Resource Management Techniques – MC9242 II MCAMC9242 RESOURCE MANAGEMENT TECNI!UES " T # C $ % % $ UNIT I "INEAR #ROGRAMMING MO&E"S 9 Mathematical Formulation - Graphical Solution of linear programming models – Simplex method – Artifcial variable Techniques- ariants of Simplex method UNIT II TRANS#ORTATION AN& ASSIGNMENT MO&E"S 9 Mathematical formulation of transportation problem- Methods for fnding initial basic feasible solution – optimum solution - degenerac! – Mathematical formulation of assignment models – "ungarian Algorithm – ariants of the Assignment problem UNIT III INTEGER #ROGRAMMING MO&E"S 9 Formulation – Gomor!#s $%% method – Gomor!#s mixed integer method – &ranch and bound technique' UNIT I' SCE&U"ING () #ERT AN& C#M 9 (et)or* +onstruction – +ritical %ath Method – %ro,ect -valuation and .evie) Technique – .esource Anal!sis in (et)or* Scheduling UNIT ' !UEUEING MO&E"S 9 +haracteristics of /ueuing Models – %oisson /ueues - 0M 1 M 1 23 4 0F$F5 1 6 1637 0M 1 M 1 23 4 0F$F5 1 ( 1 637 0M 1 M 1 +3 4 0F$F5 1 6 1 637 0M 1 M 1 +3 4 0F$F5 1 ( 1 63 models' Tota* No+ o, #erio-s . 4/ - 2 - Resource Management Techniques – MC9242 II MCA RESOURCE MANAGEMENT TECNI!UES – MC9242 #art0A – !uestions an- Ans1ers UNIT0I 2+ 3hat is o4erations research5 5perations research is a stud! of optimi8ation techniques' $t is applied decision theor!' 5. is the application of scientifc methods7 techniques and tools to problems involving the operations of s!stems so as to provide these in control of operations )ith optimum solutions to the problem' 2+ "ist some a44*ications o, OR+ • 5ptimal assignment of various ,obs to di9erent machines and di9erent operators' • To fnd the )aiting time and number of customers )aiting in the queue and s!stem in queuing model • To fnd the mimimum transportation cost after allocating goods from di9erent origins to various destinations in transportation model • :ecision theor! problems in mar*eting7fnance and production planning and control' $+ 3hat are the 6arious t74es o, mo-e*s in OR5 • Models b! function i3 :escriptive model ii3 %redictive model iii3 (ormative model • Models b! structure i3 $conic model ii3 Analogue model iii3 Mathematical model • Models b! nature of environment i3 :eterministic model ii3 %robabilistic model 4+ 3hat are main characteristics o, OR5 • -xamination of functional relationship from a s!stem overvie)' • ;tili8ation of planned approach - < - Resource Management Techniques – MC9242 II MCA • Adaptation of planned approach • ;ncovering of ne) problems for stud! /+ Name some characteristics o, goo- mo-e*+ • The number of assumptions made should be as fe) as possible • $t should be eas! as possible to solve the problem • The number of variables used should be as fe) as possible' • $t should be more =exible to update the changes over a period of time )ithout change in its frame)or*' 8+ 3hat are the -i9erent 4hases o, OR5 • Formulation of the problem • +onstruction of mathematical modeling • :eriving the solution from the model • alidit! of the model • -stablishing the control over the solution • $mplementation of the fnal solution' :+ "ist out the a-6antages o, OR5 • 5ptimum use of managers production factors • $mproved qualit! of decision • %reparation of future managers b! improving their *no)ledge and s*ill • Modifcation of mathematical solution before its use' ;+ 3hat are the *imitations o, OR5 • Mathematical model do not ta*e into account the intangible factors such as human relations etc' can not be quantifed' • Mathematical models are applicable to onl! specifc categories of problems' • .equires huge calculations' All these calculations cannot be handled manuall! and require computers )hich bear heav! cost' 9+ 3hat is *inear 4rogramming5 >inear programming is a technique used for determining optimum utili8ation of limited resources to meet out the given ob,ectives' The ob,ective is to maximi8e the proft or minimi8e the resources 0men7 machine7 materials and mone!3 2%+ 3rite the genera* mathematica* ,ormu*ation o, "##+ 2' 5b,ective function Max or Min ? @ + 2 x 2 A + < x < A B''A + n x n <' Sub,ect to the constraints a 22 x 2 Aa 2< x < ABBBBA a 2n x n 0C@D3b 2 a <2 x 2 Aa << x < ABBBBA a <n x n 0C@D3b < BBBBBBBBBBBBBBBBBBBBBB'' - E - Resource Management Techniques – MC9242 II MCA BBBBBBBBBBBBBBBBBBBBBB'' a m2 x 2 Aa m< x < ABBBBA a mn x n 0C@D3b m E' (on-negative constraints x 2 7x < 7B'x m D F 22+ 3hat are the characteristic o, "##5 • There must be a )ell defned ob,ective function' • There must be alternative course of action to choose' • &oth the ob,ective functions and the constraints must be linear equation or inequalities' 22+ 3hat are the characteristic o, stan-ar- ,orm o, "##5 • The ob,ective function is of maximi8ation t!pe' • All the constraint equation must be of equal t!pe b! adding slac* or surplus variables • ."S of the constraint equation must be positive t!pe • All the decision