Mcqs linear prog

Reviewer in Linear Programming MCQS

1.   
Question Question Bank Bank US05FBCA01- US05FBCA01- Operations Operations ResearchResearch
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TYBCATYBCA
US05FBCA01 – OPERATIONS RESEARCHUS05FBCA01 – OPERATIONS RESEARCH
QUESTION BANKQUESTION BANK
MULTIPLEMULTIPLE CHOICE QUESTIONS CHOICE QUESTIONS
UNIT 1UNIT 1
1.1.Operations research is the application of ____________methods to arrive at the optimalOperations research is the application of ____________methods to arrive at the optimal
solutions to the problems.solutions to the problems.
A. A. economical economical C. C. a a and and b b bothboth
B.B.scientificscientific   D. artisticD. artistic
2.2.Feasible solution satisfies __________Feasible solution satisfies __________
A. A. Only Only constraints constraints C.C.
[a] and [b] both[a] and [b] both  
B. B. only only non-negative non-negative restriction restriction D. D. [a],[b] [a],[b] and and Optimum Optimum solutionsolution
3.3.In Degenerate solution value of objective function _____________.In Degenerate solution value of objective function _____________.
A. A. increases increases infinitely infinitely C. C. basic basic variables variables are are nonzerononzero
B. B. decreases decreases infinitely infinitely D.D. One or more One or more basic variables are basic variables are zerozero  
4.4.Minimize Z = Minimize Z = ____________________________
A. A. –maximize(Z) –maximize(Z) C.C. -maximize(-Z)-maximize(-Z)  
B. B. maximize(-Z) maximize(-Z) D. D. none none of of the the aboveabove
5.5.In graphical method the restriction on number of In graphical method the restriction on number of constraint is __________.constraint is __________.
A. A. 2 2 C. C. not not more more than than 33
B. B. 3 3 D.D. none of the abovenone of the above  
6.6.In graphical representation the bounded region is known as _________ region.In graphical representation the bounded region is known as _________ region.
A. A. Solution Solution C. C. basic basic solutionsolution
B.B.feasible solutionfeasible solution   D. optimalD. optimal
7.7.Graphical optimal value Graphical optimal value for Z for Z can be obtained fromcan be obtained from
A.A.Corner points of feasible regionCorner points of feasible region   C. C. Both Both a a and and cc
B. B. corner corner points points of of the the solution solution region region D. D. none none of of the the aboveabove
8.8.In LPP the condition to be satisfied In LPP the condition to be satisfied isis
A. A. Constraints Constraints have have to to be be linear linear C.C. both [a ]and [b]both [a ]and [b]  
B. B. Objective Objective function function have have to to be be linear linear D. D. none none of of the the aboveabove
9.9.The solution to LPP give below is,The solution to LPP give below is,
11 22 11 22 11 22 11 22
33 1144 11,, 22 ,, 00 Max  Max Z Z x x x x subject subject to to x x x x x x x x where where x x xx= = + + − − ≥ ≥ − − + + ≥ ≥ ≥≥
  
A. A. Unbounded Unbounded solution solution C. C. Max Max Z Z = = 33
B. B. Max Max Z Z = = 14 14 D.D. Infeasible solutionInfeasible solution  
10.10.The solution to LPP give below is ______________The solution to LPP give below is ______________
11 22 11 22 11 22 11 22
3300 1155 22 2 2 2 2,, 22 2 22 2 , 0 , 0 Max  Max Z Z x x x x subject subject to to x x x x x x x x where where x x xx= = − − − − ≤ ≤ − − + + ≤ ≤ ≥≥
  
A.A.Unbounded solutionUnbounded solution   C. C. Max Max Z Z = = 1515
B. B. Max Max Z Z = = 30 30 D. D. Infeasible Infeasible solutionsolution
11.11.In the definition of LPP In the definition of LPP m stands for number m stands for number of constraints and n of constraints and n for number for number ofof
variables, then which of the following relations holdvariables, then which of the following relations hold
A. A. m m = = n n C. C. mm ≥≥ n n
B. mB. m ≤≤  n n D. D. None None of of themthem

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12.12.The linear function of variables which is to be maximized or minimized is calledThe linear function of variables which is to be maximized or minimized is called
 ________ ________
A. A. constraints constraints C.C. objective functionobjective function  
B. basic B. basic requirements requirements D. D. none none of of themthem
EXTRAEXTRA
13.13.Operation research approach isOperation research approach is
A.A.Multi-disciplinaryMulti-disciplinary   C. C. IntuitiveIntuitive
B. B. Artificial Artificial D.D. All of the aboveAll of the above  
14.14.Operation research analysts do notOperation research analysts do not  
A.A.Predict future operationPredict future operation   C. C. Collect Collect the the relevant relevant datadata
B. B. Build Build more more than than one one model model D. D. Recommend Recommend decision decision and and acceptaccept  
15.15.Mathematical model of Linear Programming is important becauseMathematical model of Linear Programming is important because  
A.A.It helps in converting the verbal description and numerical data intoIt helps in converting the verbal description and numerical data into
mathematical expressionmathematical expression  
B. B. decision decision makers makers prefer prefer to to work work with with formal formal modelsmodels
C. C. it it captures captures the the relevant relevant relationship relationship among among decision decision factorsfactors
D. D. it it enables enables the the use use of of algebraic algebraic techniquestechniques
16.16.A constraint in an LP model restrictsA constraint in an LP model restricts  
A. A. value value of of the the objective objective function function C. C. use use of of the the available available resourcesresources
B.B.value of the decision variablevalue of the decision variable  D. D. all all of of the the aboveabove
17.17.In graphical method of linear In graphical method of linear programming problem if the ios-cost line coincide with aprogramming problem if the ios-cost line coincide with a
side of region of basic feasible solutions we getside of region of basic feasible solutions we get  
A. A. Unique Unique optimum optimum solution solution C. no C. no feasible feasible solutionsolution
B. B. unbounded unbounded optimum optimum solution solution D.D.Infinite number of optimumInfinite number of optimum
solutionssolutions  
18.18.A feasible solution of LPPA feasible solution of LPP  
A.A.Must satisfy all the constraints simultaneouslyMust satisfy all the constraints simultaneously  
B. B. Need Need not not satisfy satisfy all all the the constraints, constraints, only only some some of of themthem
C. C. Must Must be be a a corner corner point point of of the the feasible feasible regionregion
D. D. all all of of the the aboveabove
19.19.The objective function for a L.P model is 3xThe objective function for a L.P model is 3x11+2x+2x22, if x, if x11=20 and x=20 and x22=30, what is the value=30, what is the value
of the objective function?of the objective function?  
A. A. 0 0 C. C. 6060
B. B. 50 50 D.D. 120120  
20.20.Maximization of objective function in LPP meansMaximization of objective function in LPP means  
A. A. Value Value occurs occurs at at allowable allowable set set decisiondecision
B.B.highest value is chosen among allowable decisionhighest value is chosen among allowable decision  
C. C. none none of of the the aboveabove
D. D. all all of of the the aboveabove
21.21.Alternative solution exist in a linear programming problem whenAlternative solution exist in a linear programming problem when  
A. A. one one of of the the constraint constraint is is redundantredundant
B.B.objective function is parallel to one of the constraintsobjective function is parallel to one of the constraints  
C. C. two two constraints constraints are are parallelparallel
D. D. all all of of the the aboveabove

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22.22.Linear programming problem involving only two varia bles can be solved byLinear programming problem involving only two varia bles can be solved by
 ______________  ______________   
A. A. Big Big M M method method C.C. Graphical methodGraphical method  
B. B. Simplex Simplex method method D. D. none none of of thesethese
23.23.The linear function of the variables which is to be maximize or minimize is calledThe linear function of the variables which is to be maximize or minimize is called
 _________  _________   
A. A. Constraints Constraints C. C. Decision Decision variablevariable
B.B.Objective functionObjective function   D. D. None None of of the the abovabov
24.24.A physical model is an example ofA physical model is an example of  
A.A.An iconic modelAn iconic model   C. C. A A verbal verbal modelmodel
B. B. An An analogue analogue model model D. D. A A mathematical mathematical modelmodel
25.25.If the value of the objective function z can be increased or decreased indefinitely, suchIf the value of the objective function z can be increased or decreased indefinitely, such
solution is called _____________ solution is called _____________   A. A. Bounded Bounded solution solution C. C. SolutionSolution
B.B.Unbounded solutionUnbounded solution   D. D. None None of of the the aboveabove
26.26.A model isA model is  
A. A. An An essence essence of of reality reality C. C. An An idealizationidealization
B. B. An An approximation approximation D.D. All of the aboveAll of the above  
27.27.The first step in formulating a The first step in formulating a linear programming problem islinear programming problem is  
A. A. Identify Identify any any upper upper or or lower lower bound bound on on the the decision decision variablesvariables
B. B. State State the the constraints constraints as as linear linear combinations combinations of of the the decision decision variablesvariables
C. C. Understand Understand the the problemproblem
D.D.Identify the decision variablesIdentify the decision variables  
UNIT 2UNIT 2
1.1.In the simplex method for solving of In the simplex method for solving of LPP number of variables can be ______________.LPP number of variables can be ______________.
A. A. Not Not more more than than three three C.C. at least twoat least two  
B. at B. at least least three three D. D. none none of of themthem
2.2.In the simplex method the variable enters tIn the simplex method the variable enters the basis if __________________.he basis if __________________.
A. A. ZZ j j – C – C j j  ≥≥  0 0 C.C. ZZjj – C – Cjj < 0 < 0
B. B. ZZ
 j j – C – C
 j j  ≤≤  0 0 D. D. ZZ
 j j – C – C
 j j = 0 = 0
3.3.In the simplex method the variable leaves the In the simplex method the variable leaves the basis if the ratio isbasis if the ratio is
A. A. maximum maximum C.C. 00  
B. B. minimum minimum D. D. none none of of themthem
4.4.The _________ variable is added to the constraint of less tThe _________ variable is added to the constraint of less than equal to type.han equal to type.
A.A.slackslack   C. artificial C. artificial
B. B. surplus surplus D. D. basic basic
5.5.For the constraint of greater than equal to type we make use of ______________For the constraint of greater than equal to type we make use of ______________
variable.variable.
A. A. slack slack C. C. artificial artificial
B.B.surplussurplus   D. D. basic basic
6.6.The coefficient of slack variable in the objective function The coefficient of slack variable in the objective function is _________________.is _________________.
A. A. -M -M C.C. 00  
B. B. +M +M D. D. none none of of themthem

