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Unformatted text preview: Overview of Lecture 15.053 February 3, 2011 ! Goals
– get practice in recognizing and modeling linear
constraints and objectives More Linear and
Nonlinear Programming Models – and nonlinear objectives
– to see a broader use of models in practice plus applications to radiation therapy
Note: Read tutorials 01 and 02 on the website.
They cover material not covered in this class. 1 2 Overview on modeling Quotes for today
Reality is merely an illusion, albeit
a very persistent one. ! ! Applications to workforce scheduling ! Everything should be made as
simple as possible, but not one bit
simpler. Modeling as an art form ! Albert Einstein Modeling as a mathematical skill Applications to radiation therapy Albert Einstein, (attributed) 3 4 Formulating as an LP
Scheduling Postal Workers
! ! Let s see if we can come up with what the
decision variables should be. ! Day
Demand ! Don t look ahead. Discuss with your neighbor how one might
formulate this problem as an LP. Each postal worker works for 5 consecutive days,
followed by 2 days off, repeated weekly. ! Mon Tues 17 Wed Thurs Fri 13 15 19 14 Sat Sun 16 11 Minimize the number of postal workers (for the
time being, we will permit fractional workers on
each day.)
5 6 For students who have a new
response card today. The linear program
Day
Demand Minimize
subject to Mon Tues 17 Wed Thurs 13 15 19 Fri Sat Sun 14 16 11 z = x1 + x2 + x3 + x4 + x5 + x6 + x7
x1 + x 4 + x5 + x6 + x7 ! 17 Mon. x5 + x6 + x7 ! 13 x1 + x 2 + Tues. x6 + x7 ! 15 Wed. x1 + x 2 + x 3 +
x1 + x 2 + x 3 + x 4 +
x1 + x 2 + x 3 + x 4 + x 5 x 2 + x3 + x4 + x5 + x6 x7 ! 19
! 14
! 16 x3 + x4 + x5 + x6 + x7 ! 11 The two digit code for our receiver is 10. Thurs.
Fri. Red:
Green:
Yellow:
Yellow: Sat.
Sun. Response not received.
Response was received.
(Multiple flash) In the process of sending
(Single flash) Polling not open xj ! 0 for j = 1 to 7
7 8 On the selection of decision variables
! A Modifications of the Model A choice of decision variables that doesn t
work ! – Let yj be the number of workers on day j. Suppose that there was a pay differential. The cost of
each worker who works on day j is cj. The new
objective is to minimize the total cost. – No. of Workers on day j is at least dj. (easy to
formulate) What is the objective coefficient for the shift that starts
on Monday for the new problem? – Each worker works 5 days on followed by 2 days off
(hard).
! Conclusion: sometimes the decision variables
incorporate constraints of the problem. 1.
2. – We will see more of this in integer programming. c1 +c2 +c3 +c4 +c5 3. – Hard to do this well, but worth keeping in mind c1 c1 +c4 +c5 +c6 +c7 9 A Different Modification of the Model
! 10 Model 2 Suppose that there is a penalty for understaffing and
penalty for understaffing. If you hire k too few
workers on day j, the penalty is 5 k2. If you hire k too
many workers on day j, then the penalty is k2. How
can we model this? Minimize 5 ! 7
i =1 d i2 + ! i =1 ei2 x1 + 7 x4 + x5 + x6 + x7 + d1 – e1 = 17 x5 + x6 + x7 + d2 – e2 = 13 x6 + x7 + d3 – e3 = 15 x7 + d4 – e4 = 19 + d5 – e5 = 14 + d6 – e6 = 16 x3 + x4 + x5 + x6 + x7 + d7 – e1 = 11 x1 + x 2 +
x1 + x 2 + x 3 +
x1 + x 2 + x 3 + x 4 + Step 1. Create new decision variables. x1 + x 2 + x 3 + x 4 + x 5 Let ej = excess workers on day j x 2 + x3 + x4 + x5 + x6 Let di = deficit workers on day j xj ! 0, dj ! 0, ej ! 0 for j = 1 to 7 11 What is wrong with this model, other than the fact that variables
should be required to be integer valued?
