İŞLETME KARAR MODELLERİ

The foundation of operations research is mostly cited to studies and applications in the U.S. Army right before The World War II. However, Leonid V. Kantorovich is recognized as the first to explore this area. In 1939, Kantorovich published a paper on a method for a production plan which he claimed to be the optimal way. This study included a linear program without a systematic solution. The systematic solution which is called “simplex algorithm” was later introduced by George B. Dantzig, who had planning experience in the U.S. Army Air Force. Tjalling C. Koopmans, who built a model for ship routes, was another contributor to the field. He and Kantorovich shared 1975 Nobel Prize in economics for their studies on the optimal allocation of resources. Industrial applications of linear programming stemmed from the oil industry after William W. Cooper and his colleagues’ publications about the efficient blending of aviation fuels. As the computational ability of the solver systems advanced rapidly, operations research started to be used widely. Operations research applications, particularly linear programming, is now a standard method used in a variety of areas and processes. Some of these are urban planning, currency arbitrage, investment, production planning, inventory control, blending and refining, manpower planning, agricultural planning, diet & nutrition, transportation and, logistics.

As also understood from the information given, the correct answer is E. All of the statements above about the historical development of operations research are correct.

Operations research is an analytical method for the management of organizations that provides mathematical solutions to decision-making problems. There are definitive steps of this approach that make it analytical:

As also understood from the information given, the correct answer is A.

In practice, linear programming has some limitations that deteriorate the model fitness to real-life situations. There is a trade-off between suitability and solvability of the problem. Linear programming provides solutions under some assumptions. These are proportionality, additivity, divisibility and certainty assumptions. As also understood from the information given, the correct answer is E. These are important to mention as it will be easier for operations researcher to evaluate how well linear programming applies to a given problem. In both the objective function and constraints, the contribution of every decision variable is proportional to its value. As a result, proportionality assumption casts out any variable that has an exponent other than 1. Consider a firm that aims to maximize its profits under the constraint of its production costs. In practice, production costs are not the same for different levels of production as higher production lowers costs by the economies of scale. To avoid a violation of the proportionality assumption, some characteristics of real-life situations such as economies of scale must be ignored. That undoubtedly makes the model less reliable. Either it is the objective or the constraints, every function is the sum of the individual contribution of the respective variables. This is called the additivity assumption, which prohibits cross-product terms in the expressions of a linear program. This assumption restricts the model design as well. For instance, a linear maximization model of aggregate revenue to be received from complementary products has to exclude cross-product terms that proxy the multiplier effect of each product over the other. In a linear program, decision variables are not limited to integer values. In fact, they are divisible. In certain situations, the divisibility assumption does not hold as some decision variables can only be an integer. Methods of integer programming, which are not in the scope of this book, overcome this obstacle. The last assumption is certainty, which means that each parameter of a linear programming model is constant in all conditions. In real applications, factors affecting the decisions are not as stable as they are strictly assumed. In the modeling phase, all these assumptions need to be considered while preserving an acceptable level of similarity between reality and what is realized.

The modeling phase is the transformation of verbal description of the problem into a mathematical model under the linear programming assumptions. This model is a system of equations and inequalities representing the objective and the constraints. The decision to be derived from the model is related to quantifiable variables that are in control. These variables of decision x₁, x₂, ... , x_n and the constants (i.e. the coefficients and right-hand side values of the inequalities) are called the decision variables and the parameters respectively. The objective function (for example, Z= 2x₁ + x₂) is an appropriate measure of performance for the problem. It can be maximized or minimized depending on the desired achievement. Finally, a constraint is a restriction on decision variables by means of equality or inequality (for example, 5x₁ + 6x₂ ≤ 30). As also understood from the information given, the statements in the options I, II and III are correct, so the correct answer is C. The statements in the option IV and V are not correct because of the fact that These variables of decision x₁, x₂, ... , x_n and the constants (i.e. the coefficients and right-hand side values of the inequalities) are called the decision variables and the parameters respectively.

In the maximization models, the objective function is alternatively named profit function. The logic behind this is the utmost goal of maximizing something is to gain a benefit or a “profit”. In the maximization models the value of Z needs to be increased as much as possible. An infinite number of iso-profit lines can be drawn for increasing values of Z. However, under the constraints of the model, the highest possible value of Z must reside in the feasible region.

As also understood from the information given, the statements in the options I, II and IV about the maximization models are correct, so the correct answer is D. The statement in the option III is not correct because of the fact that an infinite number of iso-profit lines can be drawn for increasing values of Z. However, under the constraints of the model, the highest possible value of Z must reside in the feasible region.

In the minimization models, the objective function is alternatively named cost function. The reason for this is the ultimate goal of minimizing something is to reduce loss or “cost”. In the minimization models, draw a line parallel to the objective function and near to the origin. This line should contain at least one point of the feasible region. Find the coordinates of this point by solving the equations of the drawn line and the boundary line(s) of the feasible region.

As also understood from the information given, the statements about the minimization models in the options II, III, IV and V are correct, so the correct answer is E. The statement in the option I is not correct because of the fact that in the minimization models, the objective function is alternatively named cost function.

Operations research applications, particularly linear programming is now a standard method used in a variety of areas and processes. Some of these are urban planning, currency arbitrage, investment, production planning, inventory control, blending and refining, manpower planning, agricultural planning, diet & nutrition, transportation and, logistics.

