Lecture 6 Knapsack Problem Quadratic Assignment Problem

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Lecture 6 Lecture 6 Knapsack ProblemKnapsack Problem

Quadratic Assignment ProblemQuadratic Assignment Problem

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OutlineOutline

knapsack problemknapsack problem quadratic assignment problemquadratic assignment problem

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Knapsack ProblemKnapsack Problem

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Knapsack ProblemKnapsack Problem

very useful sub-problems very useful sub-problems booksbooks

Knapsack Problems by Hans Kellerer, Ulrich Knapsack Problems by Hans Kellerer, Ulrich Pferschy, and David Pisinger (2004) Pferschy, and David Pisinger (2004)

Knapsack Problems: Algorithms and Knapsack Problems: Algorithms and Computer Implementations by Silvano Computer Implementations by Silvano Martello and Paolo Toth (1990)Martello and Paolo Toth (1990)

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Problem StatementProblem Statement nn types of products types of products

weight of type weight of type ii: : aaii

value of type value of type ii: : ppii

capacity of truck: capacity of truck: bb kg kg question: selection of products to maximize the total value in question: selection of products to maximize the total value in

a trucka truck special tpye: 0-1 knapsack problem: special tpye: 0-1 knapsack problem: ii {0, 1} {0, 1}

1 1 2 2

1 1 2 2

max ... ,. . ... ,

non-negative integers.

n n

n n

i

p p ps t a a a b

i = number of type-i products put in the truck

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Common Knapsack ProblemsCommon Knapsack Problems

project selection problemsproject selection problems capital budgeting allocation problems capital budgeting allocation problems inventory stocking problemsinventory stocking problems ……

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Multi-Dimensional Multi-Dimensional Knapsack ProblemKnapsack Problem

nn types of products types of products weight of type weight of type ii: : aaii

volume of type volume of type ii: : vvii

value of type value of type ii: : ppii

capacity of truckcapacity of truck bb kg kg c c mm33

question: selection of products to maximize the total value question: selection of products to maximize the total value in a truckin a truck

i = number of type-i products put in the truck

1 1 2 2

1 1 2 2

1 1 2 2

max ... ,. . ... ,

... ,non-negative integers.

n n

n n

n n

i

p p ps t a a a b

v v v c

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Cutting-Stock ProblemCutting-Stock Problem

knapsack problem: a sub-problem in solving knapsack problem: a sub-problem in solving cutting-stock problem by column generation cutting-stock problem by column generation

7-meter steel pipes of a given diameter7-meter steel pipes of a given diameter orderorder

150 pieces of 1.5-meter segments150 pieces of 1.5-meter segments 250 pieces of 2-meter segments250 pieces of 2-meter segments 200 pieces of 4-meter segments200 pieces of 4-meter segments

objective: minimize the amount of trim offobjective: minimize the amount of trim off

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Formulation of Formulation of the Cutting-Stock Problemthe Cutting-Stock Problem

decisions: how to cut the pipesdecisions: how to cut the pipes three types of segmentsthree types of segments

type-1 segment: 1.5 mtype-1 segment: 1.5 m type-2 segment: 2 mtype-2 segment: 2 m type-3 segment: 4 m type-3 segment: 4 m

aaijij = the number of = the number of jj type segments produced by the type segments produced by the iith th cut patterncut pattern

the the iith cut pattern: (th cut pattern: (aaii11, , aaii22, , aaii33))TT with trim loss with trim loss ttii e.g., ae.g., a11 = ( = (aa1111, , aa1212, , aa1313))TT = (0, 0, 1) and = (0, 0, 1) and tt11 = 3 = 3

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totally 15 cutting patterns totally 15 cutting patterns

Cut Patterns

x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15

1.5-meter segment

0 1 0 2 0 2 1 0 2 1 0 4 3 2 1

2-meter segment 0 0 1 0 3 2 2 2 1 1 1 0 0 0 0

4-meter segment 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0

trim loss (meter) 3 1.5 1 0 1 0 1.5 3 2 3.5 5 1 2.5 4 5.5

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xi = the # of steel pipes cut in the ith pattern

151

151 1

151 2

151 3

min ,

. . 150,

250,

200,non-negative integers

i i i

i i i

i i i

i i i

i

t x

s t a x

a x

a xx

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Solution of Solution of the Cutting-Stock Problemthe Cutting-Stock Problem

for Simplex Method, the reduced cost of the for Simplex Method, the reduced cost of the iith th variable (i.e., the variable (i.e., the iith cutting pattern) can be th cutting pattern) can be shown to relate to a Knapsack Problemshown to relate to a Knapsack Problem

max max 11aa11 + + 22aa22 + + 33aa33, ,

ss..tt.. 1.51.5aa11 + 2 + 2aa22 + 4 + 4aa33 7, 7,

where where jj = value of the = value of the jjth dual variable of th dual variable of the current basic feasible solutionthe current basic feasible solution

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Solution of Solution of the Cutting-Stock Problemthe Cutting-Stock Problem

Formulate a LP Formulate a LP

Solve the LP to get a BFSSolve the LP to get a BFS

Get dual variables Get dual variables jj of the BFSof the BFS

Solve a knapsack problem to get a Solve a knapsack problem to get a new, valuable cutting patternnew, valuable cutting pattern

any new any new cutting cutting

pattern?pattern?

