Management Science

Operational research, also known as operations research, is an interdisciplinary mathematical science that focuses on the effective use of technology by organizations.

Employing techniques from other mathematical sciences --- such as mathematical modeling, statistical analysis, and mathematical optimization--- operations research arrives at optimal or near-optimal solutions to complex decision-making problems.

Operations Research is often concerned with determining the maximum (of profit, performance, or yield) or minimum (of loss, risk, or cost) of some real-world objective

Work in operational research and management science may be characterized as one of three categories :

Fundamental or foundational work takes place in three mathematical disciplines: probability, optimization, and dynamical systems theory.

Modeling work is concerned with the construction of models, analyzing them mathematically, implementing them on computers, solving them using software tools, and assessing their effectiveness with data. This level is mainly instrumental, and driven mainly by statistics and econometrics.

Application work in operational research, like other engineering and economics' disciplines, attempts to use models to make a practical impact on real-world problems.

The major subdisciplines in modern operational research, as identified by the journal Operations Research, are:

Computing and information technologies

Decision analysis

Environment, energy, and natural resources

Financial engineering

Manufacturing, service sciences, and supply chain management

Policy modeling and public sector work

Revenue management

Simulation

Stochastic models

Transportation

History

As a formal discipline, operational research originated in the efforts of military planners during World War II. In the decades after the war, the techniques began to be applied more widely to problems in business, industry and society.

Since that time, operational research has expanded into a field widely used in industries ranging from petrochemicals to airlines, finance, logistics, and government, moving to a focus on the development of mathematical models that can be used to analyze and optimize complex systems, and has become an area of active academic and industrial research.

Applications of Management Science

The range of problems and issues to which management science has contributed insights and solutions is vast. It includes :

scheduling airlines, including both planes and crew.

deciding the appropriate place to site new facilities such as a warehouse, factory or fire station.

managing the flow of water from reservoirs.

identifying possible future development paths for parts of the telecommunications industry.

establishing the information needs and appropriate systems to supply them within the health service.

identifying and understanding the strategies adopted by companies for their information systems.

Management science is also concerned with so-called ”soft-operational analysis”, which concerns methods for strategic planning, strategic decision support, and Problem Structuring Methods (PSM). Therefore, during the past 30 years, a number of non-quantified modelling methods have been developed. These include:

stakeholder based approaches including metagame analysis and drama theory

morphological analysis and various forms of influence diagrams.

approaches using cognitive mapping

the Strategic Choice Approach

robustness analysis

Researchers in Management Science

Russell L. Ackoff- He was a pioneer in the field of operations research, systems thinking and management science. He started his career in Operations Research at the end of the 1940s.

Anthony Stafford Beer-was a British theorist, consultant and professor at the Manchester Business School. He is best known for his work in the fields of operational research and management cybernetics. Stafford Beer worked in the fields of operational research, cybernetics and management science. He had become aware of operational research while being in the army, and he was quick to identify the advantages it could bring to business.

Alfred Blumstein- He is known as one of the top researchers in criminology and operations research. Blumstein's research centers around modeling criminals,careers, deterrence, prison population, transportation analysis, drug-enforcement policy, and he developed "lambda" in criminologyas a measurement of an individual's offending frequency.

West Churchman-Churchman became internationally recognized due to his then radical concept of incorporating ethical values into operating systems. He made significant contributions in the fields of management science, operations research and systems theory.

George Dantzig-Dantzig is known for his development of the simplex algorithm, an algorithm for solving linear programming problems.

Thomas L. Magnanti- Magnanti's teaching and research interests are in applied and theoretical aspects of large-scale optimization and operations research, specifically on the theory and application of large-scale optimization, particularly in the areas of network flows, nonlinear programming, and combinatorial optimization. He has conducted research on such topics as production planning and scheduling, transportation planning, facility location, logistics, and communication systems design.

Linear Programming (LP) problems can be solved on the computer using dedicated software such as WhatsBest!, solver (Excel add-on) and many others.

There are special classes of LP problems such as the Transshipment Problem (a special class of TP).

Efficient solutions methods exist to solve the Transshipment Problem.

Transshipment Problem A network model is one which can be represented by a set of nodes, a set of

arcs, and functions (e.g. costs, supplies, demands, etc.) associated with the arcs and/or nodes.

Transshipment Problem is an example of a network problem.

Transshipment Problem

Transshipment problems are transportation problems in which a shipment may move through intermediate nodes (transshipment nodes) before reaching a particular destination node.

Transshipment problems can be converted to larger transportation problems and solved by a special transportation program.

Transshipment problems can also be solved by general purpose linear programming codes.

The network representation for a transshipment problem with two sources, three intermediate nodes, and two destinations is shown on the next slide.

