they suggest the extra amount that would be just economical to pay to buy one extra unit of the scarce resource. Shadow prices emerge as a by-product in the optimal simplex tableau. Shadow prices appear in the index row of the final simplex tableau under the columns of the slack variables. General rules concerning shadow prices 1. A resource can be either abundant or scarce. A source is said to be scarce if its entire stock is consumed in the optimal solution. Similarly, a resource is said to be abundant if it is not consumed completely. Whether resource is scarce or abundant in a LP model, the inference can be drawn directly by studying the “product mix column” and the “solution column” of the optimal simplex tableau (i.e. by studying the stub portion of the optimal tableau.) 2. As a general rule, presence of the slack variable and positive slack means that the corresponding resource is not used completely, thus is a abundant whereas a zero slack (i.e. slack variable being absent in the product mix column of the optimal tableau) means that the entire amount of the resource is consumed in the manufacture and the corresponding resource is scarce. 3. Shadow prices do not change so long the scarce resource remains effective constraints on manufacture. This implies that the shadow prices are valid only within a certain range. That is if the availability of the scarce resource is increased beyond a certain limit, then the other constraints may become binding and as a result the shadow prices will change.Simplex method : An algorithm for solving LP problems starts with zero solution (i.e. zero value of decision variables), evaluates solution, effects changes to improve value of the objective function and continues doing so until an optimal solution obtained.Simplex tableau : A table used to keep track of the calculations made at each iteration of the simplex procedure and to provide for tableau revision.Slack variable: A variable used to convert a less than or equal to constraint into equality constraint by adding it to the left side of the constraint.Surplus variable : A variable used to convert a greater than or equal to constraint into equality by subtracting it from left side of the constraint.Unrestricted Variable : While converting primal statement into dual statement, m variables & n constraints in the primal would be n variables &m constraints in the dual. but when dual problem has more variables than the constraints in the primal, a variable is added to replace additional variables (i.e. r5 = r3 – r4) which is unrestricted in sign. It is because r3 & r4 are both non-negative & therefore their differences can be greater than, less than or equal to zero. Unbounded Solution : An unbounded solution of a linear programming problem is a situation where objective function is infinite. A linear programming problem is said to have unbounded solution if its solution can be made infinitely large without violating any of its constraints in the problem. Since there is no real applied problem which has infinite return, hence an unbounded solution always represents a problem that has been incorrectly formulated. Under the Simplex Method, an unbounded solution appears when there are no positive values of Replacement Ratio i.e. Replacement ratio values are either infinite or negative. In this case there is no outgoing variable.Unrestricted Variables The variables which can assume values like = , <, or >.The constraint which is = in primal gets converted in to dual assumes signs which are unrestricted & hence these variables are called Unrestricted Variables. ................
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