Intro And Job Shop Scheduling Problem

Published Date: 02 Nov 2017

Operational research is a scientific approach to analyzing problems and making decisions. Operational Research professionals aim to provide rational bases for decision making by seeking to understand and structure complex situations and to use this understanding to predict system behavior and improve system performance. Much of this work is done using analytical and numerical techniques to develop and manipulate mathematical and computer models of organizational systems composed of people, machines, and procedures. [1]

History and development

Actually Operational Research is not a discipline with long history. It has just been around 70 years from the start and Operational Research had its origins just before the World War II when the British Air Ministry established Bawdsey Research Station to let civilian scientists determine how recently developed radar technology for Air Force and Army to controlled interception of enemy aircraft in 1937. The scientists who came from multi-discipline need to leave their own research institute and laboratory and participate in field operations, testing, and evaluation. [2] Not only were they ordered to do research and development with scarce resources in warfare, they were also deployed onsite to study the scenario and come up with ways to improve the effectiveness of the operation within the limited resource and time. This practice paved way for the present scientific, interdisciplinary approach followed for solving important problems in critical operations. During that period, a group of scientists led by a physicist P. M. S. Blackett who did contribution in convincing the authorities the need for a scientific approach to manage complex operations. P. M. X. Blackett was considered as the original operations research analyst in many different fields.

Later on, Operational Research was introduced into United States. With the lead of U. S. Navy’s Mine Warfare Operations Research Group, Operational Research was also paid great attention. At the end of World War II, the successful techniques used in warfare became professional and industrialized. Most of the companies in America and Britain started employing these techniques to mint money with minimal resources and time. In the 1950s Operational Research evolved into a profession with the formation of national societies, establishment of journals and academic departments in universities. The use of operations research expanded beyond the military to include both private companies and other governmental organizations. The petrochemical industry was one of the first to broadly embrace operations research to improve the performance of plants, develop natural resources and plan strategy.

While computers were increasingly used, the complexities of operations research techniques were growing exponentially. There was a dire need to fool proof the calculation process and speed it up. The advances in computing technology helped in solving that issue. Operations Research was considered to be theoretical to a large extent because of its ability to solve only concrete problems all the time. The use of probability theory made the tackling of uncertain situations more realistic.

Applications of Operational Research

Nowadays, the benefits of Operational Research are utilized by nearly all the fields of governments, business and industries. Though, in briefly, it is not feasible to cover all applications of Operational Research, there are still lots of the applications of Operational Research. Today, operations research plays important roles in a variety of industries such as:

Airline - Scheduling planes and crews, pricing tickets, taking reservations, and planning the size of the fleet,

Pharmaceutical – R&D management,

Logistics companies - Routing and planning,

Financial services - Credit scoring, marketing, and internal operations,

Lumber and wood products - Managing forests and cutting timber,

Local government - Deployment of emergency services, and

Policy studies and regulation - Environmental pollution, air traffic safety, AIDS, and criminal justice policy.

Here is a table shows some real cases from organizations that have done use of Operational Research so as profits and/or savings gotten thanks it.

Organization

Application

Year

Yearly Savings

The Netherlands Rijkswaterstaat

Development of the national politics of water management, including mixture of new facilities, operations' procedures and financing

1985

$15 millions

Monsanto Corp.

Production's operations optimization to obey goals with a minimum cost

1985

$2 millions

United Airlines

Programming of work turns at book offices and airports to obey the customer needs with a minimum cost

1986

$6 millions

Electrobas/CEPAL Brasil

Optimal allocation of hydraulic and thermic resources in the national energy generation system

1986

$43 millions

Citgo Petroleum Corp.

Optimization of refinement, offer, distribution and commercialization of products operations

1987

$70 millions

Electric Power Research Institute

Administration of oil and coal inventories for the electric service with the aim of balancing inventories' costs and risks of remainings.

1989

$59 millions

Texaco Inc.

Optimization of available ingredients mixture in order to gasoline products fulfill the sales and quality requests

1989

$30 millions

IBM

Integration of a national spareparts inventory net to improve the service support

1990

$20 millions + $250 millions in minor inventory

U.S. Military Airlift Command

Rapidity in the airplanes, crew, load and passengers coordination to drive the evacuation by air in the "Desert Storm" project in the Middle Orient

1992

Victory

American Airlines

Design of a pricing, over-saleses and coordination of flights structures system to improve the usefulness

1992

$500 millions of additional revenue

AT&T

Development of a system based in PC to drive the clients of business in the design of the calls center

1993

$750 millions

Delta Airlines

Profit maximization by the assignment of the airplanes types in 2,500 national flights

1994

$100 millions

Digital Equipment Corp.

