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Unformatted text preview: MSE2008
Operational Research Techniques
Venue KKLG102 ®
Day Friday
Time 10:30AM~12:25PM
Instructor Dr. Miao Song
Ofﬁce: HWBB Tel: 28597966
Email: msong@hku.hk Course Website HKU Portal/MyWebCT Method of Assessment I Homework 10% Eli will give you one problem set which
accounts for 5% I Final Exam 90% Eli will give 4 exam questions
DYou need to choose 2~3 from the 4 questions 1121:9011 Syllabus I Linear Programming
i:i Formuiaiion oflinear programming
El Geometry of linear programming
El Simplex method
o Duality theory
a Sensitivity analysis I Network Flow El Networkﬂowformulation
i:i Minimumcostﬂow I Integer Programming
E] Formuiat‘ion of integer programming
El Branch and bound
El Cutting plane I Dynamic programming
El Formulation of dynamic programming
El Shortest path
El inventory control #2112011 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 11219011 Applications of LP I Microeconomics and Management
UPIanning
[JProduction
EiTransportation
El I Approximation to Nonlinear Models 1.9112011 17 David’s Tool Corporation (DTC) I Manufacturer of slingshots kits and stone shields Stone Resources
Kits Shields Stone Gathering Time 100 hours Stone Smoothing 60 hours Delivery Time 50 hours Demand Profit (Shekels) 112112011 19 Agenda I Terminologies of Linear Programming
I A Production Problem
I A Scheduling Problem I Piecewise Linear Convex Objective
Functions 112112011 18 A Production Problem I A firm produces n different goods using m different
resources I bi, i= : the available amount of the ith
resource I cij—m 1,...,n : the unit profit of thejth good
I am: 1,._.,n : the maximum demand ofjth good I c1311: ,...,m,j=1,...,n : all 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 Q 3S5 0:43 $6.18: WWW acct: Egg? Jﬂﬁwmmﬂ mg. L 23: . ‘ ‘1 59 Ski do S319 h! X._.+r\
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v. T. mu quuﬁkv mow 0533‘ iv $33 P Sqoxw +3 IMSE2008 Operational Research Techniques ReVieW Of LeCtUre 01/21 Lecture 01/28 ® I What is Linear Programming? . I Formulation of LP
LP F ormUIatlon EIDecision variables Geometry of Linear Programs UObjective function
DConstraints
Miao Song I A production problem
Dept of Industrial & Manufacturing l A SChedUling PFObiem (t0 be COHtinued) Systems Engineering 112512011 Scheduling Postal Workers LP for the Scheduling Problem mm x1+x2 +x3+x4 +x5+x6 +x7
I Each postal worker works for 5 consecutive days, followed by 2 days off,
repeated weekly. x1+x2 +x5+x6 +2:7 213 “x2 “M + x7 215
xl+x2 +x3+x4 +x5 214 I Minimize the number of postal workers (for x2 + x3+x4 +154ch6 216
the time being, we will permit fractional
workers on each day.) 8.1;. x1+x4 +Jcs+x6 «Hr7 217 x+xz +x3+x4 +2:7 219 Jr3+x4 +x5+x6+x7 211
3:1. 20 forj=1to7 1I281‘2011 3 11289011 Convex Functions in A functionﬂx) is convex if for all x and y, the line
segment on the curve joining (x,ﬂx)) to 0am)» lies above
the curve.
25 20 15 ‘10 112812011 9 The Max of Several Linear
Functions is Convex 13(36): 3
fix) = x/3
f3(x) = 5 —x/2 ﬁx) = maX{fi(x)afz(x)af3(x)} 1123:2011 1‘ Which Functions Are Convex? ﬁzxxu ﬁx) = I) r“ 1‘1 ﬁx) 3 x3, x 2 0
ﬁx) = x1”, x 2 0 Step Function whatever
1i25i2011 10 Piecewise Linear Convex
Functions I Let x be a vector of decision variables I Letﬂx) be a linear function in x for i=
1,...,m n A function of the form max{ﬁ(x) : i: 1,...,m}
is called a piecewise linear convex
function. 1I281‘2011 12 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? ﬂooxo “2812011 19 Convex Sets
I A set S is convex if for every two points in the set, the line segment joining the points
is also in the set (/y X 112812011 18 More on Convexity I is the feasible region of a linear program
convex? I Is S= {x :ﬂx) 5 0} convex ifﬂx) is convex? I Is S: {x :f(x) 2 0} convex ifﬂx) is
concave? 1l'281'2011 2D Piecewise Linear Constraints I Let x be a vector of decision variables I Letﬁx) be a linear function in x fori=
1,...,m " max{ﬁ(x)ti=1,...,m}s 0 is equivalent to
J?(X)50forany i: I min{fi(x) :i=1,...,m}2 O is equivalent to
130020 for any 1': 1,...,m 1.9812011 21 Agenda I Formulation of Linear Programming I Geometry of Linear Programming
ElSolving a 2dimensional LP El Geometric Properties of Feasible Set and
Optimal Solutions ‘UZBIZOﬁ 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? 1I2812011 22 A Two Variable LP
(a variant of the DTC example) max 2 = 3x + 5y objective
s.t. 2x+3yS1O (1) x + 2y 5 6 (2) x + y s 5 (3) x s 4 (4) y s 3 (5) x,y a 0 (6) 11'281'2011 24 Graphing the Feasible Region Constraint (1): 2x + By s 10 i An inequality constraint for a 2dimentional LP
. = : 1 , determines a unique """"" """"" """"" """"" """"" halprlane 'i 112812011 25 Graphing the Feasible Region Constraint (6): x, y 2 0 """ ":‘ """""" ‘i “““““ "1 """"" """"" “: 112312011 27 Graphing the Feasible Region Constraint (2): x + 2y 5 6 1.128120“ Graphing the Feasible Region Constraint (4): x s 4 11281'2011 26 25 How to maximize 3x + 5y? What is the vaiue of 3x + By on the green line?
What is the value of 3x + By on the blue line?
What is the value of 3x + 5y on the purple line? Can we push the isoquant line
above the purple line? 33 Summary I Formulation of Linear Programming
El Convex and Concave Funttions
El Convex Sets I:i Linearize piecewise linear objective /
constraints I Geometry of Linear Programming
1:] Geometric Method to Solve LP
‘3
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 Can we apply the geometric
method for an LP with 3
variables? 3x+5y=16 ...
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This note was uploaded on 12/09/2011 for the course IMSE 0301 taught by Professor Song during the Spring '11 term at HKU.
 Spring '11
 SONG

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