This section is addressed the simple assembly line balancing problem (abbreviation:
SALBP), in which it is necessary to minimize number of stations m for processing n partially ordered operations V = {1, 2, . . . , n} within the cycle time c.
Let the processing time pi of operation i ∈ V and cycle time c be given. However, during the life cycle of the assembly line the values pi are definitely fixed only for the subset of automated operations V \V . Another subsete V includes manual operations,e for which it is impossible to fix exact processing times during the whole life cycle of the assembly line. If j ∈ V \V , then operation time pe j can be different for different cycles of production process.
In this section, for the optimal line balance b of paced assembly line with vector p = (p1, p2, . . . , pn) of the operation times, we investigate stability of its optimality with respect to possible variations of the processing times pj of the manual operations j ∈V .e In particular, we derive necessary and sufficient conditions when optimality of the line balance b is stable with respect to sufficiently small variations of the operation times pj, j ∈ V . We show how to calculate the maximal value of independent variations of thee processing times of all the manual operations, which definitely keep the feasibility and optimality of the line balance b.
Simple Assembly Line Balancing Problem
We consider a single-model paced assembly line, which continuously manufactures homo-geneous product in large quantities (mass production). In this section, we use terminology and main notations given in survey [Bay86] and monograph [Sch99].
The assembly line under consideration is a sequence of m linearly ordered stations, which are linked by conveyor belt (or other material handling equipment). All stations of the assembly line have to perform the same set of operations repeatedly during the life cycle of the assembly line. Set of operations V , which have to be processed on the assembly line within one cycle time c, is fixed. Cycle time c is fixed as well. Each operation j ∈ V is considered indivisible: An operation j has to be completely processed on one station within one cycle time c. All the m stations start simultaneously the sequences of their operations and buffers between stations are absent.
We assume that set V includes operations of two types. On the one hand, subset V ofe set V includes all the operations, for which it is impossible to fix exact processing times for the whole life cycle of the assembly line (e.g., manual operations). On the other hand, each operation i ∈ V \V is one with operation time pe i being fixed during the life cycle of the assembly line (e.g., in the case of automated operations).
The known technological factors define a partial order on the set of operations V . Let digraph D = (V, A) with vertices V and arcs A define partially ordered set of operations V = {1, 2, . . . , n}, which have to be processed on the assembly line within cycle time c.
Without loss of generality, we assume that
V = {1, 2, . . . ,e n} and V \e V = {e n + 1,e n + 2, . . . , n},e
where 1 ≤ ˜n ≤ n. If n = 0, thene V = ∅. We will use the following notations for thee
vectors of the operation times:
p = (pe 1, p2, ..., p
en), p = (p, p) = (pe 1, p2, ..., pn).
Thus, we have V = {1, 2, . . . ,n,e n + 1, . . . , n}. Ife n = n, thene V = V .e
Simple Assembly Line Balancing Problem (SALBP) is to find an optimal balance of the assembly line for the given cycle time c, i.e., to find a feasible assignment of all the operations V into the minimal possible number m of stations. (Note that in [Bay86] and [Sch99], abbreviation SALBP-1 is used for such a problem.)
Assignment V = V1∪ V2∪ . . . ∪ Vm of operations V into m linearly ordered stations S = (S1, S2, ..., Sm)
(i.e., partition of set V into m mutually disjoint non-empty subsets Vk, k = 1, 2, ..., m) is feasible operation assignment (also called line balance) if the following two conditions hold.
Condition (I): Feasible operation assignment does not violate the precedence constraints given by digraph D = (V, A), i.e., inclusion (i, j) ∈ A implies that operation i is assigned to station Sk: i ∈ Vk, and operation j is assigned to station St: j ∈ Vt, such that 1 ≤ k ≤ t ≤ m.
