Operations Research (OR) Past Paper PDF

En g 1. What is the primary reason OR does not offer a single general solution for all problems? M A) It lacks universal algorithms. B) Each problem has unique type and complexity. oh C) OR focuses only on specific industries. D) The avail...

En g 1. What is the primary reason OR does not offer a single general solution for all problems? M A) It lacks universal algorithms. B) Each problem has unique type and complexity. oh C) OR focuses only on specific industries. D) The available tools are outdated. am Answer: B) Each problem has unique type and complexity. Explanation: The type and complexity of the mathematical model dictate the solution method. OR provides specific techniques for different types of problems, such as linear programming or ed dynamic programming. Aw 2. Which OR technique is most prominent for models with linear objective and constraint functions? ad A) Integer programming B) Nonlinear programming 01 C) Linear programming D) Dynamic programming Answer: C) Linear programming 06 Explanation: Linear programming is specifically designed for problems with linear objectives and constraints, making it a key technique in OR. 74 59 3. What distinguishes integer programming from other OR techniques? 53 A) It involves nonlinear functions. B) The variables must assume integer values. C) It deals with waiting line performance. 8 D) It decomposes models into smaller sub-problems. Answer: B) The variables must assume integer values. Explanation: Integer programming solves problems where the solution requires variables to be integers, commonly used in scheduling and resource allocation. 4. Which OR technique decomposes a model into smaller sub-problems? A) Network programming B) Dynamic programming C) Linear programming D) Heuristic methods Answer: B) Dynamic programming Explanation: Dynamic programming simplifies complex problems by breaking them into manageable sub-problems, solving each sequentially.  En g 5. Which type of programming deals with models where functions are nonlinear? M A) Linear programming B) Nonlinear programming oh C) Network programming D) Queuing models am Answer: B) Nonlinear programming Explanation: Nonlinear programming is designed for models where the objective or constraint functions are nonlinear. ed Aw 6. Why are solutions in OR not generally obtained in closed forms? A) The problems are simple. B) Solutions require iterative algorithms. ad C) OR lacks precise methods. D) The models are all nonlinear. 01 Answer: B) Solutions require iterative algorithms. Explanation: OR uses algorithms that iteratively apply fixed computational rules to approach the optimal solution. 06 74 7. What are heuristics or metaheuristics used for in OR? 59 A) To find an exact solution for linear models. B) To simplify already solved models. 53 C) To seek good solutions for complex models. D) To analyze performance of waiting lines. Answer: C) To seek good solutions for complex models. 8 Explanation: Heuristics and metaheuristics are intelligent search rules used for problems too complex for traditional optimization algorithms. 8. What do queuing models study? A) The cost of production processes. B) The performance of waiting lines. C) The scheduling of tasks. D) The advertising effectiveness of a product. Answer: B) The performance of waiting lines. Explanation: Queuing models analyze waiting lines using probability and stochastic methods to determine metrics like average waiting time and service utilization.  En g 9. How does simulation estimate performance measures in OR? M A) By solving the mathematical model directly. B) By mimicking the behavior of the real system. oh C) By assuming linear relationships. D) By simplifying constraints. am Answer: B) By mimicking the behavior of the real system. Explanation: Simulation imitates real systems to estimate performance when analytical solutions are difficult. ed Aw 10. Which department focuses on quality control standards and machine capacity in OR? A) Sales Department B) Materials Department ad C) Production Department D) Operations Research Team 01 Answer: C) Production Department Explanation: The production department considers factors like production capacity, in-process inventory, and quality control. 06 74 11. What is the main responsibility of the Materials Department in OR? 59 A) Managing sales forecasts. B) Controlling raw material stock and delivery schedules. 53 C) Determining machine capacity. D) Optimizing advertising strategies. Answer: B) Controlling raw material stock and delivery schedules. 8 Explanation: The materials department focuses on raw materials, including stock availability, delivery schedules, and storage limitations. 12. What should OR analysts do before applying sophisticated mathematical tools? A) Solve the problem immediately using linear programming. B) Take a bird's-eye view to identify non-technical causes. C) Start with a pre-selected optimization tool. D) Use only heuristic methods. Answer: B) Take a bird's-eye view to identify non-technical causes. Explanation: Analysts should first assess the overall situation to uncover potential non-technical reasons for the problem.  