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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.