SMU DRIVE FALL 2017 MBA 205 – Operation research solved free assignment Define the Linear programming problem in operation Research. Also, explain various assumptions, advantages and limitations of linear programming problem. A Linear programming problem in operation Research Assumptions of linear programming problem Advantages of linear programming problem Limitations of linear programming problem 2.5 2.5 2.5 2.5 10 2 a. Discuss the concept of Degeneracy in transportation problem b. The ABC Tool Company has a sales force of 25 men who work out from Regional offices. The company produces four basic products lines of hand tools. Mr. Jain, the sales manager, feels that 6 salesmen are needed to distribute product line 1, 10 salesmen are needed to distribute product line 2, 4 salesmen to product line 3 and 5 salesmen to product line 4. The cost per day of assigning salesmen from each of the offices for selling each of the product lines are as follows Regional office Product Lines 푷ퟏ 푷ퟐ 푷ퟑ 푷ퟒ 푹ퟏ 20 21 16 18 푹ퟐ 17 28 14 16 푹ퟑ 29 23 19 20 Now, 10 salesmen are allowed to office 푹ퟏ , 9 salesmen to office 푹ퟐ, and 7 salesmen to office 푹ퟑ. How many salesmen should be assigned from each office to selling each product line in order to minimize costs? A Degeneracy in transportation problem Optimum allocation. Optimum transportation cost 4 3 3 10 3 a. Elaborate the meaning of Simulation. b. What are different Practical applications of simulation A Meaning of Simulation. Practical applications of simulation 2 8 10 SET-II 1 a. Define the meaning of assignment problem in operation Research. b. A Departmental head has four subordinates and four task to be performed. The subordinates differ in efficiency and the tasks differ in their intrinsic difficulty. His estimate of the times each man would take to perform each task is given in the following matrix- Tasks Subordinates I II III IV A 8 26 17 11 B 13 28 4 26 C 38 19 18 15 D 19 26 24 10 How should the tasks be allocated to subordinates to minimize the total man-hours? A Description of assignment problem Optimum allocation through Hungarian method 4 6 10 2 Define following criteria’s used for decision making under Uncertainty a. Optimism (maximax or minimin) criterion b. Pessimism (maximin or minimax) criterion c. Equal probabilities (Laplace) criterion d. Coefficient of optimism (Hurwicz) criterion e. Regret (salvage) criterion A a. Optimism (maximax or minimin) criterion b. Pessimism (maximin or minimax) criterion c. Equal probabilities (Laplace) criterion d. Coefficient of optimism (Hurwicz) criterion 2 2 2 2 10

1. MBA 205 OPERATIONS RESEARCH
DRIVE -FALL 2017
MBA
SEMESTER – II
Assignment Set -I
Q1. Define the Linear programming problem in operation Research. Also, explain various
assumptions, advantages and limitations of linear programming problem.
Answer:-
Linear Problem Programming
Linear Pr0blem Pr0gramming is a mathematical technique used t0 0btain an 0ptimum s0luti0n in
res0urce all0cati0n pr0blems, such as pr0ducti0n planning. It is a mathematical m0del 0r
technique f0r efficient and effective utilizati0n 0f limited rec0urses t0 achieve 0rganizati0n
0bjectives
Assumptions of LPP:
 Proportionality- The basic assumpti0n underlying the linear pr0gramming is that any
change in the c0nstraint inequalities will have the pr0p0rti0nal change in the 0bjective
functi0n. This means, if pr0duct c0ntributes Rs 20 t0wards the pr0fit, then the t0tal
c0ntributi0n w0uld be equal t0 20x1, where x1 is the number 0f units 0f the pr0duct.
 Additivity- The assumpti0n 0f additivity asserts that the t0tal pr0fit 0f the 0bjective
functi0n is determined by the sum 0f pr0fit c0ntributed by each pr0duct separately.
Similarly, the t0tal am0unt 0f res0urces used is determined by the sum 0f res0urces used
by each pr0duct separately. This implies, there is n0 interacti0n between the decisi0n
variables
 Continuity- An0ther assumpti0n 0f linear pr0gramming is that the decisi0n variables are
c0ntinu0us. This means a c0mbinati0n 0f 0utputs can be used with the fracti0nal values
al0ng with the integer values.
 Certainty- An0ther underlying assumpti0n 0f linear pr0gramming is a certainty, i.e. the
parameters 0f 0bjective functi0n c0efficients and the c0efficients 0f c0nstraint inequalities
is kn0wn with certainty.
 Finite Choices- This assumption implies that the decision maker has certain choices, and
the decision variables assume non-negative values.

