operational research - lecture: introduction to or

Operational ResearchLecture:Introduction to OR& Linear ProgrammingPatrick Meyer& Mehrdad MohammadiIMT Atlantique

Part 1 : Operational Research?

OR: Intro & LPP. Meyer & M. Mohammadi

OR 3Operational Research or Decision Science

- The systematic study of how to solve problems in business andindustry (Cambridge Dictionary) ;

- The use of mathematical models, statistics or algorithms to aidin decision making (Wikipedia) ;

- Approche scientifique pour la resolution de problemes de gestionde systemes complexes (Roadef).


OR 4Operational Research

- Scientific methods to solve optimization problems linked to the realworld ;

- Between mathematics and computer science :- continuation of algorithmics ;- elaborate structures : graphs, polyhedrons, . . .- possible inclusion of human preferences in the models.

- A toolbox for optimization problems.


OR 5ApplicationsThe traveling salesman problem (TSP)

- A salesman based in Luxembourg city has to travel to all hisclients in Luxembourg ;

- He wishes to make the shortest tour possible.


OR 6ApplicationsThe traveling salesman problem (TSP)

- Problem instance : n cities and a distance matrix ;- Solution : a tour through all the cities and returning to

Luxembourg.


OR 7Applications


OR 8Applications


OR 9Applications


OR 10Applications


OR 11Applications


OR 12Applications


OR 13Applications


OR 14Applications


Part 2 : Linear ProgrammingFormulation and graphical resolution


Linear Programming 16Goal of this part

- Formulation of “linear” optimization models :* minimize / maximize a function under some constraints* the objective function and the constraints are linear w.r.t the

variables

- Resolution techniques :graphical & intuitions on the simplex algorithm


Linear Programming 17An example

- A company building window frames 1 ;

- Wishes to build 2 new models buy using the residual capacity of its3 workshops ;

- An aluminum frame (product 1) and a wooden frame (product 2) ;

- The aluminum frame is constructed in the first workshop, and theglass is added in the third one ;

- The wooden frame is constructed in the second workshop, and theglass is added in the third one.

1. F.S. HILLIER et G.S. LIEBERMAN, Introduction to Operations Research, 6thedition, Mac Graw-Hill International Editions, Singapour, 1995.



- Unit margins, construction times and available capacities of theworkshops :



- How many frames of each type does the company have to build tomaximize the net profit?

- 3 steps :- Choice of the model’s variables ;- Formulation of the objective ;- Formulation of the constraints.


Linear Programming 20Choice of the variables

- A variable is a quantity necessary for the resolution of the problemand for which the model must determine a value ;

- A parameter is a given piece of data, which can vary from onescenario to another.

In the example the quantities to be determined are :- x1 = number of type 1 frames produced per week (aluminum) ;- x2 = number of type 2 frames produced per week (wood).


Linear Programming 21Formulation of the objective

- The objective function of an optimization problem is the choicecriterion which allows to differentiate between various possiblesolutions.

In the example the goal is tomaximize the net profit :

- max z = 3x1 + 5x2.


Linear Programming 22Formulation of the constraints

- The constraints of the problem are the relations which limit thepossible values of the variables.

In the example :

- Limits / boundaries onthe variables : x1, x2 ≥ 0(the produced quantitiescannot be negative) ;

- Capacity constraintslinked to the problem :x1 ≤ 4 (only 4 hours leftin workshop 1) ;

-

- 2x2 ≤ 12 ;- 3x1 + 2x2 ≤ 18.


Linear Programming 23Graphical resolution

- If the model contains only 2 variables, it is possible to solve theproblem graphically ;

- 3 steps :- graphical representation of the feasible region ;- graphical representation of the objective ;- determination of the optimal solution.


Linear Programming 24Graphical resolution

Condensed form of the example problem :


Linear Programming 25Graphical resolution : feasible region

- The feasible region is the set of values of the variables whichsatisfy all the constraints.

In the example this region corresponds to the set of points (x1, x2)verifying the following constraints :


Linear Programming 26Graphical resolution : feasible region

- An inequality 3x1 + 2x2 ≤ 18 corresponds to a semi-planedelimited by the straight line whose equation is 3x1 + 2x2 = 18 ;

In the example the intersection of all the constraints leads to the greyregion :

- All the points from the greyregion are feasible solutions ;

- Next step : choose amongthese solutions !


Linear Programming 27Graphical resolution : representation of the objective

- Determine the solution which maximizes z = 3x1 + 5x2 ;

- However : not possible to draw the objective function in the plane !

- Solution : Drawisometric lines of theobjective function forvarious values of z.


Linear Programming 28Graphical solution : determination of the optimal point

- Determine the point from the feasible region which lies on the“highest” isoline of the objective function.

In the example :

x∗ = (2;6)

- Observation :- The optimal solution is at a

vertex of the feasible region.


Linear Programming 29Graphical solution : determination of the optimal point

- Question : is the optimal solution always at a vertex of the feasibleregion?

In the example, consider a new objective max z ′ = 3x1 + 2x2 :

- The isolines of the objectivefunction are parallel with oneof the edges of the constraintspolytop ;

- Answer : Even if a wholeedge is optimal, an optimalsolution can always be foundat one of the vertexes of thefeasible region.


Linear Programming 30Intuitive resolution algorithms

- Determine all the vertexes of the polytop, evaluate the objectivefunction and determine a vertex which maximizes the objectivefunction ;

- Simplex algorithm :(1) Choose an initial vertex x∗ ;(2) Determine the edges which are connected to x∗ ;(3) Determine the edge along which z increases ; if no such edge

exists, STOP, x∗ is optimal ;(4) Determine the vertex y∗ situated on the opposite side of the edge,

set x∗ = y∗, and return to (2).


Linear Programming 31Simplex applied to the example


Linear Programming 32General formulation of a LP

- Consider that n products require the use of m resources ;- cj is the unit margin of product j and bi is the available quantity of

resource i ;- aij is the quantity of resource i used to produce a unit of product j .



- A linear program can be formulated as follows :



- Or as follows :

- where A is an (m × n) matrix,- b is an (m × 1) vector,- c is an (n × 1) vector- and x is an (n × 1) vector.


Part 3 : Linear ProgrammingDiscussion


Discussion 36Assumptions

- Proportionality- As the objective is linear, the contribution to the objective of any

decision variable is proportional to the value of the decisionvariable ;

- Producing twice as much of a product generates twice as muchprofit.

- Additivity- The contribution of a variable to the objective is independent of the

values of the other variables ;

- One notebook is worth 750 USD independent of how many desktopcomputers are produced.



- Divisibility- It is possible to take any fraction of any variable ;

- What does it mean to produce 1.62 notebooks?

- Is rounding reasonable? Yes if the quantity produced is huge(paperclips), no if it is small (battleships).

- If divisibility does not hold : integer programming.

- Certainty- LP allows no uncertainty about the numbers ;

- The number of assembly hours is given precisely and is certainlyavailable ;



- All the assumptions are rarely met exactly in practice ;

- But : Model still useful ;

- Gives useful managerial insight :

The information “The chip inventory is more than sufficient” holds,even if the proportionality assumptions are not satisfied completely.


Part 4 : What now?


What now? 40Autonomous work

- ExercisesModel decision problems as linear programsSolve certain of them graphically

- Case studyRead about the two topicsWatch videosDecide which of the two topics you would like to study


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