Companies are constantly concerned with making decisions that allow them to optimize their results so that the greatest benefit is obtained. This is why Operations Research is an important tool for management when making any decision. It is therefore necessary to apply quantitative methods that support decision-making, in order to eliminate the strong degree of empiricism that Cuban managers present when making decisions.
The objective of this article is to apply an economic inventory model to respond to an existing problem in a production company that facilitates and improves decision-making. With the application of this tool, a proposed solution to a real insufficiency detected is decided.
Keywords: decision making, operations research, inventories.
decision-making-inventories-operations-investigationIntroduction.
In today's highly competitive and complex socioeconomic environment, traditional decision-making methods have become relatively ineffective as those responsible for directing the activities of companies and institutions face complicated and dynamic situations that require creative and practical solutions. supported by a solid quantitative basis.
Companies are constantly concerned with making decisions that allow them to optimize their results so that the greatest benefit is obtained. The palpable difficulty of making these decisions has led man to search for a tool or method that allows him to make the best decisions according to the available resources and the objectives he pursues.
Business decision tools such as mathematical models have been applied to a wide range of decision-making situations within various areas of management. At present, the use of mathematical models to interpret and predict dynamics and controls in managerial decision-making has increased. These applications include decisions on inventories, sales forecasts, impact predictions, effect of advertising campaigns, strategies to protect inventory shortages, to determine optimal portfolio investment strategies, etc.
The objectives of every company must always be to achieve leadership in its branch, controlling the efficiency and effectiveness of all its components through methods that allow finding the optimal relationships that best operate the system, given a specific objective.
Given the tremendous progress that has occurred in almost all sciences in recent decades, it is no longer feasible to want to know a little about everything, but rather to specialize in some branch of science. The problems that arise in organizations cannot easily be solved by a single specialist. On the contrary, they are multidisciplinary problems, whose analysis and solution require the participation of several specialists. These interdisciplinary groups necessarily require a common language to be able to understand and communicate, where Operations Research becomes that communication bridge.
The focus of Operations Research is the same as the scientific method. In particular, the process begins with careful observation and formulation of the problem and continues with the construction of a scientific (usually mathematical) model that attempts to abstract the essence of the real problem. At this point, the hypothesis is proposed that the model is a sufficiently precise representation of the essential characteristics of the situation so that the conclusions (solutions) obtained are also valid for the real problem. This hypothesis is verified and modified by appropriate tests.
So, in a way, Operations Research includes creative scientific investigation of the fundamental properties of operations. However, there is more than this. In particular, Operations Research is also concerned with the practical administration of the organization. Thus, to be successful, you must also provide positive and clear conclusions that the decision maker can use when needed.
Cuban organizations, in particular, due to the urgent economic revitalization to which the country is obliged, face new goals: to resist, survive and be competitive in the new environment. To do this, they must raise their levels of productivity, efficiency and profitability, which is why their members question whether their companies are prepared to meet these goals in the new conditions, a reflection that leads them to analyze and review them. Hence the importance of applying mathematical methods that allow making decisions in any area of the company.
Many companies show a high degree of empiricism in decision-making by their managers and workers, which causes inadequacies in the performance of business functions, so it is considered appropriate to apply an inventory model that facilitates and improves decision-making. decisions in the organization under study.
Theoretical-conceptual references.
Among the theoretical aspects considered relevant for the present work, the analysis of operations research, the models and tools for its application, in addition to business decisions, especially those of inventories, stand out. These issues will be addressed below.
Operations research.
There are many definitions offered from Operations Research, the following definitions referred to in Delgado Landa (2008) are a useful basis for an initial understanding of the nature of IO.
Mores-Kimball (1943): Scientific method by which executive management has a quantitative basis for operations decisions under its control.
Ackoff-Sasieni (1968): The application of the scientific method by interdisciplinary teams to problems that involve the control of organized systems (man and machine) to provide the solutions that best meet the purpose of the organization as a whole.
Wagner (1969): Scientific approach to solving problems in executive management).