Condition (II): Cycle time c is not violated for each station Sk, k ∈ {1, 2, . . . , m}, i.e., sum of the processing times of all the operations assigned to station Sk (also called station time), has to be not greater than cycle time c:
X
i∈Vk
pi ≤ c. (1.45)
For SALBP, line balance b is optimal when it uses the minimal number of m stations (optimality) and when the Condition (I) and Condition (II) are satisfied for b (feasibil-ity). The following claim may be easily proven by polynomial reduction of NP-complete partition problem to the decision version of SALBP with two stations and with A = ∅ (see, e.g., [Sch99]).
Constructing an optimal line balance for SALBP is binary NP-hard problem even for the case of two stations used in the optimal line balance, S = (S1, S2), empty precedence constraints, A = ∅, and fixed processing times of all the operations V processed on the assembly line, V = ∅.e
For the sake of simplicity, we will use the following notation p(Vk) = X
i∈Vk
pi,
but only for the original vector p = (p1, p2, ..., pn) of the operation times (in order to avoid misunderstanding). As it was already mentioned, set V includes operations of two types: Subset V of operations with variable processing time (manual operations) ande
subset V \V of operations with fixed processing time (automated operations). We assumee that, if j ∈V , then operation time pe j is given non-negative real number: pj ≥ 0, however the value of this operation time can vary during life cycle of the assembly line and can even be equal to zero. Zero operation time p0j will mean that operation j ∈ VkT
V will bee
processed (e.g., by an additional worker) in such a way that processing operation j will do not increase station time for Sk for the new vector p = (˜p0, ¯p) = (p01, p02, ..., p0˜n, pn+1˜ , ..., pn) of the operation times:
X
i∈Vk
p0i = X
i∈Vk\{j}
p0i.
Obviously, the latter equality is only possible if p0j = 0. If i ∈ V TV , then operatione time pi is given real number fixed during the life cycle of the assembly line. We assume that pi > 0 for each operation i ∈ V TV . As far as the processing time of the automatede operation is fixed, one can consider only automated operations, which have strictly positive processing times. Indeed, an operation with fixed zero processing time (e.g., due to using an additional machine for processing this operation) has no influence on the solution of SALBP under consideration.
In contrast to usual stochastic settings of SALBP, we do not assume any probability distribution known in advance for the random processing times of the manual operations.
Moreover, this section does not deal with algorithms for constructing optimal line balance in this type of uncertainty. It is assumed that optimal line balance b is already constructed for the fixed vector p = (p1, p2, ..., pn) of the operation times. Our aim is to investigate stability of the concrete line balance b, which is optimal for the given operation times pi, i ∈ V , with respect to independent variations of the processing times of all the manual operations or a portion of the manual operations.
More precisely, we investigate stability radius of an optimal line balance. Stability radius may be interpreted as a maximal possible error of the processing times pj of all the manual operations j ∈V , which definitely keep the optimality of the line balance b.e Next, we give some motivation for stability analysis of the optimal line balance and then formal definition of stability radius.
Motivation for Stability Analysis
Assembly line balancing problem arises when a new assembly line must be installed, and the internal demands and properties of the assembly line have to be well estimated since assembly line has to be used for a long time (long life cycle). Cycle time c may be defined on the basis of customer demands in the finished products. More precisely, the value of c may be calculated as the ratio of available operating time of the assembly line and production volume for the same calendar interval. One of the common mathematical problems at the stage of assembly line design is SALBP. This problem may arise also when cycle time c of acting assembly line has to be changed because of changing customer demands in the finished product.
In the real-world assemble lines, processing times of some operations may be known exactly and fixed for a long time (e.g., if operation has to be done by fully-automated machine or by semi-automated machine). Modern machines and robots are able to work permanently at a constant speed for a long time. However, in some cases it is not realistic
to assume constant operation times, e.g., if an operation has to be done by a human operator with rather simple tools. In the case of a human work, operation time is subject to physical, psychological and other factors. Moreover, due to the learning of operators, the operation times during the first days (weeks, months) of a life cycle of the assembly line may differ considerably from the processing times of the same operations during the later days (weeks, months). Also, some workers can leave the plant, and new workers with lower or higher skills have to replace them.
