Monthly Archives: May 2018

Production Planning

This is one of those problems where we eliminate a lot of the parameters and are still left with a version that in NP-Complete.  I guess I would have started with the stripped-down version first, since adding extra parameters keeps the problem NP-Complete.

The problem: Production Planning.  This is problem SS21 in the appendix.

The description: Given a positive number n of time periods, and for each period i, several non-negative parameters: demand (ri), production capacity (ci), setup cost (bi), incremental production cost coefficient (pi), and inventory cost coefficient (hi).  We’re also given an overall bound B.  Can we find a positive production amount (xi)  for each period such that:

  • xi ≤ ci for all i (we never produce more than the production capacity)
  •  The inventory on hand for each period (Ii =\sum_{j=1}^i(xj -rj)  is non-negative
  • \sum_{i=1}^n(pixi+hiIi) + \sum_{x_{i}>0}bi ≤ B?

Example: There are a lot of variables here, but here’s what it’s really saying:

Suppose we have a factory that makes some product, and the factory’s schedule for making the product is split into time periods.  In each time period, we can only make a certain amount (ci) of the product, and it costs a certain amount to use the factory at all during a time period (bi) and then an additional amount to make each instance of the product (pi).   Each time period also has a demand (ri) that has to be met.  We can get ahead of the demand by producing more than we need (leaving Ii extra around at time i), but also at a cost (hi).

Suppose I have 3 time periods:

Time Period Production Capacity Demand Production Cost Inventory Cost Production Setup
1 5 3 2 1 6
2 5 3 4 3 2
3 1 2 20 20 5

Since at time period 3, we have less production capacity than demand, we will have to carry over inventory from a previous time period.  There are basically four strategies to meet the demands:

  1. Produce 4 units at time period 1, 3 at time 2, and 1 at time 3.  At time 1, this costs 6 to set up production, 4×2 to make the goods, and 1×1 to store the extra good (So 15).  At time period 2, it costs 2+3×4+1×3 = 17, and time period 3, it costs 5+1×20+0x20 = 25, for 57 total cost.
  2. Produce 3 units at time period 1, 4 at time 2, and 1 at time 3.  At time 1, this costs 6+3×2+0x1 = 12.  At time period 2, it costs 2+4×4+1×3 = 21. At time period 3, it costs 5+1×20+0x20  25.  This is a total cost of 58.
  3. Produce 5 units at time period 1, 3 at time period 2, and 0 at time period 3.  This would cost 6+5×2+2×1 = 18 at time 1, 2+3×4+2×3 =20, and 0 at time 3, for a cost of 38.
  4. Produce 3 units at time period 1, 5 at time period 2, and 0 at time period 3.  This would cost 6+3×2+0x1 =12 at time 1, 2+5×4+2×3 = 28 at time 2, and 0 at time 3, for a total time of 40.

So option 3 is the best choice.  But if we raised the inventory cost of time period 1 significantly (say, to 50), then option 3 would cost 114, and option 4 would still cost 40, and be best.

Reduction: The paper by Florian, Lenstra, and Rinnooy Kan say they use Knapsack, but by our definitions, the problem is really Sum of Subsets.    So we’re given a set of integers A= {a1 .. an} and a target B.    We will build a production planning instance with n+1 time periods numbered from 1 to n.  The demand of each period is B.  The b costs (to set up production) are 1 at all periods.   The capacity at time period 0 is n*B, and the capacity at time period i > 0 is ai.  The production cost at time period 0 is 0, the production cost at time period i is \frac{a_i -1}{a_i}.  All inventory costs are 0.   The cost bound B’ is equal to the SOS bound B + 1.

Since it costs 1 to make any amount of goods (up to our capacity limit) in time 0, we might as well make all of it that we can.  This gives us n*B goods, and will be enough to supply the demands at all of the other time periods except for the last, leaving B left over to be supplied in the last time period.  Notice that our production cost fraction is built in such a way that if in time period i, we make ai goods, we pay exactly ai cost.  If we build less than ai goods in time period i (but more than 0), we pay more than ai cost in that time period.  So the only to get a total cost of B’ while meeting the demand is to make exactly ai of the good in each time period that makes goods.  This corresponds to a subset of A that sums to exactly B.

Difficulty: 5.  If you do use this in a class, I’d give them a simpler version where you don’t even mention the inventory costs, and where the b cost is always 1, and where the demands are always equal in each time period, since that’s what the reduction does.

Staff Scheduling

This one has no reference in G&J, but I think it’s easy enough for me (or a student) to figure out.

The problem: Staff Scheduling.  This is problem SS20 in the appendix.

