3. Summary of the publications 14
3.4. Decorous combinatorial lower bounds for row layout problems
3.4.1. Combinatorial lower bounds
Our approach for calculating lower bounds for the DRFLP is related to the following problem:
Definition 3.4.1. Given a set of jobsJ with processing timesqk∈R+and weightswk∈R+, k∈ J, one looks for an assignment of start times tk ∈R+ of the jobs J to u∈N parallel identical machines such that no two jobs overlap on one machine and such that the sum of the weighted completion times Pk∈JwkCk with Ck =tk+qk is minimized. For constant u we denote this problem by Pu||PwkCk.
The scheduling problem Pu||PwkCk is weakly N P-hard, see, e. g., [78]. It is well known that the unweighted case, i. e., Pu||PCk withwk= 1 fork∈J, can be solved to optimality in polynomial time by the Shortest Processing Time rule (SPT), where one processes the jobs in increasing order of their processing time.
Our aim is to compute lower bounds for the weighted distances of department i ∈ [n] to departmentsS ⊆[n]\ {i}. Letp∈Rn denote the center positions of the departments and let r ∈ {1,2}n denote the assignment of the departments to the rows. Further, letWci(S) denote the objective value of a feasible double-row layout with departments {i} ∪S, S ⊆[n]\ {i},which minimizes Pj∈S(wij +wji)|pi−pj|. Adding the additional constraint pi=pj for somej ∈S, i. e., i lies directly opposite to j, then the corresponding optimization problem is denoted by Wc(i,j)(S). It turns out that:
Proposition 3.4.2. Let (n, w, `) be a DRFLP instance and let i ∈ [n], S ⊆ [n]\ {i}. Then Wci(S) = minj∈SWc(i,j)(S).
In the following, we determine two different lower bounds for Wc(i,j)(S) given some DRFLP instance. In both variants we interpret the optimization problem for computing Wc(i,j)(S) as a scheduling problem P4||PwkCk with weights wk = wik +wki, k ∈ S\ {j}. The departments correspond to the jobs in the P4||PwkCk and the lengths of the departments to the processing times, i. e., qk = `k, k ∈ S\ {j}. Given a feasible solution of Wc(i,j)(S), then, as illustrated in Figure 3.4.1, machine 1 and machine 2 of the scheduling problem correspond to row 1 in this solution and machine 3 and machine 4 to row 2. Additionally, we have to take into account that in the scheduling problem the completion times of the jobs are considered while in the DRFLP one measures center-to-center distances between the departments.
Thus we are able to use methods from the scheduling literature to compute lower bounds for the DRFLP. All lower bound calculations have in common that we sort the jobs in S\ {j} by some given order. Respecting some machine-dependent non-availability times from zero to a= (a1, . . . , a4)∈R4+∪ {∞} (i. e., no job on machinekmay start before ak,k= 1, . . . ,4), the
i j
s1 s2 s3
s4 s5
s6 machine 2
machine 4
machine 1 machine 3 time 0
Figure 3.4.1: Visualization of the connection of the DRFLPand parallel machine scheduling on four machines. Here departments i and j lie opposite and we have to arrange departments{s1, . . . , s6}. In the lower bound calculations we will partially adjust the start of the jobs (departments) at a machine by half the length ofi(see gray area) or half the length ofj. In the scheduling problem one considers the completion times of the jobs while in theDRFLP one measures the center-to-center distances between the departments.
jobs are assigned in a greedy manner. Whenever a machine becomes idle and is available one assigns the next unscheduled job in the list non-preemptively. Our basic algorithm is summarized in Algorithm 3.4.1.
Algorithm 3.4.1:Basic(S = (s1, . . . , s|S|), `S, a)
Input :parallel machine scheduling problem with ordered jobs S= (s1, . . . , s|S|), processing times `S∈R|+S|, non-availability times from zero to a= (a1, . . . , a4) on the 4 machines
Output :completion times Csk, sk∈S,asCbasic(S, `S, a).
1 Initialize (¯`1,`¯2,`¯3,`¯4)←(a1, . . . , a4).
2 fork= 1, . . . ,|S|do
Choose ¯m∈arg min{`¯o:o∈ {1,2,3,4}}.
`¯m¯ ←`¯m¯ +`Ssk. Csk ←`¯m¯.
3 returnCsk,sk∈S.