Reorganization of all suppliers's chain among suppliers, plants, centers of distribution, potential places and market areas

1995

$800 millions

China

Selection and optimum programming of mass projects to obey with future energy needs of the country

1995

$425 millions

Practical and Complex Problem: Job Shop Scheduling Problem

Operational research is a discipline that deals with the application of advanced analytical methods to help make better decisions. For example, in the real life, we often meet with such situation:

If you were own a clothes shop to serve customers with their customized orders, you’ll receive orders in different time and you need to complete the order one by one and then send the clothes to customers as soon as possible.

Or you own a factory and manufacture different types of productions batch by batch according to the different needs from customers. You need to arrange the machines to dealing with the different orders one by one and as soon as possible.

There is no doubt that such scheduling a custom job shop or different machines to perform operations on jobs has it challenges. We call this kind of problem as job shop scheduling problem or job shop problem. Job shop scheduling is a widely studied and difficult combinatorial optimization problem. It is defined as that a number of jobs have to be done and every job consists of using a number of machines for a certain amount of time. The problem is to find the best planning to do all the jobs on all the different machines in the shortest period of time. Although a job can have any number of operations, the most common formulation of the job shop problem specifies that each job has exactly 'n' operations, one on each machine.

Although easily stated, the job shop problem turns out to be one of the hardest problems in the area of combinatorial optimization. This is illustrated by the fact that a classical benchmark problem of 10 jobs and 10 machines remained unsolved for more than twenty years. It was posed in 1963 by Fisher and Thompson and solved by Carlier and Pinson in 1986. The job shop problem was proven NP-hard in the strong sense by Garey, Johnson, and Sethi. Some very special cases of the problem can be solved in polynomial time, but their immediate generalizations are NP-hard. These results are summarized in Table 1 where m is the number of machines and n the number of jobs, l(j) the number of operations of job j and p(i) the processing time of operation i.

Actually job shop scheduling problem is quite a large topic. Simply can we divided into two broad sections: the static problem and the dynamic problem. The static case is the case that all the information is available initially and it does not change during the time. But in the real cases, the jobs arrive on a continuous basis, which is called dynamic job shop scheduling problem and this kind of case is more complicated.[]Simulation is a tool that has been commonly used to assist with systems analysis. Simulation is still widely used and is often use in combination with other techniques that we have examined - including linear programming, expert systems, and neural networks.[]

Example by using simulation

Now we considered a situation that customers accepting service by a teller at a bank as a simple example to show the concept of simulation. Customers arrive at the bank and wait for the service by the teller and then be severed and leave after all done. When customers arrive and the teller happen to be serving others, they need to queue in line in front of the counter. Here is the table of data of the time customers arrive and the service time for each. Our objective is to simulate the above system to determine the percent of time the teller is idle and the average time a customer need to stay in the bank.

Customer No.

Time of Arrival (min)

Service Time (min)

1

3.2

3.8

2

10.9

3.5

3

13.2

4.2

4

14.8

3.1

5

17.7

2.4

6

19.8

4.3

7

21.5

2.7

8

26.3

2.1

9

32.1

2.5

10

36.6

3.4

Since the simulation is the dynamic portrayal of the changes in the state of a system over time, the states of the system must be defined. For this example, they can be defined by the status of the teller (busy or idle) and by the number of customers at the bank. The state of the system is changed by a customer arriving to the bank, and the completion of service by the teller and subsequent departure of the customer. To illustrate a simulation, we will determine the state of the system over time by processing the events corresponding to the arrival and departure of customers in a time-ordered sequence.

The manual simulation of this example corresponding to the values in the above table is summarized in the table below by customer number. It is assumed that initially there are no customers in the system, the teller is idle, and the first customer is to arrive at time 3.2.

