At the end of this chapter, we consider maximum objective function Fmax(p, x) instead of linear objective function (2.1) considered in Sections 2.1 - 2.4:
Fmax(p, x) = max
i∈N{pixi}. (2.30)
The corresponding min-max (bottleneck) Boolean programming problem is to find an optimal vector (optimal solution) xp = (xp1, xp2, . . . , xpn) ∈ X with
Fmax(p, xp) = min{Fmax(p, x) : x ∈ X}. (2.31) In this section, we derive an explicit expression for the stability radius of optimal solution xp and show that it may be calculated by solving at most polynomially many instances of problem (2.31) of about the same size as the given instance. Both Defini-tion 2.1 and DefiniDefini-tion 2.2 may be similarly rewritten using maximum funcDefini-tion (2.30) instead of linear function (2.1). Let %bw0(x, p) denote stability radius of optimal solution x = xp ∈ X of the problem (2.31) with min-max criterion. All the other notations used for problem (2.31) are assumed to be the same as that used for the problem (2.2) with min-sum criterion.
Let Boolean bottleneck programming problem (2.31) with objective vector p ∈ Rn+ be given and its optimal solution xp be available. First, we consider the case of calculating stability radius %bw0(xp, p) when all the objective coefficients of the problem (2.31) are unstable, i.e., w = n. In this case, equality xp = ˜xp holds.
Recall that V (xp) denotes the set of indices of 1’s in the vector xp (see definition (2.18) at page 83). For each index j ∈ V (xp) we can solve the modified instance of the given problem (2.31) in which component xj of each feasible solution x ∈ X is required to be 0.
Let x[j] = (x[j]1 , x[j]2 , . . . , x[j]n ) denote an optimal solution of such modified instance of the problem (2.31) with set {x : x ∈ X, xj = 0} of the feasible solutions:
Fmax(p, x[j]) = min{Fmax(p, x) : x ∈ X, xj = 0}.
Let real number b[j] be the corresponding optimal objective value:
b[j] = max
i∈N{pix[j]i }.
We denote
%j = b[j]− pj
2 . (2.32)
If equality xj = 1 holds for all feasible solutions x ∈ X of the given instance of the problem (2.31) (i.e., if set of the feasible solutions of the modified instance of the problem (2.31) is empty), then we define b[j]= %j = ∞.
Theorem 2.12 If all the n objective coefficients of the problem (2.31) are unstable (w = n), then
%bw0(xp, p) = min{%j : j ∈ V (xp)}.
Proof. First, we prove inequality
%bw0(xp, p) ≥ min{%j : j ∈ V (xp)}. (2.33) To this end, we consider a vector δ = (δ1, δ2, . . . , δn) ∈ Rn with |δi| ≤ min{%j : j ∈ V (xp)} for all indices i = 1, 2, . . . , n.
It is clear that any feasible solution x ∈ X with xj = 1 for all indices j ∈ V (xp) has always a value of the objective function greater than or equal to Fmax(p + δ, xp).
Therefore, it is suffices to consider only feasible solutions x ∈ X which have equality xj = 0 for some indices j ∈ V (xp). For such a feasible solution, let k ∈ V (xp) be such an index that xk = 0 and pk≥ pj for all indices j ∈ V (xp) with xj = 0. Note that xj = 1 for all indices j ∈ V (xp) with pk < pj. For any feasible solution x ∈ X, this implies inequality:
Fmax(p + δ, x) ≥ max{pi + δi : i ∈ V (xp), pi > pk}. (2.34) Furthermore, it follows from inequality Fmax(p, x) ≥ b[k] and equality (2.32) that
Fmax(p + δ, x) ≥ b[k]− min{%j : j ∈ V (xp)} ≥ b[k]− %k= (b[k]+ pk)/2. (2.35) For any index i ∈ V (xp) with pk ≥ pi we obtain
pi+ δi ≤ pk+ min{%j : j ∈ V (xp)} ≤ pk+ %k ≤ (b[k]+ pk)/2. (2.36) Using the lower bounds on Fmax(p + δ, x) defined in (2.34) and (2.35), as well as inequalities (2.36), we obtain
Fmax(p + δ, x) ≥ max{(b[k]+ pk)/2, max{pi+ δi : i ∈ V (xp), pi > pk}}
≥ max{pi+ δi : i ∈ V (xp)} = Fmax(p + δ, xp).