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7.7.The coefficient of artificial variable in the objective function of maximization problem isThe coefficient of artificial variable in the objective function of maximization problem is
 _________. _________.
A.A.-M-M   C. C. 00
B. B. +M +M D. D. none none of of themthem
EXTRAEXTRA
8.8.The role of artificial variables in the The role of artificial variables in the simplex method issimplex method is  
A. A. to to aid aid in in finding finding an an initial initial solutionsolution
B. B. to to find find optimal optimal dual dual prices prices in in the the final final simplex simplex tabletable
C. C. to to start start with with Big Big M M methodmethod
D.D.all of theseall of these  
9.9.For a minimization problem, the objective function coeffFor a minimization problem, the objective function coefficient for an artificial variable isicient for an artificial variable is  
A.A.+ M+ M C. ZeroC. Zero
B. B. -M -M D. D. None None of of thesethese
10.10.For maximization LPP, the simplex method is terminated For maximization LPP, the simplex method is terminated when all valueswhen all values
A.A.cj –zjcj –zj≤≤ 0 0   C. C. cj cj –zj –zj = = 00
B. B. cj cj –zj–zj≥≥  0 0 D. D. zjzj ≤≤ 0 0
11.11.If any value in b If any value in b - column of final simplex - column of final simplex table is negative, then the solutable is negative, then the solution istion is  
A. A. unbounded unbounded C. C. optimal optimal
B.B.infeasibleinfeasible   D. D. None None of of thesethese
12.12.To convertTo convert≥≥ inequality constraints into equality constraints, we must inequality constraints into equality constraints, we must  
A. A. add add a a surplus surplus variablevariable
B. B. subtract subtract an an artificial artificial variablevariable
C.C.subtract a surplus variable and an add artificial variablesubtract a surplus variable and an add artificial variable  
D. D. add add a a surplus surplus variable variable and and subtract subtract an an artificial artificial variablevariable
13.13.In the optimal simplex table cj In the optimal simplex table cj –zj = 0 value indicates–zj = 0 value indicates  
A. A. unbounded unbounded solution solution C.C. alternative solutionalternative solution  
B. B. cycling cycling D. D. None None of of thesethese
14.14.At every iteration of simplex method, for minimization problem, a variable in the currentAt every iteration of simplex method, for minimization problem, a variable in the current
basis is replaced with another variable that hasbasis is replaced with another variable that has  
A. A. a a positive positive cj cj –zj –zj value value C. C. cj cj –zj –zj = = 00
B.B.a negative cj –zj valuea negative cj –zj value   D. D. None None of of thesethese
15.15.A variable which does not appear in the basis variable (B) column of A variable which does not appear in the basis variable (B) column of simplex table issimplex table is  
A. A. never never equal equal to to zero zero C. C. called called basic basic variablevariable
B.B.always equal to zeroalways equal to zero   D. D. None None of of thesethese
16.16.To formulate a problem for solution by the simplex method, we must add artificialTo formulate a problem for solution by the simplex method, we must add artificial
variable tovariable to  
A. A. only only equality equality constraints constraints C.C. both A & Bboth A & B  
B. B. only only > > constraints constraints D. D. None None of of thesethese
17.17.If all xij values in tIf all xij values in the incoming variable column of the simplex table are he incoming variable column of the simplex table are negative, thennegative, then  
A.A.solution is unboundedsolution is unbounded   C. C. there there exist exist no no solutionsolution
B. B. there there are are multiple multiple solution solution D. D. None None of of thesethese
18.18.If an artificial variable is present in the basic variable column of optimal simplex table,If an artificial variable is present in the basic variable column of optimal simplex table,
then the solution isthen the solution is  
A. A. unbounded unbounded C. C. optimal optimal
B.B.infeasibleinfeasible   D. D. None None of of thesethese

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19.19.If for a given solution, If for a given solution, a slack variable is equal to zero, thena slack variable is equal to zero, then  
A.A.the solution is optimalthe solution is optimal   C. C. there there exist exist no no solutionsolution
B. B. the the solution solution is is infeasible infeasible D. D. None None of of thesethese
20.20.Linear programming problem involving more than two variables can be solved byLinear programming problem involving more than two variables can be solved by
 ______________  ______________   
A. A. Simplex Simplex method method C. C. Matrix Matrix minima minima methodmethod
B. Graphical B. Graphical method method D. D. None None of of thesethese
UNIT 3UNIT 3
1.1.The TP The TP is said is said to be unto be unbalanced if balanced if _______________._______________.
A.A.
≠≠
∑ ∑ ∑∑
i i jj
a a bb  
C. m=nC. m=n
B.B.
∑ ∑ ∑∑i i jj
a a = = bb  
D. D. m+n-1= m+n-1= no. no. of of allocated allocated cellcell
2.2.The initial solution of a transportation problem can be obtained by applying any knownThe initial solution of a transportation problem can be obtained by applying any known
method. However, the only condition is thatmethod. However, the only condition is that
A.A.Rim condition should be satisfiedRim condition should be satisfied   C. C. one one of of the the XXijij< 0< 0
B. B. cost cost matrix matrix should should be be square square D. None D. None of of themthem
3.3.In non-degenerate solution number of allocated cell is____________.In non-degenerate solution number of allocated cell is____________.
A. A. Equal Equal to to m+n-1 m+n-1 C. C. Equal Equal to to m+n+1m+n+1
B.B.Not equal to m+n-1Not equal to m+n-1   D. D. Not Not equal equal to to m+n+1m+n+1
4.4.From the following methods ___________ is a method to obtain initial solution toFrom the following methods ___________ is a method to obtain initial solution to
Transportation Problem.Transportation Problem.
A.A.North-WestNorth-West   C. HungarianC. Hungarian
B. B. Simplex Simplex D. D. Newton Newton RaphsonRaphson
5.5.The Penalty in VAM represents difference between _________ cost of respectiveThe Penalty in VAM represents difference between _________ cost of respective
row / column.row / column.
A. A. Two Two Largest Largest C. C. largest largest and and smallestsmallest
B.B.smallest twosmallest two   D. D. none none of of themthem
6.6.Number of basic allocation in any row or column Number of basic allocation in any row or column in Assignment Problem can bein Assignment Problem can be
A.A.Exactly oneExactly one   C. C. at at least least oneone
B. B. at at most most one one D. D. none none of of themthem
7.7.North – West corner refers to ____________.North – West corner refers to ____________.
A.A.top left cornertop left corner   C. C. both both of of themthem
B. top B. top right right corner corner D. D. none none of of themthem
8.8.The ______________ method's solution for transportation problem is sometimes anThe ______________ method's solution for transportation problem is sometimes an
optimal solution itself.optimal solution itself.
A. A. NWCM NWCM C. C. LCM LCM
B.B.VAMVAM   D. D. Row Row MinimaMinima
9.9.In Assignment Problem the value of decision variable xIn Assignment Problem the value of decision variable x
ijijis____.is____.
A. A. no no restriction restriction C.C. one or zeroone or zero  
B. B. two two or or one one D. D. none none of of themthem

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10.10.If number of sources is not equal tIf number of sources is not equal to number of destination in Assignment problem then o number of destination in Assignment problem then itit
is called ___________.is called ___________.
A.A.unbalancedunbalanced   C. unsymmetricC. unsymmetric
B. B. symmetric symmetric D. D. balanced balanced
11.11.The _____ method used to obtain optimum The _____ method used to obtain optimum solution of travelling salesman problem.solution of travelling salesman problem.
A. A. Simplex Simplex C. C. dominance dominance
B.B.HungarianHungarian   D. graphicalD. graphical
EXTRAEXTRA  
12.12.The initial solution of a transportation problem can be obtained by applying any knownThe initial solution of a transportation problem can be obtained by applying any known
method. However, the only condition is thatmethod. However, the only condition is that  
A. A. the the solution solution be be optimaloptimal
B.B.the rim condition are satisfiedthe rim condition are satisfied  
C. C. the the solution solution not not be be degeneratedegenerate
D. D. all all of of the the aboveabove
13.13.The dummy source or destination in a transportation problem is added tThe dummy source or destination in a transportation problem is added too  
A.A.satisfy rim conditionsatisfy rim condition  
B. B. prevent prevent solution solution from from becoming becoming degeneratedegenerate
C. C. ensure ensure that that total total cost cost does does not not exceed exceed a a limitlimit
D. D. all all of of the the aboveabove
14.14.The occurrence of degeneracy while solving a transportation problem means thatThe occurrence of degeneracy while solving a transportation problem means that  
A. A. total total supply supply equals equals total total demanddemand
B.B.the solution so obtained is not feasiblethe solution so obtained is not feasible  
C. C. the the few few allocations allocations become become negativenegative
D. D. none none of of the the aboveabove
15.15.An alternative optimal solution to a minimization transportation problem exists wheneverAn alternative optimal solution to a minimization transportation problem exists whenever
opportunity cost corresponding to unused routes of transportation is:opportunity cost corresponding to unused routes of transportation is:  
A. A. positive positive and and greater greater than than zerozero
B.B.positive with at least one equal to zeropositive with at least one equal to zero  
C. C. negative negative with with at at least least one one equal equal to to zerozero
D. D. all all of of the the aboveabove
16.16.One disadvantage of using North-West Corner Rule to find initial solution to theOne disadvantage of using North-West Corner Rule to find initial solution to the
transportation problem is thattransportation problem is that  
A. A. it it is is complicated complicated to to useuse
B.B.it does not take into account cost it does not take into account cost of transportationof transportation  
C. C. it it leads leads to to degenerate degenerate initial initial solutionsolution
D. D. all all of of the the aboveabove
17.17.The solution to a transportation problem with m-rows and n-columns is feasible ifThe solution to a transportation problem with m-rows and n-columns is feasible if
number of positive allocations arenumber of positive allocations are  
A. A. m+n m+n C.C. m + n -1m + n -1  
B. B. m m x x n n D. D. all all of of the the aboveabove
18.18.The calculation of opportunity cost in the MODI The calculation of opportunity cost in the MODI method is analogous to amethod is analogous to a  
A.A.cj –zj value for non-basic cj –zj value for non-basic variable columns in the simplex methodvariable columns in the simplex method  
B. B. value value of of a a variable variable in in b-column b-column of of the the simplex simplex methodmethod
C. C. variable variable in in xb-columnxb-column
D. D. all all of of the the aboveabove

