Waiting in lines is a part of our everyday life. Waiting in lines may be due to overcrowded, overfilling or due to congestion. Any time there is more customer demand for a service than can be provided, a waiting line forms. We wait in lines at the movie theater, at the bank for a teller, at a grocery store. Wait time is depends on the number of people waiting before you, the number of servers serving line, and the amount of service time for each individual customer. Customers can be either humans or an object such as customer orders to be process, a machine waiting for repair. Mathematical analytical method of analyzing the relationship between congestion and delay caused by it can be modeled using Queuing analysis. Queuing theory provides tools needed for analysis of systems of congestion. Mathematically, systems of congestion appear in many diverse and complicated ways and can vary in extent and complexity.
A waiting line system or queuing system is defined by two important elements: the population source of its customers and the process or service system. The customer population can be considered as finite or infinite. The customer population is finite when the number of customers affects potential new customers for the service system already in the system. When the number of customers waiting in line does not significantly affect the rate at which the population generates new customers, the customer population is considered infinite. Customer behavior can change and depends on waiting line characteristics. In addition to waiting, a customer can choose other alternative.… Read the rest
Management science is the science for managing and involves decision making. It utilizes what is controllable, and tries to predict what is uncontrollable in order to archive a specific objective. Science is a continuous search; it is a continuing generation of theories, models, concepts, and categories. Management science uses analytical methods to solve problems in areas such as production and operations, inventory management, and scheduling. Typical management science approach is to build a model for the problem being studied, such a model is often a mathematical model. Practical problems are often unstructured and lack clarification in definition of problem which makes mathematical modeling a challenge. Therefore modeling of a problem is important phase in problem solving technique. Once model is built, algorithms are used to solve problem. Various techniques are devised to model problem and solve it for possible solutions.
Linear programming is one of the widely used modeling techniques. Linear programming problems consist of an objective function (also know as cost function) which has to be minimized or maximized subject to a certain number of constraints. The objective function consists of a certain number of variables. The constraints are linear inequalities of the variables used in the objective function. This technique is closely related to linear algebra and uses inequalities in the problem statement rather than equalities. A linear programming problem can fall in three categories: infeasible, unbounded and an optimal solution. In an infeasible problem values of decision variables do not satisfy constraint condition. A problem is unbounded if the constraints do not sufficiently restrain the objective function so that for any given feasible solution, another feasible solution can be found that makes further improvement to the objective function.… Read the rest
Transportation and assignment models are special purpose algorithms of the linear programming. The simplex method of Linear Programming Problems(LPP) proves to be inefficient is certain situations like determining optimum assignment of jobs to persons, supply of materials from several supply points to several destinations and the like. More effective solution models have been evolved and these are called assignment and transportation models.
The transportation model is concerned with selecting the routes between supply and demand points in order to minimize costs of transportation subject to constraints of supply at any supply point and demand at any demand point. Assume a company has 4 manufacturing plants with different capacity levels, and 5 regional distribution centres. 4 x 5 = 20 routes are possible. Given the transportation costs per load of each of 20 routes between the manufacturing (supply) plants and the regional distribution (demand) centres, and supply and demand constraints, how many loads can be transported through different routes so as to minimize transportation costs? The answer to this question is obtained easily through the transportation algorithm.
Similarly, how are we to assign different jobs to different persons/machines, given cost of job completion for each pair of job machine/person? The objective is minimizing total cost. This is best solved through assignment algorithm.
Uses of Transportation and Assignment Models in Decision Making
The broad purposes of Transportation and Assignment models in LPP are just mentioned above. Now we have just enumerated the different situations where we can make use of these models.
Transportation model is used in the following:
- To decide the transportation of new materials from various centres to different manufacturing plants.
… Read the rest
Mathematical model is an idealized representation expressed in mathematical symbols and expressions. A mathematical model of a business problem might be in the form of a set of equations and related mathematical expressions that describe the essence of the problem. An economic order quantity model is given by: EOQ = √2AO/C where A – annual requirement, O – ordering cost and C – carrying cost. A linear programming model is given by objective function: Say Maximize Z = 10a + 12b, subject to 2a + b < = 60, 3a + 4b < = 120, a, b > = 0, where a and are units of products A and B, respectively to be produced to maximize total contribution given individual contribution of Rs.10 per unit of A and Rs.12 per unit of B, the resource constraints being that resource 1 and 2 are available respectively to the extent of 60 and 120 units only. An internal rate (IRR) model is given by: -I + Cft/(1 + k)t = 0, where ‘k’ is the IRR to be found, while ‘I’ is the initial investment, CFt refers to periodic cash flows. To construct a mathematical model objective of the firm, variables, constants and constraints must be known, besides the relationship amongst the variables. The relationship may be linear or non-linear. The EOQ and linear programming models given above are linear relationship based, while the IRR model is non-linear based as its exponential form.
Variables in Mathematical Models
Variables are something whose magnitude can change. … Read the rest
A decision model is an idealized representation of the problem. Decision model refers to structured presentation of the problem, solution there to and stimulation of working of the solution. The mode’s purpose is to enable the decision analyst to forecast the effect of factors crucial to the solution of the problem.
Types of Models
There are different types of models. Ionic and Symbolic models are the prime two types. Ionic Models are concretized. It is a physical representation of any real life object on a different scale. Think of a prototype of a plane/car/machine/globe/idol and so on. Symbolic models re abstract models. A cost curve, a supply curve, marginal revenue curve, a production possibility curve, etc., is a symbolic model. A forecast profit and lost account is also a symbolic model. The statement symbolizes summary of financial effects of commercial activities planned over the next period. An equation is a symbolic model of variables, their relationship and effect of independent variables on the dependent variable.
Symbolic models can be classified into: 1) quantitative and qualitative, 2) standard and customized, 3) probabilistic and deterministic, 4) descriptive and optimizing, 5) static and dynamic, 6) simulative and realistic models.
- In a quantitative model all variables are expressed in numbers, while in a qualitative model some or all of the variables are given only verbal expression.
- Standard model is universalized. Computers languages and operating systems are standardized, while application programmes are customized. Nowadays customized-standardization is the order, powered by enormous computing powers of computers. Mass customization is preferred.
… Read the rest
Quantitative analysis is a scientific approach to decision making. As a first step to decision making, decision model has to be evolved. The decision model depends on two factors, namely the problem and the problem environment.
Defining the Problem
The first step in decisions making is defining the problem. The problem (i.e. the threat, opportunity, etc.) must be fully understood as to its nature, dimensions, intensity and so on. Take a labour absenteeism. It becomes a problem when it is rampant affecting work schedules. (one or two absentees, here and there is no problem). What is its nature? Deliberate absenteeism, absenteeism due to unavoidable causes, absenteeism as a mark of protest to certain managements attitudes/actions, absenteeism due to healthy/family/social reasons, etc. are indicative of the nature of the problem of absenteeism. Specific nature must be understood, since each type of absenteeism needs a different approach to solving the same. Dimensions of absenteeism comes next. How does it affect production? How does it affect inter-departmental relations, work flow, co-ordination, morale, etc? What is its impact on turnover and customer satisfaction? What is the scale of idle capacity resulting from labour absenteeism? Will motivational/coercive courses work containing absenteeism? There are but a few dimensions of the problem. How much grave the problem is? Should it be solved immediately? These indicate the intensity of the problem. The wholesome knowledge of the problems is very essential so that we can solve the problem. Not all problems can be known fully. Some keep change from time to time. … Read the rest