An Efficient Mixed-Integer Linear Programming Scheduling

6346

Ind. Eng. Chem. Res. 2009, 48, 6346–6357

An Efficient Mixed-Integer Linear Programming Scheduling Framework for Addressing Sequence-Dependent Setup Issues in Batch Plants Georgios M. Kopanos, Jose´ Miguel Laıńez, and Luis Puigjaner* Department of Chemical Engineering, UniVersitat Polite`cnica de Catalunya, ETSEIB, AV. Diagonal 647, 08028 Barcelona, Spain

Setup times and/or costs appear in a plethora of industrial and service applications. Sequence-dependent setup times and costs entail significantly complex scheduling problems. In this paper, a new continuous time precedence-based mixed-integer linear programming scheduling framework is developed for dealing with sequence-dependent setup time and/or cost issues. This model is based on the unit-specific general precedence concept, which is introduced in this paper. Several case studies are carried out in order to highlight the efficiency and the special features of the proposed model. Comparisons were made with the existing precedence-based formulations. Medium-sized scheduling problems can be solved by the proposed framework with relatively low computational effort. Introduction Setup times (costs) are one of the most common complications in scheduling problems. The setup time can be generally defined as the time required to prepare the necessary resources (e.g., processing units, people) to perform a specific task (e.g., job, operation). Therefore, the setup cost is the cost of setting up any resources used prior to executing a task. Scheduling problems involving setup times can be mainly categorized into sequence-dependent and sequence-independent setups. The setup is sequence-dependent if its duration depends on both the current task and the one immediately preceding it; the setup is sequenceindependent if its duration only depends on the current processing task. The most complicated case and the one that is addressed least in the literature is the sequence-dependent setup.1 The problem of prescribing a sequence, even for a single machine, with sequence-dependent setups and makespan as the objective is equivalent to the traveling salesman problem (TSP) and is therefore strongly NP-hard.2 Several earlier papers have reviewed research related to setups.3,4 A recent review of scheduling problems with setup times and/or costs can be found in Allahverdi et al.5 Sequence-dependent setup operations appear in a large number of industrial applications. A typical example of sequencedependent setups, given by Conway et al.,6 is the manufacture of different colors of paint. The main industries that use applications with sequence-dependent setups are stated below: • Textile industry. The setup for weaving and dyeing operations depends on the job sequence.7 • Printing industry. The cleaning and setting of the press for processing the next job depend on the ink color, paper size and types used in the previous job. • Container and bottle industry. The settings change depending on the sizes and shapes of the containers.8 • Plastic industry. Different types and colors of products require sequence-dependent setups. • Paper industry.7 • Chemical compounds manufacturing industry. The extent of the cleansing depends on both the chemical most recently processed and the chemical about to be processed. * To whom correspondence should be addressed. E-mail: [email protected].

• Chemical process industry. Cleaning of processing units, inspecting material (analytic and quality control), preheating and mixing are some examples that are associated with product changeover.9 • Pharmaceutical industry. There may be significant sequencedependent setup times depending on legal regulations. • Food processing industry.10 • Metal processing industry.11 • Tile industry.12 A more complete list of application areas for scheduling with setup times/costs as well as a discussion of the significance of reducing setup times/costs can be found in the neatly written contribution work by Allahverdi and Soroush.13 For reasons of simplicity, the majority of the works addressing scheduling problems assume that the sequence-dependent setup times (costs) are negligible or are part of the job processing time (cost). While this assumption simplifies the analysis and/ or reflects certain applications, it adversely affects the solution quality of many scheduling applications that require setups to be treated explicitly.5 Therefore, sequence-dependent setup aspects ought to be incorporated into scheduling problem representations during the optimization process in order to obtain adequate solutions. In some situations where setup time and setup cost are proportional, which typically occur when the resource idle time is the only concern, it is sufficient to consider either setup time or setup cost. However, in other situations, where the cost of changeovers between certain tasks is relatively high even though the setup time is relatively low, the setup cost is of more significance (this may occur in environments requiring highskilled labor).13 The computational time required to solve scheduling problems is a very significant issue for the process systems engineering (PSE) scientific community, and industrial practitioners. In most cases, the scheduling problem’s computational time is as important as the scheduling problem itself, as industry demands solutions that are both optimal and quickly reached. Therefore, heuristic or metaheuristic techniques are usually implemented in order to reduce the computational burden of solving the problem. For instance, genetic algorithms, simulation annealing, tabu search, particle swarm, and ant colony optimization methods have been used in a variety of scheduling problems. Each approach has unique characteristics that make it suitable