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19.19.If we were to use opportunity cost value fIf we were to use opportunity cost value for an unused cell to test optimality, or an unused cell to test optimality, it should beit should be  
A. A. equal equal to to zero zero C. C. most most positive positive numbernumber
B.B.most negative numbermost negative number   D. D. all all of of the the aboveabove
20.20.An assignment problem is considered as a particular case of a Transportation problemAn assignment problem is considered as a particular case of a Transportation problem
becausebecause
A. A. the the number number of of rows rows equals equals columnscolumns
B. B. all all xxijij= 0= 0
C. C. all all rim rim conditions conditions are are 11
D.D.all of aboveall of above  
21.21.The purpose of a dummy row or column in The purpose of a dummy row or column in an assignment problem is toan assignment problem is to  
A.A.obtain balance between total activities and total resourcesobtain balance between total activities and total resources  
B. B. prevent prevent a a solution solution from from becoming becoming degeneratedegenerate
C. C. provide provide a a means means of of representing representing a a dummy dummy problemproblem
D. D. none none of of the the aboveabove
22.22.The The Hungarian method for solving an assiHungarian method for solving an assignment problem can also be used to solvegnment problem can also be used to solve  
A. A. a a transportation transportation problem problem C. both C. both A A and and BB
B.B.a traveling salesman problema traveling salesman problem  D. D. only only BB
23.23.An optimal of an assignment problem can be An optimal of an assignment problem can be obtained only ifobtained only if  
A. A. each each row row and and column column has has only only one one zero zero elementelement
B. B. each each row row and and column column has has at at least least one one zero zero elementelement
C. C. the the data data are are arrangement arrangement in in a a square square matrixmatrix
D.D.none of the abovenone of the above  
24.24.The method used for soThe method used for solving an assignment probllving an assignment problem is em is calledcalled  
A. reduced A. reduced matrix matrix method method C.C. Hungarian methodHungarian method  
B. MODI B. MODI method method D. D. none none of of the the aboveabove
UNIT 4UNIT 4
1.1.Dynamic programming is a mathematical technique dealing with the optimization ofDynamic programming is a mathematical technique dealing with the optimization of
 _______ stage decision process _______ stage decision process..
A.A.multimulti   C. C. both both A A and and BB
B. B. single single D. D. none none of of themthem
2.2.In sequencing if smallest time for a job belongs to machine-1 then that job has to placedIn sequencing if smallest time for a job belongs to machine-1 then that job has to placed
 ___________ of the sequence. ___________ of the sequence.
A. A. in in the the middle middle C.C. in the startingin the starting  
B. B. at at end end D. D. none none of of themthem
3.3.In sequencing the time involved in moving jobs from one machine to another isIn sequencing the time involved in moving jobs from one machine to another is
 ________. ________.
A.A.negligiblenegligible   C. C. positive positive numbernumber
B. B. significant significant D. D. none none of of themthem
4.4. ___________ operation is carried ou ___________ operation is carried out on a machine at a time.t on a machine at a time.
A. A. Two Two C. C. atleast atleast oneone
B.B.only oneonly one   D. D. none none of of themthem
5.5.Processing Processing time time MMijij’s are __________of order of processing the jobs.’s are __________of order of processing the jobs.
A. A. dependent dependent C. C. negligible negligible
B.B.independentindependent D. D. none none of of themthem

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6.6.Activity which starts only after finishing other activity is called _____________.Activity which starts only after finishing other activity is called _____________.
A. A. dummy dummy C.C. successorsuccessor  
B. B. Predecessor Predecessor D. D. none none of of themthem
7.7.Burst and Merge are types of ___________ in networking.Burst and Merge are types of ___________ in networking.
A.A.eventevent C. arrowC. arrow
B. B. activity activity D. D. tools tools
8.8.Activity which does not require any resources or time is called __________.Activity which does not require any resources or time is called __________.
A.A.dummydummy   C. successorC. successor
B. B. Predecessor Predecessor D. D. none none of of themthem
9.9.Event indicates ____________ of activity.Event indicates ____________ of activity.
A. A. starting starting C.C. both A and Bboth A and B
B. B. ending ending D. D. none none of of themthem
10.10. _____________ is indicated by dotted arrow _____________ is indicated by dotted arrow..
A. A. burst burst event event C.C. dummy activitydummy activity  
B. B. merge merge event event D. D. none none of of themthem
11.11. ______ event represents beginnin ______ event represents beginning of more than one activities.g of more than one activities.
A.A.burstburst   C. C. dummy dummy
B. B. merge merge D. D. none none of of themthem
12.12.Merge event represents ___________ of two or more events.Merge event represents ___________ of two or more events.
A. A. beginning beginning C. C. splitting splitting
B.B.completioncompletion   D. D. none none of of themthem
13.13.Activity which is completed before starting new activity is called___________.Activity which is completed before starting new activity is called___________.
A. A. dummy dummy C. C. successor successor
B.B.predecessorpredecessor   D. D. none none of of themthem
EXTRAEXTRA  
14.14.The Objective of network analysis toThe Objective of network analysis to  
A.A.Minimize total project durationMinimize total project duration  
B. B. Minimize Minimize total total project project costcost
C. C. Minimize Minimize production production delays, delays, interruption interruption and and conflictsconflicts
D. D. All All of of the the aboveabove
15.15.Network models have advantage in terms of projectNetwork models have advantage in terms of project  
A. A. Planning Planning C. C. Controlling Controlling
B. B. Scheduling Scheduling D.D. All of the aboveAll of the above  
16.16.The slack for an activity is equal tThe slack for an activity is equal too  
A. A. LF-LS LF-LS C.C. LS-ESLS-ES  
B. B. EF-ES EF-ES D. D. None None of of the the aboveabove
17.17.The Another term commonly used for activity slack time isThe Another term commonly used for activity slack time is  
A. A. Total Total float float C. C. independent independent floatfloat
B. B. Free Free float float D.D. All of the aboveAll of the above  
18.18.If an activity has zero slack, it If an activity has zero slack, it implies thatimplies that  
A.A.It lies on the critical pathIt lies on the critical path  
B. B. It It is is a a dummy dummy activityactivity
C. C. The The project project progressing progressing wellwell
D. D. None None of of the the aboveabove

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19.19.A dummy activity is used in the netA dummy activity is used in the network diagram whenwork diagram when  
A. A. Two Two parallel parallel activities activities have have the the same same tail tail and and head head eventsevents
B. B. The chaiThe chain of n of activities activities may havmay have a e a common ecommon event yvent yet be et be independent independent byby
themselvesthemselves
C.C.Both A and BBoth A and B  
D. D. None None of of the the aboveabove
20.20.While drawing the network diagram , for each activity project, While drawing the network diagram , for each activity project, we should lookwe should look  
A.A.What activities precede this activity?What activities precede this activity?  
B. B. What What activities activities follow follow this this activity?activity?
C. C. What What activities activities can can take take place place concurrently concurrently with with this this activity?activity?
D. D. All All of of the the aboveabove
21.21.The critical path satisfy the condition thatThe critical path satisfy the condition that  
A.A.EE
ii= L= L
ii& E& E
jj = L = L
jj   C. C. LL
 j j-E-E
ii=L=L
ii-E-E
 j j=c(constant)=c(constant)
B. B. LL
 j j
-E-E
ii
=L=L
ii
-L-L
 j j
   D. D. All All of of the the aboveabove
22.22.If there are in jobs to be performed, one at a time, on each of m machines, the possibleIf there are in jobs to be performed, one at a time, on each of m machines, the possible
sequences would besequences would be  
A.A.(n!)(n!)
mm
   C. (n)C. (n)
mm
  
B. (m!)B. (m!)
nn
   D. D. (m)(m)
nn
  
23.23.Total elapsed time to process all jobs through tTotal elapsed time to process all jobs through two machine is given bywo machine is given by  
A.A.
n n nn
ΣΣ   MM
1j1j + +ΣΣ   MM
2j2j  
 j=1  j=1 j=1j=1
C.C.
nn
ΣΣ   (M(M
1j1j+ I+ I
1j1j))
 j=1 j=1
B.B.n n nn
ΣΣ   MM
2j2j + +ΣΣ   MM
1j1j  
j=1 j=1j=1 j=1
  
D. D. None None of of the the aboveabove
24.24.The minimum processing time on machine MThe minimum processing time on machine M11 and M and M22are related asare related as  
A. A. Min Min tt1j1j= Max t= Max t2j2j   C.C. Min tMin t1j1j≥≥ Max t Max t2j2j  
B. B. Min Min tt1j1j≤≤ Max t Max t2j2j   D. D. Min Min tt2j2j≥≥ Max t Max t1j1j  
25.25.You Would like to assign operators to the equipment tYou Would like to assign operators to the equipment that hashat has  
A. A. Most Most jobs jobs waiting waiting to to be be processedprocessed
B. B. Job Job with with the the earliest earliest due due datedate
C. C. Job Job which which has has been been waiting waiting longestlongest
D.D.All of the aboveAll of the above  
SHORT QUESTIONS (Each of 2 Marks)SHORT QUESTIONS (Each of 2 Marks)
UNIT 1UNIT 1
1.1.Define Operation research.Define Operation research.
2.2.Write down any two scopes of operation research.Write down any two scopes of operation research.
3.3.State phases for formulating the operations research problems.State phases for formulating the operations research problems.
4.4.Define Define i) i) Solution Solution ii) ii) Basic Basic solution.solution.
5.5.Define Define i) i) Unbounded Unbounded solution solution ii) ii) Optimum Optimum solutionsolution
6.6.Define feasible solution.Define feasible solution.
7.7.Define LPP in the mathematical form.Define LPP in the mathematical form.
8.8.Give any four models of operations research.Give any four models of operations research.
ExtraExtra
9.9.What is Linear programming problem?What is Linear programming problem?
10.10.Write the advantages of LPP.Write the advantages of LPP.
11.11.Write the limitations of Write the limitations of LPPLPP
12.12.How will you plot inequalities of How will you plot inequalities of a LPP?a LPP?

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UNIT 2UNIT 2
1.1.Define slack variables.Define slack variables.
2.2.Define surplus variables.Define surplus variables.
3.3.Define artificial variables.Define artificial variables.
4.4.When is Big M method useful When is Big M method useful ??
5.5.What is the condition for optimality in simplex table ?What is the condition for optimality in simplex table ?
6.6.What is the condition for entering What is the condition for entering variable in simplex table ?variable in simplex table ?
7.7.Write the standard form of LPP for Write the standard form of LPP for the following LPP:the following LPP:
1 1 22
1 1 2 2 1 1 2 2 1 1 22
MM aa xx ii mm iizz ee 11 33 22 55
SS uu bb jj ee cc tt tt oo 22 11 33 44 00 ,, 55 22 77 ,, ,, 00
 Z  Z x x xx
 x  x x x x x x x x x xx
= = ++
+ + ≤ ≤ + + ≤ ≤ ≥≥
  
8.8.Write the standard form of LPP for Write the standard form of LPP for the following LPP:the following LPP:
11 22
11 22 11 22 11 22
MM aa xx iimm iizz ee 33 55
SS uu bb jjee cc tt ttoo 22 33 44 ,, 33 22 77 ,, ,, 00
 Z  Z x x xx
 x  x x x x x x x x x xx
= = ++
+ + ≤ ≤ + + ≥ ≥ ≥≥
  
ExtraExtra
9.9.What is NER while solving LPP by Simplex method?What is NER while solving LPP by Simplex method?
10.10.What is the criterion for the What is the criterion for the entering variable and outgoing variable?entering variable and outgoing variable?
11.11.What is Replacement ratio while solving LPP by Simplex methodWhat is Replacement ratio while solving LPP by Simplex method
12.12.What is the criterion for test of optimality?What is the criterion for test of optimality?
13.13.Represent the form of a simplex tRepresent the form of a simplex table.able.
14.14.How do you transform key row in a simplex table? Or How will you find the new solutionHow do you transform key row in a simplex table? Or How will you find the new solution
in the Simplex table.in the Simplex table.
15.15.State the situation in simplex table State the situation in simplex table which will representwhich will represent
(i) Unbounded solution (ii) unique optimum solution (iii) alternate (i) Unbounded solution (ii) unique optimum solution (iii) alternate optimum solution.optimum solution.
UNIT 3UNIT 3
1.1.What is transportation problem?What is transportation problem?
2.2.Write mathematical form of tWrite mathematical form of transportation problem.ransportation problem.
3.3.What is feasible solution and non degenerate solution in What is feasible solution and non degenerate solution in transportation problem?transportation problem?
4.4.What do you mean by balanced transportation problem?What do you mean by balanced transportation problem?
5.5.What is the Assignment problem?What is the Assignment problem?
6.6.Give mathematical form of assignment problem.Give mathematical form of assignment problem.
7.7.What is travelling salesman problem?What is travelling salesman problem?
ExtraExtra
8.8.What is the difference between Assignment Problem and Transportation Problem?What is the difference between Assignment Problem and Transportation Problem?
9.9.Write steps for North-West Corner Method.Write steps for North-West Corner Method.
10.10.Write steps for Matrix Minima Method.Write steps for Matrix Minima Method.
UNIT 4UNIT 4
1.1.State Bellman’s principle of optimality in State Bellman’s principle of optimality in dynamic programming.dynamic programming.
2.2.In brief explain problem of sequencing.In brief explain problem of sequencing.
3.3.How will you allocate jobs in a sequence if two jobs on first machine have sameHow will you allocate jobs in a sequence if two jobs on first machine have same
processing time?processing time?
4.4.Write down any two assumptions used for solving sequencing problem.Write down any two assumptions used for solving sequencing problem.
5.5.Define two types of events used in Define two types of events used in network analysis.network analysis.
6.6.What is dummy activity?What is dummy activity?
7.7.What is total float?What is total float?
8.8.What is free float and What is free float and independent float?independent float?
9.9.What is successor activity?What is successor activity?

