In this note we study the complexity of the tool switching problem with non-uniform tool sizes. More speci cally, we consider the problem where the job sequence is given as part of the input. We show that the resulting tooling problem is strongly NP-complete, even in case of unit loading and unloading costs. However, we show that if the capacity of the tool magazine is also given as part of the input, the problem is solvable in polynomial time.Research; Studies; Complexity; Job; Costs; Time;

Crama, Y

Moonen, Linda

Spieksma, Frederik

Talloen, E

English

Research Papers in Economics

The Tool Switching Problem Revisited
Yves Crama
HEC Management School, University of Li ege, Boulevard du Rectorat, 7 (B31), B-4000
Li ege, Belgium, Y.Crama@ulg.ac.be
Linda S. Moonen, Frits C.R. Spieksma
Department of Applied Economics, Katholieke Universiteit Leuven, Naamsestraat 69,
B-3000 Leuven, Belgium, flinda.moonenjfrits.spieksmag@econ.kuleuven.be
Ellen Talloen
PricewaterhouseCoopers, Woluwe Garden, Woluwedal 18, B-1932 Sint-Stevens-Woluwe,
Belgium, ellen.talloen@pwc.be
In this note we study the complexity of the tool switching problem with
non-uniform tool sizes. More specically, we consider the problem where
the job sequence is given as part of the input. We show that the resulting
tooling problem is strongly NP-complete, even in case of unit loading and
unloading costs. However, we show that if the capacity of the tool magazine
is also given as part of the input, the problem is solvable in polynomial time.
1. Introduction
Consider the following problem: given is a set of jobs J, a set of tools T,
and a single machine equipped with a tool magazine of size C. The machine
is only capable of performing job j 2 J when a given subset of the tools
T(j)  T is present in the magazine of the machine. Each tool t 2 T has a
size which represents the number of slots occupied by the tool in the mag-
azine. Since the capacity required to store all tools in T in the magazine
exceeds C, tools need to be inserted in and removed from the magazine in
order to process all jobs. It is assumed that inserting and removing a tool
is only allowed in between the processing of a pair of consecutive jobs, and
that all tools in T(j) need to be present in the magazine before job j 2 J can
start. We refer to inserting a tool in the magazine as a loading operation,
and to removing a tool from the magazine as an unloading operation. The
tool switching problem is now to nd a sequence of the jobs, and an associ-
ated sequence of tool loadings, that minimizes the total number of loading
and unloading operations (L/U operations).
In this note we focus on the problem that results when the sequence of
the jobs is given; we refer to this problem as the tooling problem (TP).
The basic tooling problem assumes unit costs for L/U operations, and uni-
form tool sizes (meaning that each tool occupies a same number of slots of
the magazine). Here, we consider the case of non-uniform tool sizes; non-
uniform tool sizes relate to the situation where this number need not be the
1same for each tool. This is often the case in practice (see for instance Stecke
(1983), Jain et al. (1996) or Matzliach and Tzur (2000)). A diculty that
arises when considering non-uniform tool sizes is that the physical location
of a tool in the magazine becomes relevant. Observe that when all tools
have size 1, the only relevant decision is whether the tool is in the magazine
or not; if tool sizes are non-uniform, the location of a tool in the magazine
becomes important. For instance, removing two tools of size one from non-
adjacent slots does not create enough space to load a tool of size two. Thus,
in addition to determining which tools need to be present in the magazine,
one now must also specify its location in the magazine.
1.1 Related Work
The general tool switching problem was rst considered by Tang and Denardo
(1988). (It is interesting to note that a special case of the tool switching
problem was already considered by Belady (1966) in the context of the so-
called paging problem; the special case arises since in this application each
job needs a single tool). The tool switching problem is already NP-hard for
C = 2 (see Crama et al. (1994)). Many heuristics have been proposed for
its solution. Recent papers are from Djellab et al. (2000), Song and Hwang
(2002), Tzur and Altman (2004), and Zhou et al. (2005). The approxima-
bility of the problem is shortly discussed in Crama and van de Klundert
(1999). Solving large instances of the tool switching problem to optimality
is a challenge: Laporte et al. (2004) report on solving instances with up to