In the case of changeable operation times, it is important to know the credibility of the optimal line balance at hand with respect to possible independent variations of all or a portion of the operation times. Line balance b, which is optimal for the original vector p = (˜p, ¯p) of the operation times, may lose its optimality (and even feasibility) for some new vector of the operation times. For example, due to increasing of some operation times, line balance b may become infeasible for cycle time c since inequality (1.45) may be violated. In such a case, it is necessary to look for another line balance and to use it, if possible, for a suitable modification of production process on the assembly line.
Also, line balance b may lose its optimality with saving its feasibility. It may occur if another operation assignment, say bs, become feasible for the modified vector p0 = (˜p0, ¯p) of the operation times, and bs uses less stations than line balance b uses. Of course, each re-engineering and modification of the assembly line being in process take an additional time and other expenditure. So, assembly line modification has to be started when it is really necessary: When the income from it will be larger than the total expenditure caused by the re-engineering. Thus, an evaluation of expenditures and benefits should be conducted before deciding whether re-engineering of assembly line is necessary. However, these expenditures and benefits are difficult to evaluate before the end of the re-engineering process. In what follows, we derive some sufficient conditions for keeping the optimality of the line balance being in process. E.g., re-engineering is not necessary if the currently applied line balance remains optimal in spite of the changes of the operation times ˜p.
To test whether line balance b remains feasible for the new vector p0 = (˜p0, ¯p) of the operation times takes O(˜n) time (if station times are included in the input data) or O(n) time (otherwise). Indeed, for the new operation times we have to verify inequality (1.45) for each subset Vk, k = 1, 2, ..., m, which includes at least one manual operation with changed processing time in the new vector p0. On the other hand, in the case of feasibility of the line balance b for the new vector p0, in order to test its optimality for p0 we have again to solve NP-hard SALBP.
Intuitively, it is clear that sufficiently small changes of the operation times p1, p2, ..., p˜n may keep line balance b feasible and optimal for the new vector p0 = (˜p0, ¯p) of the operation times. In what follows, our aim is to estimate or (what is better) to calculate the largest independent variations of the operation times pi, i ∈V , which do not violate the feasibilitye and optimality of the line balance b at hand. Also note that at the stage of the design of the assembly line, there may exist a lot of optimal line balances. Using stability analysis, we can select such an optimal line balance, which feasibility and optimality are more stable with respect to possible variations of the operation times pi, i ∈V .e
Further, stability analysis of an optimal line balance will be developed. To this end, we will use the notion of stability radius, which is similar to stability radius of an optimal schedule introduced in Section 1.2 for general shop scheduling problems. If stability radius of line balance b is strictly positive, then any independent changes of the operation times
pj, j ∈ V , within the ball with this radius, definitely keep the optimality of line balancee b. However, if stability radius of b is equal to zero (i.e., if the optimality of line balance b is unstable), then some even small changes of the processing times of all or a portion of the manual operations may deprive the optimality of line balance b.
Let B denote the set of all assignments of operations V into stations S1, S2, ..., Sm (for all possible numbers m of stations: 1 ≤ m ≤ n), which satisfy Condition (I). Subset of set B of all operation assignments (line balances) which also satisfy Condition (II) for the given vector p = (p1, p2, ..., pn) of the operation times is denoted by B(p) = {b1, b2, ..., bh}.
Let subset of set B(p) of all the optimal line balances be denoted by Bopt(p).
Thus, inclusion b ∈ Bopt(p) implies that line balance b : V = V1b∪ V2b∪ ... ∪ Vmbb,
satisfies Condition (I), Condition (II), and the following optimality condition for the vector p = (p1, p2, ..., pn) of the operation times.
Condition (III). mb = min{mbk : bk ∈ B(p)}.