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ExtraExtra
10.10.Define terms: Activity, Event, Merge Event, Burst Event, Total float, Free float,Define terms: Activity, Event, Merge Event, Burst Event, Total float, Free float,
Independent Independent float, float, Critical Critical pathpath
11.11.State Rules for Network Diagram.State Rules for Network Diagram.
12.12.Write disadvantages of Network tWrite disadvantages of Network techniques.echniques.
13.13.What is Dynamic Programming What is Dynamic Programming Problem?Problem?
14.14.State the principle of optimality in State the principle of optimality in dynamic programming.dynamic programming.
15.15.Write assumptions of sequencing Write assumptions of sequencing problems.problems.
LONG QUESTIONS (>=3 Marks)LONG QUESTIONS (>=3 Marks)
UNIT 1UNIT 1
1.1.Explain the history of operations research.Explain the history of operations research.
2.2.Write the algorithm to Write the algorithm to solve LPP using Graphical method for maximization of profit.solve LPP using Graphical method for maximization of profit.
3.3.Give the limitations of Give the limitations of operations research.operations research.
4.4.Note down the applications of operations research.Note down the applications of operations research.
5.5.Write down meanings of operations research.Write down meanings of operations research.
6.6.A person requires at least 10 and 12 units of chemicals A and B respectively, for hisA person requires at least 10 and 12 units of chemicals A and B respectively, for his
garden. A liquid product contains 5 and 2 units of A and B respectively per bottle. A drygarden. A liquid product contains 5 and 2 units of A and B respectively per bottle. A dry
product contains 1 and 4 units of A and B respectively per box. If the liquid product salesproduct contains 1 and 4 units of A and B respectively per box. If the liquid product sales
for Rs. 30 per bottle, dry product sales for Rs. 40 per box. How many of each should befor Rs. 30 per bottle, dry product sales for Rs. 40 per box. How many of each should be
purchased in order to minimize the cost and purchased in order to minimize the cost and meet the requirements? Formulate the L.P.P.meet the requirements? Formulate the L.P.P.  
7.7.A firm manufactures two types of products A and B and sells them at a profit of Rs. 200A firm manufactures two types of products A and B and sells them at a profit of Rs. 200
on type A and Rs. 300 on type B. each product is processed on two machines G and H.on type A and Rs. 300 on type B. each product is processed on two machines G and H.
type A requires 1 minute of processing time on G and 2minutes on H; Type B requires 1type A requires 1 minute of processing time on G and 2minutes on H; Type B requires 1
minute on G and 1 minute on H. the machine G is available for not more than 6 hours, 40minute on G and 1 minute on H. the machine G is available for not more than 6 hours, 40
minutes while H is available for 10 hours during any working day. Formulate this problemminutes while H is available for 10 hours during any working day. Formulate this problem
as a linear programming problem.as a linear programming problem.
8.8.A firm manufactures headache pills in two sizes A and B. size A contains 2 grains ofA firm manufactures headache pills in two sizes A and B. size A contains 2 grains of
aspirin , 5 grains of bicarbonate and 1 grain of codeine. Size B contains 1 grain of aspirin,aspirin , 5 grains of bicarbonate and 1 grain of codeine. Size B contains 1 grain of aspirin,
8 grains of bicarbonate and 6 grains of codeine. It is found by users that it requires at8 grains of bicarbonate and 6 grains of codeine. It is found by users that it requires at
least 12 grains of aspirin, 74 grains of bicarbonate and 24 grains of codeine for providingleast 12 grains of aspirin, 74 grains of bicarbonate and 24 grains of codeine for providing
immediate effect. It is required to determine the least number of pills a patient should takeimmediate effect. It is required to determine the least number of pills a patient should take
to get immediate relief. Formulate the problem as a LPP.to get immediate relief. Formulate the problem as a LPP.
9.9.A carpenter produces two products chairs and tables. Processing of these products isA carpenter produces two products chairs and tables. Processing of these products is
done on two machines A and B. Chair requires 2 hours on machine A and 6 hours ondone on two machines A and B. Chair requires 2 hours on machine A and 6 hours on
machine B. A table machine B. A table requires 5 hours on requires 5 hours on machine A and no hours on machine A and no hours on machine There aremachine There are
16 hours of time per day available on machine A and 30 hours on machine B. Assuming16 hours of time per day available on machine A and 30 hours on machine B. Assuming
that the profit per chair is Rs. 20 that the profit per chair is Rs. 20 and Rs. 35 for table. Formulate the proand Rs. 35 for table. Formulate the problem as LPP inblem as LPP in
order to determine the number of chairs and tables to be produced so as to maximize theorder to determine the number of chairs and tables to be produced so as to maximize the
profit.profit.
10.10.A manufacturer has two machines A and B. He manufactures two products P and Q onA manufacturer has two machines A and B. He manufactures two products P and Q on
these two machines. For manufacturing product P he has to use machine A for 3 hoursthese two machines. For manufacturing product P he has to use machine A for 3 hours
and machine B for 6 hours, and for manufacturing product Q he has to use machine A forand machine B for 6 hours, and for manufacturing product Q he has to use machine A for
6 hours and machine B for 5 hours. On each unit of P he earns Rs. 14 and on each unit6 hours and machine B for 5 hours. On each unit of P he earns Rs. 14 and on each unit
of Q he earns Rs. 10. How many units of P and Q should be manufactured to get theof Q he earns Rs. 10. How many units of P and Q should be manufactured to get the
maximum profit? Each machine cannot be used for more than 2100 hours. Formulate asmaximum profit? Each machine cannot be used for more than 2100 hours. Formulate as
LPP.LPP.

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11.11.Vitamin A and B are found in foods FVitamin A and B are found in foods F11 and F and F22. One unit of food F. One unit of food F11contains three unit ofcontains three unit of
vitamin A and four unit of vitamin B. One unit of food Fvitamin A and four unit of vitamin B. One unit of food F22contains six unit of vitamin A andcontains six unit of vitamin A and
three unit of vitamin B. One unit of food Fthree unit of vitamin B. One unit of food F
11 and F and F
22 cost Rs 14 and Rs 20 respectively. The cost Rs 14 and Rs 20 respectively. The
minimum daily requirement (for person) of vitamin A and B is of 80 and100 units.minimum daily requirement (for person) of vitamin A and B is of 80 and100 units.
Assuming excess of vitamin is not harmful to health, formulate LPP to obtain optimumAssuming excess of vitamin is not harmful to health, formulate LPP to obtain optimum
mixture of food Fmixture of food F11 and F and F22 required to meet the daily demand such that the total cost is required to meet the daily demand such that the total cost is
minimized.minimized.
12.12.An electronic company is engaged in production of two components CAn electronic company is engaged in production of two components C11 and C and C22 used in used in
radio sets. The availability of different aspects and the prices are given below. Formulateradio sets. The availability of different aspects and the prices are given below. Formulate
as LPP to determine number of components Cas LPP to determine number of components C11 and C and C22 to be produced so as to maximize to be produced so as to maximize
the profit.the profit.
Resources/Constraint Resources/Constraint Components Components Total Total AvailabilityAvailability
CC11   CC 22  
Budget(Rs) Budget(Rs) 10 10 / / unit unit 40 40 / / unit unit 40004000
Machine Machine time time 3 3 hr hr / / unit unit 2 2 hr hr / / unit unit 2000 2000 hrshrs
Assembly Assembly time time 2 2 hr hr / / unit unit 3 3 hr hr / / unit unit 1400 1400 hrshrs
Profit Profit Rs.22 Rs.22 Rs. Rs. 4040
13.13.A firm can produce two types of cloth, say: A and B. Three kinds of wool are required forA firm can produce two types of cloth, say: A and B. Three kinds of wool are required for
it, say : red, green and blue wool. One unit length of type A cloth needs 4 meters of redit, say : red, green and blue wool. One unit length of type A cloth needs 4 meters of red
wool and 3 meters of green wool; whereas one unit length of type B cloth needs 3 meterswool and 3 meters of green wool; whereas one unit length of type B cloth needs 3 meters
of red wool, 2 meters of green wool and 5 meters of blue wool. The firm has only a stockof red wool, 2 meters of green wool and 5 meters of blue wool. The firm has only a stock
of 10 meters of red wool, 6 meters of green wool and 15 meters of blue wool. It isof 10 meters of red wool, 6 meters of green wool and 15 meters of blue wool. It is
assumed that the profit obtained from one unit length type A cloth is Rs. 13 and of type Bassumed that the profit obtained from one unit length type A cloth is Rs. 13 and of type B
cloth is Rs. 25. Formulate as cloth is Rs. 25. Formulate as LPP.LPP.
14.14.A company makes two type varieties, Alpha and Beta, of pens. Each Alpha pen needsA company makes two type varieties, Alpha and Beta, of pens. Each Alpha pen needs
twice as much labour time as a Beta pen. If only Beta pens are manufactured, thetwice as much labour time as a Beta pen. If only Beta pens are manufactured, the
company can make 500 pens per day. The market can take only up to 150 alpha penscompany can make 500 pens per day. The market can take only up to 150 alpha pens
and 250 Beta pens per day. If alpha and Beta pens yield profits of Rs. 8 and Rs. 5and 250 Beta pens per day. If alpha and Beta pens yield profits of Rs. 8 and Rs. 5
respectively per pen, determine the number of Alpha and Beta pens to be manufacturedrespectively per pen, determine the number of Alpha and Beta pens to be manufactured
per day so as to maximize the per day so as to maximize the profit. Formulate as L.P.P.profit. Formulate as L.P.P.
15. 15. Graphical Graphical MethodMethod
11 22 11 22
11 22 11 22
11 22 11 22
22 11 22 11 22
Solve the following LLP by graphical method.Solve the following LLP by graphical method.
((11)) MMaaxxiimmiizzee 2255 2200 ((22) M) Maaxxiimmiizzee 33 22
SSuubbjjeecct t ttoo 1166 1122 110000 SSuubbjjeecct t ttoo 22 33 99,,
88 1166 8800,, 55 2200,,
22,, ,, 00.. ,, 00..
 z  z x x x x z z x x xx
 x  x x x x x xx
 x  x x x x x xx
 x  x x x x x x x xx
= = + + = = ++
+ + ≤ ≤ − − + + ≤≤
+ + ≤ ≤ − − + + ≤≤
≥ ≥ ≥ ≥ ≥≥
  