12 More Comments on Model 2. What is wrong with Model 2? Difficulty: The feasible region permits feasible solutions that
do not correctly model our intended constraints. Let us call
these bad feasible solutions.
The good feasible solutions are ones in which d1 = 0 or e1 = 0
or both. They correctly model the scenario. 1. The constraints should have inequalities.
2. The constraints don t make sense.
3. The objective is incorrect. (Note: it is OK
that it is nonlinear) Resolution: All optimal solutions are good. 4. It s possible that ej and dj are both positive. Illustration of why it works: 5. Nothing is wrong. 10 + 10 + 0 + 0 + 0 + d1 – e1 = 17 e1 = 4 and d1 = 1 is a bad feasible solution.
e1 = 3 and d1 = 0 are good feasible solution.
For every bad feasible solution, there is a good feasible
solution whose objective is better. 13 More on the model 14 Nonlinear Programs ! Usually: every feasible solution to our
math model (LP) corresponds to a
feasible solution to our managerial
model. Such solutions are good. ! An optimization problem with a single objective
and multiple constraints. ! Occasionally, we will permit bad
feasible solutions provided that the
LP solver will never select one. The
LP solver won t select it because for
every bad feasible solution, there is a
good solution with lower cost. ! Linear programs are a special case. ! We will see this technique more in this
lecture, and in other lectures as well.
15 16 Examples of
Nonlinear Objective
Functions ! Min Max Min ! ! " 7 ( x j )2
j =1
5 Cos (e j ) 7
j =1 ! ! dj 7
j =1 On Nonlinear Programs Examples of
Nonlinear Constraints
7
j =1 But they usually
can be solved if
the objective is
to minimize a
convex function,
and the
constraints are
linear. 5 Cos (e j ) 7
j =1 " xj ( x j )2 ! 30 In general, nonlinear programs are incredibly
hard to solve. Sometimes they are impossible to
solve. dj 7
j =1 = 13.76
t
righ he
the for t
d
Fin rithm lem
o
alg t prob
righ x j ! 13 17 Convex functions of one variable www.natasafety1st.org/posters/2001_Mission_Im... 18 Which functions are convex? A function f(x) is convex if for all x and y, the
line segment on the curve joining (x, f(x)) to
(y, f(y) lies on or above the curve.
f(x) = x2 25 f(x) = x3 for x ! 0 f(x) = x.5 Step Function whatever 20 f(x) 15
10
5 f(x) = x 0
0 5 x 10
19 Yes No 20 Other enhancements Time for a mental break 21 Math Programming and Radiation Therapy
! Math Programming and Radiation Therapy An important application area for optimization ! 22 Thanks to Rob Freund and Peng Sung for some
of the following slides ! High doses of radiation (energy/unit mass) can
kill cells and/or prevent them from growing and
dividing
– True for cancer cells and normal cells http://www.youtube.com/watch?
v=GMPaArG4CcM&feature=related
! 23 Radiation is attractive because the repair
mechanisms for cancer cells is less efficient than
for normal cells 24 Radiation Imaging
! Radiation Delivery Recent improvements in
imaging ! Improvements of delivery
of radiation ! Relatively new field:
tomotherapy – MRI
– CT Scan
– other IMRT www.ottawahospital.on.ca/sc/cancer Optimizing the Delivery of
Radiation Therapy to Cancer
patients, by Shepard, Ferris,
Olivera, and Mackie, SIAM
Review, Vol 41, pp 721744,
1999.