Hereafter bk, k ∈ {1, 2, ..., h}, means the following line balance:
V = V1bk ∪ V2bk ∪ ... ∪ Vmbk
bk.
Since line balance b is contained in the set B(p), we have b = br ∈ B(p) for some index r ∈ {1, 2, ..., h}. However, as a matter of convenience, index r will be omitted for the optimal line balance b, which stability will be investigated.
Note that in both definitions of set B and set B(p), number m of stations is not fixed.
Namely, for each line balance bk from the set B(p), inequalities mb ≤ mbk ≤ n must hold. And number of stations in an operation assignment from set B has to belong to set {1, 2, . . . , n}.
The main questions under consideration may be formulated as follows. How much can be modified the components of the vector ˜p simultaneously and independently from each other that the given line balance b remains feasible and optimal?
Let line balance b be optimal for the given non-negative real vector p = (˜p, ¯p) = (p1, p2, ..., pn) ∈ R+n of the operation times, i.e., b ∈ Bopt(p). The formal definition of stability radius of an optimal line balance b may be introduced as follows.
Definition 1.4 Open ball Oρ(˜p) with radius ρ ∈ R1+ and center ˜p ∈ R˜n+ in the space R˜n with the maximum metric is called a stability ball of the line balance b ∈ Bopt(p), if for each vector p∗ = ( ˜p∗, ¯p) of the operation times with ˜p∗ ∈ Oρ(˜p) ∩ R˜n+ operation assignment b remains feasible and optimal. The maximal value ρb(p) of the radius ρ of stability ball Oρ(˜p) of the optimal line balance b is called stability radius denoted by ρb(p).
It should be noted that in Definition 1.4 vector ˜p = (pn+1˜ , pn+2˜ , ..., pn) of the processing times of the automated operations and the complete vector p = (˜p, ¯p) = (p1, p2, ..., pn) of the operation times are fixed, while vector p∗ = (p∗1, p∗2, ..., p∗n˜) may vary within the intersection of the open ball Oρ(˜p) ⊂ Rn with the space Rn+˜ of non-negative real vectors.
Stability radius ρb(p) is equal to the minimal upper bound of independent variations εi
of the processing times pi of all the manual operations i ∈ V , which definitely keep thee optimality of the line balance b. In other words, inclusion b ∈ Bopt(p∗) must hold, if inequalities max{0, pi− εi} ≤ p∗i ≤ pi+ εi hold for each operation i ∈ V .
To illustrate the above notations, we use the following example of SALBP.
Example 1.6 We suppose that c = 10, ˜n = 4, n = 8 and p = (2, 3, 3, 9, 5, 3, 7, 2).
Thus, V = {1, 2, 3, 4} is set of the manual operations, and V \e V = {5, 6, 7, 8} is set ofe the automated operations. Digraph D = (V, A) is represented in Figure 1.7 where the manual operation times (which can be changed during the life cycle of the assembly line) are written in the usual form over the vertices, while the fixed operation times pi for automated operations i ∈ V \V are written in the bold.e
The following line balance b = br ∈ B(p) :
V1b = {1, 3, 5}, V2b = {2, 6}, V3b= {4}, V4b = {7, 8}
is optimal since b uses minimal possible number mb = 4 of stations. Indeed, the sum of all operation times pi, i ∈ V , is equal to p(V ) = 34, and number of stations in any line balance cannot be less than
dp(V )
c e = d34 10e = 4,
where dae denotes the smallest integer greater than or equal to a.
Figure 1.7: Digraph D = (V, A) and operation times
Let Vekb denote the subset of manual operations of set Vkb, and ¯Vkb denote the subset of automated operations of set Vkb. Thus, for each index k ∈ {1, 2, ..., mb}, we have Vkb =Vekb∪ ¯Vkb. The following remark will be used in the proofs of the main results in the rest of this section.
