Download X11 = Space leased at the beginning of month 1 for period of 1 month

Question 1
This can happen for the following reasons:
1. Management described some constraints incorrectly.
2. Formulation of problem is incorrect.
3. At least one of the constraints is too restrictive.
Question 2
Range of optimality:
The range, in which we can change the coefficients of objective function (one at a
time) without changing the optimal solution of a given problem.
Significance:
In real-world problems mostly cost coefficients don’t remain same and vary over
the time, so this range helps management to know that how long their production plan
will remain the same for different values of cost (coefficients).
Yes, optimal values do not remain same, when we change the value of
coefficients while keeping the same values for decision variables, the optimal value is
unchanged.
Question 3 (a)
Decision Variables:
y = Number of units of Y model for initial period.
z = Number of units of Z model for initial period.
L = Loan taken in the start of initial period for a duration of 3 months.
Objective Function:
(
)
Max. (58-50)y + (120-100)z - 0.03L
Functional Constraints:
L < 10,000
(Maximum Loan Constraint)
3000 + (58-50)y + (120-100)z ≥ 2(L+0.03L)
12y + 25z < 2500
y + 2z < 150
y
> 50
z > 25
Non Negativity Constraints:
y, z, L, > 0
(b)
Yes, it can be solved by simplex method because it is fulfilling all the criteria of
Linear Programming Model (Proportionality, Additivity, Certainty, and Divisibility).
Proportionality:
Profit contribution of each product and cost of loan are proportional to overall
profit. And similarly contribution of each production is proportional to available
resources.
Additivity:
Overall profit is a sum of individual profits. And similarly RHS of constraints are
sum of activities on left hand side.
Divisibility:
Values of decision variables are real, meaning can be non-integer too. Like, loan
taken can be floating point number.
Certainty:
All the coefficients in the objective function and in the constraints have
deterministic values, not the probabilistic ones.
Question 4 (a)
No, it is not a basic feasible solution. Because, in basic feasible solution, number
of basic variables (having +ve values) are equal to the number of constraints. But in the
given problem we have 4 basic variables and 3 constraints.
(b)
Yes, we can give an initial basic feasible solution by performing a couple of
simplex iterations. 1st of all reformulate this question by introducing slack and artificial
variables.
Max. z - 5x1- (3M+2)x2 - 4x3 – (5M+1)x4 = -52M
s.t.
2x1 + 5x3 + s1 = 34
3x2 + 5x4 + A = 52
4x1 + 2x4 + s2 = 40
x1, x2, x3, s1 s2, A > 0
Row
Basic
X1
(0)
-5
Z
(1)
2
S1
(2)
0
A
(3)
4
S2
X2
-3M-2
0
3
0
X3
4
5
0
0
X4
-5M-1
0
5
2
S1
0
1
0
0
S2
0
0
0
1
A
0
0
1
0
Column of X4 is pivot column and 2nd row is pivot row so the new tableau will be
Row
Basic
X1
X2
X3
X4
S1
S2
A
(0)
-5
-7/5
4
0
0
0
M+1/5
Z
(1)
2
0
5
0
1
0
0
S1
(2)
0
3/5
0
1
0
0
1/5
X4
(3)
4
-6/5
0
0
0
1
-2/5
S2
RHS
-52M
34
52
40
RHS
52/5
34
52/5
96/5
So, initial basic feasible solution is (0,0,0,52/5,34,96/5)
Question 5 (a)
Yes, these problems can be solved using simplex method but the complexity will
increase. Because as we increase the supply or demand nodes number of decision
variables will increase exponentially, plus all the constraints are of equality so we have to
introduce artificial variables equal the number of constraints. And due special nature of
Transportation Problem, specialized algorithms exist which can solve these problems
fairly quickly.
(b)
1. All the constraints’ coefficients are 1
2. In constraints, relatively a few decision variables exist (remaining have
zero coefficients).
3. All variables appear at least twice in the constraints (once in supply
constraint and 2nd time in demand constraints)
(c)
Total supply should be equal to total demand.
(d)
i) Yes! Because, it satisfies all the supply and demand constraints.
ii) If there are m supply constraints and n demand constraints then total
constraints will be (m+n-1). And as we get (m+n-1) number of basic variables
from north-west corner rule so yes it generates a basic feasible solution.
iii) No, its just starts with initial basic feasible solution which may or may not be
optimal. Because it does not consider the cost (associated with every supply node
to every demand node) at all.