( ( ))
11 22 11 22
11 22 11 22
11 22 11 22
11 22 11 22
33 MMaaxxiimmiizzee 66 88 ((44) M ) Maaxxiimmiizzee 55 77
   SSuubbjjeecctt ttoo 55 1100 6600,, SSuubbjjeecct t ttoo 44,,
44 44 4400,, 1100 77 3355,,
,, 00.. ,, 00..
 Z  Z x x x x z z x x xx
 x  x x x x x xx
 x  x x x x x xx
 x  x x x x x xx
= = + + = = ++
+ + ≤ ≤ + + ≤≤
+ + ≤ ≤ + + ≤≤
≥ ≥ ≥≥
  

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( ( ) ) ( ( ))
11 22 11 22
11 22 11 22
11 22 11
11 22 22
11 22
5 M5 M aaxxiimm iizzee ZZ 330000 xx 440000 xx 6 M6 M aax Zx Z 4400 xx 3300 xx
SSuubbjjeecctt tto 5o 5 xx 22 xx 118800,, ssuubbjjeecctt ttoo 33xx xx 3300
33xx 33xx 113355 xx 88
xx ,, xx 00.. xx 1122
xx ,, xx 00..
= = + + = = ++
+ + ≤ ≤ + + ≤≤
+ + ≤ ≤ ≤≤
≥ ≥ ≤≤
≥≥
  
( ( ))
11 22 11 22
11 22 11 22
11 22 11 22
22 11 22
11 22
77 MMaaxxiimmiizzee 33 55 ((88))MMaaxxiimmiizzee 22 33
   SSuubbjjeecctt ttoo 22 2200,, SSuubbjjeecct t ttoo 11, ,
1155,, 33 66,,
88,, ,, 00..
,, 0.0.
 Z  Z x x x x z z x x xx
 x  x x x x x xx
 x  x x x x x xx
 x  x x x xx
 x  x xx
= = + + = = ++
+ + ≤ ≤ + + ≤≤
+ + ≤ ≤ + + ≤≤
≤ ≤ ≥≥
≥≥
  
ExtraExtra
16.16.State the different scope of State the different scope of operation research.operation research.
17.17.What are various phases of What are various phases of operation research?operation research?
18.18.What is model? List out the model.What is model? List out the model.
19.19.Write the steps for solving Linear Programming Problem by Graphical method. State itsWrite the steps for solving Linear Programming Problem by Graphical method. State its
limitations.limitations.
20.20.The ABC Company has been a producer of picture tubes for television sets and certainThe ABC Company has been a producer of picture tubes for television sets and certain
printed circuits for radios. The company has just explained into full scale production andprinted circuits for radios. The company has just explained into full scale production and
marketing of AM and AM-FM rmarketing of AM and AM-FM radios. adios. It has built a new plant that can oIt has built a new plant that can operate 48 hoursperate 48 hours
per week. Production of an AM radio in the new plant will require 2 hours and productionper week. Production of an AM radio in the new plant will require 2 hours and production
of an AM-FM radio will require 3 hours. Each AM radio will contribute Rs. 40 to profitsof an AM-FM radio will require 3 hours. Each AM radio will contribute Rs. 40 to profits
while an AM-FM radio will contribute Rs. 80 to profits. The marketing departments, afterwhile an AM-FM radio will contribute Rs. 80 to profits. The marketing departments, after
extensive research, have determined that a maximum of 15 AM radios and 10 AM-FMextensive research, have determined that a maximum of 15 AM radios and 10 AM-FM
radios can be sold each week. Formulate the LPP.radios can be sold each week. Formulate the LPP.
21.21.Sudhakant has two iron mines. The production capacities of the mines are different. TheSudhakant has two iron mines. The production capacities of the mines are different. The
iron ore can be classifies into good, mediocre and bad varieties after certain process. Theiron ore can be classifies into good, mediocre and bad varieties after certain process. The
owner has decided to supply 12 or more tons of good iron, 8 or more tons of mediocreowner has decided to supply 12 or more tons of good iron, 8 or more tons of mediocre
iron and 24 or more tons of bad iron per week. The daily expense is Rs.2000 and that ofiron and 24 or more tons of bad iron per week. The daily expense is Rs.2000 and that of
the second mine is Rs.1600. The daily production of each type of iron is given in thethe second mine is Rs.1600. The daily production of each type of iron is given in the
table.table.
Mine Mine Daily Daily productionproduction
Good Good Mediocre Mediocre BadBad
I I 6 2 6 2 44
II 2 2 12II 2 2 12
Formulate the LPPFormulate the LPP
22.22.Solve the following LP problems graphicallySolve the following LP problems graphically
1) 1) Minimize Z = 3 Minimize Z = 3 xx
11+ + 2 2 xx
22  
s.t. s.t. 5 5 xx
11+ + xx
22  ≥≥   1010
xx
11+ + xx
22  ≥≥   66
xx11+ + 4 4 xx22  ≥≥   1212
xx
11, , xx
22  ≥≥   00
2) 2) Maximize Z = 30 Maximize Z = 30 xx11+ 20 x+ 20 x22  
s.t. s.t. 3 3 xx
11+ + 3 3 xx
22  ≥≥   4040
3 3 xx11+ + xx22  ≥≥   4040
2 2 xx11+ + 5 5 xx22  ≥≥   4444
xx11, , xx22  ≥≥   00

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3) Max z=3x3) Max z=3x
11+5x+5x
22  
xx
11+ 2x+ 2x
22  ≤≤  2000  2000
xx
11+ 2x+ 2x
22≤≤  1500  1500
xx
22≤≤  600  600
xx
11, x, x
22≥≥ 0 0
23.23.A company produces two articles X and Y. These are two departments through which theA company produces two articles X and Y. These are two departments through which the
articles are processed assembly and finishing. The potential capacity of the assemblyarticles are processed assembly and finishing. The potential capacity of the assembly
department is 48 hours a week and that of the finishing department is 60 hours a week.department is 48 hours a week and that of the finishing department is 60 hours a week.
Production of each of X requires 2 hours of assembly and 4 hours of finishing. Each unitProduction of each of X requires 2 hours of assembly and 4 hours of finishing. Each unit
of Y requires 4 hours in assembly and 2 hours in finishing department. If profit is Rs.6 forof Y requires 4 hours in assembly and 2 hours in finishing department. If profit is Rs.6 for
each unit of X and Rs.7 for each unit of Y, find out the number of units of X and Y to beeach unit of X and Rs.7 for each unit of Y, find out the number of units of X and Y to be
produced each week to obtain maximum profit. (use graphical method).produced each week to obtain maximum profit. (use graphical method).
UNIT 2UNIT 2
1.1.
( ( ))
11 22
11 22 11 22 11 22
1 M1 M aaxxiimm iizzee 33 55
SSuubbjjeecctt ttoo 44,, 33 22 1188,, ,, 00
 Z  Z x x xx
 x  x x x x x x x x x xx
= = ++
+ + ≤ ≤ + + ≤ ≤ ≥≥
  
2.2.
( ( ))
11 22
11 22 11 22 11 22
22 MM aaxxiimm iizzee 77 55
SSuubbjjeecctt ttoo 22 66,, 44 33 1122,, ,, 00
 Z  Z x x xx
 x  x x x x x x x x x xx
= = ++
+ + ≤ ≤ + + ≤ ≤ ≥≥
  
3.3.
( ( ))
11 22
11 22 11 22 11 22
3 M3 Maaxxiimmiizzee 55 77
SSuubbjjeecctt ttoo 44,, 1100 77 3355,, ,, 00
 Z  Z x x xx
 x  x x x x x x x x x xx
= = ++
+ + ≤ ≤ + + ≤ ≤ ≥≥
  
4.4.
5.5.
( ( ))
11 22
11 22 11 22 11 22
55 MMaaxxiimmiizzee 33 44
SSuubbjjeecctt ttoo 66,, 22 44 2200,, ,, 00
 Z  Z x x xx
 x  x x x x x x x x x xx
= = ++
+ + ≤ ≤ + + ≤ ≤ ≥≥
6.6.
( ( ))
11 22
11 22 11 22 11 22
66 MM aaxxiimmiizzee 55 33
SSuubbjjeecctt tto 3o 3 55 1155,, 55 22 1100,, ,, 00
 Z  Z x x xx
 x  x x x x x x x x x xx
= = ++
+ + ≤ ≤ + + ≤ ≤ ≥≥
  
BIG M MethodBIG M Method
7.7.
( ( ))
11 22
11 22 11 22 11 22
11 MM aaxxiimm iizzee 33
SSuubbjjeecctt ttoo 22 22,, 33 33,, ,, 00
 Z  Z x x xx
 x  x x x x x x x x x xx
= = −−
+ + ≥ ≥ + + ≤ ≤ ≥≥
  
8.8.
( ( ))
11 22 33
11 22 33 11 22 11 22
22 MM aaxxiimm iizzee 33 44
SSuubbjjeecctt ttoo 33 22,, 55 22 33,, ,, 00
= = − − ++
− − + + + + ≥ ≥ − − − − ≤ ≤ ≥≥
 Z  Z x x x x xx
 x  x x x x x x x x x x x xx
  
9.9.
( ( ))
11 22 33
11 22 33 11 22 22 33 11 22 33
3 M3 Maaxxiimmiizzee 55 22
SSuubbjjeecctt ttoo 22 22 22,, 33 44 33,, 33 55 ,, ,, 00
= = − − −−
+ + − − ≥ ≥ − − ≥ ≥ + + ≥ ≥ ≥≥
 Z  Z x x x x xx
 x  x x x x x x x x x x x x x where where x x x x xx
  
10.10.
( ( ))
11 22 33
11 22 33 11 2 3 2 3 11 22 33
4 M4 Maaxxiimmiizzee 22 99
SSuubbjjeecctt ttoo 44 22 55,, 33 22 44,, ,, ,, 00
= = − − − − −−
+ + + + ≥ ≥ + + + + ≥ ≥ ≥≥
 Z  Z x x x x xx
 x  x x x x x x x x x x x x x x x xx
  
( ( ))
11 22
11 22 11 22 11 22
44 MMaaxxiimmiizzee 33 22
SSuubbjjeecctt ttoo 22 55,, 33,, ,, 00
 Z  Z x x xx
 x  x x x x x x x x x xx
= = ++
+ + ≤ ≤ + + ≤ ≤ ≥≥