variables are of positive t!pe 2$+ 3hat are the characteristics o, canonica* ,orm o, "##5 <NO' =%:> $n canonical form7 if the ob,ective function is of maximi8ation t!pe7 then all constraints are of C t!pe' Similarl! if the ob,ective function is of minimi8ation t!pe7 then all constraints are of D t!pe' &ut non-negative constraints are Dt!pe for both cases' 24+ A ?rm manu,actures t1o t74es o, 4ro-ucts A an- ( an- se**s them at 4ro?t o, Rs 2 on t74e A an- Rs $ on t74e (+ Each 4ro-uct is 4rocesse- on t1o machines M2 an- M2+T74e A requires 2 minute o, 4rocessing time on M2 an- 2 minutes on M2 T74e ( requires 2 minute o, 4rocessing time on M2 an- 2 minute on M2+ Machine M2 is a6ai*a@*e ,or not more than 8 hours 4% minutes 1hi*e machine M2 is a6ai*a@*e ,or 2% hours -uring an7 1orAing -a7+ Bormu*ate the 4ro@*em as a "## so as to maCimiDe the 4ro?t+ <MA) =%:> Maximi8e 8 @<x 2 AEx < Sub,ect tot the constraints4 x 2 A x < C GFF <x 2 A x < C HFF x 2 7x < D F 2/+ A com4an7 se**s t1o -i9erent 4ro-ucts A an- ( E maAing a 4ro?t o, Rs+4% an- Rs+ $% 4er unit on themEres4ecti6e*7+The7 are 4ro-uce- in a common 4ro-uction 4rocess an- are so*- in t1o -i9erent marAetsE the 4ro-uction 4rocess has a tota* ca4acit7 o, $%E%%% man0hours+ It taAes three hours to 4ro-uce a unit o, A an- one hour to 4ro-uce a unit o, (+ The marAet has @een sur6e7e- an- com4an7 o9icia* ,ee* that the maCimum num@er o, units o, A that can @e so*- is ;E%%% units an- that o, - G - Resource Management Techniques – MC9242 II MCA ( is 22E%%% units+ Su@Fect to these *imitationsE 4ro-ucts can @e so*- in an7 com@ination+ Bormu*ate the 4ro@*em as a "## so as to maCimiDe the 4ro?t Maximi8e 8 @GFx 2 AEFx < Sub,ect tot the constraints4 Ex 2 A x < C EF7FFF x 2 C IFFF x < C 2<FFF x 2 7x < D F 28+ 3hat is ,easi@i*it7 region5 <MA) =%;> +ollections of all feasible solutions are called a feasible set or region of an optimi8ation model' Or A region in )hich all the constraints are satisfed is called feasible region' 2:+ 3hat is ,easi@i*it7 region in an "# 4ro@*em5 Is ti necessar7 that it shou*- a*1a7s @e a con6eC set5 A region in )hich all the constraints are satisfed is called feasible region' The feasible region of an >%% is al)a!s convex set' 2;+ &e?ne so*ution A set of variables x27x<B'xn )hich satisfes the constraints of >%% is called a solution' 29+ &e?ne ,easi@*e so*ution5 <MA) =%:> An! solution to a >%% )hich satisfes the non negativit! restrictions of >%%#s called the feasible solution 2%+ &e?ne o4tima* so*ution o, "##+ <MA) =%9> An! feasible solution )hich optimi8es the ob,ective function of the >%%#s called the optimal solution 22+ State the a44*ications o, *inear 4rogramming • Jor* scheduling • %roduction planning K production process • +apital budgeting • Financial planning • &lending • Farm planning • :istribution • Multi-period decision problem $nventor! model Financial model Jor* scheduling 22+ State the "imitations o, "#+ - L - Resource Management Techniques – MC9242 II MCA • >% treats all functional relations as linear • >% does not ta*e into account the e9ect of time and uncertaint! • (o guarantee for integer solution' .ounding o9 ma! not feasible or optimal solution' • :eals )ith single ob,ective7 )hile in real life the situation ma! be di9icult' 2$+ 3hat -o 7ou un-erstan- @7 re-un-ant constraints5 $n a given >%% an! constraint does not a9ect the feasible region or solution space then the constraint is said to be a redundant constraint' 24+ &e?ne Un@oun-e- so*ution5 $f the feasible solution region does not have a bounded area the maximum value of ? occurs at infnit!' "ence the >%% is said to have unbounded solution' 2/+ &e?ne Mu*ti4*e O4tima* so*ution5 A >%% having more than one optimal solution is said to have alternative or multiple optimal solutions' 28' 3hat is s*acA 6aria@*e5 $f the constraint as general >%% be M@ t!pe then a non negative variable is introduced to convert the inequalities into equalities are called slac* variables' The values of these variables are interpretedas the amount of unused resources' 2:+ 3hat are sur4*us 6aria@*es5 $f the constraint as general >%% be N@ t!pe then a non negative is introduced to convert the inequalities into equalities are called the surplus variables' 2;+ &e?ne (asic so*ution5 Given a s!stem of m linear equations )ith n variables0mMn3'The solution obtained b! setting 0n-m3 variables equal to 8ero and solving for the remaining m variables is called a basic solution' 29' &e?ne non &egenerate (asic ,easi@*e so*ution5 The basic solution is said to be a non degenerate basic solution if (one of the basic variables is 8ero' $%+ &e?ne -egenerate @asic so*ution5 A basic solution is said to be a degenerate basic solution if one or more of the basic variables are 8ero' $2+ 3hat is the ,unction o, minimum ratioO • To determine the basic variable to leave • To determine the maximum increase in basic variable - H - Resource Management Techniques – MC9242 