Transshipment Problem Network Representation

22

33

44

55

66

77

11

c13

c14

c23 c24

c25

c15

s1

c36

c37

c46

c47

c56

c57

d1

d2

INTERMEDIATE NODES

SOURCES DESTINATIONS

s2

Transshipment Problem

Linear Programming Formulation

xij represents the shipment from node i to node j

Min SScijxij

i j

s.t. Sxij < si for each source (origin) i j

Sxik - Sxkj = 0 for each intermediate i j node k (conservation of flow)

Sxij > dj for each destination j i

xij > 0 for all i and j (nonnegativity)

Useful Excel/Solver Functions: Sumproduct

SUMPRODUCTMultiplies corresponding components in the given arrays,

and returns the sum of those products.

Syntax: SUMPRODUCT(array1,array2,array3, ...)• Array1, array2, array3, ...   are 2 to 30 arrays whose

components you want to multiply and then add.

Example : Thomas & Washburn

Thomas Industries and Washburn Corporation supply three firms (Zrox, Hewes, Rockwright) with customized shelving for its offices. They both order shelving from the same two manufacturers, Arnold Manufacturers and Supershelf, Inc.

Currently weekly demands by the users are 50 for Zrox, 60 for Hewes, and 40 for Rockwright. Both Arnold and Supershelf can supply at most 75 units to its customers.

Additional data is shown on the next slide.

Example : Thomas & Washburn

Because of long standing contracts based on past orders, unit costs from the manufacturers to the suppliers are:

Thomas Washburn Arnold 5 8 Supershelf 7 4

The cost to install the shelving at the various locations are:

Zrox Hewes Rockwright Thomas 1 5 8

Washburn 3 4 4

Find the quantities to be shipped from each source to each destination to minimize the total shipping cost.

Example 1: Thomas & Washburn

Network Representation

ARNOLD

WASHBURN

ZROX

HEWES

-75

-75

+50

+60

+40

5

8

7

4

15

8

3 4

4

Arnold

SuperShelf

Hewes

Zrox

Thomas

Wash-Burn

Rock-Wright

1

2

3

4

5

6

7

Supply nodes Transshipment nodes

Demand nodes

Example : Thomas & Washburn

Linear Programming Formulation Decision Variables Defined

xij = amount shipped from manufacturer i to supplier j

xjk = amount shipped from supplier j to customer k

where i = 1 (Arnold), 2 (Supershelf)

j = 3 (Thomas), 4 (Washburn)

k = 5 (Zrox), 6 (Hewes), 7 (Rockwright) Objective Function Defined

Minimize Overall Shipping Costs:

Min 5x13 + 8x14 + 7x23 + 4x24 + 1x35 + 5x36 + 8x37

+ 3x45 + 4x46 + 4x47

Example: Thomas & Washburn

Constraints Defined

Amount Out of Arnold: x13 + x14 < 75

Amount Out of Supershelf: x23 + x24 < 75

Amount Through Thomas: x13 + x23 - x35 - x36 - x37 = 0

Amount Through Washburn: x14 + x24 - x45 - x46 - x47 = 0

Amount Into Zrox: x35 + x45 > 50

Amount Into Hewes: x36 + x46 > 60

Amount Into Rockwright: x37 + x47 > 40

Non-negativity of Variables: xij > 0, for all i and j.

Example: Thomas & Washburn problem via LP

The solver formulation is:

Ship From To Unit Cost0 1 Arnold 3 Thomas $50 1 Arnold 4 Washburn $80 2 SuperShelf 3 Thomas $70 2 SuperShelf 4 Washburn $40 3 Thomas 5 Zrox $10 3 Thomas 6 Hewes $50 3 Thomas 7 Rock-Wright $80 4 Washburn 5 Zrox $30 4 Washburn 6 Hewes $40 4 Washburn 7 Rock-Wright $4

The Transshipment Problem

Nodes Net Flow Supply/Demand1 Arnold 0 -752 SuperShelf 0 -753 Thomas 0 04 Washburn 0 05 Zrox 0 506 Hewes 0 607 Rock-Wright 0 40

Total Transportation Cost $0

Example: Thomas & Washburn problem via LP

The solver solution is:

Ship From To Unit Cost75 1 Arnold 3 Thomas $50 1 Arnold 4 Washburn $80 2 SuperShelf 3 Thomas $775 2 SuperShelf 4 Washburn $450 3 Thomas 5 Zrox $125 3 Thomas 6 Hewes $50 3 Thomas 7 Rock-Wright $80 4 Washburn 5 Zrox $335 4 Washburn 6 Hewes $440 4 Washburn 7 Rock-Wright $4

The Transshipment Problem

Supply (-) or Nodes Net Flow Demand (+)

1 Arnold -75 -752 SuperShelf -75 -753 Thomas 0 04 Washburn 0 05 Zrox 50 506 Hewes 60 607 Rock-Wright 40 40

Total Transportation Cost $1,150