Remark 1.2 Let us consider the line balance b ∈ Bopt(p) being in process and the modi-fied vector p0 = (˜p0, ¯p) of the operation times. If there exists subset Vkb, k ∈ {1, 2, ..., mb}, in the line balance b such that
X
i∈Vkb
p0i = 0, (1.46)
we continue to affirm that the line balance b uses mb stations for the modified vector p0 = (˜p0, ¯p) as well. We can argue this as follows. In spite of the equality (1.46) valid for the vector p0 = (˜p0, ¯p), station Sk is still exists in the assembly line with line balance b.
At least to delete station Sk may cause some additional cost and time for re-engineering the assembly line. Moreover, after deleting station Sk we will obtain another line balance, say b∗ ∈ B :
V = [
i∈{1,2,...,mb},i6=k
Vib = V1b∗∪ V2b∗ ∪ ... ∪ Vmb∗
b∗, where mb∗ = mb− 1.
Note that because of inequality pi > 0 is valid for each automated operation i ∈ V \V ,e equality (1.46) is only possible if Vkb =Vekb.
Example 1.6 (continued). For Example 1.6, we have V3b =Ve3b = {4}, and equality (1.46) holds for station S3 and the modified vector p0 = (˜p0, ¯p) with ˜p0 = (2, 3, 3, 0). Hence, for the vector p0 = (˜p0, ¯p) = (2, 3, 3, 0, 5, 3, 7, 2) of the operation times there exists another line balance which is better than line balance b since it uses less stations than b.
Indeed, manual operation 4 (with zero operation time: p04 = 0) may be assigned to another station without increasing station time. For such a moving operation 4, we need only to guarantee feasibility of a new operation assignment with respect to Condition (I).
E.g., we can use the following operation assignment b(s):
V1b(s) = {1, 3, 5}, V2b(s) = {2, 4, 6}, V3b(s) = {7, 8},
obtained from the line balance b via moving operation 4 from station S4 to station S2. It is easy to see that operation assignment b(s) satisfies both Conditions (I) and (II) for the modified vector p0 = (˜p0, ¯p) and b(s) uses fewer stations than line balance b uses:
mb(s) = 3 < 4 = mb.
It should be noted that line balance b(s) is not feasible for the original vector p of the operation times (b(s) 6∈ B(p)), since inequality (1.45) from Condition (II) is violated for station S2: p(V2b(s)) = 3 + 3 + 9 = 15 > 10 = c. Note that d(˜p, ˜p0) =| p4− p04 |= 9, and thus we can conclude that ρb(p) ≤ 9.
Moreover, we can consider another vector p00 = (2, 3, 3, 4, 5, 3, 7, 2) of the operation times for which b 6∈ Bopt(p00) and b(s) ∈ Bopt(p00). The vector p00 is closer to the original vector p than vector p0 : d(˜p, ˜p) =| p4 − p004 |= 9 − 4 = 5. Therefore, we conclude that ρb(p) ≤ 5.
Farther, we show how to modify the original vector p of the operation times when there may be no subset Vkb with valid equality (1.46) in the line balance b.
Zero Stability Radius
We derive the necessary and sufficient conditions for the case when optimality (or feasi-bility) of the line balance b ∈ Bopt(p) is unstable.
Theorem 1.8 For the line balance b ∈ Bopt(p) equality ρb(p) = 0 holds if and only if there exists subset Vkb, k ∈ {1, 2, ..., mb}, such that Vekb 6= ∅ and p(Vkb) = c.
Proof. Sufficiency (if ). We make the following assumption (called assumption IF). Let there exist subset Vkb with Vekb 6= ∅ and p(Vkb) = c.
SinceVekb 6= ∅, there exists at least one manual operation j ∈Vekb. For any small positive real ε, we set p∗j = pj+ ε . And for all other manual operations i ∈V \{j}, we set pe ∗i = pi. As a result, we obtain vector ˜p∗ = (p∗1, p∗2, ..., p∗n˜). For the complete vector p∗ = (˜p∗, ¯p) of the operation times, from equality p(Vkb) = c (see assumption IF), it follows:
X
i∈Vkb
p∗i = X
j∈Vekb
p∗j + X
i∈ ¯Vkb
pi = X
i∈Vkb
pi+ ε = c + ε > c.
