In a decision problem, the decision maker is aware of various possible events (states of nature) but has insufficient information to assign any probabilities of occurrence to them. This kind of a decision problem is termed as decision making under uncertainty. There are many unknowns and no possibility of knowing what could occur in the future to change the outcome of a decision in decision making under uncertainty.
For firms launching a new product, an important change in marketing strategy or opening first branch can be given examples of decision making under uncertainty.
In the context of the above explanations, the correct answer is C.

The Equally Likely Method indicates that all events occur equally likely if the probability of occurrence of events is unknown. Therefore, calculation is made considering that the event’s possibility occur equally. The final decision is made according to the objective (maximum or minimum) of the problem.

The correct answer is E as shown below:
The probability of occurrence of each event is 1/3.
A1 = 4*(1/3)+7*(1/3)+5*(1/3) = 5.33
A2 = 5*(1/3)+3*(1/3)+8*(1/3) = 5.33
A3 = 1*(1/3)+9*(1/3)+7*(1/3) = 5.67

If decision maker has an optimistic approach, Criterion of Optimism method can be used. In this context, if the problem is the minimization, the best outcome is the lowest value. In this case, the alternative which is the smallest value is chosen. Accordingly, the correct answer is B (vending machine 2) with the smallest value (as seen in table above; 1).

In application of the Criterion of Optimism, when the objective function is maximization, the maximum values for the alternatives are determined. The highest of these maximum values is selected.

If decision maker has a pessimistic approach, the Criterion of Pessimism can be used. The decision maker thinks that the worst outcomes for each alternative will be realized.

In application of to the Criterion of Pessimism, if the problem is maximization, the worst outcome is the lowest value. However, the best alternative (maximum value) of these worst outcomes is chosen. According to this;
The minimum value for A1 = 3
The minimum value for A2 = 5
The minimum value for A3 = 3
The minimum value for A4 = 7
The minimum value for A5 = 2
Among these values, the highest is 7, and this belongs to A4.

The decision maker is a little optimistic, a little pessimistic and the method that involves both cases is called Criterion of Realism.

First, the minimum values are determined for each alternative. These values are multiplied by the α coefficient. Then the maximum values are determined for each alternative. These values are multiplied by the (1- α) coefficient. These values of each alternative are sum up.
Among these values, the lowest is 4.80, and this belongs to A1.
A1 = 2*0,6 + 9*0,4 = 4,80

In the Minimax Regret criterion, regardless of the objective function, the smallest value is always selected from the regret matrix.

In equally likely method calculation is made considering that the event’s possibility occur equally.

In criterion of optimism (plunger), whether the objective of problem is maximization or minimization, the best outcome for each alternative is realized and decision maker select the best among these alternatives.

In criterion of pessimism, the decision maker thinks that the worst outcomes for each alternative will be realized.

Criterion of realism method allows the decision
maker to use optimistic and pessimistic approaches together.

Minimax regret is defined as the difference between the optimal and the actual pay-off.

A3 with 14 is the most appropriate decision according to equally likely method.

A3 with 9,33 is the most appropriate decision according criterion of realism.

When the decision maker does not know
the probabilities of occurrence of events, the
decision-making is called “decision making
under uncertainty”. In this sense, the correct answer is B.

There are some different methods used in decision making under uncertainty.
-Equally Likely (Laplace)
-Criterion of Optimism (Plunger)
-Criterion of Pessimism (Wald)
-Criterion of Realism (Hurwicz)
-Minimax Regret (Savage)
In this sense, the correct answer is option E.

The correct answer is given in option A. Because if decision maker has an optimistic approach, this method can be used. In this method, whether the objective of problem is maximization or minimization, the best outcome for each alternative is realized and decision maker select the best among these alternatives. If the problem is the minimization, the best outcome is the lowest value. In this case, the alternative which is the smallest value is chosen. The mathematical representation of the method is as follows:

The decision maker is a little optimistic, a little pessimistic and the method that involves both cases is called Criterion of Realism. This method allows the decision maker to use these two approaches together. So, the correct answer is A.