II MCA • To maintain the feasibilit! of follo)ing solution $2+ Brom the o4timum sim4*eC ta@*e ho1 -o 7ou i-enti,7 that "## has un@oun-e- so*ution5 To fnd the leaving variables the ratio is computed' The ratio is M@F then there is an unbounded solution to the given >%%' $$+ Brom the o4timum sim4*eC ta@*e ho1 -o 7ou i-enti,7 that the "## has no so*ution5 $f atleast one artifcial variable appears in the basis at 8ero level )ith a Ave value in the Pb column and the optimalit! condition is satisfed then the original problem has no feasible solution' $4+ o1 -o 7ou i-enti,7 that "## has no so*ution in a t1o 4hase metho-5 $f all ?, – +, C F K then atleast one artifcial variable appears in the optimum basis at non 8ero level the >%% does not possess an! solution' $/' 3hat -o 7ou un-erstan- @7 -egenerac75 The concept of obtaining a degenerate basic feasible solution in >%% is *no)n as degenerac!' This ma! occur in the initial stage )hen atleast one basic variable is 8ero in the initial basic feasible solution' $8+ 3rite the stan-ar- ,orm o, "## in the matriC notation5 $n matrix notation the canonical form of >%% can be expressed as Maximi8e ? @ +P0ob, fn'3 Sub to AP M@ b0constraints3 and P N@ F 0non negative restrictions3 Jhere + @ 0+27+<7B''+n37 A @ a22 a2< B'' a2n P @ x2 b @ b2 a<2 a<<B'' a<n 7 x< 7 b< ' ' ' ' ' ' am2 am<B' amn xn bn $:+ &e?ne @asic 6aria@*e an- non0@asic 6aria@*e in *inear 4rogramming+ - Q - Resource Management Techniques – MC9242 II MCA A basic solution to the set of constraints is a solution obtained b! setting an! n variables equal to 8ero and solving for remaining m variables not equal to 8ero' Such m variables are called basic variables and remaining n 8ero variables are called non-basic variables+ $;+So*6e the ,o**o1ing "# 4ro@*em @7 gra4hica* metho-+ <MA) =%;> MaCimiDe D G8C 2 H4C 2 Su@Fect tot the constraints. C 2 H C 2 I / C 2 J ; C 2 EC 2 J % $9+ &e?ne unrestricte- 6aria@*e an- arti?cia* 6aria@*e+ <NO' =%:> • ;nrestricted ariable 4A variable is unrestricted if it is allo)ed to ta*e on positive7 negative or 8ero values • Artifcial variable 45ne t!pe of variable introduced in a linear program model in order to fnd an initial basic feasible solutionR an artifcial variable is used for equalit! constraints and for greater-than or equal inequalit! constraints UNIT0II 2+ &e?ne trans4ortation 4ro@*em+ $t is a special t!pe of linear programming model in )hich the goods are shipped from various origins to di9erent destinations' The ob,ective is to fnd the best possible allocation of goods from various origins to di9erent destinations such that the total transportation cost is minimum' $+ &e?ne the ,o**o1ing. Beasi@*e so*ution A set of non-negative decision values xi, 0i@27<7B'mR ,@27<Bn3 satisfes the constraint equations is called a feasible solution' 4+ &e?ne the ,o**o1ing. @asic ,easi@*e so*ution A basic feasible solution is said to be basic if the number of positive allocations are mAn-2'0 m-origin and n-destination3'$f the number of allocations are less than 0mAn-23 it is called degenerate basic feasible solution' /+ &e?ne o4tima* so*ution in trans4ortation 4ro@*em A feasible solution is said to be optimal7 if it minimi8es the total transportation cost' 8+ 3hat are the metho-s use- in trans4ortation 4ro@*em to o@tain the initia* @asic ,easi@*e so*ution+ - I - Resource Management Techniques – MC9242 II MCA • (orth-)est corner rule • >o)est cost entr! method or matrix minima method • ogel#s approximation method :+ 3rite -o1n the @asic ste4s in6o*6e- in so*6ing a trans4ortation 4ro@*em+ • To fnd the initial basic feasible solution • To fnd an optimal solution b! ma*ing successive improvements from the initial basic feasible solution' ;+E 3hat -o 7ou un-erstan- @7 -egenerac7 in a trans4ortation 4ro@*emO 0(5 #FQ3 $f the number of occupied cells in a m x n transportation problem is less than 0 mAn-23 then the problem is said to be degenerate' 9+ 3hat is @a*ance- trans4ortation 4ro@*emK un@a*ance- trans4ortation 4ro@*em5 Jhen the sum of suppl! is equal to demands7 then the problem is said to be balanced transportation problem' A transportation problem is said to be unbalanced if the total suppl! is not equal to the total demand' 2%+ o1 -o 7ou con6ert an un@a*ance- trans4ortation 4ro@*em into a @a*ance- one5 The unbalanced transportation problem is converted into a balanced one b! adding a dumm! ro) 0source3 or dumm! column 0destination3 )hichever is necessar!' The unit transportation cost of the dumm! ro)1 column elements are assigned to 8ero' Then the problem is solved b! the usual procedure' 22+ EC4*ain ho1 the 4ro?t maCimiDation trans4ortation 4ro@*em can @e con6erte- to an equi6a*ent cost minimiDation trans4ortation 4ro@*em+ <MA) =%;> $f the ob,ective is to maximi8e the proft or maximi8e the expected sales )e have to convert these problems b! multipl!ing all cell entries b! -2'(o) the maximi8ation problem becomes a minimi8ation