En g 13. Why is teamwork important in OR? M A) OR analysts work independently of clients. B) Solutions are rooted in people and collaboration. oh C) Technology eliminates the need for teamwork. D) Analysts and clients rarely interact. am Answer: B) Solutions are rooted in people and collaboration. Explanation: OR relies on teamwork, where analysts and clients work together to ensure practical and effective solutions. ed Aw 14. What is the purpose of validating an OR model? A) To simplify the constraints. B) To ensure the model meets its intended purpose. ad C) To select the best algorithm. D) To guarantee the model is optimal. 01 Answer: B) To ensure the model meets its intended purpose. Explanation: Validation ensures the model accurately represents the real-world system it was designed to analyze. 06 74 15. What is the final phase of an OR study? 59 A) Problem definition B) Model construction 53 C) Solution validation D) Implementation of the solution Answer: D) Implementation of the solution 8 Explanation: The final phase involves translating the model’s results into actionable instructions for the people administering the system.  En g 1. What is the main purpose of the graphical solution method in a two-variable linear programming (LP) problem? M A) To handle large-scale problems effectively. oh B) To develop a general simplex algorithm. C) To provide concrete foundations for understanding LP concepts. am D) To solve problems with more than two variables. Answer: C) To provide concrete foundations for understanding LP concepts. Explanation: The graphical solution method is used as a foundational approach to understand LP ed concepts. While practical problems rarely involve just two variables, this method helps build an understanding for more advanced techniques like the simplex algorithm. Aw 3. What are the two steps in the graphical solution method for LP problems? ad A) Defining constraints and implementing the simplex algorithm. B) Determining the feasible solution space and identifying the optimum solution. 01 C) Setting up the equations and solving them iteratively. D) Maximizing profit and minimizing costs. Answer: B) Determining the feasible solution space and identifying the optimum solution. 06 Explanation: The graphical solution method involves (1) finding the feasible region that satisfies all constraints and (2) determining the best solution (optimum point) within that region. 74 59 1. Why is a computer necessary for solving typical LP models? A) LP problems are too simple to solve manually. 53 B) Most LP problems have thousands of variables and constraints. C) Computers guarantee optimal solutions. 8 D) Manual methods are not well-documented. Answer: B) Most LP problems have thousands of variables and constraints. Explanation: In practical applications, LP models often involve a large number of variables and constraints, making manual computation infeasible. 2. Which software is particularly appealing to spreadsheet users for solving LP problems? A) AMPL B) Python C) Excel Solver D) MATLAB Answer: C) Excel Solver Explanation: Excel Solver is widely used because of its integration with spreadsheets, making it accessible and user-friendly for solving LP problems.  En g 3. What is AMPL in the context of LP? M A) A spreadsheet-based tool. B) An algebraic modeling language. oh C) A type of financial planning software. D) A manual optimization technique. am Answer: B) An algebraic modeling language. Explanation: AMPL is a higher-order algebraic modeling language designed for defining and solving complex LP problems. ed 4. What is the role of input data cells in a spreadsheet LP model? Aw A) Represent the variables and constraints. B) Provide names for the variables. C) Store raw data used in calculations. ad D) Output the solution to the LP problem. Answer: C) Store raw data used in calculations. Explanation: Input data cells contain the necessary raw data, such as coefficients for variables, 01 used in the model's calculations. 06 74 7. What is the penalty mentioned in the Single-Period Production Model example? 59 A) Penalty for exceeding production capacity. B) Penalty for undelivered items. 53 C) Penalty for low-quality products. D) Penalty for using outdated materials. Answer: B) Penalty for undelivered items. 8 Explanation: The contract imposes a penalty for failing to deliver the items as ordered. 9. In the Multiple-Period Production-Inventory Model, how does production cost vary? A) It remains constant across all periods. B) It varies depending on the month. C) It depends on the storage cost. D) It is based on labor availability. Answer: B) It varies depending on the month. Explanation: The production cost is estimated to vary across months, ranging from $45 to $55 per window.  En g 10. What is the primary goal of the Bus Scheduling Model? M A) Maximize the number of buses in use. B) Reduce maintenance costs. oh C) Minimize the number of buses needed to meet demand. D) Optimize fuel efficiency. am Answer: C) Minimize the number of buses needed to meet demand. Explanation: The objective is to determine the minimum number of buses required to handle transportation needs. ed 13. Which department is involved in the production of clothing in the Single-Period Production Model? Aw A) Marketing and distribution B) Cutting, insulating, sewing, and packaging ad C) Sales and accounting D) Shipping and logistics Answer: B) Cutting, insulating, sewing, and packaging 01 Explanation: The clothing company’s production involves these four specific departments. 06 14. Why is spreadsheet modeling useful in LP applications? 74 A) It guarantees optimal solutions. B) It is an accessible input and output medium for LP models. 59 C) It eliminates the need for constraints. D) It avoids iterative calculations. 