10.1021/ie801127t CCC: $40.75  2009 American Chemical Society Published on Web 06/02/2009

Ind. Eng. Chem. Res., Vol. 48, No. 13, 2009

6347

Figure 1. Current and proposed precedence-based scheduling frameworks.

for specific problems. Though the aforementioned methods generate feasible solutions very quickly, they cannot guarantee optimality, which is actually only achieved in rare cases. Therefore, the industry ends up operating in suboptimal performance conditions, with the resulting loss of competitive advantage and large amounts of potential benefits. In other words, the industry’s value-chain is shrunk. With the advances in mathematical and computer sciences, exact methods seem to be closer than ever to solving industrial problems. In the PSE community, a plethora of MILP mathematical models and representations have been developed to cope with the problem under study. A clear review can be found in the work by Meńdez et al.14 Scheduling optimization approaches can be roughly classified into discrete and continuous time formulations. In discrete time models, schedule events can only take place at certain predefined time points, while in continuous time models, schedule events can occur at any time during the production time horizon. Discrete time model structures are usually simpler and faster to solve because the model complexity depends on the predefined time intervals; however, suboptimal schedules are likely to be generated because of restrictions in timing decisions. Continuous time models do not have this drawback but they are generally more computationally expensive. Continuous time representation strategies based on the precedence relationships between batches to be processed have been developed to deal with the scheduling problem. In these formulations, model variables and constraints enforcing the sequential use of shared resources are explicitly employed, and therefore sequence-dependent setups can be treated straight-forwardly.14 The three different precedencebased approaches that can be found in the literature are (i) the immediate precedence, (ii) the unit-specific immediate precedence, and (iii) the general precedence. Immediate (or local) precedence (IP) explores the relation between each pair of consecutive orders in the production schedule time horizon without taking into account whether the orders are assigned to the same unit. Unit-specific immediate precedence (USIP) is based on the immediate precedence concept. The difference is that it only takes into account the immediate precedence of the orders that are assigned to the same processing unit. General (or global) precedence (GP) generalizes the precedence concept by exploring the precedence relations of each batch, taking into account all the remaining batches and not only the immediate predecessor.15 The last approach results in a lower number of binary variables and, compared with the other two approaches, it significantly reduces the computational effort on average. However, it cannot cope with sequence-dependent setup issues explicitly (especially if there are changeover times greater than a batch processing time), as it is clearly demonstrated in the motivating example. Moreover, scheduling models based on the GP notion cannot be used to address problems with sequence-dependent changeover costs because the global-sequencing variables are active for all the batch pairs assigned to the same unit. To address this limitation, we developed a new precedence-based

scheduling formulation, the unit-specific general precedence (USGP), which is presented in detail here. Figure 1 shows the precedence-based frameworks for the short-term scheduling in batch processes. For reasons of clarity, the current precedence-based formulations are included in the Appendix. The current manufacturing trend of producing small batches or unit products to satisfy demand and avoid inventory makes the scheduling problems of sequence-dependent setup times more relevant.16 Thus, rigorous scheduling models that can deal with these issues explicitly and efficiently need to be developed. The current work aims at being a step toward smoothing the gap between scheduling theory and practice. This article presents a novel precedence-based mathematical formulation for the noncycling scheduling of single-stage batch plants with several units in parallel. Some of the most significant features of this work and the proposed MILP scheduling precedence-based representation model are listed. (1) The continuous time domain representation does not rely on the definition of time slots or time events. (2) The sequencing-assignment decision variables seem to reduce computational time; the model size increases, however, the exploration tree significantly reduces, as the case studies show. (3) The binary immediate sequencing variables allow the product sequence-dependent setup time and/or cost aspects to be dealt with more efficiently than the existing precedence-based continuous time formulations in scheduling problems. (4) The general precedence framework is unable to tackle sequence-dependent setup time and/or cost issues and problems correctly considering forbidden processing sequences. (5) The extended general precedence notion can cope explicitly with sequence-dependent issues and forbidden processing sequences. A hybrid precedence-based scheduling framework has been developed, which combines the concepts of global and local precedence. Problem Statement In this paper, we address the scheduling problem in singlestage multiproduct batch plants with different processing units in parallel. Batch to unit assignment and batch sequencing in order to meet a production goal constitutes the understudy scheduling problem. Sequence-dependent setup times, which greatly increase the complexity of the scheduling problem, are explicitly considered. The main assumptions of the proposed model include the following: (i) Only single-stage product orders are considered (see Figure 2). (ii) An equipment unit cannot process more than one batch at a time. (iii) To begin another task in a processing unit, the current task must have been completed, that is, the operation mode is non-preemptive. (iv) Processing times, unit setup times and product sequence-dependent setup times and/or costs and due dates are deterministic.