2. MBA 205 OPERATIONS RESEARCH
Advantages of LPP:
1. Utilized t0 analyze numer0us ec0n0mic, s0cial, military and industrial pr0blem.
2. Linear pr0gramming is best suitable f0r s0lving c0mplex pr0blems.
3. Helps in simplicity and Pr0ductive management 0f an 0rganizati0n which gives better
0utc0mes.
4. Pr0vides a way t0 unify results fr0m disparate areas 0f mechanism design.
5. M0re flexible than any 0ther system, a wide range 0f pr0blems can be s0lved easily.
Limitations of LPP:
1. T0 specify an 0bjective functi0n in mathematical f0rm is n0t an easy task.
2. T0 determine the relevant values 0f the c0-efficient 0f c0nstraints inv0lved in LP is a main
pr0blem.
3. The assumpti0ns 0f LP are als0 unrealistic. It assumes that fact0ry pr0p0rti0n remains
c0nstant. In additi0n f0r it, the relati0nship between input and 0utput, pr0ducti0n and c0st,
and pr0ducti0n and t0tal revenue are assumed t0 be linear.
4. All these assumpti0ns imply c0nstant returns t0 scale and perfect c0mpetiti0n in the
market. But in fact the relati0ns are n0t always linear and imperfect c0mpetiti0n prevails
in the market.
5. It is a very c0mplex meth0d as it uses mathematical techniques extensively. LP m0dels
presents a trial and err0r s0luti0ns and it is difficult t0 find 0ut really 0ptimal s0luti0ns t0
vari0us business pr0blems.

3. MBA 205 OPERATIONS RESEARCH
Q.2
a. Discuss the concept of Degeneracy in transportation problem
b. The ABC Tool Company has a sales force of 25 men who work out from Regional offices.
The company produces four basic products lines of hand tools. Mr. Jain, the sales
manager, feels that 6 salesmen are needed to distribute product line 1, 10 salesmenare
needed to distribute product line 2, 4 salesmen to product line 3 and 5 salesmento product
line 4. The cost per day of assigning salesmenfrom each of the offices for selling each of the
product lines are as follows
Regional office Product Lines
20 21 16 18
17 28 14 16
29 23 19 20
Now, 10 salesmen are allowed to office, 9 salesmen to office , and 7 salesmen to office .How
many salesmen should be assigned from each office to selling each product line in order to
minimize costs?
Answer:-
Degeneracy in Transportation Problem:
A basic s0luti0n t0 an m-0rigin, n destinati0n transp0rtati0n pr0blem can have at the m0st m+n-
1 p0sitive basic variables (n0n-zer0), 0therwise the basic s0luti0n degenerates. It f0ll0ws that
whenever the number 0f basic cells is less than m + n – 1, the transp0rtati0n pr0blem is a
degenerate 0ne. The degeneracy can devel0p in tw0 ways:
 Case 1 - The degeneracy devel0ps while determining an initial assignment via any 0ne 0f
the initial assignment meth0ds discussed earlier. T0 res0lve degeneracy, y0u must
augment the p0sitive variables by as many zer0-valued variables as is necessary t0
c0mplete the required m + n – 1 basic variable. These zer0-valued variables are selected
in such a manner that the resulting m + n – 1 variable c0nstitutes a basic s0luti0n. The
selected zer0 valued variables are designated by all0cating an extremely small p0sitive
value ε t0 each 0ne 0f them. The cells c0ntaining these extremely small all0cati0ns are
then treated like any 0ther basic cells.
 Case 2 - The degeneracy devel0ps at the iterati0n stage. This happens when the selecti0n
0f the entering variable results in the simultane0us drive t0 zer0 0f tw0 0r m0re current
(pre-iterati0n) basic variables. T0 res0lve degeneracy, the p0sitive variables are
augmented by as many zer0-valued variables as it is necessary t0 c0mplete m+n-1 basic
variables. These zer0-valued variables are selected fr0m am0ng th0se current basic
variables, which are simultane0usly driven t0 zer0. The rest 0f the pr0cedure is exactly the
same as discussed in case

4. MBA 205 OPERATIONS RESEARCH
Solution to the problem:
Since the total availability (i.e 26 salesman) at regional offices R1, R2 and R3 exceeds the total
requirement (i.e 25 salesman) a product lines P1, P2, P3 and P4, by (26-25) = 1, therefore, the
problem is unbalanced. We, therefore, add a dummy product line 5 with its requirement of 1
salesman and he cost per day of assigning salesman from each of the offices to selling each of the
product lines is zero.
The initial solution is obtained by using Vogel Approximation method as shown below:
Using this, we get the optimal solution to this problem and verify that the solution is optimal. It
may also be noted from the table that the opportunity cost is zero in cell (R2, P4) and (R3,P4) and
therefore, we shall have alternative optimal solution.
The transportation cost associated with his solution is:
Total cost = 21(4) + 16(1) + 18(5) + 17(6) + 14(3) + 23(6) + 0(1) = Rs. 472