Stop. Current BFS is Stop. Current BFS is optimumoptimum

NoNo

Add a cutting pattern (i.e., Add a cutting pattern (i.e., new column) to the LPnew column) to the LP

YesYes

Resolve the LP to get Resolve the LP to get a new BFSa new BFS

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Solution of Solution of the Knapsack Problem the Knapsack Problem

dynamic program: more appropriatedynamic program: more appropriate branch and bound: not as appropriatebranch and bound: not as appropriate

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Door Assignment in a Door Assignment in a Distribution CenterDistribution Center

a DC with two-inbound and two-outbound doorsa DC with two-inbound and two-outbound doors cross-docking operations to handle goods of two suppliers and two cross-docking operations to handle goods of two suppliers and two

retailersretailers objective: to minimize the total goods-distanceobjective: to minimize the total goods-distance

Retailer 1 Retailer 2Supplier 1 77 00Supplier 2 44 66

Door a Door bDoor A 22 33Door B 33 22

Distances Among the DoorsDistances Among the Doors

Amount of Goods from Amount of Goods from Suppliers to RetailersSuppliers to Retailers

Door A

Door B

Door a

Door b

InboundDoors

OutboundDoors

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Door Assignment in a Door Assignment in a Distribution CenterDistribution Center

goods-distance for supplier goods-distance for supplier 1 1 Door Door AA, supplier 2 , supplier 2 Door Door BB, retailer 1 , retailer 1 Door Door aa, retailer 2 , retailer 2 Door Door bb

Door A

Door B

Door a

Door b

InboundDoors

OutboundDoors

Door a Door b

Door A 22 33

Door B 33 22


Retailer 1 Retailer 2

Supplier 1 77 00

Supplier 2 44 66

Amount of Goods from Amount of Goods from Suppliers to RetailersSuppliers to Retailers3

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2A

B

a

b

46

7A

B

a

b

total goods-distance total goods-distance = (2)(7) + (3)(4) + (2)= (2)(7) + (3)(4) + (2)(6)(6)formulate an optimization formulate an optimization

to find the door assignment to find the door assignment that minimizes the total that minimizes the total

goods-distancegoods-distance

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Formulation of Formulation of the Door Assignment Problemthe Door Assignment Problem

1, if supplier is assigned to inbound door , 1,2; , ;

0, . .,iji j

x i j A Bo w

1, if retailer is assigned to outbound door , 1,2; , ;

0, . .,iji j

y i j a bo w

Door a Door b

Door A 22 33

Door B 33 22


Retailer 1 Retailer 2

Supplier 1 77 00

Supplier 2 44 66

Amount of Goods from Amount of Goods from Suppliers to RetailersSuppliers to Retailers

yy11aa yy11bb yy22aa yy22bb

xx11AA 1414 2121 00 00

xx11BB 2121 1414 00 00

xx22AA 88 1212 1212 1818

xx22BB 1212 88 1818 1212

Cost Coefficients Cost Coefficients ccijklijkl, for , for xxijij = = yyklkl = 1 = 1

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Formulation of Formulation of the Door Assignment Problemthe Door Assignment Problem

yy11aa yy11bb yy22aa yy22bb

xx11AA 1414 2121 00 00

xx11BB 2121 1414 00 00

xx22AA 88 1212 1212 1818

xx22BB 1212 88 1818 1212

1 1 2 2 1 2 1 2

1 1 2 2 1 2 2 2

. .1, 1, 1, 1,1, 1, 1, 1,

{0,1}, 1,2; , ; {0,1}, 1,2; , .

A B A B A A B B

a b a b a a a b

ij ij

s tx x x x x x x xy y y y y y y yx i j A B y i j a b

1 1 1 1 1 1 1 1

2 1 2 1 2 1 2 1

2 2 2 2 2 2 2 2

14 21 21 14min 8 12 12 8

12 18 18 12 ,

A a A b B a B b

A a A b B a B b

A a A b B a B b

x y x y x y x yx y x y x y x yx y x y x y x y

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four sets four sets SS11, , TT11, , SS22, , TT22, , SS11 and and TT1 1 of of mm items, items, SSnn and and TTnn of of nn items items

two groups of assignments two groups of assignments the pairing of items in the pairing of items in SS11 and and TT1 1

the pairing of items in the pairing of items in SS22 and and TT2 2

cost of assigning item cost of assigning item ii SS11, , jj TT11, , kk SS22, , ll TT22 = = ccijklijkl

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1 1 1 1

1

1

1

1

min ,

. .