25 jobs and 25 tools, which are the largest solved instances we are aware of.
For a xed sequence of the jobs, Tang and Denardo (1988) proposed a proce-
dure called KTNS (Keep Tools Needed Soonest), and gave an ad-hoc proof
of its correctness (thereby generalizing Belady's result) for the basic TP.
Crama et al. (1994) provided an alternative proof of the correctness using
interval matrices, which allowed a generalization when an arbitrary setup
cost bt is given for each tool t 2 T. These results have been further gener-
alized to the case of changeover costs of the form dik when tool i is inserted
directly after removing tool k by Privault and Finke (1995). They showed
that in case of these changeover costs problem TP can still be solved in
polynomial time by formulating it as a min-cost ow problem. All these
results apply to uniform tool sizes.
So far, relatively little attention is paid to the tool switching problem with
non-uniform tool sizes. Matzliach and Tzur (2000) show that this problem
is NP-complete by a reduction from Partition, thereby resolving an open
question posed by Crama (1997). A drawback of their reduction is that the
physical location of the tools in the magazine is ignored; moreover, their
2result does not imply strong NP-completeness. Tzur and Altman (2004)
propose a heuristic for solving the tool switching problem with a xed job
sequence with non-uniform tool sizes, KSTNS (Keep Smallest Tools Needed
Soonest), which is a modication of the KTNS-procedure proposed by Tang
and Denardo (1988). Unlike the KTNS-procedure, KSTNS is not guaran-
teed to produce an optimal solution.
In literature, two types of magazines are described. One type is the so-
called straight magazine which essentially is a row of consecutive slots. This
type of magazine is very common in PCB assembly. The other type is a
round magazine, which is commonly used in the metal-based industry (see
Tzur and Altman (2004)). Our results hold for both cases.
1.2 Results
We present two results in this note:
 TP is strongly NP-complete, even with unit loading and unloading
costs. We use a reduction from 3-Partition in which the physical loca-
tion of the tools in the magazine plays a crucial role (see Section 2).
 For a xed value of C, TP is solvable in polynomial time. It turns
out that when C is part of the input, a shortest path computation on
a network involving O(jTjCC!) nodes gives the minimum number of
L/U operations (see Section 3).
2. Complexity
In this section we show that TP with a straight magazine is NP-complete in
the strong sense, even in case of unit loading and unloading costs. We use
a reduction from 3-Partition, which is known to be strongly NP-complete
(Garey and Johnson (1979)). The problem 3-Partition can be dened as
follows:
3-Partition
Given a set S of 3n elements, a positive integer B (we assume that B
is even), and integral weights wk  0 for all elements k 2 S such that
B=4 < wk < B=2 and such that
P
k2S wk = nB, the problem is to partition
S into n disjoint triples T1;:::;Tn such that, for 1  i  n;
P
k2Ti wk = B.
Now, given an instance I of 3-Partition, we dene an instance I0 of TP
with the following characteristics:
3 The capacity of the tool magazine C = 2n(B=2 + 1).
 The number of jobs equals 2n + 2.
 The number of tools equals 9n, and their sizes are given in Table 1.
 All loading operations and all unloading operations have cost equal to
1.
 The sets T(j), j = 1;:::;2n + 2, are specied as follows:
- Job 1 needs tools t1;:::;t2n (each with size B=2 + 1).
- Job j needs tools t1;:::;t2n j+1, (with size B=2+1), tools t2n+1;:::;t2n+j 1
(with size B=2), and tools t4n+1;:::;t4n+j 1 (with size 1), for
j = 2;:::;2n + 1.
- Job 2n+2 needs tools t4n+1;:::;t6n (with size 1), and tools t6n+1;:::;t9n
(with sizes corresponding to the weights from problem instance
I from 3-Partition).
Table 1: Tool sizes for problem instance I0
Tools Size
t1;:::;t2n B=2 + 1
t2n+1;:::;t4n B=2
t4n+1;:::;t6n 1
t6n+1;:::;t9n wk, k 2 S
We can now formulate the decision problem corresponding to TP:
Decision problem: Does there exist a solution to instance I0 of problem
TP with no more than 13n L/U operations?
In order to show NP-completeness, we show that a YES-instance of 3-
Partition corresponds to a YES-instance of TP, and vice versa.
 First we assume that we have a YES-instance for 3-Partition, meaning
that there exists a solution consisting of n triples, each with weight
equal to B. Consider the following solution to TP: to perform job 1,
we load tools t1;:::;t2n consecutively in the tool magazine. For job
2 we remove t1 and replace it by placing t4n+1 in slot 1 and t2n+1 in
