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Unformatted text preview: MSE2008
Operational Research Tech n iq u es
Venue KKLG102 6)
Day Friday
Time 10:30AM~12:25PM
Instructor Dr. Miao Song
Ofﬁce: HWBB Tel: 28597966
Email: [email protected] Course Website HKU PortallMyWebCT Method of Assessment I Homework 10% El will give you one problem set which
accounts for 5% I Final Exam 90% E! I will give 4 exam questions
EIYou need to choose 2~3 from the 4 questions 1I21r'2011 3 Syllabus I Linear Programming
El Formulation of linear programming
Cl Geometry of linear programming
a Simplex method
I] Duality theory
D Sensitivity analysis
I Network Flow
CI Network flow formulation
:1 Minimum cost ﬂow
I Integer Programming C! Formulation ofinteger programming
[3 Branch and bound
El Cutting plane I Dynamic programming
El Formulation ofdynamic programming
El Shortest path
El Inventory control 1.0112011 References I D. Bertsimas and J. Tsitsiklis, introduction
to Linear Optimization, Athena Scientific I D. Bertsikas, Dynamic Programming and
Optimal Control Vol. 1, Athena Scientiﬁc 11212011 Terminology I Decision variables III in general, these are the quantities you can
control to improve your objective which should
completely describe the set of decisions to be
made I Constraints in Limitations on the values of decision variables
I Objective Function EIValue measure used to rank alternatives El Seek to maximize or minimize this objective [2i Examples: maximize proﬁt, minimize cost 112112011 13 More on Linear Programs I A linear program must have linearlobjectives and
linear equalities and linear inequalities to be
consrdered a linear program I Are the following problems linear programs? max 3:, o 7
mm x‘
SI. 3xl + 4x2 2. 3
51. x = 3
xI — 2x4 = 2
Ix3 2 0 I it is not a linear program if any of the linear inequalities
are strict, as in x > 0 11211‘2011 15 Linear Programs I A linear function is a function of the form f(x1~x2="'9xza)=cixi +sz2+m+cnxn i‘?
= Z ext in]
8.9., 3x1+ 2x2 — 5x4
I A mathematical program is a linear program (LP) if
the objective is a linear function and the constraints are linear equaiities or inequalities
e.g., 3x1+ 2x2 s 5x4 3 7, x, "— 5x5 = 3 “2112011 14 History of LP I World War II DPian expenditures and returns to reduce costs
to the army and increase losses to the enemy I George Dantzig, Simplex Method, 1947
I Leonid Khachiyan, Ellipsold Method, 1979 I Narendra Karmarkar, Interior Point
Method, 1984 U21I2D11 16 Applications of LP I Microeconomics and Management
1:1 Planning
DProduction
[JTransportation
:3... I Approximation to Nonlinear Models 112112011 17 David’s Tool Corporation (DTC) I Manufacturer of slingshots kits and stone shields Stone Resources
Kits Shields 100 hours 2hours 60 hours schours 1I211'2Cl11 19 Agenda I Terminologies of Linear Programming
I A Production Problem
I A Scheduling Problem I Piecewise Linear Convex Objective
Funcﬁons “2112011 18 A Production Problem I A firm produces n different goods using 111 different
resources I 52,, i= 1,....m : the available amount of the ith
resource I cj,j= : the unit profit of thejth good
I a},j= 1,...,n : the maximum demand ofjth good I a”, i=1,...,m,j=1,...,n:ay units of the ith resource
is required to produce one unit of the jth good I What is the LP to maximize the profit? 112112011 20 Agenda I Terminologies of Linear Programming
I A Production Problem
I A Scheduling Problem I Piecewise Linear Convex Objective
Functions 1121f2011 21 On the Selection of Decision
Variables I Letyj denote the number of workers on dayj El How to formulate the constraint that # of number
of workers on dayj is at least a}? [1 How to formulate the constraint that each worker
works 5 days on and followed by 2 days off? I Sometimes the decision variables incorporate
constraints of the problem.
El Hard to do this well, but worth keeping in mind
I] We will see more of this in integer programming 112112011 23 Scheduling Postal Workers I Each postal worker works for 5
consecutive days, followed by 2 days off,
repeated weekly. I Minimize the number of postal workers (for
the time being, we will permit fractional
workers on each day.) “21.12011 22 Some Modiﬁcations of the Model I Suppose that there was a pay differential.