53 Answer: B) It is an accessible input and output medium for LP models. Explanation: Spreadsheets serve as a user-friendly platform for defining, solving, and visualizing LP 8 problems. 15. What is the objective of investment LP applications mentioned in the text? A) Maximizing production efficiency. B) Optimizing capital budgeting and portfolio selection. C) Minimizing storage costs. D) Balancing dietary requirements. Answer: B) Optimizing capital budgeting and portfolio selection. Explanation: LP applications in investment include tasks like capital budgeting, bond investment strategy, and stock portfolio selection.  En g 1. What is a key requirement for applying the Simplex Method? M A) All variables must be unrestricted. B) All constraints must be equations with nonnegative right-hand sides. oh C) All solutions must be non-integer values. D) All constraints must be inequalities. am Answer: B) All constraints must be equations with nonnegative right-hand sides. Explanation: The Simplex Method requires constraints to be in equation form with nonnegative right-hand sides to ensure the feasibility of solutions. ed Aw 2. How is a "less than or equal to" (≤) inequality converted into an equation? A) By adding a surplus variable. B) By subtracting a surplus variable. ad C) By adding a nonnegative slack variable. D) By dividing by a nonnegative variable. 01 Answer: C) By adding a nonnegative slack variable. Explanation: A slack variable is added to the left-hand side of a ≤ inequality to convert it into an equation, representing unused resources. 06 74 3. In the Reddy Mikks model, what does the slack variable s1s_1 represent? 59 A) The excess amount of a resource used. B) The unused amount of a resource. 53 C) The difference between two variables. D) The minimum required value of the objective function. Answer: B) The unused amount of a resource. 8 Explanation: In the Reddy Mikks model, s1 represents the slack or unused portion of resource M1 4. How is a "greater than or equal to" (≥) inequality converted into an equation? A) By adding a surplus variable. B) By subtracting a nonnegative surplus variable. C) By multiplying by -1. D) By dividing by the largest coefficient. Answer: B) By subtracting a nonnegative surplus variable. Explanation: A surplus variable is subtracted from the left-hand side of a ≥ inequality to convert it into an equation, indicating excess over the required minimum.  En g 5. What is the primary advantage of the Simplex Method over the graphical method? M A) It works only for two-variable problems. B) It investigates an infinite number of solutions simultaneously. oh C) It is suitable for problems with more than two variables. D) It guarantees non-integer solutions. am Answer: C) It is suitable for problems with more than two variables. Explanation: The graphical method is limited to two-variable problems, while the Simplex Method can handle problems with more than two variables. ed Aw 6. How does the Simplex Method handle unrestricted variables? A) By setting them equal to zero. B) By splitting them into two nonnegative variables. ad C) By ignoring their values. D) By converting them into slack variables. 01 Answer: B) By splitting them into two nonnegative variables. Explanation: Unrestricted variables are expressed as the difference of two nonnegative variables, allowing the Simplex Method to handle them effectively. 06 74 7. What does mm represent in the context of the Simplex Method? 59 A) The number of variables. B) The number of constraints (equations). 53 C) The total number of corner points. D) The maximum number of iterations. Answer: B) The number of constraints (equations). 8 Explanation: In the Simplex Method, mm represents the number of constraints in the LP model. 8. What does n - m represent in the Simplex Method? A) The number of basic variables. B) The number of slack variables. C) The number of non-basic variables set to zero. D) The total number of equations. Answer: C) The number of non-basic variables set to zero. Explanation: n−m represents the number of non-basic variables that are set to zero during each iteration.  En g 9. What is the maximum number of corner points in an LP problem with nn variables and mm constraints? M A) C(n,m)=n×m oh B) C(n,m)=n!/m!(n−m)! C) C(n,m)=n+m am D) C(n,m)=n! Answer: B) C(n,m)=n!/m!(n−m)! Explanation: The number of corner points is given by the combination formula, as the solution ed space is determined by the intersections of mm constraints out of nn variables. Aw 10. How does the Simplex Method handle solutions? A) It solves all equations simultaneously. ad B) It targets one variable at a time. C) It allows simultaneous increases in variables. 01 D) It randomly selects variables to optimize. Answer: B) It targets one variable at a time. Explanation: The Simplex Method operates iteratively, targeting one variable at a time to improve 06 the solution. 74 11. What is the path followed by the Simplex algorithm? 59 A) Straight lines between all feasible points. 53 B) Direct jumps to the optimum solution. C) Connections between corner points of the solution space. D) Random paths within the solution space. 