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Linear Programming:VARIANTS OF THE SIMPLEX METHOD

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Operations Research (MTH601)
142
Maximize
Z = x1 + 2x2 + 3x3
Subject to
x1 + 2x2 + 3x3 = 15
2x1 + x2 + 5x3 = 20
x1 + 2x2 + x3 + x4 = 10
x1, x2, x3, x4 > 0
9.
Solve the problem by two-phase method.
Maximize
Z = x1 + x2
Subject to
3x1 + 2x2 < 20
2x1 + 3x2 < 20
x1 + 2x2 > 2
x1 , x2 > 0
VARIANTS OF THE SIMPLEX METHOD
In this section we present certain complications encountered in the application of the simplex method and
how they are resolved. These are called the variants of simplex method. We can illustrate the typical cases through
numerical examples. The following variants are being considered.
1.
Minimization
2.
Inequality in the wrong direction
3.
Degeneracy
4.
Unbounded solution
5.
Multiple solutions
6.
Non-existing feasible solution
7.
Unrestricted variables
Minimization:
Sometimes we come across problems in which the objective function has to be minimized instead of
maximizing. This situation can be tackled easily in either of the two ways. One is to make the following minor
changes in the simplex method. The new entering basic variable should be the non-basic variable that would
decrease rather than increase the value of Z at the fastest rate when this variable is increased. Similarly the test for
the fastest rate when this variable is increased. Similarly the test for optimality should be whether Z can increased.
Similarly the test for optimality should be whether Z can be decreased rather than increased by increasing any non-
basic variable.
The second method is to change the problem into an equivalent problem involving maximization and
proceed with the steps of the regular simplex method. The change is effected by maximizing the negative of the
original objective function. Minimizing any function f (x1, x2, x3, ..., xn) subject to set of constraints is completely
equivalent to maximizing -f (x1, x2, ..., xn) subject to the same set of constraints. For example if we want to minimize
a function Z = 5x1 +7x2 - 8x3, it is equivalent to maximizing a function Z = -5x1 - 7x2 + 8x3.
Inequality in the wrong direction.
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Table of Contents:
  1. Introduction:OR APPROACH TO PROBLEM SOLVING, Observation
  2. Introduction:Model Solution, Implementation of Results
  3. Introduction:USES OF OPERATIONS RESEARCH, Marketing, Personnel
  4. PERT / CPM:CONCEPT OF NETWORK, RULES FOR CONSTRUCTION OF NETWORK
  5. PERT / CPM:DUMMY ACTIVITIES, TO FIND THE CRITICAL PATH
  6. PERT / CPM:ALGORITHM FOR CRITICAL PATH, Free Slack
  7. PERT / CPM:Expected length of a critical path, Expected time and Critical path
  8. PERT / CPM:Expected time and Critical path
  9. PERT / CPM:RESOURCE SCHEDULING IN NETWORK
  10. PERT / CPM:Exercises
  11. Inventory Control:INVENTORY COSTS, INVENTORY MODELS (E.O.Q. MODELS)
  12. Inventory Control:Purchasing model with shortages
  13. Inventory Control:Manufacturing model with no shortages
  14. Inventory Control:Manufacturing model with shortages
  15. Inventory Control:ORDER QUANTITY WITH PRICE-BREAK
  16. Inventory Control:SOME DEFINITIONS, Computation of Safety Stock
  17. Linear Programming:Formulation of the Linear Programming Problem
  18. Linear Programming:Formulation of the Linear Programming Problem, Decision Variables
  19. Linear Programming:Model Constraints, Ingredients Mixing
  20. Linear Programming:VITAMIN CONTRIBUTION, Decision Variables
  21. Linear Programming:LINEAR PROGRAMMING PROBLEM
  22. Linear Programming:LIMITATIONS OF LINEAR PROGRAMMING
  23. Linear Programming:SOLUTION TO LINEAR PROGRAMMING PROBLEMS
  24. Linear Programming:SIMPLEX METHOD, Simplex Procedure
  25. Linear Programming:PRESENTATION IN TABULAR FORM - (SIMPLEX TABLE)
  26. Linear Programming:ARTIFICIAL VARIABLE TECHNIQUE
  27. Linear Programming:The Two Phase Method, First Iteration
  28. Linear Programming:VARIANTS OF THE SIMPLEX METHOD
  29. Linear Programming:Tie for the Leaving Basic Variable (Degeneracy)
  30. Linear Programming:Multiple or Alternative optimal Solutions
  31. Transportation Problems:TRANSPORTATION MODEL, Distribution centers
  32. Transportation Problems:FINDING AN INITIAL BASIC FEASIBLE SOLUTION
  33. Transportation Problems:MOVING TOWARDS OPTIMALITY
  34. Transportation Problems:DEGENERACY, Destination
  35. Transportation Problems:REVIEW QUESTIONS
  36. Assignment Problems:MATHEMATICAL FORMULATION OF THE PROBLEM
  37. Assignment Problems:SOLUTION OF AN ASSIGNMENT PROBLEM
  38. Queuing Theory:DEFINITION OF TERMS IN QUEUEING MODEL
  39. Queuing Theory:SINGLE-CHANNEL INFINITE-POPULATION MODEL
  40. Replacement Models:REPLACEMENT OF ITEMS WITH GRADUAL DETERIORATION
  41. Replacement Models:ITEMS DETERIORATING WITH TIME VALUE OF MONEY
  42. Dynamic Programming:FEATURES CHARECTERIZING DYNAMIC PROGRAMMING PROBLEMS
  43. Dynamic Programming:Analysis of the Result, One Stage Problem
  44. Miscellaneous:SEQUENCING, PROCESSING n JOBS THROUGH TWO MACHINES
  45. Miscellaneous:METHODS OF INTEGER PROGRAMMING SOLUTION