9 operation research rama murthi

Op"erations Research This page intentionally left blank Copyright © 2007, 2005 New Age International (P) Ltd., Publishers Published by New Age International (P) Ltd., Publishers All rights reserved No part of this ebook may be reproduced in any form, by photostat, microfilm, xerography, or any other means, or incorporated into any information retrieval system, electronic or mechanical, without the written permission of the publisher All inquiries should be emailed to rights@newagepublishers.com ISBN (13) : 978-81-224-2944-2 PUBLISHING FOR ONE WORLD NEW AGE INTERNATIONAL (P) LIMITED, PUBLISHERS 4835/24, Ansari Road, Daryaganj, New Delhi - 110002 Visit us at www.newagepublishers.com PREFACE I started my teaching career in the year 1964 I was teaching Production Engineering subjects till 1972 In the year 1972 I have registered my name for the Industrial Engineering examination at National Institution of Industrial Engineering, Bombay Since then, I have shifted my field for interest to Industrial Engineering subjects and started teaching related subjects One such subject is OPERATIONS RESEARCH After teaching these subjects till my retirement in the year 2002, it is my responsibility to help the students with a book on Operations research The first volume of the book is LINEAR PORGRAMMING MODELS This was published in the year 2003 Now I am giving this book OPERATIONS RESEARCH, with other chapters to students, with a hope that it will help them to understand the subject easily I hope this will help my teacher friends to teach the subject well I thank Mr N.V Jagdeesh Babu, Assistant professor of Mechanical Engineering for proof reading the script Anantapur Date: 12.1.2007 P Rama Murthy This page intentionally left blank Dedicated To My Wife Usha My Daughter Vidya Grandson Yagnavalkya and My Son In Law Shankaranarayana This page intentionally left blank CONTENTS Preface Chapter Title Page number Historical Development 1-21 Linear Programming models (Resource allocation models) 22-43 Linear Programming models (Solution by Simplex method) 44-140 Linear Programming - II (Transportation Problem) 141-211 Linear Programming III (Assignment Model) 212-254 Sequencing Model 255-294 Replacement Model 295-353 Inventory Control 354-445 Waiting line theory or Queuing Model 446-484 10 Theory of Games or Competitive Strategies 485-563 11 Dynamic Programming 564-592 12 Decision Theory 593-616 13 Simulation 617-626 14 Introduction to Non - Linear Programming 627-634 15 Programme Evaluation and Review Technique and Critical Path Method (PERT AND CPM) 635-670 Multiple choice question and answers 671-702 Index 703-705 Quiz Papers 691 11 When the game is not having a saddle point, then the following method is used to solve the game: (a) Linear Programming method, (b) Minimax and maximin criteria, (c) Algebraic method, (d) Graphical method ( ) 12 Consider the matrix given, which is a pay off matrix of a game Identify the dominance in it B A X Y Z P Q R (a) P dominates Q (c) Q dominates R 13 Identify the unfair game: C D (a) A 0 B 0 (b) Y dominates Z (d) Z dominates Y (b) A B C –1 ( ) D –1 C D C D (c) A –5 +5 (d) A B +10 –10 B 14 If there are more than two persons in a game then the game is known as: (a) Nonzero sum game (b) Open game (c) Multiplayer game (d) Big game 15 For the pay off matrix the player A always uses: ( ) ( ) B I I –5 II –2 II 10 A (a) First strategy (b) Mixed strategy of both II and I (c) Does not play game (d) Second strategy 16 For the pay off matrix the player prefers to play ( ) B I I –7 II II –10 A (a) Second strategy (c) Keep quite (b) First strategy (d) Mixed strategy ( ) 692 Operations Research 17 For