8 Answer: C) Connections between corner points of the solution space. Explanation: The Simplex algorithm moves from one corner point to the next, improving the solution at each step. 12. Why is the simplex feasibility condition important? A) It ensures solutions are optimal. B) It guarantees that all variables are nonnegative. C) It ensures that the new solution is feasible. D) It reduces the number of equations in the problem. Answer: C) It ensures that the new solution is feasible. Explanation: The feasibility condition ensures that the new solution remains within the feasible region of the solution space.  En g 13. How is the solution space represented in the Simplex Method? M A) By a set of half-spaces. B) By the intersection of m simultaneous linear equations. oh C) By inequalities only. D) By a random selection of variables. am Answer: B) By the intersection of m simultaneous linear equations. Explanation: The solution space is represented by the intersection of mm equations and nn variables, creating a feasible region. ed Aw 14. What is the iterative nature of the Simplex Method? A) It solves all equations in one step. B) It uses multiple iterations to investigate feasible solutions. ad C) It guarantees the optimum solution in the first iteration. D) It randomly selects feasible solutions. 01 Answer: B) It uses multiple iterations to investigate feasible solutions. Explanation: The Simplex Method iterates through feasible solutions, improving the objective function at each step. 06 74 15. What is the purpose of slack and surplus variables in LP models? 59 A) To eliminate constraints. B) To convert inequalities into equations. 53 C) To represent the optimum solution. D) To simplify the objective function. Answer: B) To convert inequalities into equations. 8 Explanation: Slack variables (for ≤ constraints) and surplus variables (for ≥ constraints) are introduced to transform inequalities into equalities, making the model suitable for algebraic solutions.  En g 1. What is the primary purpose of artificial variables in the Simplex Method? M A) To simplify calculations in feasible solutions. B) To handle constraints with equalities and ≥ inequalities. oh C) To eliminate degeneracy in the solution. D) To maximize the objective function. am Answer: B) To handle constraints with equalities and ≥ inequalities. Explanation: Artificial variables are used in LP models where some constraints are equalities or have a ≥ inequality, allowing the algorithm to start with a basic feasible solution. ed Aw 2. What is the role of the penalty value MM in the M-method? A) To prevent cycling. B) To force artificial variables to zero in the optimal solution. ad C) To handle degeneracy. D) To maximize the number of iterations. 01 Answer: B) To force artificial variables to zero in the optimal solution. Explanation: In the M-method, MM is a large positive value that penalizes artificial variables in the objective function, driving them to zero as the optimal solution is approached. 06 74 3. How does the two-phase method differ from the M-method? 59 A) It uses slack variables instead of artificial variables. B) It eliminates the need for a penalty value MM. 53 C) It handles unbounded solutions more effectively. D) It does not require a Phase II for solving problems. Answer: B) It eliminates the need for a penalty value MM. 8 Explanation: The two-phase method avoids the use of MM, reducing computational errors, and separates the problem into Phase I (finding feasibility) and Phase II (optimizing the objective function). 4. What happens if one or more artificial variables are basic at the end of Phase I? A) The solution is unbounded. B) The problem is infeasible. C) Phase II starts immediately. D) Artificial variables are removed from the tableau. Answer: D) Artificial variables are removed from the tableau. Explanation: At the end of Phase I, artificial variables that are basic are removed by selecting them to leave the basis, and their columns are eliminated from the tableau.  En g 5. What is the optimality condition for a maximization problem in the Simplex Method? M A) All z-row coefficients are positive. B) All z-row coefficients are nonpositive. oh C) All z-row coefficients are nonnegative. D) All basic variables are zero. am Answer: C) All z-row coefficients are nonnegative. Explanation: In a maximization problem, the solution is optimal when all z-row coefficients are nonnegative. ed Aw 6. What does degeneracy in the Simplex Method indicate? A) The model has multiple feasible solutions. B) At least one basic variable is zero in the next iteration. ad C) The solution is unbounded. D) Artificial variables must be used. 01 Answer: B) At least one basic variable is zero in the next iteration. Explanation: Degeneracy occurs when a tie in the minimum ratio test leads to a basic variable becoming zero, potentially causing cycling. 06 74 7. What is a common cause of an unbounded solution? 59 A) Incorrect estimates of constraint coefficients. B) Too many artificial variables. 53 C) Redundant constraints. D) All variables are nonnegative. Answer: A) Incorrect estimates of constraint coefficients. 8 Explanation: An unbounded solution often arises from missing or inaccurately defined constraints in the model. 8. How does the Simplex Method handle alternative optima? A) It cycles indefinitely between solutions. B) It stops when the first optimum is found. C) It identifies all corner points of the feasible region. D) It stops when the objective function is parallel to a binding constraint. Answer: D) It stops when the objective function is parallel to a binding constraint. Explanation: Alternative optima occur when the objective function is parallel to a nonredundant binding constraint, resulting in multiple optimal solutions.  