There are four events in the problem. So, the probability of occurrence of each event is 1/4.
District 1 = 10.000*1/4 + 12.500*1/4 + 15.000*1/4 + 16.500*1/4 = 13.500 TL
District 2 = 15.000*1/4 + 17.000*1/4 + 18.000*1/4 + 24.500*1/4 = 18.625 TL
District 3 = 12.000*1/4 + 13.000*1/4 + 20.000*1/4 + 32.000*1/4 = 19.250 TL
District 4 = 21.000*1/4 + 25.000*1/4 + 27.000*1/4 + 40.000*1/4 = 28.250 TL
Since the objective of this decision problem is maximization, decision maker chooses the highest value among the alternatives. The decision maker selects district 4 with 28.250 TL.

First, the maximum values are determined for each alternative. Then, the highest value (40.000 TL) and the alternative for that value (District 4) is selected. The correct answer is D.

The alpha coefficient is determined by the decision maker as 0.3. Then 1-alpha becomes 0.7. In this case, the decision maker is optimistic with a probability of 30%, and pessimistic with a probability of 70%.
First, the minimum values are determined for each alternative. These values are multiplied by the α coefficient. Then the maximum values are determined for each alternative. These values are multiplied by the (1- α) coefficient. These values of each alternative are sum up.
Product 1 = 3*0.3 + 12*0.7 = 9.3 days
Product 2 = 1*0.3 + 18*0.7 = 12.9 days
Product 3 = 5*0.3 + 15*0.7 =12 days
Product 4 = 4*0.3 + 11*0.7 = 8.9 days
Product 5 = 2*0.3 + 10*0.7 = 7.6 days
Then, because the objective is minimization the lowest value (7.6 days) and the alternative for that value (Product 5) is selected.

The method used for the loss of opportunities when the best alternative is not selected is “Minimax Regret”. This method is based on opportunity loss, also called regret. In this sense the correct answer is option B.

This method is based on opportunity loss, also called regret. It is defined as the difference between the optimal and the actual pay-off. Regret is the amount lost when the best alternative is not selected.
This method proposed that maximum regret be minimized by choosing the best pay off for each event. The regret of each event for each alternative is calculated and regret matrix or opportunity loss table is created. This table shows the losses to be incurred if the alternative is not selected for the best outcomes of each event.
If the objective is maximization, the maximum value of each event is determined. All values of the relevant event are subtracted from this maximum value. If the objective is minimization, the minimum value of each event is determined. The smallest value of each event is subtracted from all values of relevant event. In this way, the regret matrix is obtained.
After the regret matrix has been constructed, the maximum opportunity loss (regret) for each alternative is located. Then the alternative with the smallest value
among these maximum regrets selected.
In this method, regardless of the objective function, the smallest value is always selected from the regret matrix.
According to these information, the correct answer is given in the option C.

For the values given in the Table, a regret matrix is created. Because the objective is
minimization, the minimum value of each event is selected. The smallest value of each event is subtracted from all other values in relevant column. Thus, a regret matrix is created. This table shows the regret matrix of the problem.
According to the results in Table, maximum regret of each alternatives are as follows:
Product 1 = maximum regret = 5 days
Product 2 = maximum regret = 8 days
Product 3 = maximum regret = 6 days
Product 4 = maximum regret = 4 days
Product 5 = maximum regret = 3 days
The alternative with the smallest value (product 5) is chosen among the maximum regrets. The correct answer is E.

Equally Likely (Laplace) method indicates that all events occur equally likely if the probability of occurrence of events is unknown. Therefore, calculation is made considering that the event’s possibility occur equally. The final decision is made according to the objective (maximum or minimum) of the problem.