6348

Ind. Eng. Chem. Res., Vol. 48, No. 13, 2009 Table 1. Case Study I-II and Motivating Example Data processing times (days)

Figure 2. Single-stage production scheme.

(v) Unforeseen events, such as unit breakdowns, that may disrupt the normal plant operation do not appear during the scheduling time horizon. (vi) No resource constraints except for equipment availability are taken into account. (vii) Product batch sizing is carried out beforehand, and thus batch sizes are known a priori. In a real-world batch industrial plant, assumption (iii) is frequently satisfied. This is not the case, however, for assumption (i), since there are many industrial applications that are multistage. The proposed model can be appropriately modified to deal with multistage plants, but this modification is not presented here since it is not one of the objectives of the current work. By adding a set of resource constraints similar to the ones that were reported in the work by Meńdez and Cerda´,17 assumption (vi) can be satisfied. If assumptions (iv) and/or (v) are relaxed, then uncertainty should be included in the optimization procedure. There are generally two ways of treating unexpected events: proactively (off-line) or reactively (online).18 The proposed scheduling framework can be easily adapted to both cases. Finally, assumption (vii) is in accordance with the sequential modeling strategy in scheduling problems, in which first the lot-sizing problem is solved and then, once the number and sizes of batches are known, the pure scheduling problem is resolved. This approach probably results in less optimal solutions than the monolithic approach, in which lot-sizing and scheduling are simultaneously optimized. However, the sequential schedul-

Figure 3. Illustative example of the unit-specific general precedence framework.

order

due date (day)

unit 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

15 30 22 25 20 30 21 26 30 29 30 21 30 25 24 30 30 30 13 19 30 20 12 30 17

1.538 1.500 1.607

setup times (days)

unit 2

unit 3

1.564 0.736 5.263 4.865

3.025 1.500 1.869 1.282 3.750 6.796

7.000

11.250 2.632 5.000 1.250 4.474 1.492 3.130 2.424 7.317

1.074 3.614 0.864 3.624 2.667 3.448

5.952 0.180

0.175

unit 4 1.194 0.789 0.818 2.143 1.017 3.200 3.214 1.440 2.459 3.000 5.600 6.716 1.527 2.985 0.783 3.036 2.687 1.600

4.000 4.902 0.237

ing approach is less computationally expensive, and in some cases it can be viewed as a good approximation to the industrial reality. Mathematical Model Formulation In this section, the proposed MILP scheduling model is stated and described in detail. The concept of the unit-specific general precedence (USGP), which is based on the MILP model, is also introduced and explained. In the proposed mathematical formulation, the problem’s constraints have been grouped according to the type of decision (assignment, timing, and sequencing) they are imposed on. Below we describe the USGP model. Allocation Constraints. Constraint 1 presents the unit allocation constraints for every order i. As this expression states, each order i can be assigned to only one processing unit j or to none (permitting unsatisfied demand). Yij represents the binary decision of whether to assign a product order i to a processing unit j or not. Yij is active, that is, Yij ) 1, whenever product i is allocated to unit j, otherwise it is


6349

Figure 4. Motivating example schedules: General precedence fault demonstration. Table 2. Motivating Example (sdi12i10 j ) 9), Models Characteristicsa precedence schedule IP USIP GP USGP

Figure Figure Figure Figure

4a 4a 4b 4a

OF

eqns cont. vars bin. vars nodes CPU (s)

6.681 784 6.681 351 8.472 334 6.681 1430

60 60 45 753

287 496 168 305

1863 1323 6712 56

8.68 4.71 2.39 0.30

a Solved in GAMS 22.8, CPLEX 11, in a Dell Inspiron 1520 2.0 GHz, 2GB RAM.

set to zero. Let JIi denote the set of units j that can process product i. By changing the inequality to an equality total demand satisfaction is imposed.

