Operations Research — Part 2 MCQs with Explanations

The quantitative approach to decision analysis is a scientific approach. In quantitative decision analysis, we use scientific methods to inform the decision-making process. These methods help data scientists put a value to several critical pieces of information, which include the: Problem facing the organization.

Operations Research approach is typically based on the use of Mathematical model. A mathematical model is an abstract model that uses mathematical language to describe the behaviour of a system.

In a manufacturing process, Production manager takes the decisions as to what quantities and which process or processes are to be used so that the cost is minimum and profit is maximum. Production managers ensure that manufacturing processes run reliably and efficiently. Responsibilities of the job include: planning and organising production schedules.

For analyzing a problem, decision-makers should study both its qualitative aspects and Its quantitative aspects.

A model is an essence of reality an approximation and an idealization.

Managerial decisions are based on an evaluation of quantitative data, the use of qualitative factors and results generated by formal models.

The initial solution of a transportation problem can be obtained by applying any known method. However, the only condition is that the rim conditions are satisfied.

The dummy source or destination in a transportation problem is added to satisfy rim conditions.

Degeneracy in a transportation problem occurs when there are fewer independent allocations than required, i.e., fewer than (m + n - 1) , where m is the number of rows and n is the number of columns. Option A: "Total supply equals total demand" is a necessary condition for feasibility but does not define degeneracy. Option B: "The solution is not feasible" is incorrect because degeneracy does not automatically lead to infeasibility. A valid solution can still exist despite degeneracy. Option C: "The few allocations become negative" is also incorrect because negative allocations are not allowed in a transportation problem. This would indicate an infeasible solution, which is unrelated to degeneracy. Therefore, the correct answer is Option D: None of the above, as none of the options accurately describe degeneracy. Degeneracy specifically refers to having too few independent allocations, which complicates the solution process but does not inherently make it infeasible.

An alternative optimal solution to a minimization transportation problem exists whenever opportunity cost corresponding to unused route of transportation is Positive with at least one equal to zero.

One disadvantage of using North-West Corner rule to find initial solution to the transportation problem is that It does not take into account cost of transportation. North-west is an easy way to find a solution and works efficiently most of the time. But there isn't any superority among these methods. Basically, it depends on the problem.

The solution to a transportation problem with ‘m’ rows (supplies) & ‘n’ columns (destination) is feasible if number of positive allocations are m+n-1.

If an opportunity cost value is used for an unused cell to test optimality, it should be most negative number. Opportunity costs represent the benefits an individual, investor or business misses out on when choosing one alternative over another.

During an iteration while moving from one solution to the next, degeneracy may occur when two or more occupied cells on the closed path with minus sign are tied for lowest circled value

The large negative opportunity cost value in an unused cell in a transportation table is chosen to improve the current solution because it represents per unit cost reduction.

The smallest quantity is chosen at the corners of the closed path with negative sign to be assigned at unused cell because it ensure feasible solution.

The total supply available at the plants exactly matches the total demand at the destinations, such type of problems where supply and demand are exactly equal are known as Balanced Transportation Problem.

Modified distribution method (MODI) is used to verify the optimality of the current solution of the transportation problem.

The degeneracy in the transportation problem indicates that the multiple optimal solution exist. transportation problem with m origins and n destinations has fewer than m + n – 1 positive xij (occupied cells), the problem is said to be a degenerate transportation problem.

An assignment problem is considered as a particular case of a transportation problem because the number of rows equals columns, All xij = 0 or 1 and All rim conditions are 1.

An optimal assignment requires that the maximum number of lines that can be drawn through squares with zero opportunity cost be equal to the number of rows equal to number of columns.

While solving an assignment problem, an activity is assigned to a resource through a square with zero opportunity cost because the objective is to minimize total cost of assignment. The objective of assignment problem is to assign a number of jobs to an equal number of machines so as to minimize the total assignment cost or to minimize the total consumed time for execution of all the jobs. In the multi-objective assignment problem, the objectives alone are considered as fuzzy.

The method used for solving an assignment problem is called Hungarian method. The Hungarian method is a combinatorial optimization algorithm that solves the assignment problem in polynomial time and which anticipated later primal-dual methods.

The purpose of a dummy row or column in an assignment problem is to obtain balance between total activities & total resources. Dummy rows (or columns) are added in the matrix so as to complete it to form a square matrix. The dummy rows or columns will contain all costs elements as zeroes. The Hungarian method may be used to solve the problem.

If there were n workers & n jobs there would be n! solutions.

An assignment problem can be solved by Simplex method and Transportation method. The simplex method is a method for solving problems in linear programming. This method, invented by George Dantzig in 1947, tests adjacent vertices of the feasible set (which is a polytope) in sequence so that at each new vertex the objective function improves or is unchanged. The Transportation Method of linear programming is applied to the problems related to the study of the efficient transportation routes i.e. how efficiently the product from different sources of production is transported to the different destinations, such as the total transportation cost is minimum.

For a salesman who has to visit n cities (n-1)! is the way of his tour plan.

The assignment problem requires that only one activity be assigned to each resource, is a special case of transportation problem and can be used to maximize resources.

An assignment problem is a special case of transportation problem, where Number of rows equals number of columns, all rim conditions are 1and values of each decision variable is either 0 or 1.

Every basic feasible solution of a general assignment problem, having a square pay-off matrix of order, n should have assignments equal to 2n-1.

To proceed with the MODI algorithm for solving an assignment problem, the number of dummy allocations need to be added are n-1. The Modified Distribution Method or MODI is an efficient method of checking the optimality of the initial feasible solution.

The Hungarian method for solving an assignment problem can also be used to solve a travelling salesman problem. The Hungarian method is a combinatorial optimization algorithm that solves the assignment problem in polynomial time and which anticipated later primal-dual methods.

If the number of assigned cells is equal to the number of rows/columns, then it is an optimal solution.

The objective functions and constraints are linear relationship between variables. This means that no variables can be multiplied with each other. This formulation is called the Standard form.

Assignment problem helps to find a maximum weight identical in nature in a weighted Bipartite graph. The assignment problem is a well-known graph optimization problem defined on weighted-bipartite graphs. The objective of the standard assignment problem is to maximize the summation of the weights of the matched edges of the bipartite graph. In the standard assignment problem, any node in one partition can be matched with any node in the other partition without any restriction. In this paper, variations of the standard assignment problem are defined with matching constraints by introducing structures in the partitions of the bipartite graph, and by defining constraints on these structures.

All the parameters in the linear programming model are assumed to be Constraints. For a problem to be a linear programming problem, the decision variables, objective function and constraints all have to be linear functions. If the all the three conditions are satisfied, it is called a Linear Programming Problem.

The solution need not be in whole numbers.

Graphic method can be applied to solve a LPP when there are only two variable. This method is used to solve a two variable linear program. If you have only two decision variables, you should use the graphical method to find the optimal solution. A graphical method involves formulating a set of linear inequalities subject to the constraints. Then the inequalities are plotted on a X-Y plane.

If the feasible region of a LPP is empty, the solution is infeasible. A linear program is infeasible if there exists no solution that satisfies all of the constraints -- in other words, if no feasible solution can be constructed. Since any real operation that you are modelling must remain within the constraints of reality, infeasibility most often indicates an error of some kind. Simplex-based LP software like lp_solve efficiently detects when no feasible solution is possible.

The variables whose coefficient vectors are unit vectors are called basic variables. The basic variables can be defined as the m variables which can take any value other than zero. Moreover, if the variables satisfy the non-negativity condition of the LP model, the basic solution created by them is called the basic feasible solution. The remaining variables are known as the non-basic variables.