slots 2;:::; 1
2B + 1. For job 3, we replace t2 by tools t2n+2 in slots
B=2 + 2;:::;B + 1, and t4n+2 in slot B + 2. Notice that the two
tools t2n+1 and t2n+2 with size B=2 are placed next to each other. In a
similar way we deal with jobs 4;:::;2n+1. For the nal job (job 2n+2)
4we unload all tools with size B=2 from the magazine, leaving n empty
series of slots of size B. We can load the required tools (t6n+1;:::;t9n)
into the magazine in such a way that each empty series of slots of size
B contains 3 tools. This is possible since we know that there exists a
solution to the instance of 3-Partition. The number of L/U operations
corresponding to this solution is equal to 2n (for job 1) +3  2n (for
jobs 2;:::;2n + 1) +2n + 3n (for job 2n + 2) = 13n, so we conclude
that we have a YES-instance for TP.
 Now let's assume that we have a YES-instance for TP, so we have a
solution to TP with no more than 13n L/U operations. In order to
process job 1, we must load tools t1;:::;t2n in the magazine; then, for
processing job j, for j = 2;:::;2n + 1, we must unload tool t2n j+2
(i.e., a tool of size B=2+1) and we must load tools t2n+j 1 (size B=2)
and t4n+j 1 (size 1) in the slots which have just be emptied. This
requires at least 2n+32n = 8n L/U operations, no matter how the
tools are placed in the magazine. Then, for job 2n+2, we must unload
all tools t2n+1;:::;t4n of size B=2 and we must load all tools of size
w1;:::;w3n. This amounts to 5n L/U operations for the last job.
Hence, if a solution with cost 13n exists, this solution cannot involve
any additional L/U operations beyond those which have just been
listed, meaning that we are not allowed to displace any other tool in
the process.
In particular, this analysis implies that when job 2n + 1 is processed,
the magazine contains 2n tools of size B=2 and 2n tools of size 1, and
each tool of size B=2 is adjacent to at least one tool of size 1. Let us
call a tool of size B=2 isolated if it is adjacent to two tools of size 1 (as
shown in conguration (b) in Figure 1), or if this tool is the leftmost
or rightmost tool in the magazine. We claim that that there is no such
isolated tool in a solution with 13n L/U operations.
Figure 1: Possible ways in which the tool magazine can be lled.
After nishing job 2n + 1, there are two possibilities: either there are
no isolated tools, as shown in conguration (a) in Figure 1, or there is
at least one isolated tool of size B=2, as shown in conguration (b) in
Figure 1.
5We rst analyze the second case (corresponding to conguration (b) in
Figure 1): the isolated tool of size B=2 needs to be removed from the
magazine before we can start the nal job. In this empty space, we can
add some of the tools that are required for this job. We know that at
least one of these required tools will t, since wk < B=2 for all k 2 S.
However, we also know that wk > B=4 for all k 2 S, meaning that we
will never be able to t 2 tools in this empty space. This means that,
in order to ll the magazine to contain all tools required for job 2n+2,
we need additional L/U operations, resulting in more than 13n L/U
operations in total. Hence conguration (b) cannot occur.
In the case depicted in conguration (a) in Figure 1 however, we have
the following: after nishing job 2n + 1, we remove all tools of size
B=2, leaving n empty spaces of size B. Since we have a solution to TP
using 13n L/U operations, we know that we can add the required 3n
tools using only 3n L/U operations. Now, since B=4 < wk < B=2 for
all k 2 S, we know that each empty space of size B can t more than
2 tools, but never 4. This means that, in order to ll the magazine
using at most 3n L/U operations, each of these spaces must contain
exactly 3 tools. This means that the problem instance corresponding
to 3-Partition must be a YES-instance.
The above analysis shows that the tooling problem is strongly NP-complete.
We leave it to the reader to verify that a similar reasoning applies to a round
layout of the tool magazine. However, in the next section we show that if
the capacity of the tool magazine is xed (i.e., if C is given in the input of
the problem), the problem is solvable in polynomial time.
3. Fixed value of C
In this section we consider the tooling problem for a xed value of the tool
magazine capacity. In other words, we assume that the capacity C is given
in the input of the problem. As we will show in this section, the resulting
problem is solvable in polynomial time.
First we introduce some notation. Given are the set T containing all the
tools, plus C dummy tools of size 1, which represent empty slots in the tool