and it can be solved b! the usual algorithm 22+ &etermine @asic ,easi@*e so*ution to the ,o**o1ing trans4ortation 4ro@*em using *east cost metho-+ <MA) =%9> A ( C & SU##") # 2 2 2 4 $% ! $ $ 2 2 /% R 4 2 / 9 2% &eman- 2% 4% $% 2% 2$+ &e?ne transshi4ment 4ro@*ems5 - S - Resource Management Techniques – MC9242 II MCA A problem in )hich available commodit! frequentl! moves from one source to another source or destination before reaching its actual destination is called transshipment problems' 24+ 3hat is the -i9erence @et1een Trans4ortation 4ro@*em K Transshi4ment #ro@*em5 $n a transportation problem there are no intermediate shipping points )hile in transshipment problem there are intermediate shipping points 2/+ 3hat is assignment 4ro@*em5 An assignment problem is a particular case of a transportation problem in )hich a number of operations are assigned to an equal number of operators )here each operator performs onl! one operation7 the overall ob,ective is to maximi8e the total proft or minimi8e the overall cost of the given assignment' 28+ EC4*ain the -i9erence @et1een trans4ortation an- assignment 4ro@*ems5 Trans4ortation 4ro@*ems Assignment 4ro@*ems 23 suppl! at an! source ma! be a Suppl! at an! source )ill an! positive quantit!' be 2' <3 :emand at an! destination ma! :emand at an! destination be a positive quantit!' )ill be 2' E3 5ne or more source to an! number 5ne source one destination' of destination' 2:+ &e?ne un@oun-e- assignment 4ro@*em an- -escri@e the ste4s in6o*6e- in so*6ing it5 $f the no' of ro)s is not equal to the no' of column in the given cost matrix the problem is said to be unbalanced' $t is converted to a balanced one b! adding dumm! ro) or dumm! column )ith 8ero cost' 2;+ EC4*ain ho1 a maCimiDation 4ro@*em is so*6e- using assignment mo-e*5 The maximi8ation problems are converted to a minimi8ation one of the follo)ing method' 0i3 Since max 8 @ min0-83 0ii3 Subtract all the cost elements all of the cost matrix from the "ighest cost element in that cost matrix' - 2F - Resource Management Techniques – MC9242 II MCA 29+ 3hat -o 7ou un-erstan- @7 restricte- assignment5 EC4*ain ho1 7ou shou*- o6ercome it5 The assignment technique7 it ma! not be possible to assign a particular tas* to a particular facilit! due to technical di9iculties or other restrictions' This can be overcome b! assigning a ver! high processing time or cost 0it can be 63 to the corresponding cell' 2%+ o1 -o 7ou i-enti,7 a*ternati6e so*ution in assignment 4ro@*em5 Sometimes a fnal cost matrix contains more than required number of 8eroes at the independent position' This implies that there is more than one optimal solution )ith some optimum assignment cost' 22+ 3hat is a tra6e*ing sa*esman 4ro@*em5 A salesman normall! must visit a number of cities starting from his head quarters' The distance bet)een ever! pair of cities are assumed to be *no)n' The problem of fnding the shortest distance if the salesman starts from his head quarters and passes through each cit! exactl! once and returns to the headquarters is called Traveling Salesman problem' 22+ &e?ne route con-ition5 The salesman starts from his headquarters and passes through each cit! exactl! once' 2$+ Gi6e the areas o, o4erations o, assignment 4ro@*ems5 Assigning ,obs to machines' Allocating men to ,obs1machines' .oute scheduling for a traveling salesman' 24+ o1 -o 7ou con6ert the un@a*ance- assignment 4ro@*em into a @a*ance- one5 <MA) =%;> Since the assignment is one to one basis 7 the problem have a square matrix' $f the given problem is not square matrix add a dumm! ro) or dumm! column and then convert it into a balanced one 0square matrix3' Assign 8ero cost values for an! dumm! ro)1column and solve it b! usual assignment method' UNIT0III - 22 - Resource Management Techniques – MC9242 II MCA 2+ &e?ne Integer #rogramming #ro@*em <I##>5 <&EC =%:> A linear programming problem in )hich some or all of the variables in the optimal solution are restricted to assume non-negative integer values is called an $nteger %rogramming %roblem 0$%%3 or $nteger >inear %rogramming 2+ EC4*ain the im4ortance o, Integer 4rogramming 4ro@*em5 $n >%% the values for the variables are real in the optimal solution' "o)ever in certain problems this assumption is unrealistic' For example if a problem has a solution of I21< cars to be produced in a manufacturing compan! is meaningless' These t!pes of problems require integer values for the decision variables' Therefore $%% is necessar! to round o9 the fractional values' $+ "ist out some o, the a44*ications o, I##5 <MA) =%9> <&EC =%:> <MA) =%:> • $%% occur quite frequentl! in business and industr!' • All transportation7 assignment and traveling salesman problems are $%%7 since the decision variables are either ?ero or one' • All sequencing and routing decisions are $%% as it requires the integer values of the decision variables' • +apital budgeting and production scheduling