Hiller-Lieberman (1974): Optimal decision-making, and its modeling, in deterministic and probabilistic systems that have their origin in real life. These applications in government, business, engineering, economics, and the natural and social sciences are characterized primarily by the need to allocate limited resources. In these situations, scientific analysis, such as that provided by IO, can provide important information.
Gross (1979). Branch of mathematics applied to the decision-making process.
Hillier, Lieberman, Shamblin, Stevens, Taha, Tierauf, Grosse, Sasieni, to mention some of the great specialists in operations research (IO), give a series of definitions that could well be summarized as follows:
It is a scientific approach to decision making. It can be said that the OR uses a planned approach (scientific method) and an interdisciplinary group to represent, through symbolic models, the functional relationships that occur in reality, which provides a quantitative basis for decision-making. Applies tools that seek to obtain the optimal result from the use of scarce resources.
Main tools of Operations Research.
When talking about tools in IO, it refers to the different theoretical models (such as economic models of inventories and queuing theory), and to other disciplines (such as mathematics, administration, economics, etc.), which are used as instruments of usual job for the Operations Research professional. It should be clear, however, that more types of models and other disciplines are added every day.
Below in Table 1 is a list, not exhaustive, of different types of models that could be considered as tools of operations research is presented.
Table 1. Main tools of operations research. Source: Delgado Landa (2008). .
| Main tools of Operations Research |
| 1. Linear programming models. |
| 2. Multi-criteria models: multi-objective and multi-attribute |
| 3. Networks and linear programming for transportation. |
| 4. Decision-making models under conditions of uncertainty. |
| 5. Bayesian Models. |
| 6. Waiting lines (Queue Theory). |
| 7. Optimization models with networks for planning, execution and control of
Projects. |
| 8. Markov chains for the replacement of fixed assets. |
| 9. Deterministic inventory models. |
| 10. Probabilistic inventory models. |
| 11. Models of dynamic programming and game theory. |
| 12. Simulation models to obtain expert information. |
| 13. Heuristic models of self-learning and self-correction. |
In the same way, Operations Research is considered, itself as a tool at the service of other disciplines. It is well known that Business Administration has been benefiting greatly from Operations Research now that a whole revolution has started with the use of Strategic Planning, Reengineering and Total Quality programs, to name a few.
Taking into account that the models are tools within the IO, it is necessary to understand that: a decision model should be considered as a vehicle to summarize a decision problem in such a way that it makes possible the identification and systematic evaluation of all the decision alternatives of the problem. A decision is then reached by selecting the alternative that is judged to be the best among all the available options.
A solution to a model, however, if accurate, will not be useful unless the model itself provides an adequate representation of the true decision situation.
The decision model must contain three elements:
- Decision alternatives, from which a selection is made. Constraints, to exclude infeasible alternatives. Criteria for evaluating and classifying feasible alternatives.
Business decision making
Decision-making is the process by which a choice is made between alternatives or ways to solve different life situations, these can be presented in different contexts: at work, family, sentimental, that is, at all times make decisions, the difference between each of these is the process or the way in which they are reached. It basically consists of choosing an alternative among those available, in order to solve a current or potential problem (even when there is no evidence of a latent conflict).
Decision-making is also considered as the creative act of the choice, based on a set of possible decisions, in which quantitative factors are combined with the heuristic capacities of the men who make decisions.
To make a decision no matter its nature, it is necessary to know, understand, analyze a problem, in order to solve it; In some cases, because they are so simple and everyday, this process is carried out implicitly and is solved very quickly, but there are other cases in which the consequences of a bad or good choice can have repercussions in life and if it is in a context labor in the success or failure of the company, for which it is necessary to carry out a more structured process that can give more security and information to solve the problem.
Decision making is important because by using good judgment it indicates that a problem or situation is deeply valued and considered to choose the best way forward according to the different alternatives and operations.
In making decisions, considering a problem and reaching a valid conclusion means that all the alternatives have been examined and that the choice has been correct.
Economic inventory model.
