Operations Research :- Assignments
UNIT-I
Linear Programming
Q.1. Explain the properties of Liner Programming. [4]
Q.2. Answer the questions related to the model below: [4]
max. 3 x1 + 2 x2
st 2 x1 + 2 x2 ≤ 5
2 x1 + x2 ≤ 4
x1 + 2 x2 ≤ 4
x1, x2 ≥ 0
a. Use the graphical solution technique to find the optimal solution to the model.
b. Use the simplex algorithm to find the optimal solution to the model.
c. For which objective function coefficient value ranges of x1 and x2 does the solution remain optimal?
d. Find the dual of the model.
Q.3. what is the allowable range for the objective function coefficient of x2 in which current solution
(basis) remains optimal? (Hint: Use the relation of Duality-Sensitivity)
Q.4. Explain maximization and minimization models with example. [4]
Q.5. Explain mathematical model of single period production model. [4]
UNIT-II
Duality in Linear Programming
Q.1.Explain Primal dual relationship. [4]
a) Weak Duality Property
b) Optimality Property
Q.2. Explain the Steps involved in the dual simplex method. [4]
Q.3. Write down the steps for solving Revised Simplex Method in Standard Form [4]
Q.4. What is the principle of duality in linear programming? [4]
Q.5. What is an economic interpretation of duality?
UNIT-III
The Transportation Problem and Assignment Problem
Q.1. Consider the following transportation table for a minimization problem. [4]
a) A basic feasible solution for the given transportation is given as BV: {x11, x13, x21, x24, x32, x33}.
Find the values of the basic variables. Prove that this solution is not optimal.
b) Find the optimal solution using transportation simplex method starting from the basic feasible
solution given in part a.
c) Find the range of values of the c24 (the cost related to x24ı) for which the current basis remains
optimal.
[4]
Q.2. A shoe company forecasts the following demands during the next three months: 200, 260, 240. It
costs $7 to produce a pair of shoes with regular time labor (RT) and $11 with overtime labor (OT).
During each month regular production is limited to 200 pairs of shoes, and overtime production is
limited to 100 pairs. It costs $1 per month to hold a pair of shoes in inventory. Formulate a balanced
transportation problem to minimize the total cost of meeting the next three months of demand on time .
Q.3. Explain the following terms: [4]
a) North West corner method,
b) Least cost method,
Q.4. Write & Explain MODI method algorithm steps. [4]
Q.5.Define Stepping Stone Method. [4]
UNIT-IV
Game Theory and Dynamic Programming
Q.1. Discuss 2 person 0 sum games model in game theory. Write payoff function of the same. [4]
Q.2. Solve the 2 x 2 game. Consider the following game and solve it using graphical method.
Q.3. Explain Forward and backward recursion. [4]
Q.4. Explain Dynamic Programming (DP) Applications [4]
a) Knapsack
b) Equipment replacement
Q.5. Solve. [4]
Knapsack Max weight : W = 10 (units)
Total items : N = 4
Values of items : v [] = {10, 40, 30, 50}
Weight of items : w [] = {5, 4, 6, 3}
UNIT-V
Integer Programming Problem and Project Management
Q.1. Explain in detail Integer Programming Method [4]
a) Binary Variables
b) Logical Constraints
Q.2. Explain Cutting Plane Algorithms with example. [4]
Q.3. Discuss the rules for drawing network analysis with example. [4]
(Example 1For preparation of Paneer (Cottage Cheese))
Q.4. What is CPM & PERT What are the objectives of CPM & PERT? [4]
Q.5. Explain in details Scheduling and Crashing of Jobs. [4]
UNIT-VI
Decision Theory and Sensitivity Analysis
Q.1. Explain Decision Making Under Certainty Uncertainty and Risk with Examples. [4]
Q.2. Solve- [4]
If a firm having a contract to build a dam across a river requiring 300,000 cubic meters of
gravel, found two feasible sources whose characteristics are given below:
Q.3. What is goal programming distinguish it from linear programming? [4]
Q.4. Discuss the steps in model formulation (algorithm): goal programming. [4]
Q.5. Explain The weights method, & The preemptive method. [4]
Linear Programming
Q.1. Explain the properties of Liner Programming. [4]
Q.2. Answer the questions related to the model below: [4]
max. 3 x1 + 2 x2
st 2 x1 + 2 x2 ≤ 5
2 x1 + x2 ≤ 4
x1 + 2 x2 ≤ 4
x1, x2 ≥ 0
a. Use the graphical solution technique to find the optimal solution to the model.
b. Use the simplex algorithm to find the optimal solution to the model.
c. For which objective function coefficient value ranges of x1 and x2 does the solution remain optimal?
d. Find the dual of the model.