5. MBA 205 OPERATIONS RESEARCH
Q3.
a. Elaborate the meaning of Simulation.
b. What are different Practical applications of simulation?
Answer:-
a) Simulation
Simulati0n is a representati0n 0f real-life situati0ns. It is a meth0d in which a replica 0f a
real-w0rld pr0cess 0r system is devel0ped 0ver a peri0d 0f time. The simulated m0del acts
in the same manner as the selected physical 0r abstract pr0cess 0r system behaves in reality.
Simulati0n used f0r training purp0ses are divided int0 three categ0ries, which are as
f0ll0ws:
 Live Simulation - Refers t0 a simulati0n in which equipment is used t0 imitate a real
system. F0r example, testing the battery 0f a car with the help 0f an electrical tester.
 Virtual Simulation- Refers t0 a simulati0n in which real pe0ple 0perate 0n simulated
systems. F0r example, a pil0t flying a simulated jet.
 Constructive Simulation - Refers t0 a simulati0n in which simulated pe0ple 0perate 0n
simulated systems.
b) Practical applications of Simulation:
1. Simulation in the education sector
Simulati0n is widely used f0r educati0nal purp0ses. Simulati0n m0dels are used t0 create
a real-w0rld envir0nment in a classr00m that helps students t0 understand vari0us key
c0ncepts.
2. Simulation in the medical sector
In the medical sect0r, simulati0n m0dels are devel0ped t0 teach therapeutic and diagn0stic
pr0cedures and vari0us medical c0ncepts. Simulati0n m0dels are als0 devel0ped f0r
pr0viding training 0n bl00d draw, lapar0sc0pic surgery and trauma care.
3. Simulation in the entertainment sector
In the entertainment sect0r, simulati0n has been pr0ved t0 be very effective in vari0us
fields, which are as f0ll0ws:

6. MBA 205 OPERATIONS RESEARCH
 Computer and video games- Refer t0 simulati0n games that represent a real envir0nment.
These games represent interacti0ns between playable characters and envir0nment
realistically. S0me 0f the p0pular simulati0n games are The Sims, C0mmand and C0nquer,
SimCity, Black and White, Tiger W00ds
 Film- Refers t0 0ne 0f the m0st imp0rtant applicati0ns 0f simulati0n, in the field 0f
entertainment. S0me 0f the films in which simulated m0dels have been used are Finding
Nem0, Live Free 0r Die Hard, 300, Up, Ir0n Man, Wall-E, etc.
 Theme park rides- Refer t0 0ne 0f the m0st rapidly gr0wing fields 0f entertainment.
Simulat0r m0dels are used here t0 rec0rd the m0ves 0f rides. S0me 0f the p0pular
simulated rides are S0arin’ 0ver Calif0rnia, The Amazing Adventures 0f Spiderman,
Missi0n Space and The Simps0ns Ride.
4. Simulation in manufacturing
The field 0f manufacturing has a wide applicati0n 0f simulati0n m0dels. Practical use are
listed bel0w:
 Determining the thr0ughput under average and peak l0ads
 Calculating system cycle time
 Making efficient utilisati0n 0f res0urces, such as men, raw material, m0ney and machines
 Identifying b0ttlenecks and sh0rtc0mings in the pr0ducti0n pr0cess
 S0lving the pr0blems 0f queuing in a w0rk envir0nment
 Identifying staffing requirements
 Determining the effectiveness 0f scheduling and c0ntr0l systems

7. MBA 205 OPERATIONS RESEARCH
Assignment Set -II
Q1.
A Define the meaning of assignment problem in operation Research.
B A Departmental head has four subordinates and four task to be performed.
The subordinates differ in efficiency and the tasks differ in their intrinsic
difficulty. His estimate of the times each man would take to perform each task is
given in the following matrix-
How should the tasks be allocated to subordinates to minimize the total man-hours?
Answer:-
Assignment Problem in Operation Research:
An assignment pr0blem is a special type 0f transp0rtati0n pr0blem. In an assignment pr0blem, the
same number 0f facilities (s0urces 0f supply) needs t0 be all0cated t0 the same number 0f j0bs
(p0ints 0f destinati0ns) s0 that the transp0rtati0n c0st is minimized 0r the pr0fit is maximized. An
assignment pr0blem can 0ccur while assigning:
 Machines t0 fact0ry 0rders
 Salespe0ple t0 sales territ0ries
 Teachers t0 classes
 P0lice vehicles t0 patr0lling areas
 The f0ll0wing are s0me 0f the basic assumpti0ns made 0f an assignment pr0blem:
 The number 0f assignees and the number 0f tasks are the same.
 Each assignee is t0 be assigned 0nly 0ne task.
Tasks Subordinates
I II III IV
A 8 26 17 11
B 13 28 4 26
C 38 19 18 15
D 19 26 24 10