Equally likely method indicates that all events occur equally likely if the probability of occurrence of events is unknown. The mathematical representation of the method is as follows: n: number of events related to the problem. Possibility of occurrence of each event = P (O_j ) = 1/n , j = 1, 2, 3 …n.

In a decision problem, the decision maker is aware of various possible events but has insufficient information to assign any probabilities of occurrence to them. This kind of a decision problem is termed as decision making under uncertainty. There are many unknowns and no possibility of knowing what could occur in the future to change the outcome of a decision in decision making under uncertainty. For firms launching a new product, an important change in marketing strategy or opening first branch can be given examples of decision making under uncertainty.

Generally, five different methods can be used for decision making under uncertainty. These methods are: Equally Likely (Laplace), Criterion of Optimism (Plunger), Criterion of Pessimism (Wald), Criterion of Realism (Hurwicz), Minimax Regret (Savage).

If decision maker has a pessimistic approach, this method can be used. The decision maker thinks that the worst outcomes for each alternative will be realized. However, when choosing from alternatives, it selects the best alternative for the objective of the problem. If the problem is maximization, the worst outcome is the lowest value. However, the best alternative (maximum value) of these worst outcomes is chosen.

Criterion of realism allows the decision maker to use both criterion of pessimism and criterion of realism approaches together. There is a coefficient indicating the level of optimism of the decision maker. This coefficient is symbolized by alpha (α). When decision maker is pessimistic, the 1- α coefficient is used. If the problem is minimization, the best value is the smallest value and this value is multiplied by alpha coefficient. The worst value is the highest value and this value is multiplied by 1-α coefficient. Then, these values are sum up. Finally, the alternative with the best value (minimum value) is selected. The mathematical representation of the method is as follows: Min_i {α*Min (0_ij)_j + (1 - α)*Max (0_ij)_j}.

In the criterion of realism approach there is a coefficient indicating the level of optimism of the decision maker. This coefficient is symbolized by alpha (α). When decision maker is pessimistic, the 1- α coefficient is used. When the objective function is maximization, the highest values of the alternatives are multiplied by alpha and the smallest values of the alternatives are multiplied by 1- α.

The Minimax Regret (Savage) method is based on opportunity loss, also called regret. It is defined as the difference between the optimal and the actual pay-off. Regret is the amount lost when the best alternative is not selected.

Criterion of realism allows the decision maker to use optimism and pessimism approaches together. There is a coefficient indicating the level of optimism of the decision maker. This coefficient is symbolized by alpha (α). When decision maker is pessimistic, the 1- α coefficient is used. The alpha is between 0 and 1. When the alpha is equal to 1, the decision maker is optimistic. When it is equal to 0, the decision maker is pessimistic. The alpha coefficient is determined by the decision maker.

Minimax regret method is based on opportunity loss, also called regret. It is defined as the difference between the optimal and the actual pay-off. Regret is the amount lost when the best alternative is not selected.

Equally Likely (Laplace) method indicates that all events occur equally likely if the probability of occurrence of events is unknown. Therefore, calculation is made considering that the event’s possibility occur equally. The final decision is made according to the objective (maximum or minimum) of the problem.

There are three events in the problem. So, the probability of occurrence of each event is 1/3.
Product 1 = 3*1/3 + 8*1/3 + 12*1/3 = 23/3 days
Product 2 = 1*1/3 + 3*1/3 + 18*1/3 = 22/3 days
Product 3 = 5*1/3 + 9*1/3 + 15*1/3 = 29/3 days
Product 4 = 4*1/3 + 7*1/3 + 11*1/3 = 22/3 days
Product 5 = 2*1/3 + 6*1/3 + 10*1/3 = 18/3 days
Since the objective of this decision problem is minimization, decision maker chooses the lowest value among the alternatives. The decision maker selects product 5 with 18/3 (6 days).

When the objective function is minimization, the maximum values for the alternatives are determined. The smallest of these maximum values is selected. When the objective function is maximization, the minimum values for the alternatives are determined. The highest of these minimum values is selected.

