operations research management science ( 경영과학 ) 2014 년 1 학기 담당교수 : 윤 석 훈...

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Operations ResearchManagement Science ( 경영과학 )

2014 년 1 학기

담당교수 : 윤 석 훈

산업공학개론

담당교수 : 윤석훈산업공학개론 (2014/S) 2

Operations Research?

The beginning of the activity called operations research has generally been attributed to the military services early in World War II. Because of the war effort, there was an urgent need to allocate scarce resources to the various military operations and to the activities within each operation in an effective manner.



Therefore, the British and then the U.S. military management called upon a large number of scientists to apply a scientific approach to dealing with this and other strategic and tactical problems. In effect, they were asked to do research on (military) operations. These teams of scientists were the first OR teams.



By developing effective methods of using the new tool of radar, these teams were instrumental in winning the Air Battle of Britain. Through their research on how to better manage convoy and antisubmarine operations, they also played a major role in winning the Battle of the North Atlantic. Similar efforts assisted the Island Campaign in the Pacific.


World War II


Submarine

A Russian Navy Thphoon-class submarine

http://upload.wikimedia.org/wikipedia/commons/e/e1/Typhoon3.jpg


Submarine


Under the sea

http://upload.wikimedia.org/wikipedia/commons/b/b0/Oceanic_divisions.svg


Missile


Missile

1) Targeting and/or guidance

2) Flight system

3) Engine

4) Warhead

http://upload.wikimedia.org/wikipedia/commons/b/be/USAF_B-2_Spirit.jpg


Elevator Problem


Federal Express

Federal Express (FedEx) is the world’s largest express transportation company. Every working day, it delivers more than 6.5 million documents, packages, and other items throughout the United States and more than 220 countries and territories around the world. In some cases, these shipments can be guaranteed overnight delivery by 10:30 A.M. the next morning.


Federal Express

The logistical challenges involved in providing this service are staggering. These millions of daily shipments must be individually sorted and routed to the correct general location (usually by aircraft) and then delivered to the exact destination (usually by motorized vehicle) in an amazingly short period of time. How is all this possible?


Federal Express

Operations research (OR) is the technological engine that drives this company. Ever since its founding in 1973, OR has helped make its major business decisions, including equipment investment, route structure, scheduling, finances, and location of facilities. After OR was credited with literally saving the company during its early years, it became the custom to have OR represented at the weekly senior management meetings and, indeed, several of the senior corporate vice presidents have come up from the outstanding FedEx OR group.


Federal Express

FedEx has come to be acknowledged as a world-class company. It routinely ranks among the top companies on Fortune Magazine’s annual listing of the “World’s Most Admired Companies.” It also was the first winner (in 1991) of the prestigious prize now known as the INFORMS Prize, which is awarded annually for the effective and repeated integration of OR into organizational decision making in pioneering, varied, novel, and lasting ways.


Scheduling

MC1

MC2

B D E C A

B D E C A

MC1

MC2

BDECA

BDECAIdle time

Minimize makespan

Optimal Schedule


Location Problem


Optimization

4. Shortest Path Problem

2. Minimum Cost Network Problem

1. Traveling Salesman Problem

3. Maximal Flow Problem

source sink


Transportation Problem

source destination

The transportation problem is concerned with distributing any commodity from any group of supply centers, called sources, to any group of receiving centers, called destinations, in such a way as to minimize the total distribution cost.


Transportation Problem

..ts 111211 sxxx n

222221 sxxx n

mmnmm sxxx 21sup

ply…

112111 dxxx m

222212 dxxx m

nmnnn dxxx 21

dem

and

…

0 1,... , 1,...,ijx i m j n cij: unit transportation cost from i to j

xij: number of items from i to j

1 1

minimize m n

ij iji j

c x


Continental Airlines

Continental Airlines is a major U.S. air carrier that transports passengers, cargo, and mail. It operates more than 2,000 daily departures to well over 100 domestic destinations and nearly 100 foreign destinations.



Airlines like Continental face schedule disruptions daily because of unexpected events, including inclement weather, aircraft mechanical problems, and crew unavailability. These disruptions can cause flight delays and cancellations. As a result, crews may not be in position to service their remaining scheduled flights. Airlines must reassign crews quickly to cover open flights and to return them to their original schedules in a cost-effective manner while honoring all government regulations, contractual obligations, and quality-of-life requirements.



To address such problems, an OR team at Continental Airlines developed a detailed mathematical model for reassigning crews to flights as soon as such emergencies arise. Because the airline has thousands of crews and daily flights, the model needed to be huge to consider all possible pairings of crews with flights. Therefore, the model has millions of decision variables and many thousands of constraints. In its first year of use (mainly in 2001), the model was applied four times to recover from major schedule disruptions (two snowstorms, a flood, and the September 11 terrorist attacks). This led to savings of approximately $40 million. Subsequent applications extended to many daily minor disruptions as well.