The description: Given a collection C of m-tuples, each tuple consisting only of 0’s and 1’s, and each tuple having the same amount (k) of 1’s.  These represent possible schedules we can give a worker.  We’re also given a “requirement” m-tuple consisting of m non-negative integers, and a number N of workers.  Can we map each c in C to a non-negative integer (representing the number of workers we assign to that schedule) such that the total number of all integers is N or less, and the requirement tuple is met?

Example: I think this is easier to explain with an example.  Suppose we had 4-tuples (corresponding to 4 “work periods”) and C is the tuples:

0 0 1 1
1 1 0 0
1 0 0 1
0 1 1 0
0 1 0 1

Our requirement tuple is:

2 1 4 3

Then one possible schedule would be:

  • 3 people with the first schedule (0,0,1,1)
  • 2 people with the third schedule (1,0,0,1)
  • 1 person with the fourth schedule (0,1,1,0)

..for a total of 6 workers. I think that’s the best you can do, since the first and third time periods have no schedules in common, and there is a total requirement of 6 workers for those time periods.

Reduction: G&J don’t give a reference, but they do suggest to use X3C.   So we’re given a set X of 3q elements, and a collection C of 3-element subsets of X.  The set C’ that we’ll build will be a set of “3q-tuples”, one position corresponding to each element in X.  There will be one tuple for each triple in C.  The tuples will have 1’s in the 3 positions corresponding to the elements in the triple in C, and 0’s everywhere else.  The R-tuple will be all 1’s, and N will be equal to q.

The Staff Scheduling instance we’ve built needs to choose q different tuples to be successful, and that set of tuples will correspond to a solution to the original X3C instance.

Difficulty: 4, assuming I didn’t miss anything.  If you’ve previously explained X3C to your students, this would be a good homework problem.

Timetable Design

We’ve now entered the “Miscellaneous” section of the Sequencing and Scheduling chapter.

The problem: Timetable Design.  This is problem SS19 in the appendix.

The description: (I’m using the description in the paper by Even, Itai, and Shamir, because I think it’s clearer than G&J’s definition): We’re given a set H (of hours to schedule), a collection of N “teachers”, each teacher is available for some subset of the H hours.  We’re also given a collection of M “Classes”, each class is available to be taught for some subset of the H hours. We’re also given an NxM matrix R of requirements:  Rij is the number of hours teacher i needs to teach class j for.   Can create a function f: TxCxH->{0,1} such that f(t,c,h) is 1 if teacher T is teaching class c at time h (and 0 otherwise)?  The function needs to respect the following rules:

  • If f(i,j,h) is 1, then h is in Ti and Cj  (we only schedule teachers and classes when they’re available)
  • The sum of f(i,j,h) over all h = Rij  (Each teacher teaches each class the number of hours they’re supposed to
  • The sum of f(i,j,h) over all i <= 1  (Each class has at most 1 teacher in each time period)
  • The sum of (i,j,h) over all i <= 1 (Each teacher can only teach 1 class per time slot).

Example:  Suppose H was {1,2,3,4,5}

We have 3 teachers:

  • T1 is available at {1,2,3}
  • T2 is available at {3,4,5}
  • T3 is available at {1,3,5}

..and 3 classes:

  • C1 is available at {2,4}
  • C2 is available at {1,5}
  • C3 is available at {2,3}

If R was:

1 1 0
1 0 1
0 1 1

(Teachers are rows, so T1 needs 1 hour of C1, 1 hour of C2, and none of C3, this won’t be satisfiable, since neither T2 nor T3 can teach C3 during hour 2, and someone has to.  If we change C3 to be available at {2,3,5}, then this becomes satisfiable:

  • T1 can only teach C1 at time 2, and C2 at time 1
  • T2 can only teach C1 at time 4, and C3 can be either time 3 or 5.  Let’s pick 5.
  • T3 can teach C2 at times 1 or 5, but time 1 is taken by T1, so we schedule it at time 5.  Similarly, C3 is available at times 2 and 3, and T3 is also available at time 3, so we can schedule that class then.

Reduction: Even, Itai, and Shamir use 3SAT.  So we’re given a set X of literals (x1 through xl, and the negation ~x1 through ~xl), and a set D of k clauses (D1 through Dk).  Define pi to be the number of times variable i (so the literal xi or the literal ~xi)  shows up in the formula.  Each variable i will get 5*pi classes denoted Ciab, where a runs from 1 to pi, and b runs from 1 to 5.  There will be 3 hours to schedule the classes.   They organize the teachers for the classes in a widget that looks like this (from the paper, p. 693):

The idea is that the diagonal lines are teachers who have to teach the 2 classes connected by the line (So, C12 and C13 for example) and the crosses are the 2 ways they can teach it (so h2 and h3).  The bottom arrows are teachers who can have 3 hours of availability and 3 classes to teach.  Each variable gets pi teachers who teach Cq3 and Cq4 (where q runs from 1 to pi) during h1 and h2.  If Cq3 is taught at time h1, the variable is treated as being set to positive, otherwise, it’s negative.

