Introduction
A waiting line is the resulting effect on a system when the demand for a service exceeds the capacity to provide that service. This system is formed by a set of entities in parallel that provide a service to the transactions that randomly enter the system. Depending on the system in question, the entities can be cashiers, machines, traffic lights, cranes, etc., while the transactions can be: customers, parts, cars, boats, etc. Both the service time and the inputs to the system are phenomena that generally have associated sources of variation that are outside the control of the decision maker, in such a way that it is necessary to use stochastic models that allow the study of this type of systems.
A waiting line can be modeled as a stochastic process in which the random variable is defined as the number of transactions in the system at a given time; the set of values that this variable can take is {0, 1, 2,…, N \ and each of them has an associated probability of occurrence.
objective
The objective is to determine what level of service, either by number of entities or by the speed of them, to provide to minimize the total cost of the system. This cost is made up of both the service cost and the cost of the wait.
Structure of a waiting line system
Single channel waiting line
Each customer must go through a channel, a station to take and fill the order, to place the order, pay the bill and receive the product. As more customers arrive, they form a waiting line and wait for the station to clear to take and fill the order.
Waiting line model and linear programming
Arrival distribution
To determine the probability distribution for the number of arrivals in a given period, the Poisson distribution can be used.
/ = Average or average amount of occurrence in an interval
e = 2.17828
X = number of occurrences in the interval
Waiting line model and linear programming
Service time is the time a client spends on installation after the service has started.
You can use the exponential probability distribution to find the probability that the service time is less than or equal to a time t.
e = 2.17828
μ = average number of units that can be served per period
Waiting line model and linear programming
Waiting line discipline
Way in which units waiting for service are ordered to receive it.
First come first serve
Last in, first out
Attention first to highest priority
Steady state operation
Generally the activity gradually increases to a normal or stable state. The beginning or beginning period is known as the transitional period, which ends when the system reaches steady state or normal operation.
Single Channel Waiting Line Models with Poisson Arrivals and Exponential Service Times
The following are the formulas that can be used to determine the steady state operating characteristics for a single channel queue.
The purpose of the formulas is to show how information about the operational characteristics of the waiting line can be given.
Waiting line model and linear programming
How can the waiting line operation be improved?
Operational characteristics for the system with the mean service rate increased to μ = 1.25 clients per minute.
Waiting line model and linear programming
Waiting line model and linear programming
Waiting line model and linear programming
Economic analysis of waiting lines
Before an economic analysis of a waiting line can be carried out, a total cost model must be developed, which includes the cost of waiting and the cost of service.
Cw = cost of waiting per period for each unit
L = average number of units in the system
Cs = service cost per period for each channel
K = number of channels
The general form of the cost curves in the economic analysis of waiting lines is that the cost of the service increases as the number of channels increases; but with more channels, the service is better. As a result, lead time and cost decrease as the number of channels increases. The number of channels that will provide a good approximation to the minimum total cost design can be found by evaluating the total cost for various design alternatives.
Waiting line model and linear programming
Linear programming
Introduction
Linear programming is a relatively recent mathematical technique (20th century), which consists of a series of methods and procedures that allow solving optimization problems in the field, especially, of the Social Sciences.
We will focus on this topic on those simple linear programming problems, those with only 2 variables, two-dimensional problems.
For systems with more variables, the procedure is not so simple and they are solved by calling
Simplex method (devised by GBDanzig, American mathematician in 1951).
Recently (1984) the Indian mathematician established in the United States, Narenda Karmarkar, has found an algorithm, called the Karmarkar algorithm, which is faster than the simplex method in certain cases. Problems of this type, in which a large number of variables intervene, are implemented in computers.
Linear programming is an important field of optimization for several reasons. Many practical problems in operations research can be posed as linear programming problems.
Some special cases of linear programming, such as network flow problems and goods flow problems, were considered in the development of mathematics important enough to generate by themselves much research on algorithms specialized in their solution.
A series of algorithms designed to solve other types of optimization problems constitute particular cases of the broader technique of linear programming. Historically, the ideas of linear programming have inspired many of the central concepts of optimization theory such as duality, decomposition, and the importance of convexity and their generalizations.
Similarly, linear programming is widely used in microeconomics and business administration, either to maximize revenues or minimize costs of a production system. Some examples are food mixing, inventory management, portfolio and finance management, allocation of human resources and machine resources, planning of advertising campaigns, etc.
Others are:
- Optimization of the combination of commercial figures in a linear water distribution network Optimal use of the resources of a hydrographic basin, for a year with inflows characterized by corresponding to a certain frequency Support for decision-making in real time, for operation of a system of hydraulic works; Solution of transportation problems.
Steps to solve a linear programming problem
- Choose the unknowns Write the objective function based on the data of the problem Write the constraints in the form of a system of inequalities Find out the set of feasible solutions by graphing the constraints Calculate the coordinates of the vertices of the field of feasible solutions (if are few) Calculate the value of the objective function in each of the vertices to see in which of them presents the maximum or minimum value according to the problem asks us (we must take into account here the possible non-existence of a solution if the enclosure is not bounded).
Linear programming example
A department store orders pants and sports jackets from a manufacturer.
The manufacturer has 750 m of cotton fabric and 1000 m of polyester fabric to manufacture. Each trouser requires 1 m of cotton and 2 m of polyester. For each jacket you need 1.5 m of cotton and 1 m of polyester.
The price of the trousers is set at € 50 and the jacket at € 40.
What number of trousers and jackets must the manufacturer supply to the stores in order for them to achieve a maximum sale?
1Election of the unknowns.
x = number of pants
y = number of jackets
2 Target function.
f (x, y) = 50x + 40y