As also understood from the information given, all of the areas in the options are among the ones in which operations research applications, particularly linear programming, are now used as a standard method for decision making, so the correct answer is E.

The foundation of operations research is mostly cited to studies and applications in the U.S. Army right before The World War II. However, Leonid V. Kantorovich is recognized as the first to explore this area. In 1939, Kantorovich published a paper on a method for a production plan which he claimed to be the optimal way. This study included a linear program without a systematic solution. The systematic solution which is called “simplex algorithm” was later introduced by George B. Dantzig, who had planning experience in the U.S. Army Air Force.

As also understood from the information given, the correct answer is B.

Information about the other names in the options:

Tjalling C. Koopmans, who built a model for ship routes, was another contributor to the field. He and Kantorovich shared 1975 Nobel Prize in economics for their studies on the optimal allocation of resources.

Industrial applications of linear programming Stemmed from the oil industry after William W. Cooper and his colleagues’ publications about the efficient blending of aviation fuels. As the computational ability of the solver systems advanced rapidly, operations research started to be used widely.

Sometime in the midst of the last century, Chester Barnard, a retired telephone executive and author of The Functions of the Executive, imported the term “decision making” from the lexicon of public administration into the business world.

A linear expression appears below; the powers of its variables x and y are equal to one.

2x – 3y

A linear constraint requires a linear expression to be equal to a number, to be less/greater than or equal to a number.

x + y = 4

x – 3y ≤ 2

x ≥ y

The last expression stated above is called a non-negativity constraint, which requires the value x to be greater or equal to zero. A linear program either minimizes or maximizes a linear expression subject to defined linear constraints.

As also understood from the information given, the correct answer is E. All expressions in the options are non-negativity constraints.

An objective function is the function of a linear programming model that is desired to be maximized or minimized. As also understood from the information given, the correct answer is A.

Information related to the other terms in the options as follows:

An optimal solution is a feasible solution that reaches the most favorable value of the objective function.

A mathematical model is an abstraction of the analyzed system using mathematical expressions.

Linear models consist of constraints and an objective in the form of a linear expression.

A non-negativity constraint requires the value x to be greater or equal to zero.

A linear expression appears below; the powers of its variables x and y are equal to one.

2x – 3y

A linear constraint requires a linear expression to be equal to a number, to be less/greater than or equal to a number.

x + y = 4 

x – 3y ≤ 2 

x ≥ y  . 

The correct answer is B.

Operations research is an analytical method for the management of organizations that provides mathematical solutions to decision-making problems. There are definitive steps of this approach that make it analytical: 1. Problem definition 2. System analysis for the problem and data gathering 3. Model construction 4. Model solution and testing 5. The decision of implementing the solution. So, the correct answer is D.

In both the objective function and constraints, the contribution of every decision variable is proportional to its value. As a result, proportionality assumption casts out any variable that has an exponent other than 1. Consider a firm that aims to maximize its profits under the constraint of its production costs. In practice, production costs are not the same for different levels of production as higher production lowers costs by the economies of scale. To avoid a violation of the proportionality assumption, some characteristics of real-life situations such as economies of scale must be ignored. That undoubtedly makes the model less reliable. The correct answer is A.

The last assumption is certainty, which means that each parameter of a linear programming model is constant in all conditions. In real applications, factors affecting the decisions are not as stable as they are strictly assumed. In the modeling phase, all these assumptions need to be considered while preserving an acceptable level of similarity between reality and what is realized. The correct answer is C.

The problem is related to the number of workers and therefore it is about manpower planning. The correct answer is A.

The decision variables are The quantities, x1 for bonds and x2 for stocks. The objective function seeks to maximize total yields of the holdings and is thus expressed as Maximize Z = (5 × 0.25)x1 + (10 × 0.50)x2 A constraint of the model reflects the liquidity requirements at the end of the year x1 + 0.8x2 ≥ 80 Another constraint is about the size of the portfolio and related to prices of the securities 5x1 + 10x2 ≤ 1000 Percentage limits of the funds to be invested in stocks should not be less than 20% and more than 40%. That means the money spent on stocks cannot exceed the %40 of the total amount of money spent on both types of security. On the other side, the money spent on stocks cannot drop down under %20. These constraints are 10x2 ≤ 0.40 (5x1 + 10x2 ) 10x2 ≥ 0.20 (5x1 + 10x2 ) Simplified expressions of these are 6x2 – 2x1 ≤ 0 and 8x2 – x1 ≥ 0. The correct answer is B.

In the maximization models, the objective function is alternatively named profit function. The logic behind this is the utmost goal of maximizing something is to gain a benefit or a “profit”. The correct answer is D.

In the minimization models, the objective function is alternatively named cost function. The reason for this is the ultimate goal of minimizing something is to reduce loss or “cost”. The correct answer is B.

In the minimization models, draw a line parallel to the objective function and near to the origin. This line should contain at least one point of the feasible region. Find the coordinates of this point by solving the equations of the drawn line and the boundary line(s) of the feasible region. The correct answer is B.

Reality Linear programming is only effective if the model you use reflects the real world. Every model relies on certain assumptions and they may be invalid: you assume, for example, that tripling production will triple sales, but in reality it saturates the market. Linear equations sometimes give results that don’t make sense in the real world, such as a result indicating that you should contract to build 23.75 battleships for the Navy to maximize profits how will you deal with the 0.75 in practical terms? Skilled linear programmers can tweak models and equations to deal with these problems, however. The correct answer is C.