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OPERATIONS RESEARCH
EDITION 2013
OPERATIONS RESEARCH
SNO TOPICPAGE NOS
1 ASSIGNMENT PROBLEMS 3 – 15
2 TRANSPORTATION PROBLEMS 16 – 33
3LINEAR PROGRAMMING
PROBLEMS34 – 53
4 NETWORK ANALYSIS 54 – 73
5 QUEUEING THEORY 74 - 80
6 GAMES THEORY 81 - 91
7 INVENTORY MANAGEMENT 92 - 105
IND
EX
MAN JOB
A B C D E
1 11 17 8 16 20
2 9 7 12 6 15
3 13 16 15 12 16
4 21 24 17 28 26
5 14 10 12 11 15
1. An automobiles dealer wishes to put five repairmen to five different jobs. The repairmen have somewhat different kinds of skills and they exhibit different levels of efficiency from one job to another. The dealer has estimated the number of man-hours that would be required for each job-man combination. This is given in the table form as follows:
Find the optimum assignment that will result in minimum man-hours needed.
3
TypistRate per hour
(Rs)
No of pages Typed per
hourJobs No of pages
A 5 12 P 200
B 6 14 Q 176
C 3 8 R 150
D 4 10 S 300
E 4 11 T 180
2. A Solicitor’s firm employ typists for piece-meal work on an hourly basis. There are five typists available and their charges and speeds are different. Accordingly to an earlier understanding only one job is given to one typist and typist is paid for for full hour even if he works for a fraction of an hour. Find the least cost allocation.
4
ContractorRoadways
R1 R2 R3 R4 R5
C1 18 19 14 25 NB
C2 17 18 13 22 24
C3 19 21 18 20 26
C4 17 22 15 24 27
C5 14 21 10 NB 25
3. GBMC have decided to carry out repairs of five main roads in the city before the onset of the monsoon. Five contractors have submitted quotations as under. It is the policy of GBMC to award not more than one contact to a contractor. If the objective of the GBMC is to minimize the total cost, how should they award the contracts? How much expenditure should be budgeted?
5
CompanyProposals
1 2 3 4 5
A 50 85 100 75 80
B 80 85 95 X 90
C 70 80 85 75 80
D X 90 95 70 85
E 85 80 90 80 90
4. The government solicits five different proposals with the intent of giving one job to each of the companies. The bid amounts in thousands of rupees are given below with an X denoting no bid submitted. Find the optimal assignment to companies such that the total cost is minimum?
6
ConsultantsJobs
1 2 3 4 5 6
A 7 7 3 6 10 11
B 8 9 - 5 8 10
C 9 10 11 13 13 8
D 6 6 8 - 12 13
E 5 5 9 10 10 12
F 8 4 10 12 9 -
5. MM consultants have six consultants available to be assigned to do six jobs for clients. However because of technical deficiencies in particular area, consultant B cannot do the job 4 and consultant F cannot do job 6. The cost incurred to complete the available assignments are given the table below:
7
Room
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 - -
6. An ad agency wants to allocate its five managers to five rooms of different sizes and shapes. Each room has its own advantages and disadvantages. The managers were asked to rank their preferences amongst the rooms 301,302,303,304 and 305.Find out as to which manager should be assigned to which room, so that their total preference ranking is minimum.
8
Days LeasingPortfolio
managementPrivate
Mutual FundEquity
Research
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
7. Schedule the training seminars in five working days of the week so that the number of students unable to attend is kept to the minimum.
9
WorkersMachine
M1 M2 M3 M4 M5 M6
W1 1200 300 600 - 500 900
W2 400 1100 - 500 - 300
W3 800 200 1000 900 700 500
W4 - 700 800 600 1200 1000
W5 500 800 900 400 600 -
8. Five workers are available to work with the machines and the respective cost in Rs associated with each worker-machine assignment is given below. A sixth machine is available to replace one of the existing machines and associated cost are given below:
•Determine the allocation of workers to old machines at minimum cost.
•Determine whether the new machine can be accepted or not.
•Determine the savings, if any, in cost. 10
9. A departmental store agency runs five stores located at different parts of Mumbai. Each store is administered by a manager appointed by agency. The agency reimburses the car travel expenses incurred by managers in commuting to work from their residence to the sores to which they are assigned. The basis of reimbursements is:
A fixed sum of Rs 300/- per month for repairs/maintenance. A variable amount at the rate of Rs 1.60 per kilometer of travel incurred during the month is paid extra.
All stores work for 25 days in a month.
The distance in kilometers from a managers residence to the stores is given in the table which follows.
•Find the optimal assignment of the managers to the stores so that the monthly expenditure to be incurred by the agency on car travel of managers is minimum.
•Manager A requests the agency to assign him Store S1, which is closest to his residence, as he has been medically advised not to take long journeys. If the agency agrees to this request, how should the present assignments be changed and how much extra will it cost the agency?
11
Managers
Stores (Distance in kms)
S1 S2 S3 S4 S5
A 4 10 12 18 17
B 7 16 16 22 18
C 8 6 9 19 21
D 11 12 15 12 13
E 9 14 19 18 14
12
Marketing Manager
Zones
1 2 3 4 5 6
A 71 8 85 80 76 78
B 79 83 67 74 72 83
C 73 70 81 82 76 89
D 91 94 84 89 81 80
E 88 89 77 87 67 74
10. The Marketing Director of a multinational company is faced with the problem of assigning five Senior marketing managers to six zones. From past experience he knows that the efficiency percentage by sales depends a lot on marketing manager –zone combination given in the table below:
As a adviser to the company, recommend which zone should be manned by a junior manager so as to maximize the overall efficiency of the company.
13
Delhi-Jaipur
Flight No Departure Arrival
101 7 A.M. 8 A.M.
102 8 A.M. 9 A.M.
103 1.30 P.M. 2.30 P.M.
104 6.30 P.M. 7.30 P.M.
11. An airline that operates seven days in a week has time table shown below. Crews must have a minimum layover of five hours between flights. Obtain the pairing of flights that minimize layover time away from home. For any given pair
the crew will be based at the city that results in smaller layover.
14
Jaipur-Delhi
Flight No Departure Arrival
201 8A.M. 9.15A.M.
202 8.30 A.M. 9.45 A.M.
203 12 Noon 1.15 P.M.
204 5.30 P.M. 6.45 P.M.
For each pair also mention the place where the crew should be based.
15
Plant
Warehouse
SupplyD1 D2 D3 D4
01 6 5 1 3 100
02 4 8 7 2 125
03 6 3 9 5 75
Demand 70 90 80 60 300
Solve the following transportation problems for the optimum cost:
1.