We then need to add more jobs to make sure that we make one literal true in each clause.  This is done by creating a teacher with 3 classes (one for each literal in the clause).  If the literal is the qth occurrence of literal xi, the teacher is assigned to class Ciq2 if the literal is true, and Ciq5 if the literal is false.

If there is a satisfiable solution to the SAT instance, the literal that makes each clause true will define a way to set the teacher at time h1 and h2, and that starts a chain reaction that schedules everything.

If there is a feasible timetable, we look at the Ciq3 and Ciq4 classes to see how we have allocated the teachers to those tasks, which tells us how to set the variables to make each clause true.

Difficulty: 9.  I’m having a hard time following all of the classes and teachers here.  I wonder if there is a much easier way to do this if we reduce from some other problem.

 

Job-Shop Scheduling

Are you sick of all of these flow-shop problems where we have to split up the tasks of a job into exactly one thing on each processor, progressing through the processors in the same order?  Then today’s problem is for you!

The problem: Job-Shop Scheduling.  This is problem SS18 in the appendix.

Description: We’re given a set of m processors and a set J of jobs.  Each job j in J consists of a chain of tasks, but different jobs can have different numbers of tasks.  The subtasks need to be processed in order, and each task has a processor assignment associated with it, and it’s possible that the subtasks can be assigned to the same processor multiple times, and/or skip some processors entirely.  The only requirement is that two consecutive tasks need to be on different processors (otherwise, we can just merge those tasks together). If we’re given a deadline D, can we schedule all jobs to complete all of their tasks by time D?

Example: Suppose we have three processors and 4 jobs:

  1. Job 1: 1 time on P1, then 2 time on P2, then 3 time on P3, then 2 time on P2
  2. Job 2: 2 time on P1, then 3 time on P2, then 2 time on P1
  3. Job 3: 1 time on P1, then 2 time on P3, then 3 time on P1, then 2 time on P3.
  4. Job 4: 1 time on P4, and that’s it.

Then we can get everything done by time 8 with the following schedule:

P1 1 2 2 3 3 3 2 2
P2 3 1 1 2 2 2 1 1
P3 4 3 3 1 1 1 3 3

Reduction: Garey, Johnson, and Sethi use 3-Partition.  So we’re given a set of 3n elements A, and a bound B.  The Job-Shop instance will have 2 processors and 3n+1 jobs labeled C0 through C3n.  Job C0 has 2n tasks, alternating B time on each processor (so, B time on P1, then another B time on P2, then another B time on P1, and so on).  The rest of the tasks get 0 time on P1 and a1 time on P2.  The deadline is 2nB, which is the time it takes to schedule all of C0.  The paper leaves out the “straightforward” task of showing this reduction actually solves the problem.  I believe them that it is straightforward, let’s take a closer look:

If A has a 3-Partition, then we have a collection of N sets of 3 elements that all add up to B.  Each one of these sets can fit on P2 while C0 is running on P1 (since C0 uses P1 for exactly B time for each task on P1), so we can fit all of the C tasks into the “holes” left by C0.

If we have a schedule that meets the deadline, we know that C0 must be scheduled continually, which leaves the B-sized holes on P2.  For all of the A tasks to be scheduled by the deadline, they have to fit in those holes, which gives us a partition of A.

Difficulty: 5.  The hard part, of course, is coming up with the conversion.  3-Partition isn’t the easiest problem to work with, but the nice features of it (especially that you know that any subset of A that adds to B has exactly 3 elements) make it a good fit here.

I wonder if it’s necessary to have C0 spend B time on both processors.  I understand the need to have C0 running on P2 to separate the partition sets, but wouldn’t it also work (and to my mind, be easier) if C0 took just 1 time (instead of B) each time it ran on P2?

I also wonder if there isn’t an easier reduction that goes straight from Flow-Shop Scheduling.  The reduction can take the Flow-Shop instance, and map it directly to the Job-Shop instance (just because the Job-Shop problem allows you to reuse, reorder, or skip processors with tasks, doesn’t mean it’s required).  Unless I’m missing something, this is a difficulty 2 reduction.  I think the reason they do the 3-Partition reduction in the paper is because they want to show it’s NP-Complete in the case where there are only 2 processors, and the Flow-Shop Scheduling reduction uses 3 processors.