cherrypit.princeton.edu/mri.gif
www.spiralock.com/images/Mrimachine%201.JPG 25 http://www.psl.wisc.edu/wpcontent/themes/default/images/tomo.gif 26 Conventional Radiotherapy Use of Multileaf Collimaters
! http://pages.cs.wisc.edu/
~ferris/papers/sirevcancer.pdf multileaf
collimator
– blocks radiation
– turns a large
beam into a
focused beam welchcancercenter.org Relative Intensity of Dose Delivered 27 28 Conventional Radiotherapy Conventional Radiotherapy
! In conventional radiotherapy
– 3 to 7 beams of radiation
– radiation oncologist and physicist work
together to determine a set of beam angles and
beam intensities
– determined by manual trialanderror
process Relative Intensity of Dose Delivered
29 30 Optimization approach: the decision
variables Goal: maximize the dose to the tumor while
minimizing dose to the critical area, whatever that
means. ! First, discretize the space
– Divide up region into a 2D (or 3D) grid of pixels Critical Area
Tumor area wp = intensity weight assigned to beamlet p
for p = 1 to n;
31 32 Some linear constraints
! Create the beamlet data for each of
p = 1, ..., n possible beamlets. Dijp = unit dose delivered to pixel (i, j) by beamlet p. Determine objective function(s) and
constraints that limit the solution. Dij = dosage delivered to pixel (i, j) What suggestions do
you have on how the
objective might be
formulated. Should
we consider additional
constraints? Dij = ! p=1 Dijp w p
n wp ! 0 Or what questions do
you have? Imagine
that there is an
expert on radiation
therapy that can
answer any question. for all p
33 34 Lower bounds on radiation level for tumors. upper
bounds on radiation level for critical region. Questions and Suggestions – dosage over the tumor area will be at least a
target level !L .
– dosage over the critical area will be at most a
target level !U. Dij " ! L for ( i , j ) # T Dij " ! U for ( i , j ) # C Difficulty: these LPs are rarely feasible. 35 In an example reported in the 1999 paper, there
were more than 63,000 variables, and more than
94,000 constraints (excluding upper/lower bounds) 36 Results from LP optimization Dij ! .9 for ( i , j ) "T More LP Results Max beam weight / min beam weight " 5 .9 ! Dij ! 1.1 for ( i , j ) "T
37 What happens if the model is infeasible?
minimize ! y
Dij
( i , j ) ij Dij = ! p=1 Dijp w p
n 38 An even better model Allow the constraint for
pixil (i,j) to be violated by
an amount yij, and then
minimize the violations. minimize ! jD
! ( i (,(iji,),j()) yyijijij) 2 Dij = ! p=1 Dijp w p minimize the sum of
squared violations. n DDij " ij L ! L forf( i , j ) #jT# T
or ( i , )
ij + y ! " DDij " ij L ! L forf( i , j ) #jT# T
or ( i , )
ij + y ! " Dij ij " ! U ! U for f(or j()i#jC$ C
D " yij #
i, , ) Dij ij " ! U ! U for f(or j()i#jC$ C
D " yij #
i, , ) wp ! 0
yij ! 0 wp ! 0
yij ! 0 for all p
for all ( i , j )
39 Least
squares for all p
for all ( i , j ) This is a nonlinear program (NLP). This one can
be solved efficiently. 40 Optimal Solution with 7 beams Optimal Solution with 15 Beams 41 Optimal Solution with 71 Beams 42 Is that the end of the story on modeling?
! ! 43 No. Models can almost always be enhanced. But this was a huge step over doctors
determining the beam angles. 44 Modeling as
a design process A Closer Look at the Constraint Matrix x1 x2 x3 x4 x5 x6 x7 1 0 0 1 1 1 1 ! 17 Mon 1 1 0 0 1 1 1 ! 13 Tues 1 1 1 0 0 1 1 ! 15 Wed 1 1 1 1 0 0 1 ! 19 Thurs 1 1 1 1 1 0 0 ! 14 Fri 0
0 1
0 1
1 1
1 1
1 1
1 0
1 !
! 16
11 Sat
Sun It is cyclically repeating in both rows and columns.
45 46 ...
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 Spring '11
 JamesOrlin
 Management

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