Car Sequencing Problem

1. Car Sequencing Problem Assessed exercise 2 15% 27/11/09

21. representing the CSP as a csp

22. 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) Also called r out of s in AIJ58

23. Find a time/position for each car Some thoughts/sketch allDiff(cars) data read in (constants) constrained integer variables Need “sliding” p/q constraint on each row

24. Find a time/position for each car Some thoughts/sketch Heuristics? ½ baked! Bound v enumerated domains? Value ordering? allDiff(cars) How does search proceed? Symmetries? Decision variables? Need “sliding” p/q constraint on each row

25. What class of car do we make in a time slot/position? Alternative model? Values for time slots are “classes” of cars Constraints between time slots and the 2D array of 0/1 variables representing options over time Occurrence constraints on classes Due to AIJ58

26. AIJ , pages 139 to 151 variables c is number of classes of cars m is number of options n is number of cars to produce opt[i][j] = 1  class i requires option j (read in as input) Slots Array of constrained integer variables S[n] S[i]  {0..c-1} S[i] = j  a car of class j is in slot i Options Array 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

27. variables c is number of classes of cars m is number of options n is number of cars to produce opt[i][j] = 1  class i requires option j (read in as input) Slots Array of constrained integer variables S[n] S[i]  {0..c-1} S[i] = j  a car of class j is in slot i Options Array 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 Decision variables

28. variables c = 6 m = 5 n = 10

29. constraints Demand constraint d[i] is demand (number of cars) for class i For all i: occurrence of i in S is d[i]

30. spelling!

32. constraints Capacity, p/q, constraint p[j] is the number of cars with option j that can be produced 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 generally O[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

33. constraints The 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 x then 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

34. constraints The link constraint If a car of class j is being produced in time slot i and that class of car requires an option x then option x is required in time i Turn it around by 90 degrees S[i]  {0,4,5}  O[i][0] = 1 S[i]  {2,3,5}  O[i][1] = 1 S[i]  {0,4}  O[i][2] = 1 S[i]  {0,1,3}  O[i][3] = 1 S[i]  {2}  O[i][4] = 1 Do this for all time slots and all options This gets more propagation This maintains arc-consistency

35. constraints I wonder if this one produces most propagation The link constraint S[i]  {0,4,5}  O[i][0] = 1 S[i]  {2,3,5}  O[i][1] = 1 S[i]  {0,4}  O[i][2] = 1 S[i]  {0,1,3}  O[i][3] = 1 S[i]  {2}  O[i][4] = 1 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  S[i] ≠ 4  O[i][2] = 0

36. The redundant constraint (page 146) constraints Another example options 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 demand The Reasoning The demand for option 3 is 4 cars (d[3] = 4). In the last 5 slots we can make at most 2 cars with this option Consequently in the previous 5 slots we MUST make at least 2 cars • O[0][3] + O[1][3] + O[2][3] + O[3][3] + O[4][3] ≥ d[3] – 2

37. constraints The 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

38. heuristics Variable ordering What are the decision variables? Dynamic Variable Ordering: sdf? AIJ58 Page 146 to 150

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40. heuristics Value ordering What is value ordering in this context?

41. heuristics Value ordering Having selected a vacant slot to consider, what class of car should we place in that slot? Suggestions?

42. Software reuse Have a look at CSPCertificate Could this be used to produce a loader? How do you know your solution is correct?

43. My code

45. So? Encode a model for car sequencing problem using toolkit primitives Incorporate “redundant” constraint identified by PvH Use heuristics of your choice Certify your solutions Perform small empirical study

46. and … Write a short report, no more than 2 pages describing your models and how they go Email me this along with code that I can run and verify Make sure you give instructions on how to run Make sure output is verifiable (standard)