problem are %%' $n fact7 an! situation involving decisions of the t!pe either to do a ,ob or not to do can be treated as an $%%' • All allocation problems involving the allocation of goods7 men7 machines7 give rise to $%% since such commodities can be assigned onl! integer and not fractional values' 4+ "ist the 6arious t74es o, integer 4rogramming5 <MA) =%:> Mixed $%% %ure $%% /+ 3hat is 4ure I##5 $n a linear programming problem7 if all the variables in the optimal solution are restricted to assume non-negative integer values7 then it is called the pure 0all3 $%%' 8+ 3hat is MiCe- I##5 $n a linear programming problem7 if onl! some of the variables in the optimal solution are restricted to assume non-negative integer values7 )hile the remaining variables are free to ta*e an! non- negative values7 then it is called A Mixed $%%' :+ 3hat is Lero0one 4ro@*em5 $f all the variables in the optimum solution are allo)ed to ta*e values either F or 2 as in Tdo# or Tnot to do# t!pe decisions7 then the problem is called ?ero-one problem or standard discrete programming problem' - 2< - Resource Management Techniques – MC9242 II MCA ;+ 3hat is the -i9erence @et1een #ure integer 4rogramming K miCe- integer integer 4rogramming' Jhen an optimi8ation problem7 if all the decision variables are restricted to ta*e integer values7 then it is referred as pure integer programming' $f some of the variables are allo)ed to ta*e integer values7 then it is referred as mixed integer integer programming' 9+ EC4*ain the im4ortance o, Integer #rogramming5 $n linear programming problem7 all the decision variables allo)ed to ta*e an! non-negative real values7 as it is quite possible and appropriate to have fractional values in man! situations' "o)ever in man! situations7 especiall! in business and industr!7 these decision variables ma*e sense onl! if the! have integer values in the optimal solution' "ence a ne) procedure has been developed in this direction for the case of >%% sub,ected to additional restriction that the decision variables must have integer values' 2%+ 3h7 not roun- o9 the o4timum 6a*ues in stea- o, resorting to I#5 <MA) =%;> There is no guarantee that the integer valued solution 0obtained b! simplex method3 )ill satisf! the constraints' i'e' '7 it ma! not satisf! one or more constraints and as such the ne) solution ma! not feasible' So there is a need for developing a s!stematic and e9icient algorithm for obtaining the exact optimum integer solution to an $%%' 22+ 3hat are metho-s ,or I##5 <MA) =%;> $nteger programming can be categori8ed as 0i3 +utting methods 0ii3 Search Methods' 22+ 3hat is cutting metho-5 A s!stematic procedure for solving pure $%% )as frst developed b! .'-'Gomor! in 2SLI' >ater on7 he extended the procedure to solve mixed $%%7 named as cutting plane algorithm7 the method consists in frst solving the $%% as ordinar! >%%'&! ignoring the integrit! restriction and then introducing additional constraints one after the other to cut certain part of the solution space until an integral solution is obtained' 2$+ 3hat is search metho-5 $t is an enumeration method in )hich all feasible integer points are enumerated' The )idel! used search method is the &ranch and &ound Technique' $t also starts )ith the continuous optimum7 but s!stematicall! partitions the solution space into sub problems that eliminate parts that contain no feasible integer solution' $t )as originall! developed b! A'"'>and and A'G':oig' 24+ EC4*ain the conce4t o, (ranch an- (oun- Technique5 - 2E - Resource Management Techniques – MC9242 II MCA The )idel! used search method is the &ranch and &ound Technique' $t starts )ith the continuous optimum7 but s!stematicall! partitions the solution space into sub problems that eliminate parts that contain no feasible integer solution' $t )as originall! developed b! A'"'>and and A'G':oig' 2/+ Gi6e the genera* ,ormat o, I##5 The general $%% is given b! Maximi8e ? @ +P Sub,ect to the constraints AP C b7 P D F and some or all variables are integer' 28+ 3rite an a*gorithm ,or Gomor7=s Bractiona* Cut a*gorithm5 2' +onvert the minimi8ation $%% into an equivalent maximi8ation $%% and all the coe9icients and constraints should be integers' <' Find the optimum solution of the resulting maximi8ation >%% b! using simplex method' E' Test the integrit! of the optimum solution' G' .e)rite each P &i L' -xpress each of the negative fractions if an!7 in the * th ro) of the optimum simplex table as the sum of a negative integer and a non-negative fraction' H' Find the fractional cut constraint Q' Add the fractional cut constraint at the bottom of optimum simplex table obtained in step <' I' Go to step E and repeat the procedure until an optimum integer solution is obtained' 2:+ 3hat is the 4ur4ose o, Bractiona* cut constraints5 $n the cutting plane method7 the fractional cut constraints cut the unuseful area of the feasible region in the graphical solution