Q.3. what is the allowable range for the objective function coefficient of x2 in which current solution
(basis) remains optimal? (Hint: Use the relation of Duality-Sensitivity)
Q.4. Explain maximization and minimization models with example. [4]
Q.5. Explain mathematical model of single period production model. [4]
UNIT-II
Duality in Linear Programming
Q.1.Explain Primal dual relationship. [4]
a) Weak Duality Property
b) Optimality Property
Q.2. Explain the Steps involved in the dual simplex method. [4]
Q.3. Write down the steps for solving Revised Simplex Method in Standard Form [4]
Q.4. What is the principle of duality in linear programming? [4]
Q.5. What is an economic interpretation of duality?
UNIT-III
The Transportation Problem and Assignment Problem
Q.1. Consider the following transportation table for a minimization problem. [4]
a) A basic feasible solution for the given transportation is given as BV: {x11, x13, x21, x24, x32, x33}.
Find the values of the basic variables. Prove that this solution is not optimal.
b) Find the optimal solution using transportation simplex method starting from the basic feasible
solution given in part a.
c) Find the range of values of the c24 (the cost related to x24ı) for which the current basis remains
optimal.
[4]
Q.2. A shoe company forecasts the following demands during the next three months: 200, 260, 240. It
costs $7 to produce a pair of shoes with regular time labor (RT) and $11 with overtime labor (OT).
During each month regular production is limited to 200 pairs of shoes, and overtime production is
limited to 100 pairs. It costs $1 per month to hold a pair of shoes in inventory. Formulate a balanced
transportation problem to minimize the total cost of meeting the next three months of demand on time .
Q.3. Explain the following terms: [4]
a) North West corner method,
b) Least cost method,
Q.4. Write & Explain MODI method algorithm steps. [4]
Q.5.Define Stepping Stone Method. [4]
UNIT-IV
Game Theory and Dynamic Programming
Q.1. Discuss 2 person 0 sum games model in game theory. Write payoff function of the same. [4]
Q.2. Solve the 2 x 2 game. Consider the following game and solve it using graphical method.
Q.3. Explain Forward and backward recursion. [4]
Q.4. Explain Dynamic Programming (DP) Applications [4]
a) Knapsack
b) Equipment replacement
Q.5. Solve. [4]
Knapsack Max weight : W = 10 (units)
Total items : N = 4
Values of items : v [] = {10, 40, 30, 50}
Weight of items : w [] = {5, 4, 6, 3}
UNIT-V
Integer Programming Problem and Project Management
Q.1. Explain in detail Integer Programming Method [4]
a) Binary Variables
b) Logical Constraints
Q.2. Explain Cutting Plane Algorithms with example. [4]
Q.3. Discuss the rules for drawing network analysis with example. [4]
(Example 1For preparation of Paneer (Cottage Cheese))
Q.4. What is CPM & PERT What are the objectives of CPM & PERT? [4]
Q.5. Explain in details Scheduling and Crashing of Jobs. [4]
UNIT-VI
Decision Theory and Sensitivity Analysis
Q.1. Explain Decision Making Under Certainty Uncertainty and Risk with Examples. [4]
Q.2. Solve- [4]
If a firm having a contract to build a dam across a river requiring 300,000 cubic meters of
gravel, found two feasible sources whose characteristics are given below:
Q.3. What is goal programming distinguish it from linear programming? [4]
Q.4. Discuss the steps in model formulation (algorithm): goal programming. [4]
Q.5. Explain The weights method, & The preemptive method. [4]
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