1. Submodular Optimization Methods for Scheduling with Controllable Processing Times Natalia Shakhlevich University of Leeds, U.K. Akiyoshi Shioura Tohoku University, Sendai, Japan Vitaly Strusevich University of Greenwich, London, U.K.

2. This Talk Illustrates the use of methods of Submodular Optimization for a bicriteria single machine scheduling problem to minimize the maximum processing cost and the total compression cost The problem is interpreted as a Make-or- Buy Production Planning Problem

3. Make-or-Buy Decision Making If the decision-maker (a production manager) realizes that the existing production capabilities are insufficient to fulfill all orders internally or if the cost of work-in-process of an order is too high, the order can be partly subcontracted

4. Make-or-Buy Decision Making Subcontracting incurs additional cost that can be either compensated by quoting realistic deadlines for all orders or balanced by a reduction in internal production expenses

5. Make-or-Buy Decision Making The make-or-buy decisions should be taken to determine which part of each order is manufactured internally and which is subcontracted Closely related to the popular time-cost trade-off project management problems

6. Notation and Model N = {1,…, n} set of orders (jobs) to be processed on a single machine (internal manufacturing) u j processing time of order j p j actual processing time of order j (internal manufacturing) l j lower bound on processing time of order j (a mandatory part for internal manufacturing)

7. Notation and Model h j subcontracting time of order j ujuj ljlj pjpj hjhj u j = p j + h j l j ≤ p j ≤ u j subcontracted manufatured internally

8. Notation and Model A schedule can be given by the split-values p j and h j and by a sequence φ according to which the orders are processed by the machine The completion time of order φ(k) sequenced in position k of permutation φ is C φ(k) = C φ(k-1) + p φ(k), where for completeness C φ (0) =0 The whole order φ(k) becomes available to the customer at time C φ(k) (the subcontractor is able to complete the required work h φ(k) by time C φ(k) )

9. Notation and Model Producing an order j ∈ N incurs the following two costs: work-in-process cost at the main production facility f j (C j ) subcontracting cost α j h j, where all α j ≥0 Measures cost for completing j ∈ N at time C j Each f j is a non-decreasing piecewise linear function of m j pieces; L – the total number of the linear pieces

10. Notation and Model Producing an order j ∈ N incurs the following two costs: work-in-process cost f j (C j ) subcontracting cost α j h j Functions to be minimized: maximum work-in-process cost F = max{f j (C j )|j ∈ N} total subcontracting cost K= ∑ j ∈ N α j h j

11. Notation and Model Functions to be minimized: maximum work-in-process cost F = max{f j (C j )|j ∈ N} total subcontracting cost K= ∑ j ∈ N α j h j Bicriteria Model: find a set of Pareto optimal points with respect to the functions F and K Single Criterion Model: minimized one of the functions, provided that the other is bounded from above

12. In This Talk 1|p j =u j -h j |(F, K) Can be reformulated in terms of scheduling with controllable processing times Hoogeveen & Woeginger (2002), O(L 2 (n 4 +logL)) We reduce the problem to a polynomial number of parametric LP problems over a submodular polyhedron intersected with a box We show that such an LP problem can be solved in O(n 2 ) time by establishing a link between its region and a base polyhedron with a special rank function

13. t fj(t)fj(t) f1f1

14. fj(t)fj(t) f1f1 f2f2 t

15. t fj(t)fj(t) f1f1 f2f2 f3f3

16. t fj(t)fj(t) f1f1 f2f2 f3f3 S 1 consists of all break-points of all piecewise linear functions f j (t)

17. t fj(t)fj(t) f1f1 f2f2 f3f3 S 2 consists of intersection points of linear pieces

18. t fj(t)fj(t) f1f1 f2f2 f3f3 S 1 consists of all break-points of all piecewise linear functions f j (t) S 2 consists of intersection points of linear pieces S 3 consists of intersection points with

19. t fj(t)fj(t) f1f1 f2f2 f3f3 O(L 2 ) stripes can be found in O(L 2 log L ) time

20. t fj(t)fj(t) f1f1 f2f2 f3f3 y'y' y '' Order 1 Order 2 Order 3

21. y'y' y '' Order 1 Order 2 Order 3 Induces deadlines on C j such that f j (C j )≤ y