This establishes the desired inequality (2.33).
Next, we prove opposite inequality
%bw0(xp, p) ≤ min{%j : j ∈ V (xp)}. (2.37) Obviously, if min{%j : j ∈ V (xp)} = ∞, we are already done with the proof.
So, we have to prove inequality (2.37) for the case when min{%j : j ∈ V (xp)} < ∞.
To this end, we will show that for any real number % strictly greater than min{%j : j ∈ V (xp)}, there exists a vector δ ∈ Rn such that |δi| ≤ % for each index i ∈ N , while Fmax(p + δ, x) < Fmax(p + δ, xp) for some feasible solution x ∈ X.
To be more specific, we assume that % > %k for some index k with xpk = 1 and let x = x[k]. We consider the vector δ where δk = %. For i 6= k, we assume that δi = −%k, if x[k]i = 1 and pi ≥ (b[k]+ pk)/2, and we assume δi = 0 otherwise.
On the one hand, we obtain
Fmax(p + δ, xp) ≥ pk+ δk> pk+ %k = (b[k]+ pk)/2.
The last equality follows from (2.32). On the other hand, we have Fmax(p + δ, x[k]) = max{pi+ δi : i ∈ N, x[k]i = 1}.
Due to the above definition of δ, if x[k]i = 1 and pi ≥ (b[k]+ pk)/2, then pi+ δi = pi− %k ≤ Fmax(p, x[k]) − (b[k]− pk)/2 = (b[k]+ pk)/2.
As a result we obtain Fmax(p + δ, x[k]) < Fmax(p + δ, xp). Hence, inequality (2.37) holds, which completes the proof.
3 To compute value b[j]for index j ∈ V (xp), we just need to solve the instance of problem (2.31) with objective vector ˜p = (˜p1, ˜p2, . . . , ˜pn) ∈ Rn+, where ˜pj is equal to a value g which is strictly greater than the largest value of p1, p2, . . . , pn and ˜pi = pi for all other indices i ∈ N , i 6= j. If the optimal objective value of this problem instance turns out to be g, then it follows that xj = 1 for each feasible solution x ∈ X and so b[j] = ∞. Otherwise, the optimal objective value is exactly b[j]. It therefore follows from Theorem 2.12 that when all the objective coefficients are unstable, stability radius%bw0(xp, p) can be calculated by solving |V (xp)| instances of problem (2.31). It is clear that |V (xp)| = Pni=1xpi. Thus, the following corollary is proven.
Corollary 2.10 If w = n, then stability radius %bw0(xp, p) can be calculated by solving
Pn
i=1xpi instances of problem (2.31).
Stable and Unstable Objective Coefficients
Let w be any arbitrary positive integer less than or equal to n. Without loss of generality, we assume that pw+1≤ pw+2 ≤ . . . ≤ pn. Next, we show how to compute stability radius
%bw0(xp, p) as the minimum of certain value%bj, j ∈ V (xp), which are defined below.
Instead of the value b[j] used in Theorem 2.12 for calculating stability radius%bw0(xp, p) with w = n, we use here the value b[j,m] defined for each index j ∈ V (xp), j ≤ w, and index m as follows.
Let b[j,m] denote the optimal objective value of the modified instance of the given problem (2.31) with the same objective vector p and the additional restrictions on the set of feasible solutions: xj = 0 and xi = 0 for all indices i with m < i ≤ n. If this modified instance does not have a feasible solution, then we define b[j,m] = ∞.