This method is based on opportunity loss, also called regret. It is defined as the difference between the optimal and the actual pay-off. Regret is the amount lost when the best alternative is not selected. This method proposed that maximum regret be minimized by choosing the best pay off for each event. The regret of each event for each alternative is calculated and regret matrix or opportunity loss table is created. This table shows the losses to be incurred if the alternative is not selected for the best outcomes of each event.

Product 1 = 3*0.3 + 12*0.7 = 9.3 days
Product 2 = 1*0.3 + 18*0.7 = 12.9 days
Product 3 = 5*0.3 + 15*0.7 =12 days
Product 4 = 4*0.3 + 11*0.7 = 8.9 days
Product 5 = 2*0.3 + 10*0.7 = 7.6 days
Then, because the objective is minimization the lowest value (7.6 days) and the alternative for that value (Product 5) is selected.

This method proposed that maximum regret be minimized by choosing the best pay off for each event. The regret of each event for each alternative is calculated and regret matrix or opportunity loss table is created. This table shows the losses to be incurred if the alternative is not selected for the best outcomes of each event.

According to the results, maximum regret of each alternatives are as follows:
District 1 = maximum regret = 23.500
District 2 = maximum regret = 15.500
District 3 = maximum regret = 12.000
District 4 = maximum regret = 0
The alternative with the smallest value (District 4) is chosen among the maximum regrets.

According to the results, maximum regret of each alternatives are as follows:
Product 1 = maximum regret = 5 days
Product 2 = maximum regret = 8 days
Product 3 = maximum regret = 6 days
Product 4 = maximum regret = 4 days
Product 5 = maximum regret = 3 days
The alternative with the smallest value (product 5) is chosen among the maximum regrets.

For firms launching a new product, an important change in marketing strategy or opening first branch can be given examples of decision making under uncertainty. These decisions could be influenced by such factors as the reaction of competitors, changes in customer demand, technological changes, economic shifts, government regulations and the situations beyond the control of decision makers.

In a decision problem, the decision maker is aware of various possible events (states of nature) but has insufficient information to assign any probabilities of occurrence to them. This kind of a decision problem is termed as decision making under uncertainty. There are many unknowns and no possibility of knowing what could occur in the future to change the outcome of a decision in decision making under uncertainty.

Decision situations in which the chance of occurrence of each state of nature is known or can be estimated are defined as decisions made under risk. Such decision-making problems are also known as probabilistic or stochastic decision problems.

In the decision making process under risk, a probability value must be determined for the chance of realization of all states of nature in the strategy table and the sum of all probabilities of states of nature must be equal to 1. After these possibilities are added to the strategy table, the decision maker should determine which approach will be used to select the best course of action. These approaches are Expected Value (EV) (or Expected Monetary Value (EMV)), Expected Opportunity Loss (EOL), and the Maximum Probability which will be explained in the following section.

In decision making at risk, the expected value criterion is commonly used when comparing alternatives. And when the decision’s consequences involve only money, we can calculate the Expected Monetary Value (EMV).

In decision making, instead of expected monetary value (EMV), it can also be used the expected opportunity loss (EOL) criterion, which is another approach based on a decision will be taken, also known as expected regret.

In this criterion, the decision maker confronts the various possible states of nature in a decision under risk and; he or she chooses the alternative that is best for the most likely state of nature, rather than calculating in all states of nature. With another saying, it states that the decision maker should ignore all possible events except the one most likely to occur, and should select the best possible result (maximum gain or minimum loss) in the given circumstances.

The decision tree is a graphical technique that represents all the elements in the decision problem with various geometric symbols. It provides all the elements and details of the decision problem, as well as graphically, and performs the expected value calculations on the tree and provides the solution of the problem at the same time.

The expected value of perfect information (EVPI) is the difference between the expected value under certainty and the expected value under risk.