8. MBA 205 OPERATIONS RESEARCH
It sh0uld be n0ted that an assignment pr0blem is quite different fr0m a transp0rtati0n pr0blem
because 0f the f0ll0wing tw0 characteristics:
 The c0st matrix 0f an assignment pr0blem is a square matrix; in a transp0rtati0n pr0blem, the
c0st matrix can be in a rectangular shape.
 The 0ptimum s0luti0n f0r the pr0blem is 0btained when there is 0nly 0ne assignment in a r0w
0r c0lumn 0f the c0st matrix.
The Mathematical Model
Let cij be the c0st 0f assigning the ith res0urce t0 the jth task. We define the c0st matrix t0 be the
n × n matrix
An assignment is a set 0f n entry p0siti0ns in the c0st matrix, n0 tw0 0f which lie in the same r0w
0r c0lumn. The sum 0f the n entries 0f an assignment is its c0st. An assignment with the smallest
p0ssible c0st is called an 0ptimal assignment.
Solution of the Problem
Tasks Subordinates
I II III IV
A 8 26 17 11
B 13 28 4 26
C 38 19 18 15
D 19 26 24 10
Step 1: Identify the minimum element in each row and subtract it from every element of that row,
we get the reduced matrix
0 18 9 3
9 24 0 22
23 4 3 0
9 16 14 0

9. MBA 205 OPERATIONS RESEARCH
Step 2: Identify the minimum element in each column and subtract it from every element of that
column.
0 14 9 3
9 20 0 22
23 0 3 0
9 12 14 0
Step 3: Optimal assignment is: A→I, B → III, C →II and D→ IV
Step 4: The minimum total time for this assignment scheduled is 8 +4+19+10 or 41 man- hours.
Q.2
Define following criteria’s used for decision making under Uncertainty
a. Optimism (maximax or minimin) criterion
b. Pessimism (maximin or minimax)criterion
c. Equal probabilities (Laplace) criterion
d. Coefficient of optimism (Hurwicz)criterion
e. Regret (salvage) criterion
Answer:-
I. Optimism (maximax or minimin) criterion
Here, the decisi0n maker tries t0 achieve the largest p0ssible pr0fit (Maximax) 0rminimum
p0ssible c0st (minimin). If the entries in the pay0ff matrix are the 0ne which the decisi0n maker
wants as large as p0ssible, f0r example, pr0fits 0r sales revenue, he/she selects the alternative
that represents the maximum 0f the maximum pay0ff. In case where the entries 0f the pay0ff
matrix are 0ne which the decisi0n maker wants as small as p0ssible, he/she g0es f0r the
minimum 0f the minimum. Here, c0rresp0nding t0 the vari0us alternatives, he/she
l0cates the maximum pay0ff f0r each alternative (in case 0f pr0fit 0r revenues) and
then selects the alternative which gives the maximum 0f maxima.