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ExtraExtra
11.11.What is the standard form of What is the standard form of the LPP? State its characteristics.the LPP? State its characteristics.
12.12.Write the steps of the Write the steps of the algorithm for solving LPP by the Simplex method.algorithm for solving LPP by the Simplex method.
13.13.Explain about Big –M method for solving LPP by the Explain about Big –M method for solving LPP by the Simplex method.Simplex method.
14.14.Max z = 18xMax z = 18x
11   + + 24x24x
22  
s.t. 4xs.t. 4x11 + 2x + 2x22  ≤≤  8, 8, 2x2x11 + 5x + 5x22  ≤≤ 12 where x 12 where x11 ,x ,x22  ≥≥ 0 0
15.15.Min Min Z = Z = 12x12x11+ 20x+ 20x22  
s.t. 6xs.t. 6x
11 + 8x + 8x
22≥≥  10, 10, 7x7x
11+ 12x+ 12x
22  ≥≥ 12  12 where where xx
11,x,x
22  ≥≥ 0 0
16.16.Max Max z z = 30x= 30x
11+ 20x+ 20x
22  
s.t. 3xs.t. 3x11+ x+ x22  ≥≥  40, 40, 2x2x11 + 5x + 5x22  ≥≥ 44 where x 44 where x11 ,x ,x22  ≥≥ 0 0
UNIT 3UNIT 3
1.1.What is difference between transportation problem and assignment problem?What is difference between transportation problem and assignment problem?
2.2.Give the algorithm of NWCM to obtain basic feasible initial solution to transportationGive the algorithm of NWCM to obtain basic feasible initial solution to transportation
problem.problem.  
3.3.Give the algorithm of LCM to obtain basic feasible initial solution to transportationGive the algorithm of LCM to obtain basic feasible initial solution to transportation
problem.problem.
4.4.Give the algorithm of VAM to obtain basic feasible initial solution to transportationGive the algorithm of VAM to obtain basic feasible initial solution to transportation
problem.problem.
5.5.Write the steps for solving a Write the steps for solving a A.P. by Hungarian method.A.P. by Hungarian method.
6.6.Write a short note on travelling salesman problem.Write a short note on travelling salesman problem.
7. 7. Examples Examples of of Transportation Transportation problemproblem
1)1)
Obtain the initial solution to aboveObtain the initial solution to above
TP using northwest corner method.TP using northwest corner method.
DD
11DD
22DD
33DD
44SupplySupply
OO
11 6 4 6 4 1 1 5 145 14
OO
22 8 9 8 9 2 2 7 167 16
OO
33 4 3 4 3 6 6 2 52 5
Dem. Dem. 6 6 10 10 15 415 4
2)2)
Obtain the initial solution to above TPObtain the initial solution to above TP
using least cost method.using least cost method.
A A B B C C D D SupplySupply
I I 6 6 3 3 5 5 4 4 2222
II II 5 5 9 9 2 2 7 7 1515
III III 5 5 7 7 8 8 6 6 88
Demand 7 12 17 9Demand 7 12 17 9
3)3)
Obtain the initial solution to above TPObtain the initial solution to above TP
using least cost method.using least cost method.
DD11DD22DD33DD44SupplySupply
OO
11 1 2 3 4 61 2 3 4 6
OO
22 4 3 2 0 84 3 2 0 8
OO33 0 2 2 1 100 2 2 1 10
Demand Demand 4 4 6 6 8 8 66
4)4)
Obtain the initial solution to above TPObtain the initial solution to above TP
using northwest corner method.using northwest corner method.
A B A B C C D SupplyD Supply
I I 1 1 5 5 3 3 3 3 3434
II II 3 3 3 3 1 1 2 2 1515
III III 0 0 2 2 2 2 3 3 1212
IV IV 2 2 7 7 2 2 4 4 1919
Demand Demand 21 25 17 21 25 17 1717

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5)5)
Obtain the initial solution to above TPObtain the initial solution to above TP
using Vogel’s approximation method.using Vogel’s approximation method.
I I II II III III IV IV SupplySupply
A A 21 21 16 16 15 15 13 13 1111
B B 17 17 18 18 14 14 23 23 1313
C C 32 32 27 27 18 18 41 41 1919
Demand Demand 6 6 10 10 12 12 1515
6)6)
Obtain the initial solution to above TPObtain the initial solution to above TP
using Vogel’s approximation method.using Vogel’s approximation method.
DD
11DD
22DD
33DD
44SupplySupply
OO
11 1 2 1 2 1 1 4 304 30
OO
22 3 3 3 3 2 2 1 501 50
OO
33 4 2 4 2 5 5 9 209 20
Demand Demand 20 40 30 20 40 30 1010
7)7)
Obtain the optimal solution to above TObtain the optimal solution to above TP.P.
I I II II III III IV SupplyIV Supply
A A 2 2 3 3 11 11 7 7 66
B B 1 1 0 0 6 6 1 1 11
C C 5 5 8 8 15 15 10 10 1010
Demand Demand 7 7 5 5 3 3 2 2 1717
8)8)
Obtain the initial solution to above TPObtain the initial solution to above TP
using northwest corner method.using northwest corner method.
a b a b c c Supply Supply
I I 10 10 9 9 8 8 88
II II 10 10 7 7 10 10 77
III III 11 11 9 7 9 7 99
IV IV 12 12 14 14 10 10 44
Demand Demand 10 10 10 10 88
9)9)
Obtain the optimal solution to above TObtain the optimal solution to above TP.P.
WarehousesWarehouses
WW11   WW22   WW33   WW44  Supply  Supply
FF11 19 30 50 10 719 30 50 10 7
FF
22   70 70 30 30 40 40 60 60 99
FF33   40 40 8 70 8 70 20 20 1818
Demand Demand 5 5 8 8 7 7 1414
10)10)
Obtain the optimal solution to Obtain the optimal solution to above TP.above TP.
Destination SupplyDestination Supply
Source I Source I II III II III IVIV
A A 19 19 14 14 23 23 11 11 1111
B B 15 15 16 16 12 12 21 21 1313
C C 30 30 25 25 16 16 39 39 1919
Demand Demand 6 6 10 10 12 12 1515
8. 8. Examples Examples of of Assignment Assignment problemproblem
Solve the following Assignment Problems.Solve the following Assignment Problems.
1)1) P Q R SP Q R S
A 22 30 21 15A 22 30 21 15
B 18 B 18 33 33 9 9 3131
C 44 25 24 21C 44 25 24 21
D 23 30 28 14D 23 30 28 14
2)2) I I II III II III IVIV
1 1 11 11 10 10 18 18 55
2 2 14 14 13 13 12 12 1919
3 3 5 3 5 3 4 24 2
4 4 15 15 18 18 17 17 99
3)3) A B C D E FA B C D E F
1 1 13 13 16 13 13 16 23 19 923 19 9
2 2 11 19 11 19 26 16 26 16 17 1817 18
3 3 12 12 11 11 4 4 9 9 6 6 1010
4 4 7 7 15 15 9 9 14 14 14 14 1313
5 5 9 9 13 13 12 12 8 8 14 14 1111
4)4) P Q R SP Q R S
A A 5 5 3 4 3 4 77
B B 2 2 3 7 3 7 66
C C 4 4 1 5 1 5 22
D D 6 6 8 1 8 1 22

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5)5)Find the assignment of salesmen to various districts which will result Find the assignment of salesmen to various districts which will result minimum cost.minimum cost.
Salesman DistrictSalesman District
1 2 3 41 2 3 4
A A 16 16 10 10 14 14 1111
B B 14 14 11 11 15 15 1515
C C 15 15 15 15 13 13 1212
D D 13 13 12 12 14 14 1515
6)6)Solve the following assignment problem so as to minimize the time (in days) required toSolve the following assignment problem so as to minimize the time (in days) required to
complete all the task.complete all the task.
person taskperson task
1 2 3 4 51 2 3 4 5
A A 6 6 5 5 8 8 11 11 1616
B B 1 1 13 13 16 16 1 1 1010
C C 16 16 11 11 8 8 8 8 88
D D 9 14 9 14 12 12 10 10 1616
7)7)A department has 5 employees and five jobs are to be performed. The time each manA department has 5 employees and five jobs are to be performed. The time each man
will take to perform each job is given in the following table below. How should the job bewill take to perform each job is given in the following table below. How should the job be
allocated one per employee, so as to mallocated one per employee, so as to minimize the total man-hours.inimize the total man-hours.
Cost matrixCost matrix
Jobs EmployeeJobs Employee
A B C D EA B C D E
1 1 9 3 10 9 3 10 13 13 44
2 2 9 9 17 17 13 13 20 20 55
3 3 5 14 5 14 8 11 8 11 66
4 4 11 11 13 13 9 9 12 12 33
5 5 12 12 8 8 14 14 16 16 77
ExtraExtra
9.9.Three fertilizers factories X, Y and Z located at different places of the country produce 6,4Three fertilizers factories X, Y and Z located at different places of the country produce 6,4
and 5 lakh tones of urea respectively. Under the directive of the central government, theyand 5 lakh tones of urea respectively. Under the directive of the central government, they
are to be distributed to 3 States A, B and C as 5, 3 and 7 lakh respectively. Theare to be distributed to 3 States A, B and C as 5, 3 and 7 lakh respectively. The
transportation cost per tones in rupees is given below:transportation cost per tones in rupees is given below:
A A B B CC
X X 11 17 11 17 1616
Y Y 15 12 15 12 1414
Z Z 20 12 20 12 1515
Find out suitable transportation pattern at minimum cost by North West Corner methodFind out suitable transportation pattern at minimum cost by North West Corner method
and Least Cost method.and Least Cost method.
10.10.Determine an IBFS by Vogel’s Approximation method.Determine an IBFS by Vogel’s Approximation method.
Source Source D1 D1 D2 D2 D3 D3 D4 D4 Supply Supply
S1 19 30 50 10 7S1 19 30 50 10 7
S2 70 30 40 60 9S2 70 30 40 60 9
S3 S3 40 8 40 8 70 20 70 20 1818
Demand Demand 5 5 8 8 7 7 1414

