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Quantitative Techniques

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Useful for students to have a complete and brief notes about the quantitative techniques. Also helpful for those pursuing MBA.

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Quantitative Techniques

  1. 1. Quantitative Techniques Deepthy Sai Manikandan
  2. 2. Topics: <ul><li>Linear Programming </li></ul><ul><li>Transportation Problem </li></ul><ul><li>Assignment problem </li></ul><ul><li>Queuing Theory </li></ul><ul><li>Decision Theory </li></ul><ul><li>Inventory Management </li></ul><ul><li>Simulation </li></ul><ul><li>Network Analysis </li></ul>
  3. 3. LINEAR PROGRAMMING
  4. 4. Linear Programming <ul><li>It is a mathematical technique for optimum allocation of scarce or limited resources to several competing activities on the basis of given criterion of optimality, which can be either performance, ROI, cost, utility, time, distance etc. </li></ul>
  5. 5. Steps <ul><li>Define decision variables </li></ul><ul><li>Formulate the objective function </li></ul><ul><li>Formulate the constraints </li></ul><ul><li>Mention the non-negativity criteria </li></ul>
  6. 6. Components & Assumptions <ul><li>Objective </li></ul><ul><li>Decision Variable </li></ul><ul><li>Constraint </li></ul><ul><li>Parameters </li></ul><ul><li>Non-negativity </li></ul><ul><li>Proportionality </li></ul><ul><li>Addivity </li></ul><ul><li>Divisibility </li></ul><ul><li>Certainity </li></ul>
  7. 7. Problem: <ul><li>An animal feed company must produce at least 200 kgs of a mixture consisting of ingredients x1 and x2 daily. x1 costs Rs.3 per kg. and x2 Rs.8 per kg. No more than 80 kg. of x1 can be used and at least 60 kg. of x2 must be used. Formulate a mathematical model to the problem. </li></ul>
  8. 8. Solution: <ul><li>Minimize Z = 3x1 + 8x2 </li></ul><ul><li>Subject to x1 + x2 >= 200 </li></ul><ul><li> x1 <= 80 </li></ul><ul><li> x2 >= 60 </li></ul><ul><li>X1 >= 0 , x2 >= 0 </li></ul>
  9. 9. Graphical Solution <ul><li>Formulate the problem </li></ul><ul><li>Convert all inequalities to equations </li></ul><ul><li>Plot the graph of all inequalities </li></ul><ul><li>Find out the feasilble region </li></ul><ul><li>Find out the corner points </li></ul><ul><li>Substitute the objective function </li></ul><ul><li>Arrive at the solution </li></ul>
  10. 10. Problem: <ul><li>Maximize Z = 60x1+50x2 </li></ul><ul><li>subject to 4x1+10x2 <= 100 </li></ul><ul><li>2x1+1x2 <= 22 </li></ul><ul><li>3x1+3x2 <= 39 </li></ul><ul><li>x1,x2 >= 0 </li></ul>
  11. 11. Solution : <ul><li>4x1+10x2=100 (0,10)(25,0) </li></ul><ul><li>2x1+x2=22 (0,22)(11,0) </li></ul><ul><li>3x1+3x2=39 (0,13)(13,0) </li></ul>0 x2 x1 10 13 22 11 13 25 E C B A D
  12. 12. <ul><li>A (0,0) = 60*0+50*0 = 0 </li></ul><ul><li>B (11,0) = 60*11+50*0 = 660 </li></ul><ul><li>C (9,4) = 60*9+50*4 = 740 </li></ul><ul><li>D (5,8) = 60*5+50*8 = 700 </li></ul><ul><li>E (0,10) = 60*0+50*10 = 500 </li></ul><ul><li>Max Z is at C (9,4) and Z = 740 </li></ul>Z = 60x1 + 50x2
  13. 13. TRANSPORTATION PROBLEM
  14. 14. Transportation Problem <ul><li>A special kind of optimisation problem in which goods are transported from a set of sources to a set of destinations subject to the supply and demand constraints. The main objective is to minimize the total cost of transportation. </li></ul>
  15. 15. Initial Basic Feasible Solution <ul><li>North West Corner Method </li></ul><ul><li>Least Cost Method </li></ul><ul><li>Vogel’s Approximation Method </li></ul><ul><li>The solution is said to be feasible when one gets (m+n-1) allotments. </li></ul>
  16. 16. Assignment Problem <ul><li>It is a problem of assigning various people, machines and so on in such a way that the total cost involved is minimized or the total value is maximized. </li></ul>
  17. 17. QUEUING THEORY
  18. 18. Queuing Theory <ul><li>A flow of customers from finite/infinite population towards the service facility forms a queue due to lack of capacity to serve them all at a time. </li></ul><ul><li>Input Output </li></ul>Serve r
  19. 19. Measures <ul><li>Traffic intensity </li></ul><ul><li>Average system length </li></ul><ul><li>Average queue length </li></ul><ul><li>Average waiting time in queue </li></ul><ul><li>Average waiting time in system </li></ul><ul><li>Probability of queue length </li></ul>
  20. 20. Queuing & cost behavior Cost of service Cost of waiting Total cost
  21. 21. DECISION THEORY
  22. 22. Decision Theory <ul><li>The decision making environment </li></ul><ul><li>Under certainity </li></ul><ul><li>Under uncertainity </li></ul><ul><li>Under risk </li></ul>
  23. 23. Decision making under uncertainity <ul><li>Laplace Criterion </li></ul><ul><li>Maxmin Criterion </li></ul><ul><li>Minmax Criterion </li></ul><ul><li>Maxmax Criterion </li></ul><ul><li>Minmin Criterion </li></ul><ul><li>Salvage Criterion </li></ul><ul><li>Hurwicz Criterion </li></ul>
  24. 24. Inventory management <ul><li>Inventory is vital to the sucessful functioning of manufacturing and retailing organisations. They may be raw materials, work-in-progress, spare parts/consumables and finished goods. </li></ul>
  25. 25. Models <ul><li>Deterministic Inventory Model </li></ul><ul><li>Inventory Model with Price breaks </li></ul><ul><li>Probabilistic Inventory Model </li></ul>
  26. 26. Basic EOQ Model Slope=0 Total cost Carrying cost Ordering cost Minimum total cost Optimal order qty
  27. 27. SIMULATION
  28. 28. Simulation <ul><li>It involves developing a model of some real phenomenon and then performing experiments on the model evolved. It is descriptive in nature and not an optimizing model. </li></ul>
  29. 29. Process <ul><li>Definition of the problem </li></ul><ul><li>Construction of an appropriate model </li></ul><ul><li>Experimentation with the model </li></ul><ul><li>Evaluation of the results of simulation </li></ul>
  30. 30. NETWORK ANALYSIS PERT CPM
  31. 31. <ul><li>A project is a series of activities directed to the accomplishment of a desired objective. </li></ul><ul><li>PERT </li></ul><ul><li>CPM </li></ul>Network Analysis / Project Management
  32. 32. CPM-Critical Path Method <ul><li>Activities are shown as a network of precedence relationship using Activity-On-Arrow (A-O-A) network construction. </li></ul><ul><li>There is single stimate of activity time </li></ul><ul><li>Deterministic activity time </li></ul>
  33. 33. Project Evaluation & Review Technique <ul><li>Activities are shown as a network of precedence relationships using A-O-A network construction. </li></ul><ul><li>Multiple time estimates </li></ul><ul><li>Probabilistic activity time </li></ul>
  34. 34. Crashing <ul><li>Crashing is shortening the activity duration by employing more resources. </li></ul><ul><li>cost slope = Cc – Cn/ Tn - Tc </li></ul>

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