To market
_____________________________________________
M1 M2 M3 M4 Supply
_____________________________________________
From P1 6 4 9 1 40
Plant P2 20 6 11 3 40
P3 7 1 0 14 50
P4 7 1 12 6 90
____________________________________________
Demand 90 30 50 30
____________________________________________
Dummy, degeneracy, non optimal…difficult [path
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Assignment
The assignment problem, is a special case of the linear programming problem. In general items, it is concerned with a one-to-one assignment. One person to one machine, one machine to one job, etc. The cost (or profit) of each person-machine, machine-job, or other assignment is known; the objective is to minimise the total cost or to maximise the total profit of the resource - job assignment.
Hungarian Assignment Method (HAM)
Step 1 Locate the smallest cost element in each row of the cost table. Now subtract this smallest
element from each element in that row. As a result, there shall be at least one zero in each row of this new table, called the Reduced Cost Table.
Step 2 In the reduced cost table obtained, consider each column and locate the smallest element in it.
Subtract the smallest value from every other entry in the column. As a consequence of this action, there would be at least one zero in each of the rows and columns of the second reduced cost table.
Step 3 Draw the minimum number of horizontal and vertical lines (not the diagonal ones) that are required to cover all the 'zero' elements. If the number of lines drawn is equal to n (the number of rows/columns) the solution is optimal, and proceed to step 6. If the number of lines drawn is smaller than n, go to step 4.
Step 4 Select the smallest uncovered (by the lines) cost element. Subtract this element from all
uncovered elements including itself and add this element to each value located at the intersection of any two lines. The cost elements through which only one line passes remain unaltered.
Step 5 Repeat steps 3 and 4 until an optimal solution is obtained.
Step 6 Given the optimal solution, make the job assignments as indicated by the 'zero' elements. This is done as follows:
(a) Locate a row which contains only one 'zero' element. Assign the job corresponding to this element to its corresponding person. Cross out the zero's, if any, in the column
corresponding to the element, which is indicative of the fact that the particular job and p«rson are no more available.
(b) Repeat (a) for each of such rows which contain only one zero. Similarly, perform the same operation in respect of each column containing only one 'zero' element, crossing out the zero(s), if any, in the row in which the element lies.
(c) If there is no row or column with only a single 'zero' element left, then select a
row/column arbitrarily and choose one of the jobs (or persons) and make the assignment.
Now cross the remaining zeros in the column and row in respect of which the assignment is made.
(d) Repeat steps (a) through (c) until all assignments are made.
(e) Determine the total cost with reference to the original cost table.
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Q1 A project consist of four (4) major jobs, for which four (4) contractors have submitted tenders. The tender amounts, in thousands of rupees, are given below.
Jobs
Contractors A B C D 1 120 100 80 90
2 80 90 110 70
3 110 140 120 100
4 90 90 80 90
Find the assignment, which minimizes the total cost of the project. Each contractor has to be assigned
one job. Compiler 349 M01
Q2 A Production supervisor is considering, how he should assign five jobs that are to be performed, to five mechanists working under him. He wants to assign the jobs to the mechanists in such a manner that the aggregate cost to perform the jobs is the least. He has following information about the wages paid to the mechanists for performing these jobs:
Jobs
Mechanist 1 2 3 4 5
A 10 3 3 2 8
B 9 7 8 2 7
C 7 5 6 2 4
D 3 5 8 2 4
E 9 10 9 6 10
Assign the jobs to the mechanists so that the aggregate cost is the least. N01 compiler 347 Q3 A Marketing Manager has 4 subordinates and 4 tasks. The subordinates differ in efficiency. The tasks also
differ in their intrinsic difficulty. His estimates of the time each subordinate would take to perform each task is given in the matrix below. How should the task be allocated one to one man so that the total man-hours are minimized ?
Tasks
I II III IV
1 16 52 34 22
Subordinates 2 26 56 8 52
3 76 38 36 30
4 38 52 48 20 Compiler 361 N04
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made bids as detailed below: Min prohibited routes Compiler 362
Bids for Routes (Rs.)