Although other airlines subsequently scrambled to apply operations research in a similar way, this initial advantage over other airlines in being able to recover more quickly from schedule disruptions with fewer delays and cancelled flights left Continental Airlines in a relatively strong position as the airline industry struggled through a difficult period during the initial years of the 21st century. This initiative led to Continental winning the prestigious First Prize in the 2002 international competition for the Franz Edelman Award for Achievement in Operations Research and the Management Sciences.


Assignment Problem

The assignment problem is a special type of linear programming problem where assignees are being assigned to perform tasks.


Hungarian Method

1. Subtract the smallest number in each row from every number in the row. (This is called row reduction.) Enter the results in a new table.

2. Subtract the smallest number in each column of the new table from every number in the column. (This is called column reduction.) Enter the results in another table.


Hungarian Method

3. Test whether an optimal set of assignments can be made. You do this by determining the minimum number of lines needed to cover (i.e., cross out) all zeros. Since this minimum number of lines equals the maximum number of assignments that can be made to zero element positions, if the minimum number of lines equals the number of rows, an optimal set of assignments is possible. (If you find that a complete set of assignments to zero element positions is not possible, this means that you did not reduce the number of lines covering all zeros down to the minimum number.) In that case, go to step 6. Otherwise go on to step 4.


Hungarian Method

4. If the number of lines is less than the number of rows, modify the table in the following way:a. Subtract the smallest uncovered number from

every uncovered number in the table.b. Add the smallest uncovered number to the

numbers at intersections of covering lines.c. Numbers crossed out but not at the intersections

of cross-out lines carry over unchanged to the next table.

5. Repeat steps 3 and 4 until an optimal set of assignments is possible.


Hungarian Method

6. Make the assignments one at a time in positions that have zero elements. Begin with rows or columns that have only one zero. Since each row and each column needs to receive exactly one assignment, cross out both the row and the column involved after each assignment is made. Then move on to the rows and columns that are not yet crossed out to select the next assignment, with preference again given to any such row or column that has only one zero that is not crossed out. Continue until every row and every column has exactly one assignment and so has been crossed out. The complete set of assignments made in this way is an optimal solution for the problem.


Hungarian Method

min 1– 1– 2 1 1


Hungarian Method

Subtract row’s minimum

min 0 0 1 0 1


Hungarian Method

Subtract column’s minimum

The minimum uncovered element is 1.


Hungarian Method

Modify the reduced matrix:

The minimum uncovered element is 1.


Hungarian Method

Optimal solution: (1,2), (2,1), (3,5), (4,4), (5,3)

The associated optimal cost is 3 – 1 + 0 + 1 + 2 = 5.


Assignment Problem

1 1

minimize n n

ij iji j

c x

..ts 11 12 1 1nx x x

21 22 2 1nx x x

1 2 1n n nnx x x

…

11 21 1 1nx x x

12 22 2 1nx x x

1 2 1n n nnx x x

…

0 or 1 1,... , 1,...,ijx i n j n cij: assignment cost from i to j

xij = 1 if assign job i to machine j


Swift & Company

Swift & Company is a diversified protein-producing business based in Greeley, Colorado. With annual sales of over $8 billion, beef and related products are by far the largest portion of the company’s business.


Swift & Company

To improve the company’s sales and manufacturing performance, upper management concluded that it needed to achieve three major objectives. One was to enable the company’s customer service representatives to talk to their more than 8,000 customers with accurate information about the availability of current and future inventory while considering requested delivery dates and maximum product age upon delivery. A second was to produce an efficient shift-level schedule for each plant over a 28-day horizon. A third was to accurately determine whether a plant can ship a requested order-line-item quantity on the requested date and time given the availability of cattle and constraints on the plant’s capacity.


Swift & Company

To meet these three challenges, an OR team developed an integrated system of 45 linear programming models based on three model formulations to dynamically schedule its beef-fabrication operations at five plants in real time as it receives orders. The total audited benefits realized in the first year of operation of this system were $12.74 million, including $12 million due to optimizing the product mix. Other benefits include a reduction in orders lost, a reduction in price discounting, and better on-time delivery.


Linear Programming

..ts 11 1 12 2 1 1n na x a x a x b

21 1 22 2 2 2n na x a x a x b

1 1 2 2m m mn n ma x a x a x b

…

0 1,...,jx j n

cj: cost of each unit increase in level of activity j

xj: level of activity j

1

minimize n

j jj

c x

aij: amount of resource i consumed by each unit of activity j

bi: amount of resource i that is available for allocation to activities


Science

Solutions

Real World

Assumed Real World

RepeatedlyManyThe Same

operations research management science ( 경영과학 ) 2014 년 1 학기 담당교수 : 윤 석 훈...

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