16
Plant
Warehouse
SupplyD1 D2 D3 D4
01 3 1 3 5 120
02 2 6 1 3 95
03 5 1 4 8 85
Demand 50 60 90 100 300
2.
17
Plant
Warehouse
SupplyD1 D2 D3 D4
01 4 5 1 2 120
02 1 3 4 5 85
03 3 7 6 3 95
Demand 70 80 50 100 300
3.
18
Plant
Warehouse
SupplyD1 D2 D3 D4
01 7 3 8 6 60
02 4 2 5 10 100
03 2 6 5 1 40
Demand 20 50 50 80 200
4.
19
Plant
Warehouse
SupplyD1 D2 D3 D4
01 4 5 3 6 50
02 3 6 7 3 70
03 1 4 1 2 80
Demand 50 40 90 20 200
5.
20
Plant
Warehouse
SupplyD1 D2 D3 D4
01 4 2 1 5 50
02 1 4 2 3 70
03 2 3 6 1 80
Demand 50 90 35 25 200
6.
21
Plant
Warehouse
SupplyD1 D2 D3 D4
01 15 24 11 12 5000
02 25 20 14 16 4000
03 12 12 22 13 7000
Demand 3000 2500 3500 4000
7.
22
Factory
Warehouse
D E F
A 5 1 7
B 6 4 6
C 3 2 5
8. A company has three factories at locations A,B and C which supplies to three warehouses located at D,E and F. Monthly
factory capacities are 10,80 and 15 units respectively. Monthly warehouse requirements are 75,20 and 50 units respectively.
Unit shipping cost in Rs are given below.
The penalty cost for not satisfying demand at the warehouse D,E and F are Rs five, Rs three, and Rs two per unit respectively.
Determine the optimal distribution for the company using any of the known algorithms.23
BankInterest rate in percentage for project
P Q R S T Max Credit(in 000)
Private Bank 20 18 18 17 17 Any Amount
Nationalized Bank
16 16 16 15 16 400
Co-operative Bank
15 15 15 13 14 250
Amount Required (in
000)
200 150 200 125
75
9. Shreyas Construction Company is interested in taking loans from banks for some of its projects P,Q,R,S, T. The rates of
interest and the lending capacity differ from bank to bank. All these projects are to be completed .The relevant details are
provided in the following table:
Advice the company as to how it should take loans so that the total interest payable will be least.
24
Month Contracted Sales
Maximum Production
Unit Cost of Production
1 20 40 14
2 30 50 16
3 50 30 15
4 40 50 17
10. A manufacturer must produce a product in sufficient quantity to meet contractual sales in next four months. The
production capacity and unit cost of production varies from month to month The product produced in one month may be held
for sale in later months, but at an estimated storage cost of Re 1 per unit per month. No storage cost is incurred for goods sold
in the same month in which they are produced. There is no opening inventory and none is desired at the end of four months.
The necessary details are given in the following table:
How much should the manufacturer produce each month to minimize total cost?
25
FactoryDestinations Availability
D1 D2 D3 D4
A 10 8(5000) 7 11 5000
B 12 13 6(4500) 10(1500) 6000
C 8(7000) 10(500) 12 14(1500) 9000
Demand 7000 5500 4500 3000
11. Given below is an intermediate table in the solution of a transportation problem. With reference to this table, answer the following questions(giving reasons):-
•Is this solution feasible?
•Is this solution optimal? If not obtain the optimal solution.
•Does the problem have an alternate optimum solution? If so, find one such solution.
•What will be the minimum decrease in cost on the routeBD1 before the company starts using this route?
26
PlantsShowroom
I II III 1V
A 90 100 120 110
B 100 105 130 117
C 111 109 110 120
D 130 125 108 113
12. A company manufacturing television sets has four plants with a capacity of 125,250,175 and 100 units respectively. The company supplies TV sets to its four showrooms which have demand of 100,400,90 and 60 units respectively. Due to the difference in the raw material cost and the transportation cost, the profits per unit(in Rs) differ which are given in the following table:-
The demand of showroom I must be supplied from Plant A. By using VAM,plan the production programme so as to maximize profit. Also determine the maximum profit.
27
13. A company has three factories which supply their products to four warehouses. Monthly capacity of the factories are 120, 200 and 180 units respectively. Monthly requirements of warehouses are 100, 140, 110 and 150 respectively. Unit shipping costs are as follows:
Factory Warehouses
P Q R S
I 15 - 30 20
II - 24 12 15
III 25 15 - 20
Shipment from I to Q, II to P and III to R is not possible due to certain unavoidable reasons. Find the optimum distribution program to minimize shipping costs.
28
14. A hotel corporation has three restaurants around the country all of which use standard drinking (disposable) cups. Three suppliers have been invited to bid on supplying the cups. Their bid are as follows:
Supplier Price Rs.per 100 Annual capacity A 9 30,000 B 10 70,000 C 11 1,35,000
The cost of transportation (In Rs. Per 100 cups) varies from each supplier to each restaurant as given below:
From Restaurant
1 2 3 A 2 4 1 B 5 3 6 C 3 2 7
The annual requirement of cups for three restaurants are 30,000, 60,000 and 1,20,000 respectively.
How many cups should be purchased from each supplier for each restaurant?
29
15. A company has three factories manufacturing the same product and five sole selling agencies in different parts of the country. Production costs differ from factory to factory and sales price from agency to agency. Find the production and distribution schedule most profitable to the company. Given the following data :
Factory Production Cost (Rs.) Agency
I20
II22
III18
Sale Price per unitRs.
Demand
Transport Cost per unit
IIIIIIIVV
31574
97836
45327
3032313429
801007545
125
Capacity 150 200 125
30
16. ABC Limited has three production shops supplying a product to five warehouses. The cost of production varies from shop to shop and the cost of transportation from one shop to a warehouse also varies. Each shop has a specific production capacity and each warehouse has a certain amount of requirement. The costs of transportation are as given below:
ShopWarehouse Max. Credit
(in ‘000)I II III IV V
ABC
653
464
476
743
584
100125175
Requirement 60 80 85 105 70
31
The cost of manufacture of the product at different production shops is:
ShopCost
Variable Cost Fixed Cost
A
B
C
14
16
15
7000
4000
5000
Find the optimum quantity to be supplied from each shop to different warehouses at minimum total cost.
32
17. A company has factories F1, F2, F3, F4 manufacturing the same product. Production and raw material costs differ from factory to factory and are given in the following table in the first two rows. The transportation costs from the factories to sale depots S1, S2, S3 are also given. The lastt two columns in the table give sale price and the total requirements at each depot. The production capacity of each factory is given in the last row.