1, ;

1, ;

1, ;

1, ;, {0,1}, 1 , ;1 , .

m m n ni j k l ijkl ij kl

mi ij

mj ij

nk klnl kl

ij kl

c x y

s t

x j

x i

y l

y kx y i j m k l n

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Another Another Quadratic Assignment ProblemQuadratic Assignment Problem

two sets two sets SS and and TT, each of , each of nn items items

the pairing of items in the pairing of items in SS and and TT as in an as in an assignment problemassignment problem

cost of pairing cost of pairing ii SS with with jj TT and and kk SS with with ll TT: : ccijklijkl

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Another Another Quadratic Assignment ProblemQuadratic Assignment Problem

1, if is assigned to ,0, . .ij

i S j To w

1 1, 1,..., ;nj ij i n

1 1, 1,..., ;ni ij j n

{0,1}, 1 , .ij i j n

, , , 1

min ,nijkl ij kli j k l

k ic

. .s t

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ExampleExample

three factories three factories aa, , bb, , cc three cities three cities AA, , BB, , C C

AA BB CC

AA 1414 88

BB 1111

CC

Distance Between the CitiesDistance Between the Cities

aa bb cc

aa 1 01 0 1515

bb 1818

cc

Quantities Shipped Among the FactoriesQuantities Shipped Among the Factories

1, if factory is assigned to city ,0, . .

, , ; , ,

iji j

o wi a b c j A B C

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ExampleExample

aAaA aBaB aCaC bAbA bBbB bCbC cAcA cBcB cCcC

aAaA 140140 8080 210210 120120

aBaB 140140 110110 210210 165165

aCaC 8080 110110 120120 165165

bAbA 252252 144144

bBbB 252252 198198

bCbC 144144 198198

cAcA

cBcB

cCcC

aa bb cc

aa 1 01 0 1515

bb 1818

cc

Quantities Shipped Quantities Shipped Among the FactoriesAmong the FactoriesAA BB CC

AA 1414 88

BB 1111

CC

Distance Between the CitiesDistance Between the Cities

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ExampleExample1, if { , , } is assigned to { , },0, . .ij

i a b c j AB Co w

1, ;

1, ;

{0,1}, { , , }, { , , }.

i ij

j ij

ij

j

i

i a b c i A B C

, , , 1, ,

1min ,2

nijkl ij kl

i j k li k j l

c

. .s t

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Linearization of Linearization of the Quadratic Assignment Problem the Quadratic Assignment Problem

non-linear objective function with terms non-linear objective function with terms such as such as ijijklkl

to linearize the non-linear term to linearize the non-linear term ijijklkl

let let ijklijkl = = ij ij klkl

need to ensure that need to ensure that ijklijkl = 1 = 1 ij ij klkl = 1 = 1

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ijklijkl = 1 = 1 ij ij klkl = 1 = 1 ijklijkl = 1 = 1 ij ij = 1 and = 1 and klkl = 1 = 1

two parts two parts ijklijkl = 1 = 1 ij ij = 1 and = 1 and klkl = 1 = 1

ij ij = 1 and = 1 and klkl = 1 = 1 ijklijkl = 1= 1

trickstricks ijkl ijkl ijij and and ijkl ijkl kl kl

ijij + + klkl 1 + 1 + ijklijkl

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drawback of the methoddrawback of the method addition of one variable and three constraints addition of one variable and three constraints

for one cross-product for one cross-product nn((nn1)/2 cross products for 1)/2 cross products for nn ij ij ’s ’s

any method to add less variables or less any method to add less variables or less constraintsconstraints

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three cross-products 2three cross-products 21122+3+311331144 of of

four variables four variables 11, , 22, , 33, , 44

the previous method: 3 new variables and 9 the previous method: 3 new variables and 9 new constraints new constraints

another method with 1 new variable and 3 another method with 1 new variable and 3 new constraintsnew constraints

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let let ww = 2 = 21122+3+311331144 = = 11(2(222+3+33344))

hope to have: hope to have: 11 = 1 = 1 ww = 2 = 222+3+33344, and , and 11

= 0 = 0 ww = 0 = 0 possible values of possible values of ww {-1, 0, 1, 2, 3, 4, 5} {-1, 0, 1, 2, 3, 4, 5}

11 = 0 = 0 ww = 0: = 0: ww 5 511

how to model how to model 11 = 1 = 1 ww = 2 = 222+3+33344??

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to model to model 11 = 1 = 1 ww = 2 = 222+3+33344

ww = 2 = 222+3+33344

ww 2 222+3+33344 and and ww 2 222+3+33344

11 = 1 = 1 ww 2 222+3+33344 and and ww 2 222+3+33344

when when 11 = 1: two valid constraints = 1: two valid constraints

ww 2 222+3+33344 and and ww 2 222+3+33344

when when 11 = 0: two redundant constraints = 0: two redundant constraints

ww 2 222+3+33344+5+55511 and and ww 2 222+3+33344+5+51155

Lecture 6 Knapsack Problem Quadratic Assignment Problem

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