The decision node, represented in a square shape on the decision tree is a point from which two or more branches emerge. Each branch from a decision node represents a possible alternative to be chosen by the decision-maker.

Bayes’ theorem provides a way for combining prior probabilities with probabilities obtained by other sources; revised or posterior probabilities.

In decision making under risk, there exist more than one state of nature and the decision maker has sufficient information to assign probabilities to the occurrence of each of these states. The probabilities can be obtained from past records or the subjective judgment of the decision maker. In other word, decision situations in which the chance of occurrence of each state of nature is known or can be estimated are defined as decisions made under risk.

The decision situations in which the chance of occurrence of each state of nature is known or can be estimated are called decision making under risk.

The probability values change from 0 to 1, where 0 means the event occurrence probability is low and 1 means this probability is high.

An anticipated value for a given investment at some point in the future is called expected value.

Expected opportunity loss, demonstrate the average additional amount the investor would have achieved by making the right decision instead of a wrong one.

In maximum probability criterion, the decision maker confronts the various possible states of nature in a decision under risk and; he or she chooses the alternative that is best for the most likely state of nature, rather than calculating in all states of nature.

The line connecting the nodes on a decision tree is called a branch.

Since the decision tree is usually initiated by the first decision, a decision node positioned at the left side of the decision tree is also the starting node.

A chance node, represented in a circle shape on the decision tree, indicates that one of a finite number of states of nature is expected to occur at this point in the process.

In decision analysis payo refers to the consequence resulting from a specific combination of a decision alternative and a state of nature.

Other options can be found in payoff section of a decision tree.

In a decision problem, occurrences are chance occurrences and all the chance occurrences are governed by probabilities. In the decision making with probability information, a strategy matrix is created and the best results are determined. Another approach used to solve such decision problems is the decision tree. A decision tree is a graphical approach that helps decision-makers see all the scenarios they may encounter. The correct answer is A.

han one state of nature and the decision maker has sufficient information to assign probabilities to the occurrence of each of these states. The probabilities can be obtained from past records or the subjective judgment of the decision maker. In other word, decision situations in which the chance of occurrence of each state of nature is known or can be estimated are defined as decisions made under risk. Such decision-making problems are also known as probabilistic or stochastic decision problems. The correct answer is D.

As known, the probability values change from 0 to 1, where 0 means the event occurrence probability is low and 1 means this probability is high. The process of determining probabilities for states of nature is very important. In the decision making process under risk, a probability value must be determined for the chance of realization of all states of nature in the strategy table and the sum of all probabilities of states of nature must be equal to 1. After these possibilities are added to the strategy table, the decision maker should determine which approach will be used to select the best course of action. The correct answer is B.

Expected (Monetary) Value criterion (EMV), Expected Opportunity Loss criterion (EOL), and the maximum probability criterion are the criterions in decision theory. The correct answer is D.

The Expected Value (EV) is an anticipated value for a given investment at some point in the future. In decision making at risk, the expected value criterion is commonly used when comparing alternatives, based on maximizing expected profit or minimizing expected loss. The expected value criterion attempts to find the expected profit maximized or the expected cost minimized. The profit or cost that will arise from each alternative is handled by certain possibilities. When the decision’s consequences involve only money, we can calculate the Expected Monetary Value (EMV). The correct answer is B.

In this criterion, the decision maker confronts the various possible states of nature in a decision under risk and; he or she chooses the alternative that is best for the most likely state of nature, rather than calculating in all states of nature. With another saying, it states that the decision maker should ignore all possible events except the one most likely to occur, and should select the best possible result (maximum gain or minimum loss) in the given circumstances. The correct answer is D.

A decision tree is composed of some components as; branches, decision nodes, chance nodes, and payoffs. The correct answer is E.