En g 9. What does an infeasible solution in an LP model signify? M A) The constraints are inconsistent. B) The solution space is unbounded. oh C) The artificial variables are nonbasic. D) All slack variables are zero. am Answer: A) The constraints are inconsistent. Explanation: Infeasibility arises when the constraints conflict, making it impossible to satisfy all of them simultaneously. ed Aw 10. What is the feasibility condition for selecting the leaving variable in the Simplex Method? A) It must have the largest z-row coefficient. B) It is the variable with the smallest nonnegative ratio. ad C) It must be an artificial variable. D) It is the variable with the highest objective coefficient. 01 Answer: B) It is the variable with the smallest nonnegative ratio. Explanation: The feasibility condition ensures that the leaving variable corresponds to the smallest nonnegative ratio, maintaining a feasible solution. 06 74 11. What happens if the Simplex Method cycles indefinitely? 59 A) It indicates alternative optima. B) It indicates degeneracy. 53 C) It indicates an unbounded solution. D) It indicates infeasibility. Answer: B) It indicates degeneracy. 8 Explanation: Degeneracy can cause the Simplex Method to cycle indefinitely, especially if the minimum ratio test ties repeatedly. 12. What type of problem occurs when constraints have nonnegative right-hand sides but inconsistent requirements? A) Unbounded solution B) Infeasible solution C) Degeneracy D) Alternative optima Answer: B) Infeasible solution Explanation: Inconsistent constraints lead to an infeasible solution, as no point satisfies all constraints simultaneously.  En g 13. What is the main objective of Phase I in the two-phase method? M A) To minimize the objective function. B) To find a starting basic feasible solution. oh C) To identify alternative optima. D) To maximize slack variables. am Answer: B) To find a starting basic feasible solution. Explanation: Phase I focuses on finding feasibility before optimizing the original objective function in Phase II. ed Aw 14. What does the removal of redundant constraints help avoid? A) Degeneracy B) Unbounded solutions ad C) Alternative optima D) Infeasibility 01 Answer: A) Degeneracy Explanation: Redundant constraints can lead to degeneracy, as they may cause unnecessary ties in the minimum ratio test. 06 74 15. When does the Simplex Method reach the optimal solution in a minimization problem? 59 A) When all z-row coefficients are nonpositive. B) When all z-row coefficients are nonnegative. 53 C) When all basic variables are zero. D) When the objective value is unbounded. Answer: A) When all z-row coefficients are nonpositive. 8 Explanation: In a minimization problem, the solution is optimal when all z-row coefficients are nonpositive, as the negative coefficients indicate further improvements.  En g 1. What is the purpose of sensitivity analysis in LP? M A) To change the structure of the LP model. B) To determine how parameter changes affect the optimal solution. oh C) To find alternate optima. D) To eliminate redundant constraints. am Answer: B) To determine how parameter changes affect the optimal solution. Explanation: Sensitivity analysis examines how changes in the parameters (e.g., objective coefficients, resource availability) impact the optimal solution within specific limits. ed Aw 2. What does post-optimal analysis focus on? A) Finding the initial basic feasible solution. B) Determining the new optimum solution when input data changes. ad C) Solving the dual problem. D) Identifying infeasible solutions. 01 Answer: B) Determining the new optimum solution when input data changes. Explanation: Post-optimal analysis helps adjust the solution when targeted input data (e.g., resource limits or cost coefficients) change. 06 74 3. In graphical sensitivity analysis, what does sensitivity to resource availability analyze? 59 A) Changes in unit cost or profit. B) Changes in the coefficients of the objective function. 53 C) Changes in the right-hand side of the constraints. D) Changes in the simplex tableau. Answer: C) Changes in the right-hand side of the constraints. 8 Explanation: Sensitivity to resource availability examines how changes in the constraints' right- hand side values impact the feasible region and optimal solution. 6. What happens if the capacity of machine 1 exceeds the feasibility range? A) The problem has no solution. B) A complete decision cannot be made with available information. C) The revenue decreases. D) The problem becomes unbounded. Answer: B) A complete decision cannot be made with available information. Explanation: Exceeding the feasibility range means additional computations are needed to determine the impact on the solution.  En g 7. What is required to determine new optimal values for variables after a resource change? M A) Immediate recalculations. B) Sensitivity analysis and additional computations. oh C) Modifying the simplex tableau. D) Increasing artificial variables. am Answer: B) Sensitivity analysis and additional computations. Explanation: Determining new values requires analyzing the sensitivity of variables to changes and performing necessary recalculations. ed Aw 9. In duality, what does the right-hand side of the primal constraints represent in the dual problem? A) Dual constraints’ coefficients. ad B) Dual objective function coefficients. C) Dual slack variables. 