The need for companies and producers to maintain inventories, resulted in the study of them, in such a way that the most economical way of maintaining them was guaranteed. A good number of mathematical models that have been developed make it possible to determine, under a set of given conditions, the optimal way to have inventories.
According to Álvarez and Valle (1987): Inventory is called a set of resources or goods in good condition, which are stored with the aim of being used in the future. These resources can be materials, equipment, money, etc.
Inventory systems can be classified according to three fundamental considerations:
- According to the number of orders and level of independence of the demand.
- Repetitive order with independent demand Non-repetitive order with independent demand Repetitive order with dependent demand
- Raw materials Products in process Finished production.
Note that the inventory of final product for one company may be that of raw material for another.
- According to the predictability of demand.
- Deterministic: if the demand is known and constant. Probabilistic: if the demand is a random variable.
All inventory must have a limit, otherwise the cost would be detrimental and economically unsustainable, due to having a large amount of idle resources.
For the management of a production system it is important to know:
- How much resources should be in inventory in the system? How often do inventories need to be replenished?
The inventory theory brings together a series of techniques that allows us to answer these questions. Only you have to know which one to use in each case, for this it is necessary to know some essential aspects to correctly apply the appropriate model.
Procedure to apply the inventory model.
- Identify the inventory model to apply.
There are two large groups of inventory according to the characteristic of demand. These groups are:
- Deterministic inventory models: They are those in which the demand is perfectly determined or known for a given period. Stochastic inventory model: They are those in which the demand is a random variable, with a known distribution function.
Taking into account the considerations defined above, for the development of this work only deterministic inventory models will be used, within this are:
- General deterministic inventory model No deficit inventory model Instant replenishment inventory model Instant replenishment no deficit inventory model
The first two are used for systems where there is production, that is, these products or raw materials arrive at the warehouse little by little, while the remaining two are used in systems where the products arrive instantly. On the other hand, models 2 and 4 do not allow a deficit, that is, orders do not accumulate, this is not the case with 1 and 3, which do admit accumulations of orders that will later be delivered.
- Identify the controlled and uncontrolled variables of the system.
In an inventory problem there are a number of variables that can be controlled by those who run the system and others that cannot be controlled.
The variables controlled in an inventory system are:
- Quantity to acquire (how much). Frequency of acquisition (when).
The uncontrolled variables can be cost variables or others.
The main uncontrolled variables in an inventory problem are:
- Cost of maintaining inventory.
This cost can be broken down into the following:
- Cost of immobilization of resources Cost of handling Storage cost (depreciation, construction, etc.) Cost of depreciation or obsolescence of inventory Administrative cost (salary, etc.) Cost of deficit: It is the cost incurred when an inventory is finished and that brings consequence, having to acquire the merchandise in question, in an unusual way, investing resources to use faster transport, extra production, etc. Launch cost: When the inventory is part of the production system The launch cost is the preparation of a new production order, which will be incorporated into said inventory. In the case that the inventory is considered as a single system, the cost per launch is the one incurred for the administrative work to make the acquisition.Production cost: It is the unit cost of production of an item that will be incorporated into inventory Demand: It can be perfectly determined for each period of time or it can be random, in which case it would be necessary to know its probabilistic distribution function to be able to make decisions. Replenishment time: It is the time elapsed since the replenishment order is delivered, until the resources are incorporated into the inventory. The replenishment time can be fixed or random.Replenishment time: It is the time elapsed since the replenishment order is delivered, until the resources are incorporated into the inventory. The replenishment time can be fixed or random.Replenishment time: It is the time elapsed since the replenishment order is delivered, until the resources are incorporated into the inventory. The replenishment time can be fixed or random.
- Apply the selected model
Taking into account the characteristics presented by the company under study, the procedures for only the inventory model that does not allow deficit will be explained , its graphic representation is shown in Figure 1.
Figure 1. Graphic representation of the deterministic inventory model that does not allow deficit. Source: Delgado Landa (2008).