Sale Price
Requirements
F1 F2 F3 F4 (Rs.) (Units)
ProductionCost/unit (Rs.)
15 18 14 13
Raw materialCost/unit (Rs.)
10 9 12 9
TransportationCost/unit (Rs.)
S1 3 9 5 4 34 80
S2 1 7 4 5 22 120
S3 5 8 3 6 31 150
ProductionCapacity in Units
100 150 50 100
Determine the most profitable production and distribution schedule and the corresponding profit. The surplus production should be taken to yield zero net profit. 33
1. A manufacturing firm produces two products A and B.Each of these products must be processed through two different machines. One machine has 12 hours and the second machine has 8 hours of available capacity. Each unit of product A requires two hours of time on both the machines. Each unit of product B requires three hours of time on the first machine and one hour on the second machine. The incremental profit of Rs 6 per unit of product A and Rs 7 per unit of product B and the firm can sell as many units of each product it can manufacture. The objective of the frm is to maximize profits. The problem is to determine how many units of product A and product B should be produced within the limits of available machine capacities .
34
Machine Chair TableAvailable hours per
week
M1 3 3 36
M2 5 2 50
M3 2 6 60
2. A firm makes two types of furniture's: chairs and tables. The profit contribution from each product as calculated by the accounting department is Rs 20 per chair and Rs 30 per table. Both products are processed on the three machines M1, M2 and M3.The time required by each product and total time available per week on each machine are as follows:
How should the manufacturer schedule his production in order to maximize profit?
35
Department
Hours requiredAvailable hours per
monthAlpha Beta
1 2 3 1500
2 3 2 1500
3 1 1 600
3. A small scale factory’s production is limited to two industrial products, Alpha and Beta. The contributions for each product have been computed by its accounting department as Rs 10 for Alpha and Rs 12 for Beta. Each product passes through three departments of the plant. The time requirements for each product and total time available are as follows:
Determine the quantities of products Alpha and Beta to be manufactured for maximum contribution to the company.
36
Product Lathe Milling Grinder
A 5 9 3
B 4 3 0
C 0 5 2
4. A jobbing shop has three machine groups, namely lathes, milling machines and grinders. It has an idle capacity of 350 hours,500 hours and 150 hours per week respectively. It is offered products A,B and C to be manufactured. Each unit of product A yields Rs 30,product B Rs 12 and product C Rs 15.The time taken by each unit of the three products on different machines are given in table below:
•How many quantity of product A,B and C must be manufactured every week to yield maximum profit?
•What capacity of each machine remains idle after making these products?
37
5. Nahar Electronics Limited manufacture transistors in two models A and B whose contribution to profit is Rs 4 and Rs 14 respectively. Each type has to be processed and completed in two main departments viz, manufacturing and assembly. Following table indicates time in hours per set of each category in each department:
Department
Product
Capacity in hours per
weekA B
Manufacturing 2 7 21
Assembly 7 2 21
Give your advice to them to manufacture quantity of each product to maximize profit.
38
6. A manufacturer makes two products P1 and P2. Both these products pass through two machines. Products P1 required 8 minutes each on first machine and 4 minutes each on second machine while product P2 requires 10 minutes each of first machine and 4 minutes each of second machine. 4 hours and 2 hours of spare capacity is available respectively on the first and second machine. The profit per unit is Rs.20/- for product P1 and Rs.10/- for product P2. How much quantity of each product be manufactured to maximize profits?
39
7. A company manufactures three products A,B, and C. The data on resources required, availability of resources and the contribution from the sale of each product are given in the table below:
Resource Requirement ResourceavailabilityA B C
Raw material (kgs)Machining time (Hrs.)Assembly time (Hrs.)Contribution per unit (Rs.)
60 6 340
100 6 4 25
40 3 230
12,000 720 500
(a) Solve the problem as linear programming problem to determine the quality of each product to be produced to maximize contribution
(b) Interpret the final simplex tableau.
40
8. A small firm manufactures two products. Both these products pass through three main processes. A study of costs and throughput rates have recently been carried out and the average process time for each product is as follows:
Product A Product B
Preparing 1 hr. 1 hr.
Milling 10 hrs. 5 hrs.
Finishing 5 hrs. 10 hrs.
In the cost study great care has been taken to separate out those costs which vary directly with production and on his basis it is estimated that product A makes a contribution to overheads and profit of Rs.10 per unit and product B makes a contribution to overhead and profit of Rs.15. Monthly capacities on the three process have also been estimated as follows:
Preparing department 500 hours
Milling department 3000 hours
Finishing department 2200 hours
The directors of this firm are anxious to determine the most profitatle product mix with the maximum utilisation of the capacity available.
41
9. Messer s Electronics Ltd manufacture transistors in three models A, B and C whose contribution to profits are Rs.80, Rs.150 and Rs.250/- per set respectively. Each type has to be processed and completed in the three main departments viz. manufacturing, assembly and packing. Following table indicates time in hours per dozen sets of each category in each department:
DepartmentCategory Capacity in
Hrs. per weekA B C
Manufacturing
Assembly & Testing
Packing
3
4
1
3.5
5
1.5
5
8
3
150
200
60The manufacturers feel that they can sell any type in any number, but they want to know how many sets of what category should be manufactured weekly so as to get the maximum gross profit. Give your advice to them on the basis of the above data.
42
10. A jobbing shop has three machine groups, namely lathes, milling machines and grinders. It has an idle capacity of 350 hours, 500 hours and 150 hours per week respectively. It is offered products A, B and C to be manufactured. Each unit of product A yields Rs.30/-, product B Rs.12/- and product C Rs.15/-. The time taken by each unit of the three products on different machines are given in table below:
Lathe Milling Grinder
Product A
Product B
Product C
5
4
5
9
3
5
3
0
2
(a) How much quantity of product A, B and C must be manufactured every week to yield maximum profit?
(b) What capacity of each machine remains idle after making these products? State clearly the assumptions, if any, made by you.