01 D) Primal surplus variables. Answer: B) Dual objective function coefficients. Explanation: In the dual problem, the right-hand side of the primal constraints becomes the 06 coefficients of the dual objective function. 74 10. What is assigned to each primal equation constraint in the dual problem? 59 A) A primal slack variable. 53 B) A dual variable. C) A surplus variable. D) A reduced cost. 8 Answer: B) A dual variable. Explanation: In the dual problem, each primal constraint is associated with a dual variable, representing the value of relaxing that constraint. 11. What happens to primal variables in the dual problem? A) They become dual constraints. B) They are eliminated. C) They become dual slack variables. D) They remain unchanged. Answer: A) They become dual constraints. Explanation: Primal variables are transposed into dual constraints in the dual problem formulation.  En g 12. In sensitivity analysis, what is the impact of changes in unit profit or cost? M A) Changes the feasibility range of resources. B) Alters the feasible region of the LP model. oh C) Modifies the coefficients of the objective function. D) Impacts the constraints' right-hand side values. am Answer: C) Modifies the coefficients of the objective function. Explanation: Changes in unit profit or cost affect the coefficients of the objective function, which may shift the optimal solution. ed Aw 14. What is the feasibility range in sensitivity analysis? A) The range of permissible changes to the simplex tableau. B) The range of changes that keep the solution feasible and optimal. ad C) The maximum number of variables that can change simultaneously. D) The total number of iterations required to solve the LP. 01 Answer: B) The range of changes that keep the solution feasible and optimal. Explanation: The feasibility range defines the limits within which changes can occur without violating constraints or affecting optimality. 06 74 15. Why is the dual problem important in LP? 59 A) It reduces the number of variables. B) It provides alternative solutions to the primal problem. 53 C) It offers insights into the value of resources and constraints. D) It eliminates the need for slack variables. Answer: C) It offers insights into the value of resources and constraints. 8 Explanation: The dual problem provides valuable information about the shadow prices of resources and the sensitivity of the primal problem to changes in constraints.  En g 1. What is the primary objective of the transportation model? M A) To maximize the number of shipments. B) To minimize total transportation costs while satisfying supply and demand. oh C) To eliminate surplus supply at all sources. D) To balance supply and demand perfectly. am Answer: B) To minimize total transportation costs while satisfying supply and demand. Explanation: The transportation model seeks to minimize the total cost of shipping goods from sources to destinations while ensuring that supply and demand constraints are met. ed Aw 2. In the transportation model, what does cijc_{ij} represent? A) The amount of goods shipped from source ii to destination jj. B) The transportation cost per unit from source ii to destination jj. ad C) The total cost of transportation between source ii and destination jj. D) The capacity of the route between source ii and destination jj. 01 Answer: B) The transportation cost per unit from source ii to destination jj. Explanation: cijc_{ij} is the cost of transporting one unit from source ii to destination jj. 06 3. What is added to the transportation model when supply exceeds demand? 74 A) A dummy source. 59 B) A dummy distribution center. C) An additional source with zero supply. 53 D) An artificial variable. Answer: B) A dummy distribution center. Explanation: A dummy distribution center is added to balance the model, receiving the surplus 8 supply with zero or specified costs. 4. Why is a very high unit transportation cost assigned to the dummy source in some cases? A) To ensure the dummy source is utilized. B) To prevent shortages at specific destinations. C) To balance excess supply. D) To minimize total transportation cost. Answer: B) To prevent shortages at specific destinations. Explanation: A high transportation cost discourages using the dummy source unless absolutely necessary, ensuring no shortage occurs. 5. How many independent equations are in a balanced transportation model with m sources and n destinations? A) m+n B) m+n−1  En g C) m+n+1 D) 2(m+n) M Answer: B) m+n−1 Explanation: In a balanced transportation model, one constraint is redundant, reducing the oh number of independent equations to m+n−1 am 6. Which method provides the worst starting solution in the transportation algorithm? ed A) Least-cost method B) Vogel approximation method Aw C) Northwest-corner method D) Hungarian method Answer: C) Northwest-corner method ad Explanation: The Northwest-corner method provides a starting solution without considering costs, often leading to the worst solution but requiring the least computation. 01 7. What is the purpose of the least-cost method in the transportation algorithm? 06 A) To balance supply and demand. B) To find the optimal solution directly. 74 C) To provide a better-quality starting solution. D) To minimize redundant constraints. 59 Answer: C) To provide a better-quality starting solution. Explanation: The least-cost method is a heuristic that prioritizes routes with lower costs to provide 53 a starting solution closer to the optimal. 