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11.11.A departmental has five employees with five jobs to be performed. The time ( in hours)A departmental has five employees with five jobs to be performed. The time ( in hours)
each men will take to perform each men will take to perform each job is given in the effeach job is given in the effectiveness matrix.ectiveness matrix.
How the jobs should be allocated, one per employee, so as to minimize the total man-How the jobs should be allocated, one per employee, so as to minimize the total man-
hours.hours.
EmployeesEmployees
 jobs  jobs 1 1 2 2 3 3 4 4 55
a a 10 5 10 5 13 13 15 16 15 16
b b 3 3 9 9 18 18 13 6 13 6
c c 10 10 7 7 2 2 2 2 22
d d 7 7 11 11 9 7 12 9 7 12
e e 7 9 7 9 10 10 4 12 4 12
12.12.A machine operator processes five types of items on his machine each week, and mustA machine operator processes five types of items on his machine each week, and must
choose a sequence for them. The set up cost per change depends on the item presentlychoose a sequence for them. The set up cost per change depends on the item presently
on the machine and the set- on the machine and the set- up cost be made according to the following table:up cost be made according to the following table:
To itemTo item
From From item item A A B B C C D D EE
1 -- 4 7 3 41 -- 4 7 3 4
2 4 -- 6 3 42 4 -- 6 3 4
3 7 6 -- 7 53 7 6 -- 7 5
4 3 3 7 -- 74 3 3 7 -- 7
5 4 4 5 7 --5 4 4 5 7 --
If he processes each type of item once and only once each week, how should heIf he processes each type of item once and only once each week, how should he
sequence the items on his machine in order to sequence the items on his machine in order to minimize the total set-up cost?minimize the total set-up cost?
13.13.A city corporation has decided to carry out road repairs on main four arteries of the city.A city corporation has decided to carry out road repairs on main four arteries of the city.
The government has agreed to make a special grant of Rs 50 lakh towards the cost withThe government has agreed to make a special grant of Rs 50 lakh towards the cost with
a condition that repairs are done at the lowest cost and quickest time. If the conditionsa condition that repairs are done at the lowest cost and quickest time. If the conditions
warrant, a supplwarrant, a supplementary token ementary token grant will algrant will also be considso be considered favorably. ered favorably. The corporationThe corporation
has floated tenders and five contractors have sent in their bids. In order to expedite work,has floated tenders and five contractors have sent in their bids. In order to expedite work,
one road will be awarded to only one one road will be awarded to only one contractor.contractor.
Cost of Repairs (Rs in lakh)Cost of Repairs (Rs in lakh)
Contractors RContractors R 11   RR 22   RR 33   RR 44   RR 55  
CC
11   9 9 14 14 19 15 13 19 15 13
CC22   7 7 17 17 20 19 18 20 19 18
CC
33   9 9 18 18 21 18 17 21 18 17
CC
44   10 10 12 12 18 18 19 19 1818
CC
55   10 10 15 15 21 21 16 16 1515
Find the best way of Find the best way of assigning the repair work to the contractors and the costs.assigning the repair work to the contractors and the costs.
If it is If it is necessary to seek supplementary grants, what should be the amount sought?necessary to seek supplementary grants, what should be the amount sought?
14.14.A company has factories at F1, F2 & F3 which supply warehouses ay W1,W2 and W3.A company has factories at F1, F2 & F3 which supply warehouses ay W1,W2 and W3.
Weekly factory capacities are 200,160 and 90 units respectively. Weekly warehousesWeekly factory capacities are 200,160 and 90 units respectively. Weekly warehouses
requirements are 180,120 and 150 units respectively. Unit shipping costs ( in rupees) arerequirements are 180,120 and 150 units respectively. Unit shipping costs ( in rupees) are
as follows:as follows:
WarehouseWarehouse
Factory Factory w1 w1 w2 w2 w3 w3 Supply Supply
F1 16 20 12 200F1 16 20 12 200
F2 F2 14 14 8 8 18 18 160 160
F3 26 24 16 90F3 26 24 16 90
Demand Demand 180 180 120 120 150 150
Determine the optimum distribution for tDetermine the optimum distribution for this company to minimize the shipping cost.his company to minimize the shipping cost.

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15.15.Determine the optimum basic feasible solution to Determine the optimum basic feasible solution to the following transportation problem.the following transportation problem.
A A B B C C AvailableAvailable
II50 50 30 30 220 220 11
IIII90 90 45 45 170 170 33
IIIIII250 200 250 200 50 50 44
Required Required 4 4 2 2 22
Hint: Find Initial Basic Feasible Solution using VAM metHint: Find Initial Basic Feasible Solution using VAM method.hod.
16.16.Determine the optimum basic feasible solution to Determine the optimum basic feasible solution to the following transportation problem.the following transportation problem.
DD
11 DD
22  DD
33   DD
44AvailableAvailable
OO11   1 1 2 2 1 1 4 4 3030
OO
22   3 3 3 3 2 2 1 1 5050
OO
33
   4 4 2 2 5 5 9 9 2020
Required Required 20 20 40 40 30 30 10 10 100 100
Hint: Find Initial Basic Feasible Solution using Lowest cost Hint: Find Initial Basic Feasible Solution using Lowest cost method.method.
17.17.Determine the optimum basic feasible solution to the following transportation problem inDetermine the optimum basic feasible solution to the following transportation problem in
which cell entries represent unit costs.which cell entries represent unit costs.
To AvailableTo Available
2 2 7 7 4 4 55
From From 3 3 3 3 1 1 88
5 5 4 4 7 7 77
1 1 6 6 2 2 1414
Required Required 7 7 9 9 18 18 3434
Hint: Find Initial Basic Feasible Solution using VAM.Hint: Find Initial Basic Feasible Solution using VAM.
18.18.Solve the transportation problem where all entries are unit costs.Solve the transportation problem where all entries are unit costs.
DD
11 DD
22  DD
33   DD
44 DD
55aaii
OO
11   73 73 40 40 9 9 79 79 20 20 88
OO22   62 62 93 93 96 96 8 8 13 13 77
OO
33   96 96 65 65 80 80 50 50 65 65 99
OO
4457 58 57 58 29 29 12 12 87 87 33
OO5556 23 56 23 87 87 18 18 12 12 55
bb j j6 8 6 8 10 10 4 4 44
Hint: Find Initial Basic Feasible Solution using Lowest cost Hint: Find Initial Basic Feasible Solution using Lowest cost method.method.
19.19.The following table gives the cost for transporting material from supply points A, B, C andThe following table gives the cost for transporting material from supply points A, B, C and
to demand points E, F, G, H and J.to demand points E, F, G, H and J.
ToTo
E E FF GG H H JJ
A A 8 8 10 12 10 12 17 17 1515
B B 15 13 15 13 18 18 11 11 99
From From C C 14 14 20 20 6 6 10 10 1313
D D 13 13 19 19 7 7 5 5 1212
The present allocation is as follows:The present allocation is as follows:
A to E 90, A to F 10, B to F 150, C to F 10, C to G 50, C to J 120, D to H 210, D to J 70.A to E 90, A to F 10, B to F 150, C to F 10, C to G 50, C to J 120, D to H 210, D to J 70.
Check if this allocation is optimum. If not, find an optimum schedule.Check if this allocation is optimum. If not, find an optimum schedule.

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20.20.The following table shows all the necessary information on the available supply to eachThe following table shows all the necessary information on the available supply to each
warehouse, the requirement of each market and the unit transportation cost for eachwarehouse, the requirement of each market and the unit transportation cost for each
warehouse to each market.warehouse to each market.
MarketMarket
I I II III II III IVIV SupplySupply
A A 5 5 2 2 4 4 3 3 2222
Warehouse Warehouse B B 4 4 8 8 1 1 6 6 1515
C C 4 4 6 6 7 7 5 5 88
Requirement Requirement 7 7 12 17 12 17 99
The shipping clerk has worked out the following schedule from experience: 12 units fromThe shipping clerk has worked out the following schedule from experience: 12 units from
A toA toIIII, 1 unit from A to, 1 unit from A toIIIIII, 9 units from A to, 9 units from A toIVIV, 15 units from B to, 15 units from B toIIIIII, 7 units from C to, 7 units from C toII  
and 1 unit from C toand 1 unit from C toIIIIII..
Check and see if the clerk has the optimum schedule. If not, find the optimum scheduleCheck and see if the clerk has the optimum schedule. If not, find the optimum scheduleand minimum total shipping cost.and minimum total shipping cost.
21.21.Determine the optimum basic feasible solution to Determine the optimum basic feasible solution to the following transportation problem.the following transportation problem.
A A B B C C AvailableAvailable
I I 50 50 30 30 220 220 11
II II 90 90 45 45 170 170 33
III III 250 250 200 200 50 50 44
Required Required 4 4 2 2 22
Hint: Find Initial Basic Feasible Solution using Lowest cost Hint: Find Initial Basic Feasible Solution using Lowest cost method.method.
22.22.Is Is xx1313 = 50, x = 50, x1414 = 20, x = 20, x2121 = 55, x = 55, x3131 = 30, x = 30, x3232 = 35 , x = 35 , x3434 = 25 an optimum solution of the = 25 an optimum solution of the
following transportation problem?following transportation problem?
ToTo AvailableAvailable
From From 6 6 1 1 9 9 3 3 7070
11 5 11 5 2 2 8 8 5555
10 10 12 12 4 4 7 7 9090
Required Required 85 85 35 35 50 50 4545
23.23.Solve following assignment problem.Solve following assignment problem.
1 1 2 3 2 3 44
II   12 12 30 21 30 21 1515
IIII   18 18 33 9 33 9 3131
IIIIII   44 44 25 24 25 24 2121
IVIV   23 23 30 28 30 28 1414
24.24.Solve following cost minimizing problem.Solve following cost minimizing problem.
JobsJobs
I I II III II III IV V IV V
A A 45 45 30 65 30 65 40 55 40 55
B B 50 50 30 25 30 25 60 30 60 30
Machines Machines C 25 C 25 20 20 15 20 15 20 4040
D D 35 35 25 30 25 30 30 20 30 20
E E 80 80 60 60 60 60 70 50 70 50

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25.25.Solve following cost minimizing problem.Solve following cost minimizing problem.
JobsJobs
I I II III II III IV V IV V
A A 2 2 9 9 2 2 7 7 11
B B 6 6 8 8 7 7 6 6 11
Machines Machines C 4 C 4 6 6 5 5 3 3 11
D D 4 4 2 2 7 7 3 3 11
E E 5 5 3 3 9 9 5 5 11
26.26.A department head has four subordinates and four tasks have to be performed.A department head has four subordinates and four tasks have to be performed.
Subordinates differ in efficiency and tasks differ in their intrinsic difficulty. Time each manSubordinates differ in efficiency and tasks differ in their intrinsic difficulty. Time each man
would take to perform each task is given in the efficiency matrix. How the tasks should bewould take to perform each task is given in the efficiency matrix. How the tasks should be
allocated to each person so as to minimize tallocated to each person so as to minimize the total man-hours?he total man-hours?
I I II III II III IVIV
A 8 A 8 26 17 26 17 1111
Tasks Tasks B 13 B 13 28 28 4 4 2626
C C 38 38 19 18 19 18 1515
D D 19 19 26 24 26 24 1010
27.27.A car hire company has one car at each of five depots a, b, c, d and e. A customerA car hire company has one car at each of five depots a, b, c, d and e. A customer
requires a car in each town, namely A, B, C, D and E. Distance (in kms) between depotsrequires a car in each town, namely A, B, C, D and E. Distance (in kms) between depots
(origins) and towns 9destinations) are given in the following matrix:(origins) and towns 9destinations) are given in the following matrix:
aa b b cc d d ee
A A 160 160 130 130 175 190 175 190 200 200
B B 135 135 120 120 130 160 130 160 175 175
C C 140 140 110 110 155 170 155 170 185 185
D 50 50 80 80 110D 50 50 80 80 110
E 55 35 70 80 105E 55 35 70 80 105
How should cars be assigned to customers so as to minimize tHow should cars be assigned to customers so as to minimize the distance travelled?he distance travelled?
28.28.A department has five employees with five jobs to be performed. The time (in hours) eachA department has five employees with five jobs to be performed. The time (in hours) each
man will take to perform man will take to perform each job is given in the effeach job is given in the effectiveness matrix.ectiveness matrix.
EmployeesEmployees  
I I II III II III IV V IV V
A A 10 10 5 5 13 13 15 16 15 16
B B 3 3 9 18 9 18 13 6 13 6
Jobs Jobs C C 10 10 7 7 2 2 2 2 22
D D 7 7 11 9 7 1211 9 7 12
E E 7 7 9 10 9 10 4 12 4 12
How should the jobs be allocated, one per employee, so as to minimize the total man-How should the jobs be allocated, one per employee, so as to minimize the total man-
hours?hours?
29.29.A solicitor’s firm employs typists for their daily work. There are five typists and theirA solicitor’s firm employs typists for their daily work. There are five typists and their
charges are different. Only one job is given to one typist. Find least cost allocation for thecharges are different. Only one job is given to one typist. Find least cost allocation for the
following data:following data:
JobsJobs  
P P Q R Q R S T S T
A 85 A 85 75 75 65 65 125 125 7575
B 90 B 90 78 78 66 66 132 132 7878
Typists Typists C 75 C 75 66 66 57 57 114 69 114 69
D 80 D 80 72 72 60 60 120 120 7272
E 76 E 76 64 64 56 56 112 112 6868