43
Write the dual linear programming problem of each of the following primals:
1. Max Z= 40 x1+60x2
Subject to
5x1+2x2 ≤20
4x1+32x2≤36
x1,x2≥ 0
2. Min Z=20x1+30x2
Subject to
2x1+3x2 ≥ 6
1x1+2x2 ≥4
1x1+1x2≥5
x1,x2≥ 0 44
3. Max Z=24x1+20x2
Subject to
5x1+4x2 ≤ 80
3x1+5x2 ≥20
x1,x2≥ 0
4. Max Z= 4x1+8x2+6x3
Subject to
2x1+4x2-1x2 ≤ 8
-1x1-1x2+1x3 ≥ -4
1x1+2x2+3x3≥ -6
4x1+1x2+2x3≤ 4
x1,x2,x3≥0
45
5. Max Z= 2x1+3x2
Subject to
2x1-3x2 ≤10
1x1+2x2 ≤6
1x1 ≥2
1x2≤4
2x1+1x2≤ 20
x1,x2≥0
6. Max Z=6x1+8x2
Subject to
1x1+2x2 ≤40
3x1+2x2 = 20
x1,x2 ≥ 0
46
7. Min Z= 20 x1+40 x2
Subject to
4x1+3x2 ≥60
4x1+5x2 ≤40
1x1+4x2 ≥20
1x1+2x2 = 16
x1,x2 ≥0
8. Max Z= 3x1+2x2+3x3
Subject to
2x1+2x2+1x3 ≤20
2x1-1x2-1x3 ≤ 12
x1,x2 ≥ 0, x3 is unrestricted
47
SENSITIVITY ANALYSIS
Rule 1:-
Any change in the objective coefficient of a basic variable:
(i) Affects only the coefficients in the index row under the columns of the non basic variables. Each new coefficient equals coefficient in the index row before change plus ∆ multiplied by the corresponding coefficient in the row of the variable whose objective function is being changed.
(ii) Affects profit.The revised profit equals profit before change plus ∆ multiplied by the coefficient in the solution column in the row of the variable whose objective coefficient is being changed,
48
Rule2:
Any change in the objective coefficient of a non basic variable:
(i) Affects only the coefficients in the index row under the column of the variable whose objective coefficient is being changed. The new coefficient equals coefficient before change minus ∆.
(ii) Does not affect profit.
Rule3:
Any change in the availability of a scarce resource:
(i) Affects the values of the solution column of the optimal simplex table. Each value equals old value plus ∆ multiplied by corresponding element under the column of the slack variable of the resource whose availability is being changed.
(ii) Affects profit. Revised profit equals old profit plus ∆ multiplied by corresponding element in the index row under the column of the slack variable of the resource whose availability is being changed.
Rule 4:
Any change in the availability of an abundant resource :
Affects only the value of the corresponding slack variable in the solution column. Revised value equals value before change plus ∆. 49
ProductTime required per unit (minutes)
M1 M2 M3
P1 4 3 2
P2 4 4 1
P3 4 3 1
Available capacity per
week (minutes)
1200 900 400
1.Pai Engineering Works manufactures three products P1, P2 and P3. All these products pass through three machines m1, M2 and M3.The time required to process each unit of the products on each machine and the available capacity of the machines is as under:
SENSITIVITY ANALYSIS
The profit per unit to be realised from the manufacture of the products are Rs 20,Rs 12 and Rs 8 respectively.
50
•How many units of each product be produced to maximize profit.
•Within what range of profit of each product the solution will remain optimal?
•Within what range of the capacity of M3 will the solution remain optimal?
•If the manufacturer can sell another product P4 which requires 3 minutes of machine M1 and 5 minutes each on machine M2 and M3, would it be worthwhile manufacturing P4.
•Discuss the effect on product mix if product P1, P2 or P3 is dropped?
•What reduction in the consumption per unit of non-basic variable will bring it into the basis?
51
A market survey has estimated that the weekly demand of the product P2 cannot exceed that of P1 by more than 100.The survey also shows that the maximum weekly demand of product P2 is limited to 250 nos.
The profit per unit is Rs 30 for product P1 and Rs 20 for product P2.
•Write the constraints and the objective function.
•Solve the problem as a LP problem.
•What are the decision variables and their values in the optimal solution?
•Classify the status of the resource.
•What are the shadow prices?
•Will the optimal value of profit improve if
1.Capacity of machine M1 is increased?
2.Capacity of machine M2 is increased?
3.Demand of product P1 is increased?
4.Demand of product P2 is increased?
52
• Which of the four resources should be given priority in allocation of new funds?
• Determine the optimum right hand side of the constraint equations resulting from the change of resource 1 from 700 to (700+ )
• Determine the feasible range for due to changes in above.
• If the marginal profit of product P1 is changed from 30 to ( 30 + ), where represents either positive or negative change, specify the range of equations to keep the current optimal solution unchanged.
53
Q.1 Draw the network for the following dependencies:
54
55
Q.2 The following list of activities must be accomplished in order to complete a construction project:
Activity Time (in weeks) Predecessors
A 3 None
B 8 None
C 4 A, B
D 2 B
E 1 A
F 7 C
G 5 E, F
H 6 D, F
I 8 G, H
J 9 I
Construct a network diagram for this project.
Find the critical path and the duration of the project.56
Q.3 The following table lists the activities of a maintenance project.
Activity Duration (in months)
1-2 2
1-3 2
1-4 1
2-5 4
3-6 5
3-7 8
4-7 3
5-8 1
6-8 4
7-9 5
8-9 3
Draw the project network.
Find the critical path and duration of the project. 57
Q.4 Draw the network for the following dependencies and identify critical path. Also find the project duration.
Activity 1-2 1-3 1-4 2-3 2-6 3-5 3-6 4-5 5-6 5-7 6-7
Duration 8 7 3 6 8 6 4 12 0 6 8
(Months)
58
Q.5 The following table gives the activities in a construction project and other relevant information:
(a) Draw the activity network of the project.
(b) Find the Critical Path
(c) Using the above information, crash or shorten the activity step by step until the shortest duration is reached.
Activity Preceding Activity
Normal Time
(Days)
Crash Time
(Days)
Normal Cost (Rs.)
Crash Cost (Rs.)
1-2
1-3
2-3
2-4
3-4
4-5
-
-
1-2
1-2
1-3, 2-3
2-4, 3-4
20
25
10
12
5
10
17
25
8
6
2
5
600
200
300
400
300
300
720
200
440
700
420
600
59
Q.6. The table below provides cost and time estimates of seven activities of a project:
(i) Draw the project network corresponding to normal time.
(ii) Determine the critical path and the normal duration and normal cost of the project.
(iii) Crash the activities so that the project completion time reduces to 9 weeks, with minimum additional cost.
Activity
Time estimates (weeks) Direct cost estimates
(Rs. in thousand)
Normal Crash Normal Crash
1-2
1-3
2-4
3-4
3-5
4-6
5-6
2
8
4
1
2
5
6
1
5
3
1
1
3
2
10
15
20
7
8
10
12
15
21
24
7
15
16
36
60
Q.7. The Basic – Time Data for the jobs in a project are as follows:
Activity
Normal Crash
Time (Days) Cost (Rs.) Time (Days) Cost (Rs.)
A
B
C
D
E
F
G
H
3
6
2
4
2
7
4
3
140
215
160
130
170
165
210
110
2
5
1
3
1
4
3
2
210
275
240
180
250
285
290
160
Total 1300 1890
61
The activity (job) dependencies are as follows:
(i) A, B, C are starting activities.
(ii) Activities D, F, E can start when A is completed.
(iii) Activity G can start after B and D are completed.
(iv) Activity H can start after C and E are completed.
(v) Activities G, F and H are the final activities.
Draw the network and indicate the critical path.
What is the total time required to complete the project?
If the project is to be completed in 9 days, what is the minimum cost to be incurred? What is the least cost schedule?
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Q.8. Normal and Crash times and costs are given below for a plant expansion project.
Activity Preceding Activity
Normal Time
(Months)
Crash Time (Months)
Normal Cost
(Rs.000)
Crash Cost (Rs.000)
A
B
C
D
E
F
G
H
-
A
-
C
A
D
D,E
B,F,G
3
6
3
2
1
5
7
4
2
4
2
1
1
3
6
3
40
200
20
20
20
150
120
160
50
300
35
32
20
190
150
195
If the Company has Rs.7,76,000 available for this project, how should the funds be allocated to minimize overall completion time, to the nearest 0.1 month? What is the minimum completion time?
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Q.9. The details of activities in a building project are given below:
Activity Preceding Activity
Normal Crash
Time (Days)
Cost (Rs.) Time (Days)
Cost (Rs.)
A
B
C
D
E
F
G
H
-
A
A
C
-
E
B,D
F,G
9
14
4
6
14
6
5
2
12000
14000
2000
44000
1600
4000
4000
12000
6
4
3
4
13
6
3
1
18000
24000
2400
56000
1800
4000
4800
14000
Find the minimum (crashed) schedule using CPM Technique.
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Q.10.List of activities for erecting a canteen in a factory is given below with other relevant details. Job A must precede all others while Job E must follow others. Apart from this, other jobs can run concurrently.
Code Normal Crash
Duration (Days) Cost (Rs.) Duration (Days) Cost (Rs.)
A
B
C
D
E
5
6
4
5
3
3000
1200
1000
1200
1600
4
2
3
3
3
4000
2000
1800
2000
1600
(i) Draw the network and identify critical path.
(ii) Crash the network fully to find out minimum duration.
(iii) If indirect costs are Rs.300 per day, find the time-cost trade-off for the project.
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Q.11.The table below shows jobs, their normal time and cost as well as crash time and cost for a project.
Job Normal Time (Days)
Cost (Rs.) Crash Time (Days)
Cost (Rs.)
1-2
1-3
2-3
2-4
3-4
3-5
4-6
5-6
6
8
4
3
Dummy
9
10
3
1400
2000
1100
800
-
900
2500
500
4
5
3
2
-
6
6
2
1900
2800
1500
1400
-
1600
3500
800
Indirect cost for the project is Rs.300/- per day.
(i) Draw the network of the project and identify the critical path.
( ii) Using the above data, find the different minimum cost schedules between normal and crash points to arrive at the optimum duration and its associated cost.
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Q.12.The data for PERT – network is displayed in the table. Determine the critical path and the expected duration of completion of the entire project. Give answers to the following:
Activity Nodes
Time Duration (Days)
a m b
1-2
1-3
1-4
2-3
2-5
3-4
3-6
4-6
5-6
2
6
6
2
11
15
3
9
4
4
6
12
5
14
24
6
15
10
6
6
24
8
23
45
9
27
16
(i) What is the probability that the project duration will exceed 60 days?
(ii) What is the chance of completing the project between 45 days and 54 days?
(iii) If it becomes known later that the duration of the three time estimates for activity 4-6 has to be revised to 14-20-32, what impact does this have on project completion time? What is the probability that the project can now be completed before 46 days?
(a) Optimistic time (m) Most likely (b) Pessimistic 67
Q.13. The following table gives the relevant data of the activities in a PERT project.
(i) Draw an arrow diagram of the project.
(ii) Calculate the expected duration and variance of the critical path.
(iii) Assess the probability that the project will take more than 41 days.
(iv) What is the probability that project will be completed in 31 days or less time.
Activity Duration (Days)
Optimistic Most Likely Pessimistic
1-2
7-8
2-3
4-5
3-5
5-8
1-6
2-4
6-7
3
4
6
3
5
1
2
2
3
6
19
12
6
11
4
5
5
9
15
28
30
15
17
7
14
8
27 68
Q.14. A project is characterized by the following activity time:
Activity Optimistic Time
To Days
Pessimistic Time
To Days
Most Likely Time To Days
1-2
1-3
2-4
3-3
3-5
4-5
4-6
5-6
1
3
4
9
1
10
5
5
5
7
8
11
5
20
13
9
3
5
6
10
3
12
6
5.5
(a) Find the critical path and the project completion time.
(b) Find the standard deviation of the distribution of the expected project length.
(c) What is the probability that the project will be completed in 35 days?
(d) What is the expected project completion time, if you are allowed to qualify the same with a chance of 95%?
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Q. 15. The part of the project network is given below:
Activity Preceding Activity Duration (months) Cost (Rs ‘000)
A - 2 70B - 3 75C - 2 50D B 4 80E C 4 60F A 3 90G D 6 90H C 4 100J E 6 120K F,G,H 2 90
Determine the cash flow requirement based on EST schedule. The project sponsor has set maximum cash flow in any month as Rs. 70,000. Reschedule the activities so that project duration is not extended.
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Q. 16. The direct cost estimates for various activities of an project network are as given below.
Activity (i-j) Duration in months Direct costs (Rs.)
1-2 13 26,00,000
1-3 12 60,00,000
2-4 2 20,00,000
3-4 8 20,00,000
2-5 15 15,00,000
4-5 2 15,00,000
Total 1,56,00,000
a. Draw network, find critical path and project duration.
b. On graph paper compile monthly and cumulative monthly cash flow requirements for early starts and late start schedules.
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c. Draw cumulative cash flow requirement.
d. The government has decided to release Rs. 1,56,00,000 in the following manner.
Rs. 69,00,000 in the first year, Rs. 68,00,000 in the second year and Rs. 19,00,000 in third year. It has also stipulated that the unspent amount would lapse and hence cannot be carried forward. Schedule the activities on graph to match release of funds. Is it possible to schedule the project without extending its project duration? If not, give reasons and suggest new time estimate for completion of this project.