Let x[j,m] be an optimal solution of the modified problem instance, and let d[j,m]denote the value of the largest pi, i > w, for which x[j,m] = 1. We define d[j,m] to be −∞ if x[j,m]i = 0 for all indices i > w. For index j ∈ V (xp), j ≤ w, we define
%bj = min
m≥wmax{b[j,m]− pj
2 , d[j,m]− pj}. (2.38)
First, we prove the following auxiliary claim.
Lemma 2.3 Let x ∈ X be an optimal solution of the modified instance of the given problem (2.31) with xj = 0 for some index j ∈ V (xp), j ≤ w. Let l be the largest index such that l > w with xl = 1 (we assume pl = −∞, if no such index exists). Then
pi+%bj ≤ max{(Fmax(p, x) + pj)/2, pl}.
Proof. We suppose that xi = 0 for all indices i > w. Then Fmax(p, x) ≥ b[j,w]. Since d[j,w] = −∞, it follows that
pj+%bj ≤ max{(b[j,w]+ pj)/2, d[j,w]} ≤ (Fmax(p, x) + pj)/2.
If xi = 1 for some index i > w, then Fmax(p, x) ≥ b[j,l] and pl ≥ d[j,l]. Therefore, pj+%bj ≤ max{(b[j,l]+ pj)/2, d[j,l]} ≤ max{(Fmax(p, x) + pj)/2, pl}.
3 Now, we can formulate and prove the following theorem.
Theorem 2.13 If there are both stable and unstable objective coefficients in the problem (2.31), then
%bw0(xp, p) = min{%bj : j ∈ V (xp)}.
Proof. First, we show that the stability radius is at least equal to min{%bj : j ∈ V (xp)}.
To this end, we consider a vector δ ∈ Rn such that δi = 0 for all indices i > w, and
|δi| ≤ min{%bj : j ∈ V (xp)} for all indices i ≤ w. Thus, we obtain
Fmax(p + δ, xp) = max
max{pi+ δi : i ∈ V (xp), i ≤ w, pi ≤ pk}, max{pi+ δi : i ∈ V (xp), i ≤ w, pi > pk}, max{pi : i ∈ V (xp), i > w, xpi = 1}, max{pi : i ∈ V (xp), i > w, xpi = 0}.
(2.39)
For any feasible solution x ∈ X, we will show that the four expressions on the right-hand side of (2.39) are all lower bounds on Fmax(p + δ, x).
Let expression {pi : i ∈ V (xp), i ≤ w, xpi = 0} take the maximum value for index i = k. Then the first expression on the right-hand side of (2.39), max{pi + δi : i ∈ V (xp), i ≤ w, pi ≤ pk}, is less than or equal to pk + min{%bj : j ∈ V (xp)} ≤ pk +%bk. Because of Lemma 2.3 this is at most max{(Fmax(p, x) + pk)/2, pl}, where pl is defined as in Lemma 2.3. Clearly, pl ≤ Fmax(p + δ, x). Furthermore, if (Fmax(p, x) + pk)/2 > pl, then Fmax(p + δ, x) ≥ Fmax(p, x) −%bk= (Fmax(p, x) + pk)/2.
To prove that the second expression on the right-hand side of (2.39) is a lower bound on Fmax(p + δ, x), it suffices to observe that if i ∈ V (xp) and pi > pk, then xpi = 1.
The third expression on the right-hand side of (2.39) is obvious lower bound on Fmax(p + δ, x).
Let expression {pi : i ∈ V (xp), i > w, xpi = 0} take the maximum value for index i = r. Then the forth expression on the right-hand side of (2.39) is equal to pr. To show that this is a lower bound on Fmax(p + δ, x), we first note that this is certainly true if it is not greater than pl. Now we suppose that pr > pl, i.e., xpi = 0 for all i > w with pi ≥ pr. Then Fmax(p, x) ≥ gr, and we have Fmax(p + δ, x) ≥ Fmax(p, x) −%br ≥ gr−%br = pr.