Company/Routes R1 R2 R3 R4
C1 4,000 5,000 − −
C2 − 4,000 − 4,000
C3 3,000 − 2,000 −
C4 − − 4,000 5,000
Each bidder can be assigned only one route. Determine the minimum cost that the BPO should incur. N06 Q5 To stimulate interest and provide an atmosphere for intellectual discussion, a finance faculty in a
management school decides to hold special seminars on four contemporary topics: leasing, portfolio
management, private mutual funds, swaps and options. Such seminars should be held once in a week in the afternoons. However, scheduling these seminars (one for each topic, and not more than one seminar per afternoon) has to be done carefully so that the number of students unable to attend is kept to a minimum. A careful study indicates that the number of students who cannot attend a particular seminar on a specific day is as follows:
Leasing Portfolio Management Private Mutual Fund Swaps & Options
Monday 50 40 60 20
Tuesday 40 30 40 30
Wednesday 60 20 30 20
Thursday 30 30 20 30
Friday 10 20 10 30
Find an optimal schedule of the seminars. Also find out the total number of students who will be missing at least one seminar. Unbal min Compiler 357 M99
Q6 Solve the following unbalanced assignment problem of minimizing total time for doing all the jobs. M85 Min unbal pad 16.16
Jobs
I II III IV V
Operators 1 6 2 5 2 6
2 2 5 8 7 7
3 7 8 6 9 8
4 6 2 3 4 5
5 9 3 8 9 7
6 4 7 4 6 8
Q7 Five workers are available to work with the machine and the respective costs (in Rs) associated with each worker-machine assignment is given below. A sixth machine is available to replace one of the existing machines and the associated costs are also stated – unbal min prohib routes
Machines
Workers M1 M2 M3 M4 M5 M6
A 12 3 6 - 5 8
B 4 11 - 5 - 3
C 8 2 10 9 7 5
D - 7 8 6 12 10
E 5 8 9 4 6 - RTP
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Q8 Five swimmers are eligible to compete in a relay team which is to consist of four swimmers swimming four different swimming styles; back stroke, breast stroke, free style and butterfly. The time taken for the five swimmers - Anand, Balu, Chandru, Deepak and Eswar - to cover a distance of 100 meters in various swimming styles are given below in minutes : seconds.' Anand swims the back stroke in 1 : 09, the breast stroke in 1 : 15 and has never competed in the free style or butterfly. Balu is a free style specialist averaging 1 : 01 for the 100 meters but can also swim the breast stroke in 1 : 16 and butterfly in 1 : 20 Chandru swims all styles - backstroke 1 :10 breaststroke 1 : 12 free style 1 : 05 and butterfly,1 : 20 Deepak swims only the butterfly 1 : 11 while Easwar swims the back stroke 1 : 20, the breast stroke 1 : 16, the free style 1 : 06 and the butterfly 1 : 10. Which swimmers should be assigned to which swimming style? Who will not be in the
Team? unbal min prohib routé N91
Q9 The cost of transporting different products to different products to different warehouses of a company is given. ‘’x’’ in the matrix denotes that the particular product is not required in the particular warehouse.
Make optimal assignments of the products and also find the cost of such assignment. Pad 16.13 min unbal prohib route
Warehouses
I II III IV V
Product A 28 19 ‘’x’’ 21 22
B 32 25 22 13 18
C 21 18 21 ‘’x’’ 17
D 22 29 21 17 28
Q10 Solve the assignment problem represented by the following effective matrix min multiple opti sol
a b c d e f
___________________________________
A 9 22 58 11 19 27
B 43 78 72 50 63 48
C 41 28 91 37 45 33
D 74 42 27 49 39 32
E 26 11 57 22 25 18
F 3 56 53 31 17 28 N99
Q11 ABC airline operating 7 days a week has given the following time-table. Crews must have minimum layover of 5 hours between flights. Obtain the pairing flights that minimize the layover time away from home. For any given pairing the crew will be based at the city that results in the smaller layover. M00
Hyderabad-Delhi Delhi-Hyderabad
________________________________ ________________________________
Flight No. Departure. Arrival Flight No. Departure. Arrival
A1 6 AM 8 AM B1 8 AM 10 AM
A2 8 AM 10 AM B2 9 AM 11 AM
A3 2 PM 4 PM B3 2 PM 4 PM
A4 8 PM 10 PM B4 7 PM 9 PM
Compiler 353
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Q12 WELLDONE Company has taken the third floor of a multistoried building for rent with a view to locate one of their zonal offices. There are five. main rooms in this floor to be assigned to five managers. Each room has its own advantages and disadvantages. Some have windows, some are closer to the washrooms or to the canteen or secretarial pool. The rooms are of all different sizes and shapes. Each of the five managers was asked to rank their room preferences amongst the rooms 301, 302, 303, 304 and 305.
Their preferences were recorded in a table as indicated below:
Manager
M1 M2 M3 M4 M5
302 302 303 302 301
303 304 301 305 302
304 305 304 304 304
301 305 303
302
Most of the manager did not list all the 5 rooms sice they were not satisfied with some of these Rooms and they have left off these from the list. Assuming that their preferences can be qualified by numbers, find out as to which manager would be assigned to which room so that their total preference is minimum.
N90, minimize ranking
Q13 The XYZ co. has 5 jobs A,B,C,D and E to be done and 5 men L,M,N,O and P to do these jobs. The number of hours each man would take to accomplish each job is given by take to accomplish each job is given by the following table. Work out the optimum assignment and the total minimum time taken.