the game given the value is: B I I II II –5 A (a) 3, (c) 19 In the game given the saddle point is: A 20 21 22 23 24 I II III I (b) –5 (d) B II –4 –3 –5 ( ) III –2 (a) –2 (b) (c) –3 (d) A competitive situation is known as: (a) Competition, (b) Marketing, (c) Game, (d) None of the above One of the assumptions in the game theory is: (a) All players act rationally and intelligently, (b) Winner alone acts rationally, (c) Loser acts intelligently, (d) Both the players believe in luck A play is played when: (a) The manager gives green signal, (b) Each player chooses one of his courses of action simultaneously, (c) The player who comes to the place first says that he will start the game, (d) The late comer says that he starts the game The list of courses of action with each player is (a) Finite, (b) Number of strategies with each player must be same, (c) Number of strategies with each player need not be same, (d) None of the above A game involving ‘n’ persons is known as: (a) Multimember game, (b) Multiplayer game, (c) n-person game, (d) Not a game ( ) ( ) ( ) ( ) ( ) ( ) Quiz Papers 693 25 Theory of Games and Economic Behaviour is published by: (a) John Von Neumann and Morgenstern (b) John Flood (c) Bellman and Neumann (d) Mr Erlang, 26 In the matrix of a game given below the negative entries are: ( ) B I I II –1 A (a) (b) (c) (d) II –1 Payments from A to B Payments from B to A Payment from players to organisers Payment to players from organisers ( ANSWERS REPLACEMENT MODEL (1 TO 30) (a) (b) (d) (b) (c) (b) (d) (d) (a) 10 (a) 11 (d) 12 (d) 13 (c) 14 (c) 15 (a) 16 (d) 17 (a) 18 (c) 19 (d) 20 (c) 21 (d) 22 (c) 23 (d) 24 (b) 25 (d) 26 (b) 27 (b) 28 (a) 29 (b) 30 (c) GAME THEORY: ( TO 26) (c) (d) (d) (a) (d) (c) (a) (c) (c) 10 (d) 11.(b) 12 (d) 13 (d) 14 (c) 15 (d) 16 (b) 17 (d) 18 (c) 19 (c) 20 (c) 21 (a) 22 (b) 23 (c) 24 (c) 25 (a) 26 (a) ) Unit – V Inventory Management and Waiting Line Models INVENTORY MODELS One of the important basic objectives of Inventory management is: (a) To calculate EOQ for all materials in the organisation, (b) To go in person to the market and purchase the materials, (c) To employ the available capital efficiently so as to yield maximum results, (d) Once materials are issued to the departments, personally check how they are used ( The best way of improving the productivity of capital is: (a) Purchase automatic machines, (b) Effective labour control, (c) To use good financial management, (d) Productivity of capital is to be increased through effective materials management ( Materials management is a body of knowledge, which helps manager to: (a) Study the properties of materials, (b) Search for needed material, (c) Increase the productivity of capital by reducing the cost of material, (d) None of the above ( The stock of materials kept in the stores in anticipation of future demand is known as: (a) Storage of materials, (b) Stock of materials, (c) Inventory, (d) Raw materials ( The stock of animals reared in anticipation of future demand is known as: (a) Live stock inventory, (b) Animal inventory, (c) Flesh inventory, (d) None of the above ( The working class of human beings is a class of inventor known as: (a) Live stock, (b) Human inventory, (c) Population, (d) Human resource inventory ( In general, the percentage of materials cost in product is approximately equal to: (a) 40 to 50 % (b) to 10 % (c) to % (d) 90 to 95% ( ) ) ) ) ) ) ) Quiz Papers 10 11 12 13 14 15 16 17 18 695 Materials management brings about increased productivity of capital by: (a) Very strict control over use of materials, (b) Increasing the efficiency of workers, (c) Preventing large amounts of capital locked up for long periods in the