Other Math Answers

Four operators A,B,C and D are available for a manager who has to get four jobsI,II,III and IV done by assuming one job to each operator.The times taken by different operators to do different jobs are given in the table below:
Operators
A B C D
I 15 13 14 13
Jobs II 11 12 11 13
III 18 12 10 11
IV 15 17 14 16
(i)Express the assignment problem above as a balanced transportation problem.
(ii)Obtain the optimal solution of the assignment problem by the Hungarian method.
(iii)Hence give an optimal solution of the equivalent balanced transportation problem.

Solve the following LPP by the two-phase simplex method:
Maximise Z=x_1+2x_2+3x_3
Subject to
x_1+2x_2+3x_3=15
2x_1+x_2+5x_3=20
x_1+2x_2+x_3+x_4=10
x_1,x_2,x_3,x_4≥0

Which of the following statements are true?Give reasons for your answers in each of the following parts.
(i)The solution x_1=1,x_2=2,x_3=x_4=x_5=0 of the equations:
2x_1+x_2+x_3+x_4=4
2x_1+x_2+2x_3+x_5=4 is a solution.
(ii)The number of basic variables in a feasible solution of a balanced transportation problem with m sources and n destinations is mn.
(iii)A pay-off matrix can posses more than one saddle-point in case of a pure strategy game problem.
(iv)(1,2) is an optimal solution of LPP
"Max. z=2x_1+4x_2
s.t. x_1+2x_2≤5
x_1+x_2≤4
x_1≥0
x_2≥0."
(v)AUB is a convex set,where A={(x_1,x_2)|x_1+x_2≤8,2x_1+x_2≥10,x_1≥0,x_2≥0} and B={(x_1,x_2)|x_1+x_2≥8,2x_1+x_2≤10,x_1≥0,x_2≥0}.

Solve the following LPP by two phase simplex method:
Maximize Z=X+2Y+3Z
Subject to X+2Y+3Z=15
2X+Y+5Y=20
X+2Y+Z+W=10
X,Y,Z,W≥0

Write down the dual of the LPP given by: Maximize 20x₁+30x₂ subject to
x₁+2x₂ ≤ 20
x₁+x₂ ≤ 12
5x₁+x₂ ≤ 40
x₁ , x₂ ≥ 0
Solve the primal LPP graphically. Use the optimal solution to primal LPP and complementary slackness condition to identify the dual variable that will have zero value in the optimal solution to the dual.

Describe how quadratic equations can be used in decision-making.

Which of the following statements true or false? Give a short proof or a counter example in support of your answers. i) The forward and backward recursive formulation in Dynamic programming techniques can result in different optimum solutions to the same problem. ii) A non-critical activity cannot have zero total float. iii) The addition of a consultant to all the elements of an assignment problem can affect the optimal solution of the problem. iv) If the primal LPP has an unbounded solution, the dual LPP cannot have a feasible solution. v) In queuing theory, if the arrivals are according to Poisson distribution with parameter λ, the inter-arrival time is exponential with parameter e^λ.

A distribution system has the following constraints:
Factory A B C
Capacity (in units): 45 15 40
Warehouse 1 2 3
Demand (in units): 25 55 20
The transportation costs per unit (in Rs.) through each route are:
To 1 2 3
From A 10 7 8
B 15 12 9
C 7 8 12
Find an initial basic feasible solution by the North-West corner method. Starting with this solution, carry out as many iterations of the u-v method as necessary to find a basic feasible solution with transportation cost less than Rs.800.

A sales manager wishes to assign four sales territories to four salespersons. The salespersons differ in their ability and skills and consequently the sales expected in each territory are different. The estimates of sales per month for each salesperson in different territories are given below:
Estimated monthly sales
Territory
1 2 3 4
Salesperson
A 20 40 45 30
B 50 40 55 40
C 45 40 42 50
D 48 50 42 45
Find the optimal assignment of the four salespersons to the four different territories and the maximum monthly sales.

A distribution system has the following constraints:
Factory: A B C
Capacity(in units): 45 15 40
Warehouse: 1 2 3
Demand(in units): 25 55 20
The transportation costs per unit (in Rs.)through each route are:
To
| || |||
From A 10 7 8
B 15 12 9
C 7 8 12
Find an initial basic feasible solution by the North-West Corner method. Starting with this solution, carry out as many iterations of the u-v method as necessary to find a basic feasible solution with transportation cost less than Rs.800.