Customer Number (1)

Arrival Time (2)

Start Service Time (3)

Departure Time (4)

Time in Queue (5)=(3)-(2)

Time in Bank

(6)=(4)-(2)

1

3.2

3.2

7.0

0.0

3.8

2

10.9

10.9

14.4

0.0

3.5

3

13.2

14.4

18.6

1.2

5.4

4

14.8

18.6

21.7

3.8

6.9

5

17.7

21.7

24.1

4.0

6.4

6

19.8

24.1

28.4

4.3

8.6

7

21.5

28.4

31.1

6.9

9.6

8

26.3

31.1

33.2

4.8

6.9

9

32.1

33.2

35.7

1.1

3.6

10

36.6

36.6

40.0

0.0

3.4

In the above table, columns (1) and (2) are taken from the first table. The start of service time given in column (3) depends on whether the preceding customer has departed the bank. It is taken as the larger value of the arrival time of the customer and the departure time of the previous customer. Column (4), the departure time, is the sum of the column (3) value and the service time for the customer given in the first table. Values for time in queue and time in bank for each customer are computed as shown in the above table. Average values per customer for these variables are 2.61 minutes and 5.81 minutes, respectively.

The above table presents a good summary of information concerning the customer but does not provide information about the teller and the queue size for the teller. To portray such information, it is convenient to examine the events associated with the situation.

The logic associated with processing the arrival and departure events depends on the state of the system at the time of the event. In the case of the arrival event, the disposition of the arriving customer is based on the status of the teller. If the teller is idle, the status of the teller is changed to busy and the departure event is scheduled for the customer by adding his service time to the current time. However, if the teller is busy at the time of an arrival, the customer cannot begin service at the current time and, therefore, he enters the queue (the queue length is increased by one). For the departure event, the logic associated with processing the event is based on queue length. If a customer is waiting in the queue, the teller status remains busy, the queue length is reduced by one, and the departure event for the first waiting customer is scheduled. However, if the queue is empty, the status of the teller is set to idle.

An event-oriented description of the bank teller status and the number of customers at the bank is given in the table below. The events are listed in chronological order. The results indicate that the average number of customers at the bank in the first 40 minutes is 1.4525 and that the teller is idle 20 percent of the time.

Event Time

Customer Number

Event Type

Number in Queue

Number in Bank

Teller Status

Teller Idle Time

0

　

Start

0

0

Idle

　

3.2

1

Arrival

0

1

Busy

3.2

7

1

Departure

0

0

Idle

　

10.9

2

Arrival

0

1

Busy

3.9

13.2

3

Arrival

1

2

Busy

　

14.4

2

Departure

0

1

Busy

　

14.8

4

Arrival

1

2

Busy

　

17.7

5

Arrival

2

3

Busy

　

18.6

3

Departure

1

2

Busy

　

19.8

6

Arrival

2

3

Busy

　

21.5

7

Arrival

3

4

Busy

　

21.7

4

Departure

2

3

Busy

　

24.1

5

Departure

1

2

Busy

　

26.3

8

Arrival

2

3

Busy

　

28.4

6

Departure

1

2

Busy

　

31.1

7

Departure

0

1

Busy

　

32.1

9

Arrival

1

2

Busy

　

33.2

8

Departure

0

1

Busy

　

35.7

9

Departure

0

0

Idle

　

36.6

10

Arrival

0

1

Busy

0.9

40

10

Departure

0

0

Idle

　

To place the arrival and departure events in their proper chronological order, it is necessary to maintain a record calendar of future events to be processed. This is done by maintaining the times of the next arrival event and the next departure event. The next event to be processed is then selected by comparing these event times. For situations with many events, an ordered list of events would be maintained which is referred to as an event file or event calendar.

There are several important concepts illustrated by the above example. We observe that at any instant in simulated time, the model is in a particular state. As events occur, the state of the model may change as prescribed by the logical-mathematical relationships associated with the events. Thus, the events define the dynamic structure of the model. Given the starting state, the logic for processing each event, and a method for specifying sample values, our problem is largely one of bookkeeping. An essential element in our bookkeeping scheme is an event calendar which provides a mechanism for recording and sequencing future events. Another point is that we can view the state changes from two perspectives: the process that the customer encounters as he seeks service (the customer's view), or the events that caused the state of the teller to change (the teller's or bank's view).

Conclusion

In this essay, we introduced the definition of operational research and went through the history and development of operational research. Job shop scheduling problem, which is one of the complex and important fields of OR, is particularly discussed. A simple example is given to illustrate how to deal with the specific situation with the method of simulation.