This establishes the desired inequality.
If we are given any real number % strictly greater than min{b %bj : j ∈ V (xp)}, then we can find a vector δ ∈ Rn such that |δi| ≤ % for each index i ≤ w, δb i = 0 for each index i > w, while Fmax(p + δ, x) < Fmax(p + δ, xp) for some feasible solution x ∈ X. The argument is quite similar to that used in the proof of Theorem 2.12, and therefore we omit the details. So, this completes the proof.
3
Similarly as Corollary 2.10 was proven on the basis of Theorem 2.12, next we prove analogous Corollary 2.11 on the basis of Theorem 2.13.
To compute value b[j,m], j ∈ V (xp), j ≤ w, m ≥ w, we need to solve the instance of problem (2.31) with objective vector ˜p∗ = (˜p∗1, ˜p∗2, . . . , ˜p∗n) ∈ Rn+, where ˜p∗i is equal to a value g for i = j and all i > m, and ˜p∗k = pk for all other indices k ∈ N . If the optimal objective value of this problem instance turns out to be g, then b[j,m] = ∞. Otherwise, the optimal objective value is exactly b[j,m] and we obtain an optimal solution x[j,m] and the corresponding value d[j,m]. In fact, to compute %bj it actually suffices to calculate b[j,m]
in order of decreasing m until a value of m is reached for which inequality b[j,m]− pj
2 ≥ d[j,m]
holds, because b[j,m] is non-increasing in m.
For j ∈ V (xp), j > w, let gj denote the optimal value of the problem instance with the objective vector p and the additional restrictions xj = 0 and xi = 0 for all indices i > w with pi ≥ pj. If this problem instance does not have a feasible solution, then we obtain gj = ∞. For j ∈ V (xp), j > w, we obtain
%bj = gj− pj. (2.40)
The calculation of gj is obvious. Thus, we proved the following claim.
Corollary 2.11 If inequalities 1 ≤ w < n hold, then stability radius %bw0(xp, p) can be calculated by solving Pni=1xpi instances of problem (2.31).
Corollaries 2.10 and 2.11 imply the following claim.
Theorem 2.14 If Boolean bottleneck programming problem (2.31) is polynomially solv-able for any objective vector p ∈ Rn+, then stability radius%bw0(xp, p) of an optimal solution xp can be calculated in polynomial time.
2.6 Comments and References
This is no surprise, since it has been shown in [RC95] and [vHW99] that, even if w = 1, computing the stability radius exactly for any ≥ 0, is NP-hard if the associated optimization problem (in our case: the scheduling problem) is NP-hard.
The traveling salesman problem was the first linear trajectory problem for which a formula for calculating the stability radius of the whole set of optimal trajectories has been derived [Leo75]. The proof of Theorem 4 can be found in [Sot93]. In the following we strengthen an upper bound for the stability radius given in [Sot93] by using the following result about the existence of an infinitely large stability radius of an -approximate solution.
Necessary and sufficient conditions for %w(x, p) = 0 have been given in [Sot93], which are valid only for ˜x 6= ˜0, where ˜0 is the zero vector in Rw+.
A concept of stability analysis for the latter problem has been developed in [GL80, GL85, Leo75, Leo76, Lib91, Tar82] and in some other papers (see [SLG95] for the ixtensive
survey). It should be noted that most results have been obtained for the stability radius of the whole set of optimal trajectories, i.e., for the largest radius of an open ball in the space of the numerical input data such that a new optimal trajectory does not arise.
Unfortunately, the set of all optimal trajectories is often unknown since its cardinality may be large. Even if the optimal trajectory is unique for the problem, this information usually is inaccessible for OR workers. On the other hand, the investigation of the stability radius of one optimal trajectory of such a problem has the following drawback: The stability radius of an optimal solution of a linear trajectory problem is equal to zero, if at least one alternative optimal solution exists. Therefore, in [KSL98, Sot93] the stability of an
-approximate solution has been investigated.
