Car Sequencing Problem

Assessed exercise 2

15%

27/11/09

representing the CSP as a csp

Line 1: 10 cars, 5 options (sunRoof, airCon, elecWin, 2TonePaint, alloys) 6 classes (subsets of options)Line 2: Number of cars with an option allowed in a block (pi)Line 3: Size of block, measured as cars (qi)Line 4: class# #cars option1 (0/1) option2 (0/1) … option5 (0/1)…Line 9: class# #cars option1 (0/1) option2 (0/1) … option5 (0/1)

ii qlength ofblock ain carspmost at Produce/ ii qp

Also called r out of s in AIJ58

]5][[]][5[][

]2][[]][2[][

]1][[]][1[][

icarOptionjoptionjicar

icarOptionjoptionjicar

icarOptionjoptionjicar

allDiff(cars)

Some thoughts/sketch

Need “sliding” p/q constraint on each rowconstrained integer variables

data read in (constants)

Find a time/position for each car

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54321option\time

]5][[]][5[][

]2][[]][2[][

]1][[]][1[][

icarOptionjoptionjicar

icarOptionjoptionjicar

icarOptionjoptionjicar

allDiff(cars)

Some thoughts/sketch

Decision variables?Need “sliding” p/q constraint on each row

Symmetries?

Heuristics?

Value ordering?

How does search proceed?

Bound v enumerated domains?

Find a time/position for each car

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54321option\time

½ baked!

Alternative model?

slots Time

Values for time slots are “classes” of cars

Constraints between time slots and the 2D array of 0/1 variablesrepresenting options over time

Occurrence constraints on classes

Due to AIJ58

What class of car do we make in a time slot/position?

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54321option\time

variablesAIJ , pages 139 to 151

c is number of classes of carsm is number of optionsn is number of cars to produce

opt[i][j] = 1 class i requires option j (read in as input)

SlotsArray of constrained integer variables S[n] S[i] {0..c-1} S[i] = j a car of class j is in slot i

OptionsArray of constrained 0/1 integer variables O[n][m] O[i][j] = 1 S[i] = x opt[x][j] = 1 O[i][j] = 0 S[i] = x opt[x][j] = 0 A car of class x is in slot i and requires option j alternatively option j produced at time i

variables

Decision variables

c is number of classes of carsm is number of optionsn is number of cars to produce

opt[i][j] = 1 class i requires option j (read in as input)

SlotsArray of constrained integer variables S[n] S[i] {0..c-1} S[i] = j a car of class j is in slot i

OptionsArray of constrained 0/1 integer variables O[n][m] O[i][j] = 1 S[i] = x opt[x][j] = 1 O[i][j] = 0 S[i] = x opt[x][j] = 0 A car of class x is in slot i and requires option j alternatively option j produced at time i

variables

c = 6m = 5n = 10

5

S

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000113

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0

43210O

constraintsDemand constraint

d[i] is demand (number of cars) for class iFor all i: occurrence of i in S is d[i]

spelling!

constraintsCapacity, p/q, constraint

p[j] is the number of cars with option j that can beproduced in a block of length q[j]

O[0][j] + … + O[q[j]-1][j] ≤ p[i]O[1][j] + … + O[q[j]][j] ≤ p[i]...More generallyO[i][j] +…+ O[i+q[j]-1][j] ≤ p[i] 1 ≤ i ≤ n-q[j]+1

Uses constraints sum and leq

Do this for all options j

r outof s in AIJ58

constraintsThe link constraint (page 144)

If a car of class j is being produced in time slot i and that class of car requires an option xthen option x is required in time i

S[i] = 2 O[i][0] = 0 O[i][1] = 1 O[i][2] = 0 O[i][3] = 0 O[i][4] = 1

Do this for all times/positions 0 ≤ i < n and all classes 0 ≤ j < c

If a car of class j is being produced in time slot i and that class of car requires an option xthen option x is required in time i

constraintsThe link constraint

S[i] {0,4,5} O[i][0] = 1 S[i] {2,3,5} O[i][1] = 1S[i] {0,4} O[i][2] = 1S[i] {0,1,3} O[i][3] = 1S[i] {2} O[i][4] = 1

Do this for all time slots and all options

This maintains arc-consistency

Turn it around by 90 degrees

This gets more propagation

constraintsThe link constraint

Are these the same i.e. logically and wrt propagation?

S[i] {0,4} O[i][2] = 1

S[i] = 0 S[i] = 4 O[i][2] = 1

S[i] ≠ 1 S[i] ≠ 2 S[i] ≠ 3 O[i][2] = 1

S[i] {0,4,5} O[i][0] = 1 S[i] {2,3,5} O[i][1] = 1S[i] {0,4} O[i][2] = 1S[i] {0,1,3} O[i][3] = 1S[i] {2} O[i][4] = 1

S[i] ≠ 0 S[i] ≠ 4 O[i][2] = 0

I wonder if this oneproduces most propagation

constraintsThe redundant constraint (page 146)

Let d[i] be the demand for option i

Example: d[3] = 1 + 1 + 0 + 2.1 + 0 + 0 = 4 p[3]/q[3] = 2/5

O[0][3] + O[1][3] + O[2][3] + O[3][3] + O[4][3] ≥ d[3] – 2

Another example

options

demand

The ReasoningThe demand for option 3 is 4 cars (d[3] = 4). In the last 5 slots we can make at most 2 cars with this optionConsequently in the previous 5 slots we MUST make at least 2 cars

constraintsThe redundant constraint

Example: d[0] = 1 + 0 + 0 + 0 + 2 + 2 = 5 p[0]/q[0] = 1/2

O[0][0] + + O[7][0] ≥ d[0] – 1 O[0][0] + + O[5][0] ≥ d[0] – 2 O[0][0] + + O[3][0] ≥ d[0] – 3 O[0][0] + O[1][0] ≥ d[0] – 4

This constraint makes a BIG difference

heuristicsVariable ordering

What are the decision variables?Dynamic Variable Ordering: sdf?

AIJ58 Page 146 to 150

This slide intentionally left blank

heuristicsValue ordering

What is value ordering in this context?

heuristicsValue ordering

Having selected a vacant slot to consider,what class of car should we place in that slot?

Suggestions?

Software reuseHave a look at CSPCertificate

Could this be used to produce a loader?

How do you know your solutio

n is correct?

My code

So?

Encode a model for car sequencing problem using toolkit primitives

Incorporate “redundant” constraint identified by PvHUse heuristics of your choiceCertify your solutionsPerform small empirical study

and …

Write a short report, no more than 2 pagesdescribing your models and how they go

Email me this along with code that I can run and verifyMake sure you give instructions on how to runMake sure output is verifiable (standard)

You must read this

And so on ….

You should read PvH’s paper

Looking at both sides

Interchanging variables with values

- to make things all different? - a permutation?