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GRAPHICAL REPRESENTATION

GRAPHICAL REPRESENTATION. LP Model. DEFINITIONS. Solution – Values of Decision variables x j (j=1,2,…n) that satisfy the constraints of the LP problem are said to constitute solution to that problem.

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GRAPHICAL REPRESENTATION

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  1. GRAPHICAL REPRESENTATION LP Model

  2. DEFINITIONS • Solution – Values of Decision variables xj (j=1,2,…n) that satisfy the constraints of the LP problem are said to constitute solution to that problem. • Feasible Solution – Values of a decision variable xj (j=1,2,..n) that satisfy the constraints of the LP and non-negativity conditions simultaneous are said to constitute the feasible solution of LP problem GRAPHICAL SOLUTION

  3. DEFINITIONS • Basic Solution – For a set of m equations in n variables, a solution obtained by setting (n-m) variables equal to zero and for solving remaining m equations in m variables is called a basic solution. The (n-m) variables whose values do not appear in solution are called non-basic variables and remaining m variables are called basic variables. • Basic Feasible Solution – A feasible solution to an LP problem which is also the basic solution is called the basic feasible solution. GRAPHICAL SOLUTION

  4. DEFINITIONS • Degenerate – A basic feasible solution is called to degenerate if value of at least one basic variable is zero. • Non-degenerate – If values of all basic variables ‘m’ are non-zero and positive. • Optimum basic feasible solution – A basic feasible solution which optimises the objective function value of the given LP problem is called basic feasible solution (BFS) • Unbounded solution – A solution which can increase or decrease the value of objective function of the LP problem indefinitely is called an unbounded solution. GRAPHICAL SOLUTION

  5. GRAPHICAL SOLUTION • Graphical solution can be developed through • Extreme point enumeration method • Iso-profit (cost) function line approach • Graphical procedure includes basic two steps • The determination of the solution space that defines the feasible solutions that satisfy all the constraints of the model. • The determination of the optimum solution from among all the points in the feasible solution space. GRAPHICAL SOLUTION

  6. Graphical Solution • The collection of all feasible solutions to an LP problem constitutes a convex set whose extreme points correspond to basic feasible solution. • There are a finite number of basic feasible solutions within the feasible solution space. • If the convex set of the feasible solutions of the system of simultaneous equations Ax=b, x>0, is a convex polyhedron, then at least one of the extreme points gives an optimal solution. • If the optimal solution occurs at more than one extreme point, the value of the objective function will be the same for all convex combinations of these extreme points. GRAPHICAL SOLUTION

  7. Polyhedron Convex - FS Non-convex – Not an LP GRAPHICAL SOLUTION

  8. Problem 1 • Maximise Z = 15x1 + 10 x2 • subject to following constraints • 4x1 + 6x2  360 • 3x1 + 0x2  180 • 0x1 + 5x2  200 • x1, x2  0 • Treat x1 as horizontal axis and x2 as vertical axis • Solution Class room GRAPHICAL SOLUTION

  9. Problem 2 • Minimise 3x1 + 5x2 • Subject to following constraints • 5x1 + x2  10 • x1 + x2  6 • x1 + 4 x2  12 • x1, x2  0 GRAPHICAL SOLUTION

  10. Problem 3 (Assignment) • A manufacturer produces two different models; X & Y of the same product. Model X makes a contribution of Rs 50 per unit and model Y Rs 30 per unit. Raw materials r1 & r2 are required for production. At least 18 kg of r2 and 12 kg of r2 must be used daily. Also at most 34 hours of labor are to be utilised. A quantity of 2 kg of r1 is needed for X and 1 kg for Y. For each X & Y, 1 kg of r2 is required. It takes 3 hours to manufacture model X and 2 hours to manufacture model Y. How many units of each model should be produced to maximise profit? GRAPHICAL SOLUTION

  11. Isoprofit (Cost) Function Line approach • Identify the feasible regions and the extreme points of the region. • Draw the iso-profit line for any arbitrary but small value of objective function. • Move this line parallel to direction of the increasing (decreasing) objective function • Feasible extreme point for which the value of iso profit (cost) is largest (smallest) is the optimal solution. GRAPHICAL SOLUTION

  12. Exceptional Cases • Infeasible Solutions – If the constraints are not satisfied simultaneously, the model has no feasible solution. • Class room example. • Unbounded Solution – In some cases the values of variables may be increased indefinitely without violating any of the constraints, meaning the solution space is unbounded in at least one direction. • Class room example. GRAPHICAL SOLUTION

  13. Exceptional Cases • Degeneracy – In some cases one of the constraints may be redundant and look superfluous. One of the constraints may be more binding than other. • Example • Alternative Optima (Multiple Optima) – When the objective function is parallel to a binding constraint (a constraint that is satisfied by as an equation by the optimal solution) then the objective function will assume the same optimal value at more than one solution point. • Example GRAPHICAL SOLUTION

  14. SENSITIVITY ANALYSIS • Consider the Reddy Mikks problem • Maximise Z = 5x1 + 4x2 • Subject to • 6x1 + 4x2  24 • x1 + 2x2  6 • -x1 + x2  2 • x2  2 • x1, x2 > 0 • The solution for this is x1 = 3, x2 = 1.5 and value of Z is 21. (Check this graphically) GRAPHICAL SOLUTION

  15. GRAPHICAL SENSITIVITY ANALYSIS • Sensitivity Analysis is the study of changes that happen on the optimum solution by making changes in the model parameters. • Two cases need to be investigated • Change in objective function coefficients • Changes in the right hand side of constraints. GRAPHICAL SOLUTION

  16. Changes in Objective Coefficient • What are objective function coefficient? Contribution per unit. • Consider Z = c1x1 + c2x2. • Change in objective coefficient changes the optimum solution to different corner point. • There will be a range of variation for c1 & c2 for which the current optimum point will not change. • Sensitivity analysis is to figure out this range. • We determine the range of optimality for the ratio c1/c2 for which the point will remain same. GRAPHICAL SOLUTION

  17. Changes in Constraints • The RHS of the constraint equation represents the limit on the availability of the resources. • Sensitivity of the optimum solution to changes in the availability of resources. • The analysis provides a single measure, called the unit worth of resource. • UWR quantifies the rate of change in the optimum value in the objective function as a result of making changes in the availability of a resource. • Refer RM example GRAPHICAL SOLUTION

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