of the problem' i'e' cut that area )hich has no integer-valued feasible solution' Thus these constraints eliminate all the non-integral solutions )ithout loosing an! integer-valued solution' 2;+A manu,acturer o, @a@7 -o**s maAes t1o t74es o, -o**sE -o** M an- -o** )+ #rocessing o, these -o**s is -one on t1o machines A an- (+ &o** M requires 2 hours on machine A an- 8 hours on Machine (+ &o** ) requires / hours on machine A an- / hours on Machine (+ There are 28 hours o, time 4er -a7 a6ai*a@*e on machine A an- $% hours on machine (+ The 4ro?t is gaine- on @oth the -o**s is same+ Bormat this as I##5 >et the manufacturer decide to manufacture x 2 the number of doll P and x < number of doll U so as to maximi8e the proft' The complete formulation of the $%% is given b! - 2G - Resource Management Techniques – MC9242 II MCA Maximi8e ? @ x 2 Ax < Sub,ect to < x 2 A L x < C2H H x 2 A L x < CEF and DF and are integers' 29+ EC4*ain Gomor7=s MiCe- Integer Metho-5 The problem is frst solved b! continuous >%% b! ignoring the integrit! condition' $f the values of the integer constrained variables are integers7 then the current solution is an optimal solution to the given mixed $%%' -lse select the source ro) )hich corresponds to the largest fractional part among these basic variables )hich are constrained to be integers' Then construct the Gomarian constraint from the source ro)' Add this secondar! constraint at the bottom of the optimum simplex table and use dual simplex method to obtain the ne) feasible optimal solution' .epeat this procedure until the values of the integer restricted variables are integers in the optimum solution obtained' 2%+ 3hat is the geometrica* meaning o, 4ortione- or @ranche- the origina* 4ro@*em5 Geometricall! it means that the branching process eliminates portion of the feasible region that contains no feasible-integer solution' -ach of the sub-problems solved separatel! as a >%%' 22+ 3hat is stan-ar- -iscrete 4rogramming 4ro@*em5 $f all the variables in the optimum solution are allo)ed to ta*e values either F or 2 as in Tdo# or Tnot to do# t!pe decisions7 then the problem is called standard discrete programming problem' 22+ 3hat is the -isa-6antage o, @ranche- or 4ortione- metho-5 $t requires the optimum solution of each sub problem' $n large problems this could be ver! tedious ,ob' 2$+ o1 can 7ou im4ro6e the e9icienc7 o, 4ortione- metho-5 The computational e9icienc! of portioned method is increased b! using the concept of bounding' &! this concept )henever the continuous optimum solution of a sub problem !ields a value of the ob,ective function lo)er than that of the best available integer solution it is useless to explore the problem an! further consideration' Thus once a feasible integer solution is obtained7 its associative ob,ective function can be ta*en as a lo)er bound to delete inferior sub-problems' "ence e9icienc! of a branch and bound method depends upon ho) soon the successive sub-problems are fathomed' UNIT0I' 2+ 3hat -o 7ou mean @7 4roFect5 - 2L - Resource Management Techniques – MC9242 II MCA A pro,ect is defned as a combination on inter related activities )ith limited resources namel! men7 machines materials7 mone! and time all of )hich must be executed in a defned order for its completion+ 2+ 3hat are the three main 4hases o, 4roFect5 • %lanning7 Scheduling and +ontrol $+ 3hat are the t1o @asic 4*anning an- contro**ing techniques in a net1orA ana*7sis5 • +ritical %ath Method 0+%M3 • %rogramme -valuation and .evie) Technique 0%-.T> 4+ 3hat are the a-6antages o, C#M an- #ERT techniques5 • $t encourages a logical discipline in planning7 scheduling and control of pro,ects • $t helps to e9ect considerable reduction of pro,ect times and the cost • $t helps better utili8ation of resources li*e men7machines7materials and mone! )ith reference to time • $t measures the e9ect of dela!s on the pro,ect and procedural changes on the overall schedule' /+ 3hat is the -i9erence C#M an- #ERT +%M • (et)or* is built on the basis of activit! • :eterministic nature • 5ne time estimation %-.T • An event oriented net)or* • %robabilistic nature • Three time estimation 8+ 3hat is net1orA5 A net)or* is a graphical representation of a pro,ect#s operation and is composed of all the events and activities in sequence along )ith their inter relationship and inter dependencies' :+ 3hat is E6ent in a net1orA -iagram5 An event is specifc instant of time )hich mar*s the starts and end of an activit!' $t neither consumes time nor resources' $t is represented b! a circle' ;+ &e?ne acti6it75 - 2H - Resource Management Techniques – MC9242 II MCA A pro,ect consists of a number of ,ob operations )hich are called activities' $t is the element of the pro,ect and it ma! be a process7 material handling7 procurement c!cle etc' 9+ &e?ne Critica* Acti6ities5 $n a (et)or* diagram critical activities are those )hose if consumer more than estimated time the pro,ect )ill be dela!ed' 2%+ &e?ne non critica* acti6ities5 Activities )hich have a provision such that the event if the! consume a specifed time over and above the estimated time the pro,ect )ill not be dela!ed are termed as non critical activities' 22+ &e?ne &umm7 Acti6ities5 Jhen t)o activities start at a same time7 the head event are ,oined b! a dotted arro) *no)n as dumm! activit! )hich ma! be critical or non critical' 22+ &e?ne -uration5 $t is the estimated or the actual time required to complete a trade or an activit!' 