The line connecting the nodes on a decision tree is called a branch. A branch is a single strategy that connects either two nodes or a node and an outcome. When a decision tree is drawn, a general approach is the direction from left to right is shown; therefore the line that comes to the right of a decision node is called the decision branch, while the line to the right of a chance node is called the chance branch. In decision making problems, decision branches are used to represent alternatives (strategies) and chance branches are used to show the states of nature (events). The chance branch is labeled with a probability which represents the decision maker’s estimate of the probability that a particular branch will be followed. The correct answer is C.

Bayes’ theorem is basically a process of revising the known possibilities of an event under the light of new knowledge. The correct answer is B.

Payoff: In decision analysis payoff refers to the consequence resulting from a specific combination of a decision alternative and a state of nature. Payoffs can be expressed in terms of profit, cost, time, distance, or any other measure appropriate for the decision problem being analyzed. The correct answer is C.

In decision making under risk, there exist more than one state of nature and the decision maker has sufficient information to assign probabilities to the occurrence of each of these states. Such decision-making is also known as probabilistic or stochastic decision making.

In the decision making process under risk, a probability value must be determined for the chance of realization of all states of nature in the strategy table and the sum of all probabilities of states of nature must be equal to 1. After these possibilities are added to the strategy table, the decision maker should determine which approach will be used to select the best course of action. These approaches are Expected Value (EV) (or Expected Monetary Value (EMV)), Expected Opportunity Loss (EOL), and the Maximum Probability.

As known, the probability values change from 0 to 1, where 0 means the event occurrence probability is low and 1 means this probability is high. The process of determining probabilities for states of nature is very important. In the decision making process under risk, a probability value must be determined for the chance of realization of all states of nature in the strategy table and the sum of all probabilities of states of nature must be equal to 1.

In maximum probability criterion, the decision maker confronts the various possible states of nature in a decision under risk and; he or she chooses the alternative that is best for the most likely state of nature, rather than calculating in all states of nature. With another saying, it states that the decision maker should ignore all possible events except the one most likely to occur, and should select the best possible result (maximum gain or minimum loss) in the given circumstances.

While maximum probability criterion has the advantage of simplicity, ignoring substantive information related to the less likely states of nature makes it a weaker decision making criterion. Because it does not mean that the state of nature which has a low probability will not occur. On the contrary, it indicates that there is a possibility of occurrence of the event, but it has a low chance of occurring according to other alternatives.

The expected value of perfect information is used to place an upper limit on what you should pay for information that will aid in making a better decision. That is if we have perfect information about a state of nature before the decision is made, how much is this information worth? The expected value of perfect information (EVPI) is the difference between the expected value under certainty and the expected value under risk.

EVPI = EVwPI - EVwoPI, where EVwPI stands for expected value with perfect information and EVwoPI stands for expected value without perfect information. So EVPI = 569 - 183 = 386.

A chance node, represented in a circle shape on the decision tree, indicates that one of a finite number of states of nature is expected to occur at this point in the process. The states of nature are shown on the tree as branches to the right of the chance nodes and the assumed probabilities of the states of nature are written above the branches.

The probability multiplication in the denominator of Bayes rule equality is called joint probability. In order to obtain any multiplication result in the denominator, the researcher needs to know what state of nature has emerged, as well as which B result is reached with this state of nature. The joint probabilities calculated in the denominator, correspond to each scenario mentioned in decision tree definitions in decision making problems.

With the formula used in Bayes’s theorem, while the result of a given event is certain, the possibility of the causes of this result is investigated. From the other perspective, in Bayes’ formula, the cause and result are displaced.

Recommended Corrections:
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Industrial applications of linear programming Stemmed from the oil industry...
Industrial applications of linear programming stemmed from the oil industry...
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This section develops The basics of linear programming, which include …
This section develops the basics of linear programming, which include …
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However, LeonidV. Kantorovich…
However, Leonid V. Kantorovich…
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:

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

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.

LINEAR PROGRAMMING BASICS
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
The correct answer is E.

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
The correct answer is C.

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. The correct answer is E.

An objective function is the function of a linear programming model that is desired to be maximized or minimized.
The correct answer is D.