Therefore, operation assignment b does not satisfy Condition (II) for the new vector p∗ = (˜p∗, ¯p). As a consequence, b 6∈ B(p∗), and hence b 6∈ Bopt(p∗). Since distance defined by formula (1.4) with Q = V between the original vector ˜p and the new vector
˜
p∗ may be as small as desired: d(˜p, ˜p∗) = ε, we conclude that ρb(p) = 0. In such a case, stability ball Oρ(˜p) may be interpreted as a single point: Oρ(˜p) = {˜p}.
Necessity (only if ). We make the opposite assumption (called assumption ONLY IF).
There is no subset Vkb, k ∈ {1, 2, . . . , mb}, such that both conditionsVekb 6= ∅ and p(Vkb) = c hold. Hence, if Vekb 6= ∅, then p(Vkb) < c.
In order to get zero stability radius for the line balance b we have to show that for any arbitrarily small real ε > 0 there exists vector ˜p ∈ Rn+˜ that d(˜p, ˜p0) = ε and b ∈ Bopt(p0) with p = (˜p, ¯p) and p0 = (˜p0, ¯p). In principle, for realization equality ρb(p) = 0 there are three possibilities (we denote them as case (i), case (ii) and case (iii)), each of which corresponds to violation Condition (I), Condition (II) or Condition (III) for the operation assignment b with modified vector p0 of the operation times. Next, we consider these possibilities in details and prove that they are impossible for sufficiently small real ε > 0.
(i) To make operation assignment b inadmissible due to violation Condition (I) for the modified vector p0 of the operation times, i.e., b 6∈ B.
However, the latter contradicts to the inclusion b ∈ Bopt(p) being assumed in Theorem 1.8.
(ii) To make operation assignment b ∈ B inadmissible for the modified vector p0 due to violation Condition (II), i.e., b 6∈ B(p0).
Since b ∈ B, cycle time c may be exceeded for the vector p = (˜p, ¯p) = (p01, p02, ..., p0n˜, p0˜n+1, . . . , pn) ∈ Rn+ (i.e., inequality (1.45) may be violated) only for some station Sk withVekb6= ∅. Due to assumption ONLY IF for each such station Sk inequality p(Vkb) < c must hold.
Let us consider successively all such stations Sk, k ∈ {1, 2, ..., mb}, with Vekb 6= ∅ and p(Vkb) < c. To violate inequality (1.45) for subset p(Vkb) (and so to violate Condition (II) for b)), the closest to ˜p vector ˜p0 may be obtained by increasing all the components
pi, i ∈Vekb, of the vector ˜p = (p1, p2, ..., pn˜) by the same value, which has to be more than the following fraction:
c − p(Vkb)
|Vekb| = δbk. (1.47)
Due to inequalities p(Vkb) < c and |Vekb| ≥ 1 the value δkb is strictly positive: δib > 0.
If we add δib to the processing time pi of each operation i ∈ Vekb : pOi = pi+ δkb, and if we set pOj = pj for all other manual operations j ∈ V \e Vekb, we obtain equality Pi∈Vb
k pOi = c for the modified vector pO = (˜pO, ¯p) with pO = (pO1, pO2, ..., pOn˜).
In order to violate Condition (II), we can continue similarly to the above proof of sufficiency. Note only that in this case, distance d(˜pO, ˜p) is equal to δkb and therefore such a distance cannot be arbitrarily small for the fixed line balance b.
In order to violate Condition (II), we can continue similarly to the above proof of sufficiency. Note only that in this case, distance d(˜pO, ˜p) is equal to δkb and therefore such a distance cannot be arbitrarily small for the fixed line balance b.