2$+ &e?ne tota* 4roFect time5 $t is time ta*en to complete to complete a pro,ect and ,ust found from the sequence of critical activities' $n other )ords it is the duration of the critical path' 24+ &e?ne Critica* #ath5 $t is the sequence of activities )hich decides the total pro,ect duration' $t is formed b! critical activities and consumes maximum resources and time' 2/+ &e?ne Noat or s*acA5 <MA) =%;> Slac* is )ith respect to an event and =oat is )ith respect to an activit!' $n other )ords7 slac* is used )ith %-.T and =oat )ith +%M' Float or slac* means extra time over and above its duration )hich a non-critical activit! can consume )ithout dela!ing the pro,ect' 28+ &e?ne tota* Noat5 <MA) =%;> The total =oat for an activit! is given b! the total time )hich is available for performance of the activit!7 minus the duration of the activit!' The total time is available for execution of the activit! is given b! the latest fnish time of an activit! minus the earliest start time for the activit!' Thus Total =oat @ >atest start time – earliest start time' - 2Q - Resource Management Techniques – MC9242 II MCA 2:+ &e?ne ,ree Noat5 <MA) =%;> This is that part of the total =oat )hich does not a9ect the subsequent activities' This is the =oat )hich is obtained )hen all the activities are started at the earliest' 2;+ &e?ne In-e4en-ent Noat5 <MA) =%:> <MA) =%;> $f all the preceding activities are completed at their latest7 in some cases7 no =oat available for the subsequent activities )hich ma! therefore become critical' $ndependent =oat @ free – tail slac*' 29+ &e?ne Inter,ering Noat5 Sometimes =oat of an activit! if utili8ed )holl! or in part7 ma! in=uence the starting time of the succeeding activities is *no)n as interfering =oat' $nterfering =oat @ latest event time of the head - earliest event time of the event' 2%+ &e?ne O4timistic5 5ptimistic time estimate is the duration of an! activit! )hen ever!thing goes on ver! )ell during the pro,ect' 22+ &e?ne #essimistic5 %essimistic time estimate is the duration of an! activit! )hen almost ever!thing goes against our )ill and a lot of di9iculties is faced )hile doing a pro,ect' 22+ &e?ne most *iAe*7 time estimation5 Most li*el! time estimate is the duration of an! activit! )hen sometimes thing go on ver! )ell7 sometimes things go on ver! bad )hile doing the pro,ect' 24+ 3hat is a 4ara**e* critica* 4ath5 Jhen critical activities are crashed and the duration is reduced other paths ma! also become critical such critical paths are called parallel critical path' 2/+ 3hat is stan-ar- -e6iation an- 6ariance in #ERT net1orA5 <NO' =%:> The expected time of an activit! in actual execution is not completel! reliable and is li*el! to var!' $f the variabilit! is *no)n )e can measure the reliabilit! of the expected time as determined from three estimates' The measure of the variabilit! of possible activit! time is given b! standard deviation7 their probabilit! distribution ariance of the activit! is the square of the standard deviation 28+ Gi6e the -i9erence @et1een -irect cost an- in-irect cost5 <NO' =%:> - 2I - Resource Management Techniques – MC9242 II MCA :irect cost is directl! depending upon the amount of resources involved in the execution of all activities of the pro,ect' $ncrease in direct cost )ill decrease in pro,ect duration' $ndirect cost is associated )ith general and administrative expenses7 insurance cost7 taxes etc' $ncrease in indirect cost )ill increase in pro,ect duration' UNIT0' 2+ &e?ne Oen-a*=s notation ,or re4resenting queuing mo-e*s+ A queuing model is specifed and represented s!mbolicall! in the form 0a1b1c3 4 0d1e3 Jhere a- inter arrival time b-service mechanism c-number of service d-the capacit! of the s!stem e-the queue discipline 2+ In a su4er marAetE the a6erage arri6a* rate o, customer is / in e6er7 $% minutes ,o**o1ing #oisson 4rocess+ The a6erage time is taAen @7 the cashier to *ist an- ca*cu*ate the customer=s 4urchase is 4+/ minutesP ,o**o1ing eC4onentia* -istri@ution+ 3hat is the 4ro@a@i*it7 that the queue *ength eCcee-s /5 Arrival rate@ L1EF min Service rate@<1Smin %robabilit! that the queue length exceeds L @ 0V3 nA< @ 0'QL3 Q @F'2EE $+ EC4*ain !ueue -isci4*ine an- its 6arious ,orms+ 0i3 F$F5 or F+FS - First $n First 5ut or First +ome First Served' 0ii3 >$F5 or >+FS - >ast $n First 5ut or >ast +ome First Served' 0iii3 S$.5 - Selection for service in