form of inventory (d) To apply the principles of capital management, ( ) We can reduce the materials cost by: (a) Using systematic inventory control techniques, (b) Using the cheap material, (c) Reducing the use of materials, (d) Making hand to mouth purchase, ( ) The basis for ABC analysis is (a) Interests of Materials manager, (b) Interests of the top management, (c) Pareto’s 80-20 rule, (d) None of the above ( ) ABC analysis depends on the: (a) Quality of materials, (b) Cost of materials, (c) Quantity of materials used, (d) Annual consumption value of materials ( ) ‘A’ class materials consume: (a) 10% of total annual inventory cost, (b) 30% of total annual inventory cost, (c) 70 to 75% of total inventory cost, (d) 90% of total annual inventory cost ( ) ‘B’ class of materials consumes % of annual inventory cost (a) 60 to 70% (b) 20 to 25% (c) 90 to 95% (d) to 8% ( ) ‘C’ class of materials consume % of annual inventory cost (a) to 10 % (b) 20 to 30% (c) 40 to 50% (d) 70 to 80% ( ) The rent for the stores where materials are stored falls under: (a) Inventory carrying cost, (b) Ordering cost, (c) Procurement cost, (d) Stocking cost ( ) Insurance charges of materials cost fall under: (a) Ordering cost, (b) Inventory carrying cost, (c) Stock out cost (d) Procurement cost ( ) As the volume of inventory increases, the following cost will increase: (a) Stock out cost, (b) Ordering cost, (c) Procuring cost, (d) Inventory carrying cost ( ) As the order quantity increases, this cost will reduce: (a) Ordering cost, (b) Insurance cost, (c) Inventory carrying cost, (d) Stock out cost ( ) 696 Operations Research 19 Procurement cost may be clubbed with: (a) Inventory carrying charges, (b) Stock out cost, (c) Loss due to deterioration, (d) Ordering cost ( ) 20 The penalty for not having materials when needed is: (a) Loss of materials cost, (b) Loss of ordering cost, (c) Stock out cost, (d) General losses ( ) 21 Losses due to deterioration, theft and pilferage come under, (a) Inventory carrying charges, (b) Losses due to theft, (c) Not any cost, (d) Consumption cost ( ) 22 Economic Batch Quantity is given by (where, C1 = Inventory carrying cost, C3 = Ordering cost, r = Demand for the product) (a) (2C1/C3)1/2, (b) (2 C3/C1 r )1/2, (c) 2C3r / C1, (d) (2C3r / C1)1/2 ( ) 23 If λ is the annual demand, C1 = Inventory carrying cost, i = rate of inventory carrying charges, p = unit cost of material in Rs., then EOQ = (a) (2C3 λ /ip)1 / 2, (b) 2C3 λ /ip, (c) (2 C3/ip λ )1/2, (d) (2 λ /C3 ip)1/2 ( ) 24 If C1 = carrying cost, C3 is the ordering cost, r = demand for the product, then the optimal period for placing an order is given by: (a) (2 C3/C1 r) 1/2 (b) (2C1 C3/r )1 / (c) ( 2C3 r/C1) /2 (d) ( 2C1 C3 r) 1/2 ( ) 25 When C1 = Inventory carrying cost, C3 = ordering cost, r = demand for the product, the total cost of inventory is given by: (a) (2C1 C3 r) (b) (2C1 C3) 1/2 1/2 (c) (2C3r/C1) (d) (2C1 C3 r )1 /2 ( ) 26 When load is the annual demand for the material, p = unit price of the material in Rs., C3 is the ordering cost, q = order quantity, then the total cost including the martial cost is given by: (b) 2C3 λ ip + λ p (a) (q/2) ip + λ /q C3 + λ p (c) (q/2) ip + λ p 27 In VED analyses, the letter V stands for: (a) Very important material, (c) Weighty materials, 28 In VED analysis, the letter D strands for: (a) Dead stock, (c) Deserved materials, 29 The VED analysis depends on: (a) Annual consumption cost of materials, (b) Unit price of materials, (c) Time of arrival of materials, (d) Criticality of materials (d) ( 2C3 q λ ip) 1/2 ( ) (b) Viscous material (d) Vital materials ( ) (b) Delayed material, (d) Diluted materials ( ) ( ) Quiz Papers 30 In FSN analysis the letter S stands for: (a) Slack materials, (b) Stocked materials, (c) Slow moving materials, (d) Standard materials 31 In FSN analysis, the letter N stands for: (a) Nonmoving materials, (b) Next issuing materials, (c) No materials, (d) None of the above 32 FSN analysis depends on: (a) Weight of the material, (b) Volume of the material, (c) Consumption pattern, (d) Method of moving materials 33 MRP stands for: (a) Material Requirement Planning, (b) Material Reordering Planning, (c) Material Requisition Procedure, (d) Material Recording Procedure 34 A system where the period of placing the order is fixed is known as: (a) q-system, (b) Fixed order system, (c) p-system, (d) Fixed quantity system 35 A system in which quantity for which order is placed is constant is known as: (a) q-System, (b) p-system, (c) Period system, (d) Bin system 36 LOB stands for: (a) Lot of Bills, (b) Line of Batches, (c) Lot of Batches, (d) Line of Balance 37 High reliability spare parts in inventory are known as: (a) Reliable spares, (b) Insurance spares, (c) Capital spares, (d) Highly reliable spares 38 The property of capital spares is: (a) They have very low reliability; (b) These can be purchased in large quantities, as the price is low, (c) These spares have relatively higher purchase cost than the maintenance spares, (d) They are very much similar to breakdown spares 39 Re-usable spares are known as: (a) Multi use spares, (b) Repeated useable stores, (c) Scrap materials, (d) Rotable spares 40 JIT stands for: (a) Just In Time Purchase, (b) Just In Time production, (c) Just In Time use of materials, (d) Just In Time order the material 41 The cycle time, selected in balancing a line must be: (a) Greater than the smallest time element given in the problem, (b) Less than the highest time element given in the problem, (c) Slightly greater than the highest time element given in the problem, (d) Left to the choice of the problem solver 697 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 698 Operations Research 42 The lead-time is the time: (a) To place orders for materials, (b) Of receiving materials, (c) Between receipt of material and using materials, (d) Between placing the order and receiving the materials ( ) 43 The PQR classification of inventory depends on: (a) Unit price of the material, (b) Annual consumption value of the material, (c) Criticality of the material, (d) Shelf life of the materials ( ) 44 The classification made on the weight of the materials is known as: (a) PQR analysis, (b) VED analysis, (c) XYZ analysis, (d) FSN analysis ( ) 45 At EOQ (a) Annual purchase cost = Annual ordering cost, (b) Annual ordering cost = Annual carrying cost, (c) Annual carrying cost = Annual shortage cost, (d) Annual shortage cost = Annual purchase cost ( ) 46 If shortage cost is infinity, (a) No shortages are allowed; (b) No inventory carrying cost is allowed, (c) Ordering cost is zero, (d) Purchase cost = Carrying cost ( ) 47 The most suitable system for a retail shop is (a) FSN Analysis, (b) ABC analysis, (c) VED analysis, (d) GOLF analysis ( ) 48 The inventory maintained to meet unknown demand changes is known as (a) Pipeline inventory, (b) Anticipatory inventory, (c) De coupling inventory, (d) Fluctuatory inventory ( ) 49 The most suitable inventory system for a Petrol bunk is (a) P-System, (b) Bin system, (c) Q-System, (d) Probabilistic model ( ) 50 The water consumption from a water tank follows (a) P-system, (b) PQ-system (c) Q-System, (d) EOQ System ( ) 51 Which of the following inventories is maintained to meet expected demand fluctuations? (a) Fluctuatory Inventory, (b) Buffer stock, (c) De-coupling inventory, (d) Anticipatory inventory ( ) 52 Which of the following increases with quantity ordered per order? (a) Carrying cost, (b) Ordering cost, (c) Purchase cost, (d) Demand ( ) 53 The ordering cost per order and average unit carrying cost are constant, and demand suddenly falls by 75% then EOQ will: (a) Decrease by 50% (b) Not change (c) Increase by 50% (d) Decrease by 40% ( ) Quiz Papers 699 54 In JIT system, the following is assumed to be zero (a) Ordering cost, (b) Transportation cost, (c) Carrying cost, (d) Purchase cost 55 Which of the following analyses neither considers cost nor value? (a) ABC, (b) XYZ, (c) HML, (d) VED ( ) ( ) ANSWERS (c) (d) (c) (c) (a) (d) (a) (c) (a) 10 (c) 11 (d) 12 (c) 13 (b) 14 (a) 15 (a) 16 (b) 17 (d) 18 (a) 19 (d) 20 (c) 21 (a) 22 (b) 23 (c) 24 (a) 25 (d) 26 (a) 27 (d) 28 (c) 29 (d) 30 (c) 31 (a) 32 (c) 33 (a) 34 (c) 35 (a) 36 (d) 37 (b) 38 (c) 39 (d) 40 (b) 41 (c) 42 (d) 43 (c) 44 (d) 45 (b) 46 (b) 47 (a) 48 (d) 49 (c) 50 (a) 51 (d) 52 (a) 53 (c) 54 (c) 55 (d) WAITING LINE MODELS OR QUEUING THEORY As per queue discipline the following is not a negative behavior of a customer: (a) Balking, (b) Reneging, (c) Boarding, (d) Collusion The expediting or follow up function in production control is an example of (a) LIFO, (b) FIFO, (c) SIRO, (d) Preemptive In M/M/S N/FIFO the following does not apply (a) Poisson arrival, (b) Limited service, (c) Exponential service, (d) Single server The dead bodies coming to a burial ground is an example of: (a) Pure Birth Process, (b) Pure Death Process, (c) Birth and Death Process, (d) Constant Rate of Arrival ( ) ( ) ( ) ( ) 700 Operations Research The system of loading and unloading of goods usually follows: (a) LIFO, (b) FIFO, (c) SIORO, (d) SBP A steady state exists in a queue if: (a) λ > µ, (b) λ < µ, (c) λ = µ, (d) λ = µ If the operating characteristics of a queue are dependent on time, then it is said to be: (a) Transient state, (b) Busy state, (c) Steady state, (d) Explosive state A person who leaves the queue by losing his patience to wait is said to be: (a) Reneging, (b) Balking, (c) Jockeying, (d) Collusion The characteristics of a queuing model is independent of: (a) Number of service stations, (b) Limit of length of queue, (c) Service Pattern, (d) Queue discipline 10 The unit of traffic intensity is: (a) Poisson, (b) Markow, (c) Erlang, (d) Kendall 11 In (M /M /1) : (∞ / FCFS) model, the length of the system Ls is given by: (a) p2/1/p (b) p/1– λ 12 13 14 15 16 ( ) ( ) ( ) ( ) ( ) ( ) (c) λ 2/(µ – λ ) (d) λ 2/µ(µ – λ ) ( ) In (M/M/1) : (∞/ FIFO) model, 1/(µ – λ ) represents: (a) Ls, Length of the system, (b) Lq length of the queue, (c) Wq Waiting time in queue, (d) Ws Waiting time in system ( ) The queue discipline in stack of plates is: (a) SIRO, (b) Non-Pre-emptive, (c) FIFO, (d) LIFO ( ) Office filing system follows: (a) LIFO, (b) FIFO, (c) SIRO, (d) SBP ( ) SIRO discipline is generally found in: (a) Loading and unloading, (b) Office filing, (c) Lottery draw, (d) Train arrivals at platform ( ) The designation of Poisson arrival, Exponential service, single server and limited queue selected randomly are represented by: (a) (M/E/S) : ( ∞ /SIRO), (b) (M/M/1) : ( ∞ /SIRO), (c) (M/M/S) : (N/SIRO), (d) (M/M/1) : ( N/SIRO) ( ) Quiz Papers 17 18 19 20 21 22 23 24 25 26 For a simple queue (M/M/1) , ρ = λ /µ is known as: (a) Poisson busy period, (b) Random factor, (c) Traffic intensity, (d) Exponential service factor With respect to simple queuing model which on of the given below is wrong: (a) Lq = λ Wq (b) λ = µ β (d) Ls = Lq + β (c) Ws = Wq + µ When a doctor attends to an emergency case leaving his regular service is called: (a) Reneging, (b) Balking, (c) Pre-emptive queue discipline, (d) Non-Pre-Emptive queue discipline A service system, where customer is stationary and server is moving is found with: (a) Buffet Meals, (b) Outpatient at a clinic, (c) Person attending the breakdowns of heavy machines, (d) Vehicle at petrol bunk In a simple queuing model the waiting time in the system is given by: (a) (Lq/ λ ) + (1/µ) (b) 1/(µ – λ ) 701 ( ) ( ) ( ) ( ) (c) µ/(µ – λ ) (d) Wq + µ ( ) This department is responsible for the development of queuing theory: (a) Railway station, (b) Municipal office, (c) Telephone department, (d) Health department ( ) If the number of arrivals during a given time period is independent of the number of arrivals that have already occurred prior to the beginning of time interval, then the new arrivals follow - distribution (a) Erlang, (b) Poisson, (c) Exponential, (d) Normal, ( ) Arrival → Service → Service → Service → Out The figure given represents: (a) Single channel single phase system, (b) Multichannel single-phase system, (c) Singlechannel multiphase system, (d) Multichannel multiphase system ( ) In queue designation A/B/S : (d/f), what does S represent? (a) Arrival Pattern, (b) Service Pattern (c) Number of service channels, (d) Capacity of the system ( ) When the operating characteristics of the queue system is dependent on time, the it is said to be: (a) Steady state, (b) Explosive state, (c) Transient state, (d) Any one of the above ( ) 702 Operations Research 27 The distribution of arrivals in a queuing system can be considered as a: (a) Death Process, (b) Pure birth Process, (c) Pure live process, (d) Sick process 28 Queuing models measure the effect of: (a) Random arrivals, (b) Random service, (c) Effect of uncertainty on the behaviour of the queuing system, (d) Length of queue 29 Traffic intensity is given by: (a) Mean arrival rate / Mean service rate, (b) λ × µ, ( ) ( ) (c) µ/ λ , (d) Number present in the queue/Number served 30 Variance of queue length is : ( ) (a) ρ = λ /µ, (b) ρ /1 – ρ , (c) λ /µ – λ , (d) ρ /(1 – ρ )2 ( ) ANSWERS (c) (d) (d) (a) (a) (c) (a) (a) (d) 10 (c) 11 (b) 12 (c) 13 (d) 14 (a) 15 (c) 16 (d) 17 (c) 18 (c) 19 (d) 20 (c) 21 (a) 22 (c) 23 (b) 24 (c) 25 (a) 26 (c) 27 (b) 28 (c) 29 (a) 30 (d) Index Abc analysis of inventory 364 Administrative decisions 594 Annual consumption cost 364 Anticipation inventory 356 Assignment algorithm 216 Assignment model 212 Assumptions made in sequencing problems 256 Balking 453 Bar chart 635 Basic assumptions 23 Basic feasible solution 145 Bathtub curve 298 Business games 485 Capital cost 300 Collusion 453 Competitive strategies 485 Completely random 451 Conditions of risk 486 Cost slope 660 Costs associated 300 Costs associated with inventory 356 Criterion of regret 601 Critical path 643 Critical Path Method (CPM) 635 Customer behaviour 453 Cycle inventories 356 Decision Decision 593 Decision Making Under Risk (DMUR) 597 Decision making under uncertainty 599 Decision theory 486 Decision theory 593 Decoupling inventories 356 Definition of inventory 354 Degeneracy in transportation problem 165 Degree of certainty 593 Demand 360 Desirable items 372 Deterministic and stochastic 564 Direct cost 658 Direct inventories 355 Direct production cost 359 Discount rate 315 Discrete 565 Discrete or continuous systems 565 Dominance in games 499 Dual problem 105 Dual simplex method 122 Duality in linear programming 105 Dummy activity 638 Dynamic programming 564 Earliest event time 654 Economic lot size with finite rate of replenishmen 390 Economic order quantity models 375 Essential items 372 Events 636 Explosive state 457 Finished goods inventories 355 Fixed time model 403 Flood’s technique 216 704 Fluctuation inventories 356 FSND analysis 373 Graphical method in L.P.P 28 Group replacement 302 Group replacement policy 332 Historical development 450 Hungarian method 216 Implied cost 157 Imputed value 107 In process inventories 355 Incremental discount 389 Indirect cost 659 Indirect inventories 355 Individual replacement policy 332 Input process 450 Integrality of items 393 Inter-arrival time 451 Inventory carrying charges 356 Inventory control 354 Inventory control 354 Isocost line 34 Isoprofit line 32 Jockeying 453 Latest allowable occurrence 656 Lead time 361 Least cost cell (or inspection) method 147 Likely time 641 Linear programming 22 Linear programming Lost-sales shortages 299 Maintenance 295 Manpower planning 344 Materials management 354 Maximin 486 Maximisation problem 179 Maximization case of transportation problem 161 Operations Ressearch Milestone chart 635 Minimax 486 Minimax principle 486 Minimization of the maximum losses 486 Mixed strategies 501 Mixed strategy games 490 Models with shortages 395 Modified distribution method of optimality test 156 Monte-carlo simulation 619 Mortality tables 333 Mortality theorem 333 Multi channel facility 454 N- jobs and two machines 257 N- person game 488 Negative exponential distribution 451 Non - negativity constraint 14 Non-linear programming 627 North- west corner method 146 Numbering of events 641 Objective function 14 Objective of inventory 360 Oddments 513 Operations research History Objectives Characteristics 10 Model 15 Scope 10 Steps in solving 12 Operating costs 300 Opportunity cost 148 Optimistic time 640 Ordering cost 358 P - system 362 Pessimistic time 640 Poisson random 451 Policy 565 Pq - system 364 Index Present worth factor 315 Preventive maintenance technique 302 Primal problem 105 Probabilistic 413 Probabilistic models 564 Production of goods 361 Programme Evaluation and Review Technique (PERT) 635 Properties of linear programming model 22 Purchase price 359 Pure strategy 490 Q - system 362 Quantity discount model 388 Queuing theory 446 Random numbers 621 Redundancy in transportation problems 198 Reneging 453 Replacement model 295 Safety stock 361 Scheduling problem 237 Scrap value 300 Selective approach system 364 Sensitivity analysis 117 Sequencing problem 255 Service channel 447 Service discipline 455 Service mechanism 453 Set up cost 358 Shadow price 51 Shortage cost or stock 357 Simulation models 618 Simulation technique 619 Single channel facility 454 705 Skip numbering 641 Slack time 656 Spare parts inventories 355 Stages 564 Standard deviation 641 State 565 Steady state 456 Stochastic models 413 Strategic decision 594 Symmetrical dual simplex 108 Trader problem 175 Transient state 457 Transportation inventories 356 Transportation model 142 Transshipment problem 192 Traveling salesman problem 281 Two phase method 84 Two-bin system 362 Two-person game 488 Types of sequencing problems 257 Unbound solutions 96 Unrestricted variables 99 Variance 641 Ved analysis 372 Vital items 372 Vogel’s approximation method 148 Waiting line theory 446 Waste inventory 356 XYZ analysis 373 Zero sum game 488 ... operations research 8 Operations Research (c) Operations Research is Research into Operations - J Steinhardt This definition does not give anything in clear about the subject of Operations Research. .. OF OPERATIONS RESEARCH After considering the objective and definitions of Operations Research, now let us try to understand what are the characteristics of Operations Research (a) Operations Research. .. QUESTIONS 10 Trace the history of Operations Research Give a brief account of history of Operations Research Discuss the objective of Operations Research "Operations Research is a bunch of mathematical