The cost of workers who start work on day
j is C}. per worker. I Suppose that one can hire part time
Workers (one day at a time), and that the
cost of a part time worker on dayj is p}. I Suppose that we need to ensure that at
least 30% of the workers have Sunday off. 112112011 24 _ 1 Y... .
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LP FormUIatlon DDecision variables Geometry of Linear Programs DObJecﬂvefuncﬁon
El Constraints
Miao Song I A production problem
Dept of industrial & Manufacturing I A scheduling problem (to be continued) Systems Engineering #2812011 Scheduling Postal Workers LP for the Scheduling Problem mm 361+):2 +x3+x4 +x5+x6 +267
I Each postal worker works for 5  >
consecutive days, followed by 2 days off, S't' xi+x4 +x5+x6 “‘7 ~17
repeated weekly. x1+x2 +x5+xﬁ +9:7 213 1 21
Wed Thu Fri Sat Sun x1+xz+x1+x6+x7 5
15 19 14 16 11 Xi+xg+X3+X4+X7219 x1+x2 +x3+x4 +x5 214 I Minimize the number of postal workers (for x2 +x3+x4 +x5+x6 216 the time being, we will permit fractional
workers on each day.) x3+x4+xs+x6+x7 211
ijOforjzlto7 11281'2011 3 1I28f2011 Some Modifications of the Model I Suppose that we need to ensure that at
least 30% of the workers have Sunday off. 112512011 A Motivating Example I Suppose that the desirable number of
workers on dayj is but it is not required.
Let sj be the “excess number of workers day
j. s]. > 0 if there are more workers on dayj
than a9; otherwise SJ. 5 0. I What is the minimum cost schedule, where
the “cost” of having too many workers on day j isﬁsj), which is a non—linear function? I What are the new decision variables?
I What is the resulting nonlinear model? 1f28I2011 Agenda I Formulation of Linear Programming
I:I Piecewise Linear Objective Functions
I:I Piecewise Linear Constraints I Geometry of Linear Programming 1J'281'2D11 On Nonlinear Functions I Occasionally a nonlinear program can be
transformed into a linear program I Rare, but useful when it occurs I In general, nonlinear programming
solvers can work well on a minimization
problem when the objective function is
convex 112512011 Convex Functions I A function f(x) is convex if for all x and y, the line
segment on the curve joining (x,f(x)) to (tngJD lies above the curve.
25 112812011 9 The Max of Several Linear
Functions is Convex f1(x) = 3
f2(x) :x/3
f3(x) = 5 *x/Z ﬁx) : maxtﬁlxlaﬂfoixH 1I28f2011 11 Which Functions Are Convex? swig ﬂxl:lxl ﬁx)le ﬂxﬁxﬁxzo
ﬂx} = x“, x 2 0 Step Function whatever
11‘2812011 10 Piecewise Linear Convex
Funcﬁons I Let x be a vector of decision variables I Letﬂx) be a linear function in x for i:
1,...,m I A function of the form max{ﬁ(x) : i= 1,...,m}
is called a piecewise linear convex
function. 112812011 12 Minimizing Piecewise Linear
Convex Function I Suppose that F is the set of feasible solutions for
some linear programming problem I Then the nonlinear optimization problem min zemax 1x) : iz 1,...,m}
st. K II] the eamble set F is equivalent to
min 2
s.t. 2312 x) for an i=1,...,m
2: mt e feasrb 6 set F 112312011 13 Concave Functions I A function ﬁx) is concave iff —f(x) is convex
I What is a piecewise linear concave function? A function of the form mi11{ﬂ(x) : i= 1,...,m} is called a piecewise linear concave function,
whereﬂx) is a linear function in xfor i :
1,...,m I Can we linearize the objective function
max 2 = min{ﬁ(x) : i= 1,...,m}? 112812011 15 Back to the Scheduling Problem... I Suppose $31.) = lsjl for any j
I How do we modify it to make it linear? I Minimize the maximum number of excess
workers needed on any day I Shortage of workers is not allowed
I Can we model it as an LP? 1i28l2011 34'» Agenda I Formulation of Linear Programming
DPiecewise Linear Objective Functions
UPEecewise Linear Constraints I Geometry of Linear Programming 1i28l'2011 15 Motivating Example I Back to the scheduling problem... I Suppose that we want the number of
workers starting on Mon, Tue, Wed, Thu,
Fri and Sat to be almost the same, e.g.,
the difference is less than 3 I What are the additional constraints? 112812011 17 Which Sets Are Convex? l ‘.I ‘i'
.9; i._ . ' it.