In our first combinatorial lower bound forWc(i,j)(S), i∈[n], S⊆[n]\{i},we fixioppositej∈S and we make use of the SPTrule, so we sort the departmentsS\ {j}by increasing length. The departments are assigned as described in Algorithm 3.4.1. Recall that theSPT rule determines an optimal solution for the P4||PCk (withwk= 1, k∈J). Then, we assign the highest weights to departments closest to iand obtain a lower bound forWc(i,j)(S). As illustrated in Figure 3.4.1, machine 1 and machine 2 are not available from 0 to `2i and machine 3 and machine 4 from 0 to
`j
2.
Definition 3.4.3. Let (n, w, `) be a DRFLP instance. Let i ∈ [n], S ⊆ [n]\ {i}, j ∈ S with Sjspt = (s1, . . . , s|S|−1) a sequence of departments in S\ {j} with length `Sjspt = (`s1, . . . , `s|S|−1) ordered by increasing lengths and let
Cspt,(i,j)(S, `) :=Cbasic(Sjspt, `Ssptj ,(`2i,`2i,`2j,`2j))
denote the completion times returned by Algorithm 3.4.1. Furthermore, let wi•0 = (w0i1 + w01i, . . . , w0i(|S|−1) +w0(|S|−1)i) be the weights wik +wki of k ∈ S \ {j}, ordered decreasingly.
Then the SPT-lower-bound is
In the special case of all weights being equal to one the SPT-distance-bound is
W(i,j)dst (S) :=|S|−1X
TheSPT-distance-bound cannot be used to derive bounds for the optimal value of theDRFLP.
However, it can be used to derive lower bounds for the (geometric) distances between the departments themselves without regarding the amount of transports.
Now we consider a second variant for computing a lower bound for theWc(i,j)(S) based on the Smith rule which has been extended to P||PwkCk in the following way: The jobs are ordered non-increasingly by their relative weights wqkk for k∈J and we assign each of the jobs using this order to the next machine that gets idle. It is proven in [62] that the Smith rule for sorting the jobs leads to a 1+2√2-approximation algorithm for theP||PwkCk. We set αKK := 1+2√2. Definition 3.4.4. Let(n, w, `)be aDRFLP instance, and leti∈[n],S ⊆[n]\ {i} and j∈S. We denote by Sjsc = (s1, . . . , s|S|−1) a sequence of departments S\ {j} with length vector`Sscj ordered non-increasingly by wik`+wki
k , k∈S\ {j}. Denote by
Csc,(i,j)(S, `) :=Cbasic(Sjsc, `Sscj ,(0,0,0,0))
the completion times returned by Algorithm 3.4.1 for this ordering. Then theSCHED1-lower-bound is In Section D a third combinatorial lower bound for Wc(i,j)(S) is presented and used in the calculation of (3.4.2). Additionally, in Section D it is shown that the combinatorial lower bounds can be extended to theMRFLP.
In the equidistant case, we can simplify the calculation of theSPT-lower-bound since an optimal solution for the P4||PwkCk can be determined by assigning the departments with the highest weights first. So for i ∈ [n] we sort the departments in S ⊆ [n]\ {i} by decreasing weights wik +wki, k ∈ S, and assign the departments in that order as close as possible to i, i. e., a department with highest weight wik+wki, k∈S,lies directly opposite i. We denote this lower bound byWisort(S).
If we know that two departmentsi, j∈[n], i < j, overlap and so lie exactly opposite due to the grid structure [15], we can determine a lower bound for the weighted distances ofi andj to the departmentsS ⊆[n]\ {i, j}. For this we order the departments inS by decreasing weight wik+wki+wjk +wkj,k∈S, and get a sequence Si,jE-spt= (s1, . . . , s|S|). With
CE-spt,(i,j)(S) =Cbasic(Si,jE-spt,(1, . . . ,1),(0,0,0,0)) we denote the completion times returned by Algorithm 3.4.1 and we set
W(i,j)E-spt(S) := |
S|
X
k=1
(wisk+wski+wjsk +wskj)(CE-spt,(i,j)
sk (S)). (3.4.3)
Proposition 3.4.6. Let (n, w,1) be aDREFLP instance. Leti, j ∈[n], i < j, and S⊆[n]\ {i, j}, then for all equidistant double-row layouts with pi =pj we have
W(i,j)E-spt(S)≤ X
k∈S
(wik+wki+wjk+wkj)|pi−pk|.