or1.nf.v3

or1.nf.v3 - |MSE2008 Operational Research Tech n iq u es...

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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: HWB-B Tel: 28597966 Email: msong@hku.hk 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... . SSE 91.3 e» _ WWW REY? .3:va 093.. l 9:3: . méwo so 56; fl. $538.3”. .Sfiv wad. 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ML W My, €353? ? :53? $750.35 f N - F. X“ We fil§?g£g‘&um E. [q 7%.? fig §§.§.% Bag €385 Strap; & m??? A $2.23 a 51ng .w 09326 553%? 3 P SN 9n... .3”. Ru\fis§§?ofi>\m U3 Hcr 5: We um. v\ ,1; ,3 .3 we (w 935:? 85920 .. .. 3 $953.3 +90 W 2532? “fig 1.430 a 09.295 osfiv 35$. EEK nimiiso yixv 175453 miymxicéésé 3305...? fig 9:09;? 093:3 <69 (:5. mwéc. c; E? gxmfiflkagwuw (Illl/Xlrrlk (S...) #1 $92 w w M £03 w mfg 5 mg- gfiwx $9.0 QC$$$Q Mv .W <s>xmx:xff«fxflxiafid 1 .3...) Mx. i. 5w MW Oas<exmed H .\ _. 3 +5? $§§r<x Rem?) q.» ? rw C§<ox~ v. 7 w" M xvii mew. 92(2 AH .33 .w 926$ ‘ .35 <10 F mu My: .PNVVow 092$.“ .er +4.5 w oat/gab s): n V .m sisn v+ w .\ ,3 m. :3 5:. Via .<_.m.x__+w Kn. .m. ESFX. . . Krvfw Xfi qu+w K. .unxw+.w. I: m 53.5» u+u x. 39% §EQ:+L m 0 WW 7.5.3? 952.33% lMSE2008 Operational Research Techniques ReVieW Of LeCture 01/21 Lecture 01/28 ® I What is Linear Programming? . I Formulation of LP LP FormUIatlon DDecision variables Geometry of Linear Programs DObJ-ecflvefuncfion 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, non-linear 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 2-dimensional 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 half-plane ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, ,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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or1.nf.v3 - |MSE2008 Operational Research Tech n iq u es...

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