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30.30.In a textile sales emporium, four salesman A, B, C and D are available to four countersIn a textile sales emporium, four salesman A, B, C and D are available to four counters
W, X, Y and Z. Each salesman can handle any counter. The service (in hour) of eachW, X, Y and Z. Each salesman can handle any counter. The service (in hour) of each
counter when manned by each salesman is given below:counter when manned by each salesman is given below:
SalesmanSalesman
A A B C B C DD
W W 41 41 72 72 39 39 5252
Counter Counter X X 22 22 29 29 49 65 49 65
Y Y 27 27 39 60 39 60 5151
Z Z 45 45 50 48 50 48 5252
How should the salesman be allocated appropriate counters so as tHow should the salesman be allocated appropriate counters so as to minimize the serviceo minimize the service
time? Each salesman must handle only one counter.time? Each salesman must handle only one counter.
UNIT 4UNIT 4
1.1.Write down the procedure for solving problem of sequencing with two mWrite down the procedure for solving problem of sequencing with two machines.achines.
2.2.State the rules for drawing network diagram.State the rules for drawing network diagram.
3.3.Write down the procedure to obtain optimum Write down the procedure to obtain optimum completion time using Critical Path method.completion time using Critical Path method.
4.4.Find the critical path and calculate the Total float and Free float for the following PERTFind the critical path and calculate the Total float and Free float for the following PERT
diagram.diagram.
5.5.A small maintenance project consists of the following A small maintenance project consists of the following 12 jobs12 jobs
JobsJobs
DurationDuration
in daysin days
JobsJobs
DurationDuration
in daysin days
JobsJobs
DurationDuration
in daysin days
1-2 1-2 2 2 3-5 3-5 5 5 6-10 6-10 44
2-3 2-3 7 7 4-6 4-6 3 3 7-9 7-9 44
2-4 2-4 3 3 5-8 5-8 5 5 8-9 8-9 11
3-4 3-4 3 3 6-7 6-7 8 8 9-10 9-10 77
Draw the arrow network of the project. Determine the Draw the arrow network of the project. Determine the critical path.critical path.
22
55
44
22
11
22
44
33
88
55
77
66
99  
22
77
1515
1313
55
1212
66
77
1616
88
99
88
1010
11
22
33
44
55
66
77
99
88
1010

23.   
Question Question Bank Bank US05FBCA01- US05FBCA01- Operations Operations ResearchResearch
PagePage2323 of of2424  
6.6.A project has the following time A project has the following time schedule:schedule:
Activity TimeActivity Time
In monthIn month
Activity TimeActivity Time
In monthIn month
Activity TimeActivity Time
In monthIn month
1-2 1-2 2 2 3-6 3-6 8 8 6-9 6-9 55
1-3 1-3 2 2 3-7 3-7 5 5 7-8 7-8 44
1-4 1-4 1 1 4-6 4-6 3 3 7-9 7-9 33
2-5 2-5 4 4 5-8 5-8 11
Construct PERT network and compute total float for each activity. Find Critical path withConstruct PERT network and compute total float for each activity. Find Critical path with
its duration.its duration.
7.7.In a machine shop 8 different products are being manufactured each requiring time onIn a machine shop 8 different products are being manufactured each requiring time on
two different machines A and B are given in two different machines A and B are given in the table below:the table below:
Product 1 Product 1 2 2 3 4 5 3 4 5 6 7 6 7 88
Machine-A Machine-A 30 30 45 45 15 15 20 20 80 80 120 120 65 1065 10
Machine Machine B B 20 20 30 50 30 50 35 35 35 35 40 40 50 50 2020
Find an optimal sequence of processing of different Find an optimal sequence of processing of different product in order to minimize the totalproduct in order to minimize the total
manufctured time for all product. Find manufctured time for all product. Find total ideal time for two machines atotal ideal time for two machines and elapsed nd elapsed time.time.
8.8.In a machine shop 6 different products are being manufactured each requiring time onIn a machine shop 6 different products are being manufactured each requiring time on
two different machines A and B are given in two different machines A and B are given in the table below:the table below:
Product 1 2 Product 1 2 3 4 5 3 4 5 66
Machine-A Machine-A 30 30 120 120 50 50 20 20 90 90 110 110
Machine Machine B B 80 80 100 100 90 90 60 60 30 30 8080
Find an optimal sequence of processing of different Find an optimal sequence of processing of different product in order to minimize the totalproduct in order to minimize the total
manufctured time for all product. Find manufctured time for all product. Find total ideal time for two machines atotal ideal time for two machines and elapsed nd elapsed time.time.
9.9.In a printing shop 7 In a printing shop 7 different books are printed and bounded on two different machines Adifferent books are printed and bounded on two different machines A
and B. Time required on two machines are gand B. Time required on two machines are given in the table below:iven in the table below:
Product 1 2 Product 1 2 3 4 5 3 4 5 6 7 6 7
Printing Printing 3 4 3 4 8 3 6 8 3 6 7 5 7 5
Binding Binding 8 6 8 6 3 7 2 3 7 2 8 4 8 4
Find an optimal sequence of processing of different Find an optimal sequence of processing of different product in order to minimize the totalproduct in order to minimize the total
manufctured time for all product. Find manufctured time for all product. Find total ideal time for two machines atotal ideal time for two machines and elapsed nd elapsed time.time.
ExtraExtra
10.10.Write similarities and differences between PERT and CPM.Write similarities and differences between PERT and CPM.
11.11.Write applications of PERT/CPM techniques.Write applications of PERT/CPM techniques.
12.12.Write uses of PERT/CPM techniques.Write uses of PERT/CPM techniques.
13.13.Write the steps for Processing n jobs through two machines.Write the steps for Processing n jobs through two machines.
14.14.State “the Principle of Optimality” and its mathematical formulation of a dynamicState “the Principle of Optimality” and its mathematical formulation of a dynamic
programming problem.programming problem.
15.15.Draw the Network Diagram for the following activities and fDraw the Network Diagram for the following activities and find the critical path.ind the critical path.
Job Job A A B B C C D D E E F F G G H H I I J J KK
Job Job time(days) time(days) 13 13 8 10 8 10 9 9 11 11 10 10 8 8 6 7 6 7 14 1814 18
ImmediateImmediate
predecessorspredecessors
- - A B A B C C B B E E D,F D,F E H E H G,I JG,I J

24.   
Question Question Bank Bank US05FBCA01- US05FBCA01- Operations Operations ResearchResearch
PagePage2424 of of2424  
16.16.A project has the following time Schedule. Construct a PERT network and computeA project has the following time Schedule. Construct a PERT network and compute
Critical Path and its duration. Also calculate Total Critical Path and its duration. Also calculate Total float,Free float.float,Free float.
Activity Activity 1-2 1-2 1-3 1-3 2-4 2-4 3-4 3-4 3-5 3-5 4-9 4-9 5-65-6
Time Time in in Weeks Weeks 4 4 1 1 1 1 1 1 6 6 5 5 44
Activity Activity 5-7 5-7 6-8 6-8 7-8 7-8 8-9 8-9 8-10 8-10 9-109-10
Time Time in in Weeks Weeks 8 8 1 1 2 2 1 1 8 8 77
17.17.A project schedule has the following characteristics. Construct the PERT network and findA project schedule has the following characteristics. Construct the PERT network and find
the critical path and time duration of the project.the critical path and time duration of the project.
ActivitActivity y 1 1 - - 2 2 1 1 - - 4 4 1 1 - - 7 7 2 2 - - 3 3 3 3 - - 6 6 4 4 - - 5 5 4 4 - - 8 8 5 5 - - 6 6 6 6 - - 9 9 7 7 - - 8 8 8 8 - - 99
Time Time 2 2 1 2 2 1 4 1 5 4 1 5 8 4 3 8 4 3 3 5 3 5
18.18.A salesman located in a A salesman located in a city A decided to travel to ccity A decided to travel to city ity B. He knew the distance oB. He knew the distance off
alternative routes from city A to city B. He then drew a highway network map as shown inalternative routes from city A to city B. He then drew a highway network map as shown in
Figure. The city of origin, A, is city 1. The destination city B, is city 10. Other cities throughFigure. The city of origin, A, is city 1. The destination city B, is city 10. Other cities through
which the salesman will have to pass through are nu mbered 2 to 9. The arrowwhich the salesman will have to pass through are nu mbered 2 to 9. The arrow
representing routes between cities and distances in kilometers are indicated on eachrepresenting routes between cities and distances in kilometers are indicated on each
route. The salesman’s problem is to find the shortest route that covers all the selectedroute. The salesman’s problem is to find the shortest route that covers all the selected
cities from A to B.cities from A to B.
Also the distance in kilometers is given below:Also the distance in kilometers is given below:
Activity Activity Distance Distance travelled. travelled. Activity Activity DistanceDistance
travelled.travelled.
1-2 1-2 4 4 3-7 3-7 44
1-3 1-3 6 6 4-5 4-5 66
1-4 1-4 3 3 4-6 4-6 1010
2-5 2-5 7 7 4-7 4-7 55
2-6 2-6 10 10 5-8 4 5-8 4
2-7 2-7 5 5 5-9 5-9 88
3-5 3-5 3 3 6-8 6-8 33
3-6 3-6 8 8 6-9 6-9 77
3-7 3-7 4 4 7-8 7-8 88
4-5 4-5 6 6 7-9 7-9 44
4-6 4-6 10 10 8-10 7 8-10 7
4-7 4-7 5 5 9-10 9-10 99
11
22
33
44
55
66
77
1010
88
99
44