22. y'y' y '' Order 1 Order 2 Order 3 Problem LP(y); A solution is a piece-wise linear function of y Solving for all stripes gives the efficient frontier

23. Submodular Systems For a set N={1,2,…,n}, let 2 N denote the set of all subsets of N A vector x=(x 1, x 2,…, x n ) ∈ X ⊆ ℝ n is called maximal in X if there is no vector z=(z 1, z 2,…, z n ) ∈ X such that x ≤ z (componentwise) For a vector x=(x 1, x 2,…, x n ) ⊆ ℝ n define x( ∅ )=0 and x(A)=∑ j ∈ A x j for a non-empty set A ∈ 2 N

24. Submodular Systems A collection D of subsets of N is called a distributive lattice if for any two sets in D their union and their intersection are both in D, i.e., X ∈ D and Y ∈ D implies X∩Y ∈ D and X ∪ Y ∈ D A set-function ψ: D → ℝ is called submodular if the inequality ψ (A  B)+ψ (A  B) ≤ ψ(A)+ψ(B) holds for all sets A,B  D

25. Submodular Systems For a submodular function ψ defined on a distributive lattice D ⊆ 2 N such that ∅ ∈ D, N ∈ D and ψ( ∅ )=0, the pair ( D,ψ) is called a submodular system on N, while ψ is called the rank function of that system.

26. Submodular Systems For a submodular system ( D,ψ) define two polyhedra P(ψ) = {x ∈ ℝ n ∣ x(A)≤ψ(A), A ∈ D } and B(ψ) = {x ∈ ℝ n ∣ x ∈ P(ψ), x(N)=ψ(N)} B(ψ) represents the set of all maximal vectors in P(ψ) Submodular Polyhedron Base Polyhedron

27. Submodular Systems A submodular polyhedron associated with the pair (2 N,ψ) is called a polymatroid, provided that the rank function ψ is monotone, i.e., ψ satisfies ψ(A)≤ψ(B) for A ⊆ B Shakhlevich & Strusevich (JoSch, 2005; Algorithmica, 2008) developed a unified approach to scheduling problems with controllable processing times based on reduction to LP problems over (generalized) polymatroids

28. Submodular Systems: 2D x 1 x 2 x 1 ≤ ψ({1}) x 2 ≤ ψ({2}) x 1 + x 2 ≤ ψ({1,2}) Polymatroid Base Polyhedron

29. LP over Base Polyhedra Base Polyhedron

30. Problem LP(y) p(N j )≤ψ(N j, y), Submodular polyhedron intersected with a box

31. Submodular Polydron with Box For a submodular system ( D,ψ) and a submodular polyhedron P(ψ) = {x ∈ ℝ n ∣ x(A)≤ψ(A), A ∈ D } introduce P(ψ) l u = {x ∈ ℝ n ∣ x ∈ P(ψ),l≤x≤u} We prove Theorem. Maximizing a linear function over P(ψ) l u is equivalent to maximizing a linear function over a base polyhedron B(ψ l u ) with the rank function ψ l u (A)=min D  D { ψ(D)+u(A\D)- l (D\A)}

32. Application to Problem LP(y) Theorem. Problem LP(y) is equivalent to maximizing the same objective function over a base polyhedron B(ψ l u ) with the rank function ψ ' (A,y)=min 1≤j≤n { ψ(N j,y)+u(A\N j )- l (N j \A)} Van Hoesel et al. (1994), O(n) = O(n 2 )

33. Algorithm To solve Problem 1|p j =u j -h j |(F, K) 1.Perform the pre-processing, i.e., find the stripes 2.For the lowest stripe determine the linear piece of each function f j, j = 1,...,n, related to that stripe. For each stripe based on the linear pieces of the functions in the previous stripe find the pieces in the current stripe. 3.For each stripe solve Problem LP(y). Step 1 of takes O(L² logL) time. Step 2 takes O(n logL) time for the lowest stripe, and O(L²n) all together. In Step 3, for each stripe Problem LP(y) can be solved in O(n²) time.

34. Conclusion Our algorithm for Problem 1|p j =u j -h j |(F, K) requires O(L² (n 2 +logL) time, factor n² less than the algorithm by Hogeveen and Woeginger (2002) The link between LP problem over a submodular polyhedron intersected with a box and over a base polyhedron is a useful tool to handle various scheduling problems with controllable processing times