10. MBA 205 OPERATIONS RESEARCH
II. Pessimism (maximin or minimax) criterion
In this criteri0n, the decisi0n maker selects the alternative representing the maximum 0f the
minimum pay0ffs in case 0f pr0fits. In the case 0f c0st 0r l0ss, he/she selects the minimum
0f the maxima.
III. Equal probabilities (Laplace) Criterion:
As the pr0babilities are unkn0wn f0r the states 0f nature, all are and c0llective) equal
pr0babilities. Since the states 0f nature are mutually exclusive equal 0ne d'
c0llectively exhaustive, the pr0babilities 0f each 0f these states pay0ff is divided by
the number 0f states 0f nature. Expected value 0f the calculated and in case 0f pr0fit,
the maximum value 0f the expected pay0ff is ch0sen; whereas, in case 0f l0sses, the
minimum value 0f the pay0ff is ch0sen.
IV. Coefficient of optimism (Hurwicz) criterion
Hurwicz intr0duced the idea 0f a c0efficient 0f 0ptimism, den0ted by a, t0 measure
the degree 0f 0ptimism f0r the decisi0n maker. The c0efficient takes a value between
zer0 and 0ne. Here, zer0 represents a c0mplete pessimistic attitude 0f the decisi0n
maker ab0ut the future; whereas, 0ne represents a c0mplete 0ptimistic attitude 0f the
decisi0n maker t0wards the future. The criteri0n suggests that an alternative that
maximises a x (maximum in the c0lumn) + (1 — a) x (minimum in the c0lumn) sh0uld
be ch0sen.
V. Regret (salvage) criterion
This criteri0n is als0 kn0wn as the 0pp0rtunity-l0ss decisi0n criteri0n 0r minimax
regret decisi0n criteri0n. This is because the decisi0n maker feels regret after ad0pting
a wr0ng c0urse 0f acti0n (alternative), which results in an 0pp0rtunity l0ss 0f the
pay0ff. The decisi0n maker theref0re wants t0 minimise the regret. In 0rder t0 devel0p
the 0pp0rtunity l0ss table, perf0rm the f0ll0wing steps:
1. Find the best pay0ff c0rresp0nding t0 each state 0f nature.
2. Subtract all 0ther entries (pay0ff values) in that r0w fr0m this value.
3. F0r each c0urse 0f acti0n (strategy 0r alternative), identify the w0rst 0r
maximum regret value. Rec0rd this number in a new r0w.
4. Select the c0urse 0f acti0n (alternative) with the smallest anticipated
0pp0rtunity-l0ss value.

11. MBA 205 OPERATIONS RESEARCH
Q3.
a. Explain the importance and utility of the replacement model in business
Organizations.
b. The maintenance cost and re-sale value per year of a machine whose purchase
price is Rs.7000 is given below-
Year 1 2 3 4 5 6 7 8
Maintenance cost (in Rs.) 900 1200 1600 2100 2800 3700 4700 5900
Re- sale value (in Rs.) 4000 2000 1200 600 500 400 400 400
When should the machine be replaced?
Answer:-
Importance of replacement Model:
In an 0rganisati0n, replacement pr0blems arise when fixed assets, such as machines, equipment,
and 0ther t00ls, need t0 be replaced due t0 reduced efficiency, failure 0r breakd0wn. S0metimes,
replacement takes place when m0re efficient equipment is available in the market 0r the
maintenance 0f the existing equipment is incurring a huge c0st 0n an 0rganisati0n. H0wever, an
0rganisati0n needs t0 decide when the replacement 0f new equipment w0uld be ec0n0mical.
Replacement m0dels help an 0rganisati0n t0 determine when t0 replace equipment in a c0st-
effective manner s0 that the 0verall pr0ductivity 0f the 0rganisati0n is n0t affected.
These m0dels help manager’s t0 answer the f0ll0wing questi0ns:
1. At what time sh0uld equipment be replaced?
2. Sh0uld the existing equipment be replaced, if the new equipment with better efficiency is
available? If yes, then when?
3. Sh0uld the time value 0f m0ney be taken int0 acc0unt while replacing the equipment?
4. What sh0uld be the replacement plan f0r equipment that is used in large quantities and may
fail rand0mly (f0r example, light bulbs)?
5. What kind 0f replacement p0licy sh0uld be in place f0r the human res0urce 0f an
0rganisati0n?
Replacement m0dels, in c0mbinati0n with vari0us mathematical techniques, help managers in
devel0ping successful replacement p0licies. An 0rganisati0n uses different types 0f equipment.
S0me equipment deteri0rates with time; f0r example, the efficiency 0f a car reduces with time. 0n
the 0ther hand, s0me equipment, such as electric bulbs, fails instantly and needs t0 be replaced
immediately. An 0rganisati0n cann0t use the same replacement p0licy f0r every kind 0f
equipment. Different equipment is replaced differently.

12. MBA 205 OPERATIONS RESEARCH
Solution of the problem
The cost of the machine (c) = 7000.
The optimal replacement period of the machine is determined below:
Year
(1)
Maintenance
cost (Rs.)
(2)
Cumulative
Maintenance
cost (Rs.)
(3)
Resale
value
(Rs.)
(4)
Depreciation
(5)= 7000-(4)
Total cost
(6)=(3+5)
Average
cost =
(6)/(1)
1 900 900 4000 3000 3900 3900
2 1200 2100 2000 5000 7100 3550
3 1600 3700 1200 5800 9500 3166.67
4 2100 5800 600 6400 12200 3050
5 2800 8600 500 6500 15100 3020
6 3700 12300 400 6600 18900 3150
7 4700 17000 400 6600 23600 3371.43
8 5900 22900 400 6600 29500 3687.50
It can be seen that the lowest average cost is 3020, which corresponds to the 5th year. Therefore,
the best time to replace the machine is after the 5th year.