magazine. We call T 0 the set of all tools in T together with all dummy
tools. Then, a magazine conguration can be described by listing at most C
elements of T0. For example, the conguration ft4;t1;td;t2;tdg corresponds
to the magazine shown in Figure 2, where td represent dummy nodes (i.e.,
empty spaces in the tool magazine). Observe that, if we have jT 0j tools,
there are at most O(jT 0jCC!) dierent magazine congurations.
6Figure 2: Possible conguration for the tool magazine.
Now we create a directed graph D = (V;A) containing n layers of vertices
(n being the number of jobs). A vertex in layer ` of D corresponds to a
feasible magazine conguration for performing job `, so there are at most
O(jT0jCC!) vertices in each layer (notice that there is a vertex only for those
magazine congurations that contain all required tools for a specic job).
The arc set A contains arcs from all vertices in layer ` to all vertices in layer
`+1 (` = 1;:::;n 1), and an arc (i;j) has a length equal to the switching
costs to go from the magazine conguration corresponding to vertex i to the
magazine conguration corresponding to vertex j. In order to nd an opti-
mal solution to the tooling problem, we need to nd a shortest path from
layer 1 to layer n in the resulting graph D. This can be done in polynomial
time in case of a xed value of the magazine capacity C.
Notice that the only thing required is to be able to specify the switch-
ing costs between two magazine congurations. Also, the method described
in this section holds for a straight tool magazine, as well as for a round
magazine.
4. Conclusion
In this paper, we consider a special case of the tool switching problem, where
the tools have non-uniform sizes, and the job sequence is given. We show
that the resulting tooling problem is strongly NP-complete, even for unit
loading and unloading costs. We also show that, in case the capacity of the
tool magazine is also given in the problem input, the problem can be solved
in polynomial time by solving a shortest path problem in a directed graph.
7References
Belady, L.A. (1966). A study of replacement algorithms for a virtual-storage com-
puter. IBM Systems Journal 5, 78-101.
Crama, Y. (1997). Combinatorial optimization models for production scheduling in
automated manufacturing systems. European Journal of Operational Research
99, 136-153.
Crama, Y., J. van de Klundert (1999). Worst-case performance of approximation
algorithms for tool management problems. Naval Research Logistics 46, 445-
462.
Crama, Y., A.W.J. Kolen, A.G. Oerlemans, F.C.R. Spieksma (1994). Minimizing
the number of tool switches on a exible machine. The International Journal
of Flexible Manufacturing Systems 6, 33-54.
Djellab, H., K. Djellab, M. Gourgand (2000). A new heuristic based on a hyper-
graph representation for the tool switching problem. International Journal of
Production Economics 64, 165-176.
Garey, M.R., D.S. Johnson (1979). Computers and intractability. A guide to the
theory of NP-completeness. W.H. Freeman and Company, New York.
Jain, S., M.E. Johnson, F. Safai (1996). Implementing setup optimization on the
shop oor. Operations Research 43, 843-851.
Laporte, G., J.-J. Salazar, F. Semet (2004). Exact algorithms for the job sequencing
and the tool switching problem. IIE Transactions 36, 37-45.
Matzliach, B., M. Tzur (2000). Storage management of items in two levels of
availability. European Journal of Operational Research 121, 363-379.
Privault, C., G. Finke (1995). Modelling a tool switching problem on a single
NC-machine. Journal of Intelligent Manufacturing 6, 87-94.
Stecke, K.E. (1983). Formulation and solution of nonlinear integer production
planning problems for exible manufacturing systems. Management Science
29, 273-288.
Song, C.Y., H. Hwang (2002). Optimal tooling policy for a tool switching problem
of a exible machine with automatic tool transporter. International Journal of
Production Research 40, 873-883.
Tang, C.S., E.V. Denardo (1988). Models arising from a exible manufacturing ma-
chine, part I: minimization of the number of tool switches. Operations Research
36, 767-777.
Tzur, M., A. Altman (2004). Minimization of tool switches for a exible manufac-
turing machine with slot assignment of dierent tool sizes. IIE Transactions
36, 95-110.
Zhou, B.H., L.F. Xi, Y.S. Cao (2005). A beam-search-based algorithm for the tool
switching problem on a exible machine. International Journal of Advanced
Manufacturing Technology 25, 876-882.
8

The tool switching problem revisited.

https://lirias.kuleuven.be/bitstream/123456789/228426/1/0552.pdf

The tool switching problem revisited.

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics