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Educational Level Secondary school, High school and College
Program Purpose Provide step by step solutions of your problems using online calculators (online solvers)
Problem Source Your textbook, etc

1. Assignment Problem (Using Hungarian Method)

2. Simplex Method (Solve linear programming problem using)
1. Simplex Method
2. BigM Method
3. Two-Phase Method
4. Dual Simplex Method
5. Integer Simplex Method (Gomory's cutting plane method)
6. Graphical Method
7. Primal to dual conversion

3. Transportation Problem using
1. North-West corner method
2. Least cost method
3. Vogel's approximation method
4. Optimal solution by MODI method

4. PERT and CPM

5. Sequencing Problems

6. Replacement and Maintenance Models
1. Model-1 : Replacement policy for items whose running cost increases with time and value of money remains constant during a period
1.1 Model-1.1
1.2 Model-1.2
1.3 Model-1.3
2. Model-2 : Replacement policy for items whose running cost increases with time but value of money changes constant rate during a period
3. Model-3 : Group replacement policy

1. Assignment Problem (Using Hungarian Method)
1. A computer centre has 3expert programmers. The centre wants 3 application programmes to be developed. The head of thecomputer centre, after studying carefully the programmes to be developed, estimates the computer time in minutes required by the experts for the application programmes as follows.
Programmers
A B C
Programmes 1 6 3 5
2 5 9 2
3 5 7 8


2. A department has five employess with five jobs to be permormed. The time (in hours) each men will take to perform ech job is given in the effectiveness matrix.
Employees
I II III IV V
Jobs A 10 5 113 15 16
B 3 9 18 13 6
C 10 7 2 2 2
D 7 11 9 7 12
E 7 9 10 4 12
How should the jobs be allocated, one per employee, so as to minimize the total man-hours?


Unbalanced Assignment Problem.

3. In the modification of a plant layout of a factory four new machines M1, M2, M3 and M4 are to be installed in a machine shop. There are five vacant places A, B, C, D and E available. Because of limited space, machine M2 cannot be placed at C and M3 cannot be placed at A. The cost of locating a machine at a place (in hundred rupess) is as follows.
Location
A B C D E
Machine M1 9 11 15 10 11
M2 12 9 -- 10 9
M3 -- 11 14 11 7
M4 14 8 12 7 8
Find the optimal assignment schedule.

4. An airline company has drawn up a new flight schedule involving five flights. To assist in allocating five pilots to the flights, it has asked them to state their preference scores by giving each flight a number out of 10. The higher the number, the greater is the preference. Certain of these flights are unsuitable to some pilots owing to domestic reasons. These have been marked with a -.
Flight Number
I II III IV V
Pilot A 8 2 - 5 4
B 10 9 2 8 4
C 5 4 9 6 -
D 3 6 2 8 7
E 5 6 10 4 3
What should be the allocation of the pilots to flights in order to meet as many preferences as possible?
 
2.1 Simplex Method
1. Use the simplex method to solve the following LP problem.
Maximize Z = 3x1 + 5x2 + 4x3
subject to the constraints
2x1 + 3x2 ≤ 8
2x2 + 5x3 ≤ 10
3x1 + 2x2 + 4x3 ≤ 15
and x1, x2, x3 ≥ 0

2. Use the simplex method to solve the following LP problem.
Maximize Z = 4x1 + 3x2
subject to the constraints
2x1 + x2 ≤ 1000
x1 + x2 ≤ 800
x1 ≤ 400
x2 ≤ 700
and x1, x2 ≥ 0
2.2 BigM Method
1. Use the penalty (Big - M) method to solve the following LP problem.
Minimize Z = 5x1 + 3x2
subject to the constraints
2x1 + 4x2 ≤ 12
2x1 + 2x2 = 10
5x1 + 2x2 ≥ 10
and x1, x2 ≥ 0

2. Use the penalty (Big - M) method to solve the following LP problem.
Minimize Z = x1 + 2x2 + 3x3 - x4
subject to the constraints
x1 + 2x2 + 3x3 = 15
2x1 + x2 + 5x3 = 20
x1 + 2x2 + x3 + x4 = 10
and x1, x2, x3, x4 ≥ 0
 
2.3 Two-Phase Method
1. Solve the following LP problem by using the two-phase simplex method.
Minimize Z = x1 + x2
subject to the constraints
2x1 + 4x2 ≥ 4
x1 + 7x2 ≥ 7
and x1, x2 ≥ 0

2. Solve the following LP problem by using the two-phase simplex method.
Minimize Z = x1 - 2x2 - 3x3
subject to the constraints
-2x1 + 3x2 + 3x3 = 2
2x1 + 3x2 + 4x3 = 1
and x1, x2, x3 ≥ 0
2.4 Dual Simplex Method
1. Solve the following LP problem by using the two-phase simplex method.
Minimize Z = x1 + x2
subject to the constraints
2x1 + 4x2 ≥ 4
x1 + 7x2 ≥ 7
and x1, x2 ≥ 0

2. Solve the following LP problem by using the two-phase simplex method.
Minimize Z = x1 - 2x2 - 3x3
subject to the constraints
-2x1 + 3x2 + 3x3 = 2
2x1 + 3x2 + 4x3 = 1
and x1, x2, x3 ≥ 0
 
2.5 Gomorys Integer Cutting Method
1. Solve the following integer programming problem using Gomory's cutting plane algorithm.
Maximize Z = x1 + x2
subject to the constraints
3x1 + 2x2 ≤ 5
x2 ≤ 2
and x1, x2 ≥ 0 and are integers.

2. Solve the following integer programming problem using Gomory's cutting plane algorithm.
Maximize Z = 2x1 + 20x2 - 10x3
subject to the constraints
2x1 + 20x2 + 4x3 ≤ 15
6x1 + 20x2 + 4x3 ≤ 20
and x1, x2, x3 ≥ 0 and are integers.
2.6 Graphical Method
1. Use graphical method to solve following LP problem.
Maximize Z = x1 + x2
subject to the constraints
3x1 + 2x2 ≤ 5
x2 ≤ 2
and x1, x2 ≥ 0

2. Use graphical method to solve following LP problem.
Maximize Z = 2x1 + x2
subject to the constraints
x1 + 2x2 ≤ 10
x1 + x2 ≤ 6
x1 - x2 ≤ 2
x1 - 2x2 ≤ 1
and x1, x2 ≥ 0
 
2.7 Primal to dual conversion
1. Write the dual to the following LP problem.
Maximize Z = x1 - x2 + 3x3
subject to the constraints
x1 + x2 + x3 ≤ 10
2x1 - x2 - x3 ≤ 2
2x1 - 2x2 - 3x3 ≤ 6
and x1, x2, x3 ≥ 0

2. Write the dual to the following LP problem.
Minimize Z = 3x1 - 2x2 + 4x3
subject to the constraints
3x1 + 5x2 + 4x3 ≥ 7
6x1 + x2 + 3x3 ≥ 4
7x1 - 2x2 - x3 ≤ 10
x1 - 2x2 + 5x3 ≥ 3
4x1 + 7x2 - 2x3 ≥ 2
and x1, x2, x3 ≥ 0
 
Examples
1. A Company has 3 production facilities S1, S2 and S3 with production capacity of 7, 9 and 18 units (in 100's) per week of a product, respectively. These units are tobe shipped to 4 warehouses D1, D2, D3 and D4 with requirement of 5,6,7 and 14 units (in 100's) per week, respectively. The transportation costs (in rupees) per unit between factories to warehouses are given in the table below.
D1 D2 D3 D4 Capacity
S1 19 30 50 10 7
S2 70 30 40 60 9
S3 40 8 70 20 18
Demand 5 8 7 14 34

Find initial basic feasible solution for given problem by using
(a) North-West corner method
(b) Least cost method
(c) Vogel's approximation method
(d) obtain an optimal solution by MODI method
if the object is to minimize the total transportation cost.


2. Find an initial basic feasible solution for given transportation problem by using
(a) North-West corner method
(b) Least cost method
(c) Vogel's approximation method
D1 D2 D3 D4 Supply
S1 11 13 17 14 250
S2 16 18 14 10 300
S3 21 24 13 10 400
Demand 200 225 275 250


3. A company has factories at F1, F2 and F3 which supply to warehouses at W1, W2 and W3. Weekly factory capacities are 200, 160 and 90 units, respectively. Weekly warehouse requiremnet are 180, 120 and 150 units, respectively. Unit shipping costs (in rupess) are as follows:
W1 W2 W3 Supply
F1 16 20 12 200
F2 14 8 18 160
F3 26 24 16 90
Demand 180 120 150 450

Determine the optimal distribution for this company to minimize total shipping cost.


4. Find an initial basic feasible solution for given transportation problem by using
(a) North-West corner method
(b) Least cost method
(c) Vogel's approximation method
P Q R S Supply
A 6 3 5 4 22
B 5 9 2 7 15
C 5 7 8 6 8
Demand 7 12 17 9 45
 
Examples
1. An assembly is to be made from two parts X and Y. Both parts must be turned on a lathe Y must be polished where as X need not be polished. The sequence of acitivities, together with their predecessors, is given below
Activity Description Predecessor Activity
A Open work order -
B Get material for X A
C Get material for Y A
D Turn X on lathe B
E Turn Y on lathe B,C
F Polish Y E
G Assemble X and Y D,F
H Pack G
Draw a network diagram of activities for the project.



2. An established company has decided to add a new product to its line. It will buy the product from a manufacturing concern, package it, and sell it to a number of distributors that have been selected on a geographical basis. Market research has already indicated the volume expected and the size of sales force required. The steps shown in the following table are to be planned.
Activity Description Predecessor Activity Duration (days)
A Organize sales office - 6
B Hire salesman A 4
C Train salesman B 7
D Select advertising agency A 2
E Plan advertising campaign D 4
F Conduct advertising campaign E 10
G Design package - 2
H Setup packaging campaign G 10
I Package initial stocks J,H 6
J Order stock from manufacturer - 13
K Select distributors A 9
L Sell to distributors C,K 3
M Ship stocks to distributors I,L 5
(a) Draw an arrow diagram for the project.
(b) Indicate the criticla path.
(c) For each non-critical activity, find the total and free float.
 
Examples
1. There are seven jobs, each of which has to go through the machines A and B in the order AB. Processing times in hours are as follows.
Job 1 2 3 4 5 6 7
Machine A 3 12 15 6 10 11 9
Machine B 8 10 10 6 12 1 3
Decide a sequence of these jobs that will minimize the total elapsed time T. Also find T and idle time for machines A and B.



2. Find the sequence that minimizes the total time required in performing the following job on three machines in the order ABC. Processing times (in hours) are given in the following table.
Job 1 2 3 4 5
Machine A 8 10 6 7 11
Machine B 5 6 2 3 4
Machine C 4 9 8 6 5
 
Model-1.1
1. A firm is considering the replacement of a machine, whose cost price is Rs 12,200 and its scrap value is Rs 200. From experience the running (maintenance and operating) costs are found to be as follows:
Year12345678
Running Cost2005008001,2001,8002,5003,2004,000

When should the machine be replaced?
 
Model-1.2
1. The data collected in running a machine, the cost of which is Rs 60,000 are given below:
Year12345
Resale Value42,00030,00020,40014,4009,650
Cost of spares4,0004,2704,8805,7006,800
Cost of labour14,00016,00018,00021,00025,000

Determine the optimum period for replacement of the machine.
 
Model-1.3
1. Machine A costs Rs 45,000 and its operating costs are estimated to be Rs 1,000 for the first year increasing by Rs 10,000 per year in the second and subsequent years. Machine B costs Rs 50,000 and operating costs are Rs 2,000 for the first year, increasing by Rs 4,000 in the second and subsequent years. If at present we have a machine of type A, should we replace it with B? if so when? Assume that both machines have no resale value and their future costs are not discounted.
 
Model-2
Replacement policy for items whose running cost increases with time but value of money changes constant rate during a period
1. An engineering company is offered a material handling equipment A. It is priced at Rs 60,000 includeing cost of installation. The costs for operation and maintenance are estimated to be Rs 10,000 for each of the first five years, increasing every year by Rs 3,000 in the sixth and subsequent years. The company expects a return of 10 percent on all its investment. What is the optimal replacement period?
Year12345678910
Running Cost10,00010,00010,00010,00010,00013,00016,00019,00022,00025,000


2. A company is buying mini computers. It costs Rs 5 lakh, and its running and maintenance costs are Rs 60,000 for each of the first five years, increasing by Rs 20,000 per year in the sixth and subsequent years. If the money is worth 10 percent per year, What is the optimal replacement period?
 
Model-3
Group replacement policy
1. A computer contains 10,000 resistors. When any resistor fails, it is replaced. The cost of replacing a resistor individually is Rs 1 only. If all the resistors are replaced at the same time, the cost per resistor would be reduced to 35 paise. The percentage of surviving resistors say S(t) at the end of month t and the probability of failure P(t) during the month t are as follows:
t0123456
P(t)00.030.070.200.400.150.15
What is the optimal replacement plan?

2. The following mortality rates have been observed for a certain type of fuse:
t012345
P(t)00.050.100.200.400.25
There are 1,000 fuses in use and it costs Rs 5 to replace an individual fuse. If all fuses were replaced simultaneously it would cost Rs 1.25 per fuse. It is proposed to replace all fuses at fixed intervals of time, whether or not they have burnt out, and to contiune replacing burnt out fuses as they fail. At what time intervals should the group replacement be made? Also prove that this optimal policy is superior to the straight forward policy of replacing each fuse only when it fails.
 

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