8 8. Which method provides the best-quality starting solution in the transportation algorithm? A) Northwest-corner method B) Least-cost method C) Vogel approximation method D) Simplex method Answer: C) Vogel approximation method Explanation: The Vogel approximation method aims to provide the best starting solution by considering cost differences and penalties. 9. In a transportation model, what does xij represent? A) The supply available at source i B) The demand required at destination j  En g C) The amount shipped from source ii to destination j D) The unit transportation cost between ii and j M Answer: C) The amount shipped from source ii to destination j Explanation: xij denotes the quantity of goods transported along the route between source I and oh destination j am 10. What is the main advantage of the transportation model over other LP models? ed A) It eliminates the need for artificial variables. B) It uses fewer basic variables. Aw C) It has a unique solution for all cases. D) It is always balanced. Answer: A) It eliminates the need for artificial variables. ad Explanation: The transportation model inherently ensures a basic feasible solution without needing artificial variables. 01 11. In non-traditional transportation models, which application involves balancing production 06 and inventory? A) Tool sharpening service 74 B) Production-inventory control C) Warehouse distribution 59 D) Logistics optimization Answer: B) Production-inventory control 53 Explanation: Non-traditional applications include using transportation models for production- inventory control, optimizing the flow of resources over time. 8 12. How does the transportation model represent its structure? A) Using nodes for constraints and arcs for variables. B) Using nodes for sources and destinations, and arcs for routes. C) Using equations for nodes and variables for arcs. D) Using matrices for all components. Answer: B) Using nodes for sources and destinations, and arcs for routes. Explanation: The model represents sources and destinations as nodes and the transportation routes as arcs between them.  En g 13. What happens if demand exceeds supply in a transportation model? M A) A dummy destination is added. B) A dummy source is added. oh C) The model becomes infeasible. D) Surplus variables are introduced. am Answer: B) A dummy source is added. Explanation: A dummy source is introduced with high costs to balance the excess demand and ensure feasibility. ed Aw 14. What is the main limitation of the Northwest-corner method? A) It requires too many computations. B) It disregards transportation costs. ad C) It cannot balance supply and demand. D) It is unable to handle dummy variables. 01 Answer: B) It disregards transportation costs. Explanation: The Northwest-corner method focuses solely on fulfilling supply and demand, ignoring costs, which often leads to suboptimal solutions. 06 74 15. Why is one constraint in the transportation model redundant? 59 A) Because it ensures balanced supply and demand. B) Because the total supply equals the total demand. 53 C) To minimize the number of variables. D) To simplify the simplex tableau. Answer: B) Because the total supply equals the total demand. 8 Explanation: In a balanced transportation model, the total supply and total demand are equal, making one constraint redundant.  En g 1. What is the primary objective of the transportation algorithm? M A) To balance supply and demand. B) To minimize transportation costs by finding the optimal solution. oh C) To assign workers to jobs at minimum cost. D) To maximize transportation efficiency. am Answer: B) To minimize transportation costs by finding the optimal solution. Explanation: The transportation algorithm iteratively improves the starting solution to achieve the minimum transportation cost while satisfying supply and demand constraints. ed Aw 2. What is the purpose of the multipliers ui and vj in the transportation algorithm? A) To compute the nonbasic coefficients in the z-row. B) To determine the supply and demand at each node. ad C) To balance the transportation tableau. D) To eliminate redundant constraints. 01 Answer: A) To compute the nonbasic coefficients in the z-row. Explanation: The multipliers ui and vj are used to calculate the nonbasic coefficients cij in the z- row for determining the entering variable. 06 74 3. How is the entering variable in the transportation algorithm determined? 59 A) By finding the variable with the smallest cost. B) By checking the simplex feasibility condition. 53 C) By computing the z-row coefficients using the method of multipliers. D) By eliminating surplus variables. Answer: C) By computing the z-row coefficients using the method of multipliers. 8 Explanation: The entering variable is determined by calculating the z-row coefficients for the nonbasic variables using the multipliers ui and vj 4. What does the simplex feasibility condition ensure in the transportation algorithm? A) That all variables remain nonnegative. B) That the optimal solution is achieved. C) That the leaving variable is selected appropriately. D) That all constraints are satisfied. Answer: C) That the leaving variable is selected appropriately. Explanation: The simplex feasibility condition identifies the leaving variable by ensuring the solution remains feasible after the basis changes.  