(Note: Assume the funds for an activity are required uniformly throughout its time duration)
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Q. 17. The MCC is bidding on a contract to install a new gas pipeline. It has identified the following activities along with expected time in months.
Activity Duration (Months)
Monthly cash flow required in millions Rs.
1-2 4 41-3 7 22-4 3 42-6 3 23-4 2 63-5 2 34-5 2 35-6 3 4
i. Draw an arrow diagram and find earliest and latest start of jobs.
ii. Use graph paper to compile data of monthly and cumulative cash flow requirement taking early start schedule and late start schedule for all jobs?
iii. Develop a schedule that will give a near uniform monthly cash flow requirement on this project.
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Q.1 Customers arrive at a banking office window being manned by a single individual at a rate of 25 per hour. The time required to serve a customer has exponential distribution with a mean of 120 seconds. Find all operating characteristics of the queueing system.
Q.2 On an average 10 customers arrive per hour at the stamp counter of a post-office, the number of arrivals being poisson distributed. The clerk takes on an average 3 minutes per customer, the time following a negative exponential distribution. Customers follow FIFO queue discipline and any number may join the system.
(A) Find :
i) the probability of the clerk being idle,
ii) the average number of customers in the system and queue,
iii) the mean time spent by a customer in the queue and system.
(B) If in part (A) the clerk adopts a “Work-to-Rule” policy and takes on an average 4 minutes per customer. Find the waiting time of a customer in the queue and system now.
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Q.3 On average customers arrive at a check-out point in a supermarket every 3 minutes. The single cashier is capable of serving on average 30 customers per hour. Service times and inter-arrival times follow a negative exponential distribution.
(a) What is the probability of a customer arriving and having to wait for service?
(b) What is the probability of a customer arriving and finding at least one customer already at the check out?
(c) What is the average number of customers at the check-out at any moment?
(d) What is the length of time that a customer would expect to spend in the system?
(e) What is the average number of customers at the check-out who are not being served?
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Q.4 A departmental store has a single cashier. During the rush hours, customer arrive at the rate of 20 customers per hour. The average number of customers that can be processed by the cashier is 24 per hour. Assume that the conditions for use of the single channel queueing model apply. Find:
(a) The utilization parameter.
(b) The probability that the queueing system is idle.
(c) The average time that the cashier is free on a 10-hour working day.
(d) The expected number of customers in the store.
(e) The expected number of customers waiting for cashier’s service.
(f) The average length of queues that have at least one customer.
(g) The expected time a customer would spend in the queue.
(h) The expected time a customer would spend in the store.
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Q. 5 In a bank with a single server, there are two chairs for waiting customers. On an average one customer arrives every 10 minutes and each customer takes 5 minutes for getting served. Making suitable assumptions, find:
(a) Probability that an arrival will get a chair to sit down.
(b) Probability that an arrival will have to stand.
(c) Expected waiting time of a customer.
Q.6 A coin-operated telephone is installed in a canteen for the use of the staff. On average 8 people per hour use the phone and their calls last 3 minutes. The staff association think that enough use is made of the phone to justify the installation of a second instrument, but the telephone company say that they will only do this when they are convinced that the staff would have to wait on average for at least 3 minutes to use the phone. Assuming that all calculations are based on simple queueing theory, what rate of use will have to be achieved before the need for a second telephone is justified?
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Q. 7 On average 96 patients per 24 hours require the service of an emergency clinic. Also, on average, a patient requires 10 minutes of active attention. Assume that the facility can handle only one emergency at a time. Suppose that it costs the clinic Rs.400 per patient treated to obtain an average servicing time of 10 minutes and that each minute of decrease in this average time would cost Rs.50 per patient treated. How much would have to be budgeted by the clinic to decrease the average size of the queue from 1 1/3 patients to ½ patient.
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Q. 8 In the Central Railway Station, 15 computerised reservation counters are available. A customer can book his/her ticket in any train on any day in any one of these computerized reservation counters. The average time spent by each clerk is 5 minutes. Average arrivals per hour during three types of activity periods have been calculated and customers have been surveyed to determine how long they are willing to wait during each type of period.
Type of period Arrivals per hour Customer’s Acceptable waiting time
Peak
Normal
Low
110
60
30
15 minutes
10 minutes
5 minutes
Making suitable assumptions on this queueing process, determine how many counters should be kept open during each type of period.
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Q. 9 A duplicating machine in an office is used by the secretaries to make copies. The average rate of service is 10 jobs/hour. Generally duplicating jobs come up randomly at 7 jobs/hour, yet queues have been noticed at the machine. If the secretaries time cost Rs.30 per hour, determine:
Q. 10 The XYZ Corporation is considering leasing 1 of 2 possible duplicating machines. The mark I is capable of duplicating 20 jobs per hour at a cost of Rs.500 per day. Alternatively, the Mark II can duplicate 24 jobs per hour at the cost of Rs.800 per day. The duplicating centre is open 10 hours a day with an average arrival of 18 jobs per hour. The duplication is performed by employees randomly arriving from various departments whose average hourly wage is Rs.50. Should the company lease Mark I or Mark II?
(a) Machine utilization.
(b) Percentage of time a secretary coming for duplication has to wait.
(c) Average time spent in the system.
(d) Average cost per day towards waiting for and operating the machine.
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Games Theory deals with such problems where actions and interactions of competing firms gives rise to conditions of business conflict (i.e., competitive situations).
Typical examples of competitive situations are:
(i) Firms trying to snatch each other’s market share.
(ii) Military attacks.
(iii) Selection of best advertising media.
Terminology of Games Theory
(i) Players :
The participants in the game who act as decision-makers are called players.
(ii) Strategies :
A finite number of possible courses of action available to a player are called strategies.
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(iii) Play :
A play occurs when each player selects one of his strategies.
(iv) Pay-off :
Every combination of strategies of players determines an outcome called pay-off.
(v) Pay-off matrix :
The gains resulting from a game if presented in the form of a table is called pay-off matrix.
(vi) Maximin :
A maximum element among the row minima is called maximin.
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(vii) Minimax :
A minimum element among the column maxima is called minimax.
(viii) Saddle Point :
A saddle point is that element of the matrix which represents the maximin value of a player and minimax value of his opponent.
(ix) Value of the game :
The value of the game is the expected gain to a player if he and his opponent use their best strategies.
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TYPES OF GAMES
Two-person game Multiple-person game
Two-person zero sum game Two-person non-zero sum game
Pure strategy game
Mixed strategy game
2 x 2 game 2 x n ormx2 game
mxn game
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PURE STRATEGY GAMES
Pure strategy games are those in which players stay with one strategy throughout the game.
1. Find the optimal strategies of X and Y in the following game. Also find value of the game.
Y
1 2 3 4 5
1 9 3 1 8 0
2 6 5 5 6 7
X 3 -2 4 3 3 8
4 5 6 2 2 1
5 0 1 3 4 3
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PURE STRATEGY GAMES
2. Solve the following games to determine the optimal strategies. Also obtain the value of the game.
N
3 1 2 0
6 7 4 6
M 4 5 4 6
6 2 2 1
Management
10 13 15
Union 6 9 16
3 6 12
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A company management and the labour union are negotiating a new 3 year settlement. Each one has the following four strategies:
(i)Hard and aggressive bargaining.
(ii)Reasoning and logical bargaining.
(iii)Legalistic approach.
(iv)Conciliatory approach
The cost to the company for every paid of strategy choices are given below:
Union Strategies
Company strategies
I II III IV
I 20 16 12 30
II 18 15 8 10
III 35 6 10 8
IV 2 10 11 6
What strategies should the two sides adopt? What will be the cost of the settlement to the company?
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Mixed Strategy Games
Games without saddle point are called Mixed Strategy Games. Players in the mixed strategy games need to play more than one strategy to optimize their gains (or losses). The strategies to be followed and the proportion of time each strategy should be played can be determined by the players in advance of the game. Mixed strategies are advantageous since opponent is always kept guessing.
Solve the following two persons 2x2 mixer strategy games:
B N
(i) A 10 5
7 8
(ii) M 1 -4
-4 1
Y
(iii) X - 7 - 4
- 5 - 7
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Solve the following 2 persons 2 x 2 or m x 2 game by dominance method:
B
6 5 2
(i) A -1 1 -2
4 9 6
(ii) A and B play a game in which each has three coins: a 5 paise, a 10 paise and a 20 paise coin. Each player selects a coin without the knowledge of the other’s choice. If the sum of the coins is an odd amount, A wins B’s coins and if the sum is even, B wins A’s coins. Find the best strategy for each player and the value of the game.
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(iii) Two firms F1 and F2 make colour and black and white television sets. F1 can make either 300 colour sets in a month or an equal number of black & white sets and make a profit of Rs.200 per colour set and Rs.150 per black & white set. F2, on the other hand, makes either 600 colour sets or 300 colour and 300 black & white sets, or 600 black & white sets per month. It also has the same profit margin on the two sets as F1. Each month there is a market of 300 colour sets and 600 black & white sets and the manufacturers would share market in the proportion in which they manufacture a particular type of set.
Write the pay-off matrix of F1 per month. Obtain F1 and F2’s optimal strategies and value of the game.
Solve the following 2 persons 2xn or mx2 game by sub-game method.
B
(i) A 4 2
3 8
2 12
Y
(ii) X -2 0 7
5 2 -1
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Limitations of Games Theory
o Rarely the managerial decisions are taken in an environment of two parties.
o The competing parties of the game in actual situation neither have equal information nor they have equal intelligence.
o The pay-offs in the game matrix are extremely difficult to establish.
o Games theory assumes that the gain of one player is the loss of other player. Many a time it is not so. There may be situations when both parties can earn.
o Strategies in real life situations unlike those in game theory are played for a fairly long time.
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1. Using the following data, obtain the EOQ and the total cost associated with the policy of ordering quantities of that size.
Annual Demand= Rs 20,000
Ordering cost= Rs 150 per order
Inventory carrying cost= 24% of average inventory value
2. A factory follows an EOQ system for maintenance stocks of one of its component requirements. The annual demand is for Rs 24,000 units, the cost of placing order is Rs 300, the component cost is Rs 60 per unit. The inventory carrying cost is 24%.
•Find the optimal interval for placing orders, assuming a year equivalent to 360 days.
•If it is decided to place only one order per month, how much extra cost would the factory incur per year as a consequence of this decision?
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3. A manufacturing company has determined from an analysis of its accounting and production data for its certain part that (a) its demand is 9000 units per annum and is uniformly distributed over the year, (b) its cost is Rs 2 per unit, (c) the inventory carrying charge is 9% of the inventory value.
Further, it is known that the lead time is uniform and equals working days, and that the total working days in the year are 300.
Determine:
•The economic order quantity;
•The optimum number of orders per annum;
•The total ordering and holding cost associated with the policy of ordering an amount equal to EOQ;
•The re-order level;
•The number of days stock at re-order level;
•The length of the inventory cycle;
•The amount of savings that would be possible by switching to the policy of ordering EOQ determined in (a) from the present policy of ordering the requirements of this part thrice a year; and
•The increase in the total cost associated with ordering (i) 20% more, and (ii) 40% less than EOQ. 93
4. Yogesh keeps his inventory in special containers. Each container occupies 10 sq ft of store space. Only 5,000 sq ft of space is available. The annual demand for the inventory item is 9,000 containers, priced at Rs 8 per container. The ordering cost is estimated at Rs 40 per order, and the annual ordering cost amount of 25% of the inventory value.
Would you recommend to Yogesh to increase his storage space? If so, how much should be the increase?
5. A wholesaler supplies 30 stuffed dolls each day to various shops. Dolls are purchased from the manufacturer in lots of 120 each at Rs 1,200 per lot. Every order incurs a handling charge of Rs 60 plus a freight charge of Rs 250 per lot. Multiple and fractional lots can be ordered, and all orders are met the next day. The incremental cost is Rs 0.60 per year to store a doll in inventory. The wholesaler finances inventory investments by paying its holding company 2% monthly for borrowed funds.
How man dolls should be ordered at a time in order to minimize the total annual inventory cost? Assume that there are 250 weekdays in a year. How frequently should he order?
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Order size Price per unit (Rs)
Less than 1,000 500
1,000-2,999 450
3,000-4,999 400
5,000 or more 350
6. A manufacturing company has a contract to supply 5,000 units of an item per year to a dealer. For this item, the company estimates that the ordering cost is Rs 150 every time that an order is made while the carrying cost per annum is reckoned to be 20% of the unit price.
The company is negotiating with a dealer who offers to give the following quantity discount.
Recommend to the company the best inventory policy with regard to this item.
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7. A company uses 8,000 units of aproduct as raw material, costing 10 per unit. The administrative cost per purchase is Rs 40. The holding cost is 28% of the average inventory. The company is following an optimal purchase policy and places orders according to the EOQ. It has been offered a quantity discount of one per cent if it purchases its entire requirement only four times a year.
Should the company accept the offer of quantity discount of one per cent? If not, what minimum discount should the company demand?
8. A manufacturing company needs 2,500 units of a particular component every year. The company buys it at the rate of Rs 30 per unit. The order processing cost for this part is estimated at Rs 15 and the cost of carrying a part in stock comes to about Rs 4 per year.
The company can manufacture this part internally. In that case, it saves 20% of the price of the product. However, it estimates a set-up cost of Rs 250 per production run. The annual production rate would be 4,800 units. However, the inventory holding cost remains unchanged.
•Determine the EOQ and the optimal number of orders placed in a year.
•Determine the optimum production lot size and the average duration of the production run.
•Should the company manufacture the component internally or continue to purchase it frm the supplier?
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9. An aircraft uses high tensile bolts at an approximately constant rate of 50,000 numbers per year. The bolts cost is Rs 20 each and the purchase department estimated the cost at Rs 200 to place an order. The opportunity cost on working capital is 20% per year. No shortages are allowed.
How frequently should orders be placed and what is economic order quantity?
If orders could be executed only once in two months, the ordering quantity would be higher than the optimal quantity. By this what would be the percentage change in the total relevant cost?
The company finds, to its error, that the cost of placing an order was Rs 5,000 and carrying cost was 15% per year and not the earlier data, how much was the company losing per year on inventory, because of imperfect information?
Working on the new ordering cost and carrying cost, the company receives the following offer from the supplier:
Upto 20,000 pieces per order, the price is Rs 20 per unit
Above 20,000 pieces up to 30,000 Rs 19.50 per unit
Above 30,000 pieces up to 45,000 Rs 19.25 per unit
Above 45,000 pieces Rs 19 per unit
Should they make use of this offer?
If the entire requirement has to be bought in a single order, what should be justifiable unit price offer to the company? 97
The annual demand for a component is 2,08,000 units at a steady weekly rate of 4,000 units. An appropriate formula for calculating the economic batch quantity for production of a component which is being used (at the rate of s) and produced (at the rate of r per week) at the same time is
EBQ=
The initial cost of installing the line for producing the component was Rs 6,000 for a maximum production capacity of 8,000 per week. The operating cost at full capacity is Rs 100 per week for labour, Rs 600 per week for material, Rs 300 per week for variable overhead and Rs 250 per week for fixed overhead. The cost of preparing the production order, producing drawings is Rs 40 each time production is required. Storage cost including interest has been calculated at Rs 2 per unit per annum.
•Calculate the most economic quantity that should be produced each time the line is set up.
•Advice the management if it now thinks that there is an opportunity to produce special one –off order for 50,000 components for delivery in six months time. Your answer should consider quantitative and qualitative factors.
2ACo
___________
(1-s/r)ip
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11. A dealer supplies you the following information with regard to a product dealt in by him:
Annual demand : 10,000 units
Ordering cost : Rs 10 per order
Inventory carrying cost : 20% of the value of inventory per year
Price : Rs 20 per unit
The dealer is considering the possibility of allowing some back-order (stock out) to occur. He has estimated that the annual cost of back-ordering will be 25% of the value of inventory.
What should be the optimum number of units of the product he should buy in one lot?
What quantity of the product should be allowed to be back-ordered, if any?
What should be the maximum quantity of inventory at any time of the year?
Would you recommend to allow back-ordering? If so what would be the annual cost saving by adopting the policy of back-ordering?
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12. The demand for an item is deterministic and constant over time at 600 units per year. The item cost is Rs 50 per unit and the cost of placing an order is estimated to be Rs 5. The inventory carrying cost is 20% and the shortage cost is Re 1 per unit per month. Find the optimal ordering quantity if stock outs are permitted and the units can be backordered at the shortage cost indicated. What will the company lose if stock outs are permitted?
13. You are given the following information in regard to an item:
Annual usage = 20,000units
Ordering cost =Rs 160 per order
Carrying cost = 20% of the average inventory investment
Unit cost = Rs 2
Lead time =10 working days
Total working days = 250 per annum
It is observed in past that the demand during lead time has been upto maximum level of 150 units per day. Keeping this level in mind, what safety stock would you recommend? Also determine (a) the re-order level when the safety stock level suggested by you is kept in stock, (b) average level of inventory stock held, and (c) the ordering and carrying cost associated with this fixed order inventory policy?
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14. For a Fixed Order Quantity System, find out the various parameters for an item with the following data:
Annual consumption -10,000 units, cost per unit-Re 1, set up cost-Rs 12 per production run, the inventory carrying cost-Re 0.24 per unit, Past lead times:15 days,25 days, 12 days, 14 days, 30 days, 17 days.
15. Daily demand for a product AX-303 is normally distributed with mean equal to 60 units and a standard deviation of 6 units. The lead time is constant at 9 days (working). The cost of placing an order is Rs 20 and the annual ordering cost is 20% of the unit price of Rs 10. A 95% service level is desired for the customers, who place orders during the re-order period. You are requested to determine the order quantity and the re-order level for the item in question, assuming that there are 300 working days during the year.
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Motors sold
30 40 50 60 70 80 90 Total
No of days
10 20 50 30 15 5 5 135
16. CD Ltd sells an electric motor but finds that it runs out of stock on occasions and thus loses the contribution on missed sales. The following information is available:
Estimated demand Rs 12,000 per year
Purchase price Rs 100 each
Selling price Rs 155 each
Lead time 5 days guaranteed
Cost of holding motor Rs 20 per year
EOQ 1,200 motors
The company works five days in a week for 48 weeks in a year. The demand figures for the last 27 weeks are given in the table below:-
At present, CD Ltd uses a re-order level of 250 motors and does not carry any safety stock because of the guaranteed delivery time. Ideally, it wishes to satisfy customers on an average at least 95% of the time while minimizing the associated cost.
You are required
•To estimate the annual stock-out cost of using the present re-order level;
•To calculate what re-order level would enable the company to meet its 95% requirement . 102
Stock out (No of units) Number of Times
2,000 4
1,600 8
1,000 12
400 16
200 40
0 320
Total 400
17. Your company’s experience of being out-of-stock in respect to key items is as follows:
Assume that the stock out cost is Rs 100 per unit. The carrying cost of inventory is Rs 50 per unit. Determine the optimal level of stock out inventory (safety stock).
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Item Units Unit price (Rs)
Item Units Unit price(Rs)
1 700 5.00 7 6,000 0.20
2 2,400 3.00 8 300 3.50
3 150 10.00 9 30 8.00
4 60 22.00 10 2,900 0.40
5 3,800 1.50 11 1,150 7.10
6 4,000 0.50 12 410 6.20
18. Perform ABC analysis using the following data:
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THANK YOU