In [Sot93] the following theorem has been proved.
There exist a lot of papers presenting different approaches to stability analysis of discrete optimization problems, and in the last part of this section, we provide a sketch of some approaches to stability analysis, which are close to the subject of this dissertation.
A related approach to stability analysis for linear trajectory problems (such as the traveling salesman problem, the assignment problem, the shortest path problem) and some other discrete optimization problems has been initiated in [GL80, Leo75, Leo76, Lib91, SW80, Tar82] and developed in some other papers (see Sotskov et al. [SLG95] for an extensive survey). Most results have been obtained for the stability radius of the whole set of solutions (optimal trajectories), i.e. for the largest radius %(p) of an open ball in the space of the numerical input data p such that a new optimal trajectory does not arise. A formula for calculating the stability radius %(p) of the set of all solutions of the traveling salesman problem is obtained by Leontev [Leo75, Leo76] and the extreme values of %(p) are determined. Gordeev and Leontev [GL80] derive analogous results for a similar problem with a bottleneck objective function. A specific transformation of a branch-and-bound algorithm for the traveling salesman problem for calculating %(p) is suggested by Gordeev et al. [GLS83]. Gordeev [Gor89] proposes a polynomial algorithm for calculating the stability radius of the whole set of solutions of extremal problems on matroids and on the intersection of two matroids.
It should be noted that related approaches to stability analysis for the traveling salesman problem, the shortest path problem, and some others, which can be repre-sented as a binary optimization problem with a linear objective function, are developed in [GL80, Lib99, Lib91, LvdPSvdV96, Tar82].
The complexity of calculating the stability radius %(p) of a solution of a discrete optimization problem is studied in [GL85, RC95]. Ramaswamy and Chakravarti [RC95]
show that the problem of determining the arc tolerance for a discrete optimization problem is as hard as the problem itself (the arc tolerance is the maximum change, i.e. increase or decrease, of a single weight, which does not destroy the optimality of a solution). This means that in the case of the traveling salesman problem, the arc tolerance problem is NP-hard even if an optimal tour is given. Gordeev [Gor89] proved the NP-hardness of the problem of calculating %(p) for the polynomially solvable shortest path problem in a digraph without negative circuits. Sotskov et al. [SWW98] show that the stability radius of an approximate solution may be calculated in polynomial time if the number of unstable components grows rather slowly, namely as O(log N ), where N is the number of cities in the traveling salesman problem. Libura et al. [LvdPSvdV96] argue that it is rather convenient from a computational point of view to use the set of k shortest tours
when applying a stability analysis to the symmetric traveling salesman problem.
In deterministic scheduling theory the processing times are supposed to be given in advance, i.e., before applying a scheduling procedure. More general cases have been considered in stochastic scheduling (see [Pin95a]), where pi is a random variable with a known distribution of probabilities. However, in practice such functions may also be unknown. The results surveyed and developed in this paper may be considered as an attempt to initialize further investigations of scheduling problems under conditions of uncertainty.
We applied the same stability analysis for a large scale of scheduling problems: That which may be represented as linear binary programming problem and more general scheduling problems which may be represented as extremal problem on a disjunctive (mixed) graph. Of cause, complexity of the problem has to be taken into account: The stability results which seems to be appropriate for the general shop problem (see Sections 2 – 6) are rather rough for the linear binary programming problem which allows to derive more deep mathematical results and more efficient algorithms.