The inventory cycle, as can be seen in the previous figure, follows the following sequence:
- Start with inventory equal to zero. Start production with a constant ratio (r). There will be a constant consumption ratio (a), where r> a, until a certain level is reached, stopping production (interval t).Then there will be inventory consumption at a constant ratio (a) occurring during a time t, until all inventory is exhausted and the cycle will repeat again.
For the mathematical formulation of this model it is defined:
r: constant production ratio (physical units per unit of time).
a: constant demand (physical units per unit of time).
s: maximum inventory level (physical units).
Q: quantity of units to be produced in each run or batch size (physical units).
t and t: time intervals represented in the graph of figure 2.1.
In addition, the following costs will be taken as data:
c: unit cost of production (cost per physical unit).
h: cost of maintaining inventory.
k: launch cost (cost).
Based on the above, the expressions for this model can be seen in Table 2.
Table 2 Necessary formulas to apply the inventory model that does not allow deficit. Source: Delgado Landa (2008).
| Optimal batch size | |
| Time between two production runs | T = t + t == |
| Frequency of runs | f == |
| Timeslots | t ===
t = |
| Maximum inventory level | s = at = t |
| Total cost | C (T) = |
Proposal to support inventory decision making.
Characterization of the company under study.
The Company's mission is: ”To produce wheat flour, as well as cereals for breakfast, pellets for frying and corn curls, sorbets, breads and various pastries, of high quality and efficiency, to satisfy the needs of customers.
The general objective of the company is: To carry out the wheat milling process for the production and wholesale commercialization in both currencies, of Wheat Flour for human and animal consumption, as well as to produce corn and wheat cereals for breakfast, flavored and natural, and pellets for frying, sorbets, breads and fine pastries, in both currencies.
Proposed solution to the problem identified in the Wheat Mill.
- Definition of the problem.
The problem that must be solved with the highest priority in the Wheat Mill relates to the level of inventory that must remain in the warehouse.
The Company after a reorganization in the old corn mill, is now dedicated to the milling of wheat. It does not have sufficient experience in this regard, so it considers an inventory study necessary, in order to know what is the amount of tons of Wheat Flour that it has to produce in each production batch to have the least amount in storage. quantity possible with the least cost. Well, given the characteristics of the flour that can easily spoil, it is not advisable to have idle inventory and on the other hand it should not be lacking due to the importance it has for the preparation of bread.
- Construction and solution of the model.
The model to be used to solve the above problem is the Inventory Model. This allows you to fully meet the requirements of the company and decide the changes that should be made after obtaining the optimal number of bags of flour that must remain in inventory to ensure the minimum cost.
Inventory control decisions are both a problem and an opportunity for at least three different departments such as Production, Commercial and Economic-financial. Decision-making in inventory control has a huge impact on the productivity and performance of the organization because it manages the total flow of materials. Proper inventory control can minimize material stockouts, thus reducing the organization's capital. Additionally, it enables the organization to produce wheat flour in cheaper quantities, thus minimizing the overall cost of production.
A good inventory model allows:
- Reduce temporary gaps between supply and demand. Contribute to the reduction of production costs. Provide a way to "store" work; For example, make more tons of flour now, and free up work later. Provide fast customer service.
- Identification of the inventory model to apply.
Within the inventory model, taking into account that the demand is known and constant, the deterministic inventory model that does not allow deficit will be applied. It is important to clarify that this model is selected for the reason explained below:
The company produces wheat flour mainly for making bread, so orders cannot be accumulated. The lack of flour would cause serious problems for the territory that benefits from this production. In addition, the food industry sector is prioritized by the government, which justifies the fact that this product cannot be absent.
- Identification of the controlled and uncontrolled variables of the system.
The variables present in the system are stated as follows:
- Controlled variables:
Q: Quantity to produce in each production batch.
: Frequency of production.
- Uncontrolled variables:
h: cost of maintaining inventory.
According to the company's economics department, the cost of maintaining the inventory is 1.68 pesos for each tonne stored daily.
k: launch cost (cost).