random order' 0iv3 %$. - %riorit! in selection 4+ &istinguish @et1een transient an- stea-7 state queuing s7stem+ A s!stem is said to be in transient state )hen its operating characteristics are dependent on time' A stead! state s!stem is one in )hich the behavior of the s!stem is independent of time' /' &e?ne stea-7 state5 - 2S - Resource Management Techniques – MC9242 II MCA A s!stem is said to be in stead! state )hen the behavior of the s!stem independent of time' >et p n 0t3 denote the prob that there are Tn# units in the s!stem at time t' then in stead! state@N lim p n W0 t 3@F tX6 8+ 3rite -o1n the *itt*e ,ormu*a5 > s @> q AY1Z Jhere > s @ the average no' of customers in the s!stem > q @ the average no' of customers in the queue :+ I, tra9ic intensit7 o, MQMQI s7stem is gi6en to @e %+:8E 1hat 4ercent o, time the s7stem 1ou*- @e i-*e5 Tra9ic intensit! @ F'QH 0bus! time3 S!stem to be idle @ 2-F'QH @F'<G ;+ 3hat are the @asic e*ements o, queuing s7stem5 S!stem consists of the arrival of customers7 )aiting in queue7 pic* up for service according to certain discipline7 actual service and departure of customer' 9+ 3hat is meant @7 queue -isci4*ine5 The manner in )hich service is provided or a customer is selected for service is defned as the queue discipline+ 2%+ 3hat are the c*assi?cations o, queuing mo-e*s5 m [ m [ $ [6 m [ m [ $ [n m [ m [ c[6 m [ m [ c [n 22+ 3hat are the characteristic o, queuing 4rocess5 Arrival pattern of customers7 service pattern of servers7 queue discipline7 s!stem capacit!7 no' of service channels7 no' of service stage' 22+ &e?ne #oisson 4rocess5 The %oisson process is a continuous parameter discrete state process 0ie3 a good model for man! practical situations' if P0t3represents the no' of 5ccurrences of a certain in 0F7 t3 then the discrete random process \P 0t3] is called the %oisson process' if it satisfes the follo)ing postulates $' %^2 occurrence in 0t7tA_t3` @Y_t A 50_t3 $$' %^F occurrence in 0t7tA_t3` @2-Y_t A 50_t3 $$$' %^< or more occurrence in 0t7tA_t3` @50_t3 $' P0t3 is independent of the number of occurrences of the event in an! interval prior 0or3 after the interval0F7t3 ' The prob that the event occurs a specifed number of times in 0t F 7 t F At3 depends onl! on t but not on t F' 2$+ Gi6en an7 t1o eCam4*es o, #oisson 4rocess5 - <F - Resource Management Techniques – MC9242 II MCA 2' The number of incoming telephone calls received in a particular time <' The arrival of customer at a ban* in a da! 24+ 3hat are the 4ro4erties o, #oisson 4rocess5 2' The %oisson process is a mar*ov process' <' Sum of t)o independent poissen processes is a poisson process' E' :i9erence of t)o independent poisson processes is not poisson process' G' The inter arrival time of a poisson process has an exponential distribution )ith mean 21Y' 2/+ Customer arri6es at a one0man @ar@er sho4 accor-ing to a #oisson 4rocess 1ith an mean inter arri6a* time o, 22 minutes+ Customers s4en- a a6erage o, 2% minutes in the @ar@er=s chain+3hat is the eC4ecte- no o, customers in the @ar@er sho4 an- in the queue5 Given mean arrival rate 21Y @ 2<' Therefore Y @ 212< per minute' Mean service rate 21Z @ 2F' Therefore Z @ 212F per minute' -xpected number of customers in the s!stem > s @ Y1Z-Y @ 212<1212F-212< @ L customers' 28' &e?ne 4ure @irth 4rocess5 $f the death rates Z* @ F for all * @ 27 <BB )e have a pure birth process' 2Q+ 3rite -o1n the 4ostu*ates o, @irth an- -eath 4rocess5 23 p ^2 birth 0t7 t A _t3` @ Yn0t3_t A F0_t3 <3 p ^F birth in 0t7 t A _t3` @ 2 - Yn0t3_t A F0_t3' E3 p ^2 death in 0t7 t A _t3` @ Zn0t3_t A F0_t3 G3 p ^F death in 0t7 t A _t3` @ 2 - Zn0t3_t A F0_t3' 2;+ 3hat is the ,ormu*a ,or the 4ro@*em ,or a customer to 1ait in the queue un-er <mQmQ2 NQBCBS> J s @ > s 1Y' 29+ 3hat is the a6erage num@er o, customers in the s7stem un-er <mQmQe. RQBCBS>5 YZ 0Y1Z3 c 1 0c-23a0cZ - Y 3 < A Y1Z' - <2 - Resource Management Techniques – MC9242 II MCA 2%+ 3hat is the -i9erence @et1een 4ro@a@i*istic -eterministic an- miCe- mo-e*s5 %robabilistic4 Jhen there is uncertaint! in both arrivals rate and service rate are assumed to be random variables' :eterministic4 &oth arrival rate and service rate are constants' Mixed4 Jhen either the arrival rate or the service rate is exactl! *no)n and the other is not *no)n' 22+ 3hat are the assum4tions in mQmQ2 mo-e*5 0i3 -xponential distribution of inter arrival times or poisson distribution of arrival rate' 0ii3 /ueue discipline is frst come7 frst serve' 0iii3 Single )aiting line )ith no restriction no length of queue' 0iv3 Single server )ith exponential distribution of service times' 22+ #eo4*e arri6e at a theatre ticAet @ooth in 4oisson -istri@ute- arri6a* rate o, 2/Qhour+ Ser6ice time is constant at 2 minutes+ Ca*cu*ate the mean5 Y @ <L1hr Z @ 0b3HF @ EF per hour' V @ Y1Z @ <L1EF @ L1H @ F'IEE >q @ V < 1 2-V @ 0'IEE3 < 1 2 - 'IEE @ G'2LLF< Mean )aiting time@>q1Y @ G1<L @ G1<L c HF @ S'H minutes' - << -