Men bal,min,multi
L M N O P
Jobs A 16 13 17 19 20
B 14 12 13 16 17
C 14 11 12 17 18
D 5 5 8 8 11
E 5 3 8 8 10
Q14 Solve the following assignment problem and obtain the minimum cost at which all the jobs can be performed.
Min unbal multi Job (cost in ’00 Rs)
Worker 1 2 3 4 5
A 25 18 32 20 21
B 34 25 21 12 17
C 20 17 20 32 16
D 20 28 20 16 27
Q15 A solicitor’s firm employ’s typists on hourly piece rate basis for their daily work. There are 5 typists and their charges and speed are different. According to an earlier understanding , only one job is given to one typist and the typist is paid for a full hour even when he works for a fraction of an hour. Find the least cost allocation for the following data:
Typist Rate/hour (Rs) No. of pages typed per hour Job No. of pages
A 5 12 P 199
B 6 14 Q 175
C 3 8 R 145
D 4 10 S 298
E 4 11 T 178 min
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Q16 A hospital has to pay Nurses for 40 hours a week. One Nurse is assigned to one patient. The cost per hour for each of the Nurses is given below:
(a) Find the Nurse- Patient combination to minimize cost to the hospital.
(b) How much does each nurse earn per week?
Suppose that a new patient Z is admitted, and that a new nurse N is appointed. The new patient is charged Rs 40 per hour by each of the existing nurses. The new nurse charges Rs 50 per hour irrespective of the patient.
(c) What would be your revised calculations?
(d) Comment on the new solution. M10
Q17 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 lakhs towards the cost with a condition that the repairs must be done at the lowest cost and quickest time.
If conditions worsens. then a supplementary token grant will also be considered favorably. The
corporation has floated tenders and 5 contractors have sent in their bids. In order to expedite work. one road will be awarded to only one contractor
Cost of Repairs (Rs lakhs)
(i) Find the best way of assigning the repair work to the contractors and the costs.
(ii) If it is necessary to seek supplementary grants, then what should be amount sought?
(iii) Which of the five contractors will be unsuccessful in his bid?
(iv) If C1 unable to accept any work, find best assignment
Q18 A company plans to assign 5 salesman to 5 districts in which it operates. Estimates of sales revenue in thousands of rupees for each salesman in different districts are given in the following table. In your opinion, what should be the placement of the salesman if the objective is to maximize the expected sales revenue?
Q19 A firm produces 4 products. There are 4 operators who are capable of producing any of these 4 products.
The firm records 8 hours a day and allows 30 minutes for lunch. The processing time in minutes and profit for each of the products are given below:
Products
Find the optimal assignment of products to operators bal max, Patient W Patient X Patient Y
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Q20 A company has four zones open and four marketing managers available for assignment. The zones are not equal in sales potentials. It is estimated that a typical marketing manager operating in each zone would bring in the following Annual sales:
Zones Rs.
East 2,40,000
West 1,92,000
North 1,44,000
South 1,20,000
The four marketing manages are also different in ability. It is estimated that working under the same conditions, their yearly sales would be proportionately as under:
Manager M : 8 Manager N : 7 Manager O : 5 Manager P : 4 Required:
If the criterion is maximum expected total sales, find the optimum assignment and the maximum sales. N07 Compiler 365
Q21 The Marketing director of a multi unit company, Mr Ramesh Arora is faced with the problem of assigning 5 senior marketing managers to 6 zones. The efficiency table of these managers is given below-
Zones
As an advisor to the company, recommend which zone should be managed by a junior manager because of non availability of one more senior marketing manager, so as to maximize the efficiency of the company.
Max,unbal Q22 Solve the following assignment problem to maximize the production.
Machines
Q23 An airline has drawn up a new flight schedule involving 5 flights. To assist in allocating 5 pilots to the flights, it has asked them to state their preference scorers by giving each flight a number out of 10. The higher the number, greater the preference is. Certain of these flights are unsuitable to some pilots owing to domestic
reasons. These have been marked with a ‘’x’’. max,bal,prohib
Flight Number
What should be the allocation of pilots to flights in order to meet as many preferences as possible.
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Q24 A methods engineer wants to assign 4 new methods to 3 work centers. The assignment of the new methods will increase production, details of which are given below. Determine the optimal assignment, if only one method can be assigned to each work center.
Increase in production(units)
Q25 The captain of the playwell cricket team has to allot 5 middle order batting positions to five batsmen. The average run scored by each batsmen at these positions are as follows M92
Batting positions
1. Find the assignment of batsmen to positions, which would give the maximum number of runs.
2. What will be the total runs scored if batsmen T wants only the III position
3. If another bats men U with the following average runs in the batting positions as given below:
Batting position I II III IV V
Average Runs 45 52 38 50 49
Is added to the team, should he be included to play in the team?, If so who will be replaced by him?