The launch cost was necessary to estimate, knowing that:
- The salary of the person in charge of coordinating the orders is $ 640.00, the average phone bill used in this task.
It is also known that this person spends an average of one hour preparing each production run.
The launch cost calculation would be:
Launch Cost = Salary Cost + Phone Cost
Launch Cost = $ 3.33 + $ 10.07
Launch Cost = $ 13.40
Salary cost
Salary cost = × run preparation time
Salary cost = × 1 hour
Salary cost = $ 3.33
Phone cost
To determine the telephone cost, the telephone accounts were observed, determining the calls for the preparation of production orders during the months corresponding to the second half of 2007. The amounts were averaged, resulting in $ 10.07 for each batch. of production.
c: Unit cost of production (cost per physical unit).
According to the wheat flour cost sheet consulted, the cost to produce one ton is 570.20 pesos
a: Constant demand (physical units per unit of time).
The daily demand for wheat flour is 72 tons and is broken down by customers in Table 3.
Table 3. Daily demand for Wheat Flour. Source: Delgado Landa (2008).
| customers | Amount of tons required per day |
| Provincial Food Company | 60 |
| Provincial Food Wholesale Company | 10 |
| Meat | two |
| Total | 72 |
- Model application
The data necessary for the mathematical formulation of the model are the following:
r = 95 tx day.
a = 72 tx day.
Storage capacity = 165 t
Note: it is considered 24 working days a month and 24 hours a day of work.
Costs:
c = $ 570.20 xt
h: $ 1.68 xt daily
k: $ 13.40 x lot.
Based on the above, the calculations for this model are:
- Optimal batch size.
Q ===== 68.67≈ 69
The optimal batch size is 69 tons of Wheat Flour, which indicates that this quantity should be produced in each production run.
- Optimal time between runs.
T === 0.9583 days × 24 h = 23 hours
The full production run takes approximately 23 hours.
- Frequency of runs.
f === 1.0434 times a day
Ø Time intervals.
t === 0.7263 days × 24 h = 17.43 hours
Production of the batch takes approximately 17 hours 26 minutes.
t ===== 0.2316 days × 24 h = 5.56 hours
After production is finished, the product continues to be consumed from what is in inventory, which takes approximately 5 hours and 34 minutes.
- Maximum inventory level.
s = at = 72 × 0.2316 = 16.68
The maximum inventory level is 16 tons.
- Total cost.
C (T) = =
C (T) = = 41 100,40 pesos.
The total cost is $ 41,100.40 per day.
As a proposed solution to the problem under study, the following is summarized:
In each batch, 69 tons of Wheat Flour must be produced to minimize the total cost. Starting production every 23 hours. With which a maximum inventory volume of 16 tons is achieved. On the day there will be a production run and the beginning of another. With all this, the minimum possible total cost will be reached, which would be $ 41,100.40 per day.
Conclusions
The development of this research materialized in the theoretical-conceptual aspects exposed, and its practical validation, allow us to reach the following conclusions:
Decision-making essentially constitutes the choice of one of the possible alternative solutions to a current or potential problem, which requires previously that the problem under study be detected and that the required internal and external information be sought. Subsequently, the decision must be converted into a concrete action.
Operations Research provides decision-makers with quantitative bases to select the best decisions and allows raising skills to make future plans.
Operations Research uses the Scientific Method to apply tools that seek to obtain the optimal result from the use of scarce resources and solve dissimilar problems in the business context.
Decision-making in inventory control has a significant impact on the productivity and development of companies, since there must be a limit, otherwise the cost would be detrimental and economically unsustainable, due to having a large amount of idle resources.
In each batch of 69 tons of Wheat Flour must be produced to minimize the total cost. Starting production every 23 hours. With which a maximum inventory volume of 16 tons is achieved and a minimum total cost of $ 41,100.40 per day.
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Operations research
Lerner, V. S, Trujaiev, R. I (1974): Dynamic models of decision-making processes. Kishiniev.
It is known as EMPA
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