‘1‘;
:3! 1 I 7.
Id “2812011 19 Convex Sets
I A set S is convex if for every two points in the set, the line segmentjoining the points
is also in the set (/Y X “2812011 18 More on Convexity I Is the feasible region of a linear program
convex? I Is S: {x :ﬂx) S 0} convex ifﬂx) is convex? I Is S: {x :f(x) 2 0} convex ifﬂx) is
concave? 1f28r’2011 20 Piecewise Linear Constraints I Let x be a vector of decision variables I Letﬁ(x) be a linear function in x for 1':
1,...,m I max{ﬁ(x) : i=1,...,m} s 0 is equivalent to
ﬁ(x)$0 for any i: 1,...,m I min{ﬁ(x) : i= 1,...,m} 2 0 is equivalent to
1305020 for any i: 1,...,m 1128i2011 21 Agenda I Formulation of Linear Programming I Geometry of Linear Programming
USolving a 2dimensional LP El Geometric Properties of Feasible Set and
Optimal Solutions 112812011 23 Motivating Example I Back to the scheduling problem... I Suppose that we want the number of
workers starting on Mon, Tue, Wed, Thu,
Fri and Sat to be almost the same, e.g.,
the difference is less than 3 I What are the additional constraints?
I Can we linearize the constraints? “28.9011 22 A Two Variable LP
(a variant of the DTC example) max 2 = 3x + 5y objective
s.t. 2x+3ys 10 (1) x + 2y 5 6 (2) x + y s 5 (3) x s 4 (4) y s 3 (5) x,y 2 O (6) 1rzarzu11 24 Graphing the Feasible Region Constraint (‘1 )1 2x + 3y S ‘10 I An inequality constraint for a 2—dimentional LP
l determines a unique """"" """""" ””””” """"" """"" ‘1 halfplane ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, ,1 #282011 25 Graphing the Feasible Region ‘ Constraint (6): x, y 2 0 """ "" """"" '3 """""" "1 """"" ““““ ": 11281'2011 27 Graphing the Feasible Region Constraint (2): x + 2y S 6 “28.0011 Graphing the Feasible Region ' Constraint (4): x s 4 1128f2011 2E! 23 Graphing the Feasible Region Constraint (3): x + y s 5 : Aconstraint is redundant i if deleting the constraint does not increase the size of the feasible region 112312011 29 How to maximize 3x + 5y? Let’s avoid adding a 3rd dimension. Find feasible solutions such that 3x + By = p
for different values of p ___._’_______C_l_1_oos_e_pas large as possible 11282011 31 Graphing the Feasible Region Constraint (5): y s 3 x Is this constraint Is the feasible region
‘ unvex? ................... .u 112312011 30 How to maximize 3x + 5y? Is there a feasible solution s.t. 3x + 5y = O? How to improve the solution? Any feasible solution above the line 3x + 5y = 0
satisfies 3x + 5y > 0 Parallelly push the isoquant line 32 How to maximize 3x + 5y? What is the value of 3x + 5y on the green line?
What is the value of 3x + 5y on the blue line?
What is the value of 3x + By on the purple line? Can we push the isoquant line
above the purple line? 33 éummaw I Formulation of Linear Programming
El Convex and Concave Functions
El Convex Sets :1 Linearize piecewise linear objective /
constraints I Geometry of Linear Programming
El Geometric Method to Solve LP At
“a,” 112812011 35 How to maximize 3x + 5y? What is optimal solution of the LP? Why?
The optimal solution occurs at a corner point This is called the geometric method for
optimizing in 2D Can we apply the geometric
method for an LP with 3 variables? 3x+5y=16 ...
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 Spring '11
 SONG
 Linear Programming, Optimization

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