En g 5. What is the assignment model primarily used for? M A) Balancing supply and demand in transportation problems. B) Matching workers to jobs at minimum cost. oh C) Finding the shortest path in a network. D) Allocating resources to various locations. am Answer: B) Matching workers to jobs at minimum cost. Explanation: The assignment model deals with allocating workers (sources) to jobs (destinations) while minimizing the total assignment cost. ed Aw 6. How does the assignment model differ from the transportation model? A) It does not involve costs. B) It has equal supply and demand values of 1 for each source and destination. ad C) It allows for surplus and shortage in supply or demand. D) It eliminates the need for basic feasible solutions. 01 Answer: B) It has equal supply and demand values of 1 for each source and destination. Explanation: In the assignment model, every source (worker) and destination (job) has a supply and demand of exactly 1, making it a special case of the transportation model. 06 74 7. What is the purpose of the Hungarian Method? 59 A) To determine the optimal solution for the transportation model. B) To find the minimum cost assignment in the assignment model. 53 C) To balance supply and demand in a transportation problem. D) To compute the multipliers in the simplex algorithm. Answer: B) To find the minimum cost assignment in the assignment model. 8 Explanation: The Hungarian Method is a specialized algorithm for solving the assignment model efficiently, ensuring minimum cost. 8. How many basic variables are present in the starting solution of a transportation model with mm sources and nn destinations? A) m+n B) m+n−1 C) m×n D) m−n+1 Answer: B) m+n−1 Explanation: In a balanced transportation model, the starting solution contains m+n−1 basic variables due to the redundancy of one equation.  En g 9. Why is the assignment model considered a special case of the transportation model? M A) It has no constraints. B) Each source and destination has a supply or demand value of 1. oh C) It uses the Northwest-corner method for the initial solution. D) It does not involve transportation costs. am Answer: B) Each source and destination has a supply or demand value of 1. Explanation: The assignment model simplifies the transportation model by assigning a supply of 1 to each source and a demand of 1 to each destination. ed Aw 10. What happens after the basis is updated in the transportation algorithm? A) The solution is recomputed from scratch. B) The simplex feasibility condition is reapplied. ad C) The optimality condition is checked again. D) The tableau is balanced. 01 Answer: C) The optimality condition is checked again. Explanation: After updating the basis, the algorithm returns to Step 1 to check the simplex optimality condition and determine whether the solution is optimal. 06 74 59 1. What is the primary objective of the minimal spanning tree algorithm? 53 A) To determine the maximum flow in a network. B) To find the shortest route between two nodes. C) To link all nodes in a network with the smallest total length of arcs. 8 D) To identify all possible paths in a network. Answer: C) To link all nodes in a network with the smallest total length of arcs. Explanation: The minimal spanning tree algorithm connects all nodes in a network using the smallest total length of arcs, which is useful for economical design in applications like road systems. 2. What does the notation (N,A)(N, A) represent in a network model? A) The set of nodes and arcs in the network. B) The maximum flow and capacity in the network. C) The number of paths and cycles in the network. D) The shortest route and cost of the network. Answer: A) The set of nodes and arcs in the network. Explanation: In a network model, NN represents the set of nodes, and AA represents the set of arcs (connections) linking the nodes.  En g 3. What is the key difference between Dijkstra's and Floyd's algorithms? M A) Floyd's algorithm is limited to undirected networks, while Dijkstra's handles directed networks. B) Floyd's algorithm finds the shortest route between all pairs of nodes, while Dijkstra's focuses on oh one source node to all others. C) Dijkstra's algorithm is used for maximum flow problems, while Floyd's is for minimal spanning am trees. D) Floyd's algorithm is a heuristic, while Dijkstra's guarantees an optimal solution. Answer: B) Floyd's algorithm finds the shortest route between all pairs of nodes, while Dijkstra's ed focuses on one source node to all others. Explanation: Dijkstra's algorithm calculates the shortest paths from a single source node to all other nodes, while Floyd's algorithm determines the shortest paths between every pair of nodes in Aw the network. ad 4. What is the primary application of the shortest-route problem? 01 A) Maximizing network flow. B) Determining the shortest route between two nodes in a network. C) Designing a network with the smallest total arc length. 06 D) Identifying the most efficient cycle in a network. Answer: B) Determining the shortest route between two nodes in a network. 74 Explanation: The shortest-route problem focuses on finding the shortest path between two specific nodes in a network, commonly used in road and pipeline design. 59 53 5. In network optimization, what is a directed arc? A) An arc that allows flow in one direction only. 8 B) An arc that forms a cycle or loop. C) An arc with infinite capacity. D) An arc that connects a node to itself. Answer: A) An arc that allows flow in one direction only. Explanation: A directed arc allows positive flow in one direction only, and networks with such arcs are called directed networks.