In turn, stability properties of an optimization problem may be used to characterize its complexity. We can illustrate it on a job shop problem from the first part, and on a traveling salesman problem and an assignment problem from the second part. (The letter means optimal distribution of n jobs on m parallel non-identical machines in a single-stage system). The stability radius of optimal schedule for problem J//Cmax have usually strictly positive stability radius, even if it is not unique optimal schedule (see Theorem 1 and computational results in Section 6). On the other hand, it is easy to show that %ω0(x, p) = 0 if there exist at least two optimal solutions for a traveling salesman problem (or for an assignment problem). Because of this reason, the main focus in the second part of the paper was given to stability analysis of -approximate solution (see Theorems 6 and 7). As follows from [BSW96] a general shop problem with mean flow time criterion (i.e. of type ’min-sum’) becomes even more difficult for stability analysis than problem G
Cmax.
So possible trends for future research may be to investigate connections between the complexity of scheduling problems and the complexity of calculating the stability radius of an optimal schedule (see [RC95, SLG95, vHW99]). Recall that in [RC95, vHW99] it was shown that the existence of a polynomial algorithm for calculating %10(x, p) implies a polynomial algorithm for problem (6.1). In [vHW99] similar implication was proven also for the case > 0. Moreover, in [RC95] it was shown that if problem (6.1) is polinomially solvable, then %10(x, p) may be calculated in polynomial time. Thus the value %10(x, p) may be calculated in polynomial time for the assignment problem, if optimal solution x is given, while the calculating the same value for traveling salesman problem is N P -hard. Interesting subject for research may be connected with generalization of the result from [RC95]: Is it possible to find the stability radius %s(p) of an optimal schedule s in polynomial time, if s may be constructed in polynomial time?
The setting of scheduling problem in the first part of the paper is so general that it is unlikely to find simple answers to those questions, which are usual for deterministic scheduling problems. So, possible stability analysis may consist in determining classes of rather simple scheduling problems for which it is possible to find the stability radius of an optimal or an approximate solution in a suitable time, e.g., if the stability region for such
a class will be convex, an algorithm similar to that developed in the proof of Theorem 11 may be applied. Therefore also interesting topic of research is to establish that for some kinds of scheduling problems the stability region of an optimal schedule is a convex set.
Important question is connected with the determination of simple conditions (prefer-ably conditions which can be verified in polynomial time) for the validity of %s(p) = 0 similar to those derived in [KSW95] for %s(p) = ∞ for the problems J//Cmaxand J//Lmax. It is of interest also to develop simple bounds for %s(p) and %w(x, p) (see e.g. (2.2), Theo-rem 5, Corollary 2 and TheoTheo-rem 2.4). An interesting question is how a branch and bound algorithm, which is often used for NP-hard scheduling problems, can be combined with calculating (bounds on) the stability radius of an optimal or -approximate schedule (see [LvdPSvdV96, SSW97]).
Concluding we can note that the above approach to stability analysis is not the only possible one (see survey [SLG95]). For example, in [KRKvHW94] the sensitivity of a heuristic calculating algorithm with respect to the variation of the processing time of one job has been investigated.
Table 2.1: Notations for Boolean programming
Symbols Description
X Set of feasible solutions x (n-dimensional Boolean vectors): x ∈ X ⊆ {0, 1}n N Subset of natural numbers: N = {1, 2, . . . , n}
F (p, x) Linear objective function: F (p, x) =P
i∈Npixi
p Objective vector p = (p1, p2, . . . , pn) ∈ Rn+, where pi is objective coefficient xp Optimal solution (optimal vector) xp= (xp1, xp2, . . . , xpn) ∈ X ⊆ {0, 1}n:
F (p, xp) = min{F (p, x) : x ∈ X}
x -approximate solution x = (x1, x2, . . . , xn) ∈ X, if F (p, x) ≤ (1 + ) · F (p, xp) w Number of unstable objective coefficients
˜
x Vector ˜x = (x1, x2, . . . , xw) of the first w components of n-dimensional Boolean vector x ∈ {0, 1}n, 1 ≤ w ≤ n
X˜ Set of w-dimensional Boolean vectors of the first w components of feasible solutions x: ˜X = {˜x ∈ {0, 1}w : x ∈ X ⊆ {0, 1}n}
˜
p w-dimensional vector of unstable objective coefficients ˜p = (p1, p2, . . . , pw) of the objective vector p with n − w stable objective coefficients:
pw+1= ¯pw+1, pw+2= ¯pw+2, . . . , pn= ¯pn
Kw(x) Stability region of an -approximate solution x ∈ X:
Kw(x) = {˜p ∈ Rw+ : F (p, x) ≤ (1 + ) · F (p, xp)}
O%w(p) Stability ball of -approximate solution x ∈ X, if O%w(p) ∩ Rw+⊆ Kw(x)
%w(x, p) Stability radius of the -approximate solution x ∈ X
of the Boolean programming problem with linear objective function
%w0(x, p) Stability radius of the optimal solution x = xp∈ X
of the Boolean programming problem with linear objective function U Binary relation on set ˜X with maximal cardinality:
If (˜x, ˜x0) ∈ U and i ∈ {1, 2, . . . , w}, then xi= 1 implies x0i = 1 0 Zero vector in the space Rw+: 0 = (0, 0, . . . , 0) ∈ Rw+
p∗ Non-negative real number defined as follows: p∗= max{pi : 1 ≤ i ≤ n}
p Non-negative real number defined as follows:
p= maxmax{pi : 1 ≤ i ≤ w}, (1 + ) ·Pn
i=w+1pi−Pn
i=w+1pixi
∆p Non-negative real number defined as follows: ∆p= (1 + ) · F (p, xp) − F (p, x) V (x) Set of indices of 1’s in the first w components of
n-dimensional Boolean vector x ∈ X ⊆ {0, 1}n, 1 ≤ w ≤ n v(x) Cardinality of set V (x): v(x) = |V (x)|
Fmax(p, x) Maximum objective function: Fmax(p, x) = maxi∈N{pixi}
%bw0(x, p) Stability radius of the optimal solution x = xp∈ X
of the Boolean programming problem with min-max criterion x[j] Optimal solution of the modified instance of the given problem with
w = n: Fmax(p, x[j]) = min{Fmax(p, x) : x ∈ X, xj= 0}
b[j] Optimal objective value for the modified instance of the given problem with w = n: b[j] = maxi∈Npix[j]i
%j Non-negative real number defined as follows: %j= (b[j]− pj)/2 x[j,m] Optimal solution of the modified instance of the given problem:
Fmax(p, x[j,m]) = min{Fmax(p, x) : x ∈ X, xj= 0, xi = 0, i > m}
b[j,m] Optimal objective value for the modified instance of the given problem:
b[j,m]= maxi∈Npix[j,m]i
General and Job Shops with Uncertain Processing Times
This chapter deals with the general shop and job shop scheduling problems with the objective to minimize the makespan or mean flow time provided that the numerical input data are uncertain.
In [Pin95a], it was noted that one “source of uncertainty is processing times, which, typically, are not known in advance. Thus, a good model of a scheduling problem would need to address these forms of uncertainty.” In stochastic settings of scheduling problems, the random processing time of an operation is assumed to take a known probability dis-tribution. The scheduling environments that we consider in this chapter are so uncertain that all information available about the processing time of an operation is its upper and lower bounds.
To be more specific, we consider problem G/ai≤ pi≤ bi/Φ and problem J /ai≤ pi≤ bi/Φ for dealing with uncertain scheduling environments in which only lower bound ai and upper bound bi for the processing time pi of operation i ∈ Q are known before scheduling.
Such problems may arise in many practical situations since, even if no specific bounds for an uncertain processing time pi are known in advance, we can set ai = 0 and bi equal to the planning horizon. In spite of practical interest, such a type of scheduling problems was considered in a limited OR literature so far.
In this chapter, we use a mixed graph model for representing the input data, the scheduling process and the final solution. Our ‘strategy’ is to separate the ‘structural’
In this chapter, we use a mixed graph model for representing the input data, the scheduling process and the final solution. Our ‘strategy’ is to separate the ‘structural’