Q26 A manufacturing company has four zones East, West, North and South and four sales engineers A,B,C, and D respectively for assignment. The zones are not equally rich in sales potential. Therefore, it is estimated that a particular engineer operating in a particular zone will bring the following sales
East – Rs 4,20,000, West – Rs 3,36,000, North – Rs 2,94,000, South Rs 4,62,000
The sales engineers have different sales ability, working under the same conditions, their yearly sales are [proportional to 14, 9, 11 and 8 respectively. The criteria for maximum expected total sales is to be met by assigning the best sales engineer to the richest zone, the next best to the second richest zone and so on. Find the optimum assignment and the maximum sales. M98
Q 27 five star hotel has 4 banquet halls that can be used for all functions including weddings. The halls are all about the same size but the facilities in each hall differs. During the heavy marriage season, 4 parties approached the executive director to reserve a hall for wedding to be celebrated on the same day. These marriage parties were told that the first choice among these four halls would cost Rs 10,000 for the day. They were also required to indicate the second third and fourth preferences and the price that they would be willing to pay.
Marriage party A and D indicated that they woudn’t be interested in halls H3 and H4. Other relevant particulars are given below-
Decide on the allocation that will maximize the revenue to the hotel.
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Q28 A car hiring company has one car at each of the 5 depots A,B,C,D and E. A customer in each of the 5 towns V, W, X, Y and Z requires a car. The distance in Kms. Between depots (origin) and the towns (destinations) are given in the following table:
Depots
A B C D E
V 3 5 10 15 8
W 4 7 15 18 8
X 8 12 20 20 12
Z 10 10 15 25 10
Find out as to which car should be assigned to which customer so that the total distance travelled is a minimum. How much is the total travelled distance?
Q29 The cost matrix giving selling costs per unit of a product by salesman A,B,C and D in Regions R1, R2, R3 and R4 is given below:
Assign one salesman to one region to minimize selling cost.
If the selling price of the product id Rs 200 per unit and variable cost excluding the selling cost given in the table is Rs 100 per unit, find the assignment that would maximize the contribution.
What other conclusions can you make from the above?
A B C D
R1 4 12 16 8
R2 20 28 32 24
R3 36 44 48 40 N08
R4 52 60 64 56
Q30 A firm is contemplating the introduction of three products 1,2 and 3 in the 3 plants A,B and C. Only a single product is decided to be introduced in each of the three plants. The unit cost of production is given in the following matrix:
A B C (M indicates
impossible assignment)
1 8 12 M
2 10 6 4
3 7 6 5
A. How should the products be assigned so that the total unit cost is minimum?
B. If the quantity of different products to be produced is as follows then which assignments shall minimize the aggregate production cost?
C. It is expected that the selling prices of the products produced by different plants would be different. Assuming that the the quantities given at (b) above would be produced and sold, how should the products be assigned to maximize profit?
A B C
1 15 18 -
2 18 16 10
3 12 10 8
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Q31
A company has 4 territories and 4 salesmen available for assignment. Te territories are not equally rich in their sales potential. The annual total sales potential in each territory is- Territory I – Rs 60,000, Territory II – Rs 50,000, Territory III – Rs 40,000, Territory IV – Rs 30,000.Four salesman differ in ability. Estimation of their yearly sales proportion under the same working conditions are-
Salesman A B C D
Proportion 7 5 5 4
To maximize expected total sales, the intitutive answer is to assign the best salesman to the richest territory the next best salesman to the second richest and so on. Verify this answer by assignment technique RTP
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SIMULATION
Q What is simulation and what are the steps in simulation?
SIMULATION means to assume the mere appearance of without the reality. Thus the appearance is true but not real, which implies that simulation is imitation of reality. Simulation is the representation of a system by a model which will react to change in a similar way to that which is being simulated.
Simulation may be defined as "a quantitative procedure which describes a process by developing a model of that process and later conducting a series of organized trial and error experiments to predict the behaviours of the process over a period. In other words, simulation portrays how the real process would react to certain change."
Steps Involved in Simulation Process
The following are the steps involved generally in a Simulation Process.
1. Identification of the problem or system which is intended to be s!mulated.
2. Formulation of the model intended to be used.
3. Testing of the model by comparing its behaviour with the behaviour of the actual problem environment
4. Identification and collection of the data required for testing of the model.
5. Running of the simulation.
6. Analysing the results of simulation in case necessary, the solution which is being evaluated may be changed
7. Re-running of the simulation to test the new solution
7. Re-running of the simulation to test the new solution