∑Y

ij

∀i

e1

(1)

j∈JIi

Timing Constraints. The completion time Ci for batch i, when it is assigned to unit j, should be greater than the summation of its corresponding processing time ptij and setup time suij in this unit j. The maximum of the ready unit time ruj and the release order time roi are also added to this summation as the following equation states. Ci g

∑ (max[ru , ro ] + pt j

i

ij

+ suij)Yij

∀i

(2)

j∈JIi

Sequencing-Timing Constraints. Constraint 3 expresses the order sequencing constraints between two orders, i and i′. This equation is formulated as a big-M constraint. It is activated for every task i′ that is processed after task i, when both are assigned to the same unit j. In other words, this equation is considered whenever the global sequencing-allocation binary variable Xii′j is equal to 1. Ci + sdii'jSeqii'j e Ci' - pti'j - sui'j + M(1 - Xii'j) ∀i, i' * i, j ∈ (JIi ∩ JIi')

(3)

It is worth mentioning that the binary variable Seqii′j defines the immediate (local) precedence of two tasks i and i′, when both are assigned to the same unit j. If two orders i and i′ are allocated to the same processing unit j and order i′ is processed directly after order i, then Seqii′j ) 1. Hence, this formulation

allows us to consider explicitly and efficiently the sequencedependent setup times and/or costs. Sequencing-Allocation Constraints. To assess sequencingassignment binary variable Seqii′j with the sequencingassignment binary variable Xii′j, a set of additional constraints, presented below, is needed. As mentioned above, the binary variable Xii′j only stands for two products, i and i′, that are assigned to the same equipment unit j. In disjunctive programming this statement can be expressed as follows: Xii'j ⇒ [Yij ∧ Yi'j]

∀i, i' * i, j ∈ (JIi ∩ JIi')

Later, the aforementioned disjunctive programming expression is decomposed into eqs 4 and 5. It can be clearly seen that Xii′j may take the value of 1 only if both orders i and i′ are into the same unit j; otherwise, it is set to zero without exploring the sequencing of the orders further. Xii'j + Xi'ij e 1 + Yij + Yi'j

∀i, i' * i, j ∈ (JIi ∩ JIi') (4)

2(Xii'j + Xi'ij) e Yij + Yi'j

∀i, i' * i, j ∈ (JIi ∩ JIi') (5)

To explicitly tackle scheduling problems with sequencedependent setup times and/or costs, the immediate (local) precedence of every pair of orders must be assessed. We now describe our approach. Obviously, two orders i and i′ are consecutive only in the case that the sequencing-allocation binary variable is Xii′j ) 1 and when there is no other order i′′ between orders i and i′, and vice versa. In disjunctive programming this expression can be stated as follows: Xii'j ∧ ¬

∨ (Xii''j - Xi'i''j)] S Seqii'j [i''*[i,i']

∀i, i' * i, j ∈ (JIi ∩ JIi')

Figure 3 shows an illustrative example of the concept of this expression. It can be seen that if the total number of batches that follow batch i, excluding batch i′, is equal to the total number of batches that follow batch i′, excluding batch i, then batches i and i′ are consecutive. For the sake of simplicity, only one processing unit is considered. As a unique machine case is studied, the unit index j is omitted in this example. The complete

6350


Table 3. Case I. Sequence-Dependent Setup Times sdii–j (days)a order

i1

i2

i3

i4

i5

i6

i7

i8

i9

i10

i11

i12

i13

i14

i15

i16

i17

i18

i19

i20

i1 i2 i3 i4 i5 i6 i7 i8 i9 i10 i11 i12 i13 i14 i15 i16 i17 i18 i19 i20

0.2 0.5 1.1 0.5 0.2 0.9 1.5 2.5 0.8 0.2 0.9 1.8 1.5 1.3 0.7 0.6 0.7

0.3 0.9 0.7 1.0 0.0 0.5 2.0 0.6 1.0 0.7 0.8 1.5 0.9 2.0 0.7 0.5 0.5

0.8 1.3 0.2 0.0 1.3 1.1 0.4 0.5 1.3 0.6 1.3 2.0 1.3 1.5 0.9 1.1 2.0

1.5 0.9 0.5 1.3 1.0 0.0 1.3 0.8 0.8 0.3 1.1 1.5 0.9 0.5 0.8 0.5 1.4

0.6 2.5 0.7 0.8 1.0 1.4 0.5 0.6 1.1 0.9 1.3 0.4 0.6 0.4 1.4 0.4 0.0

0.5 0.2 0.4 2.0 0.5 0.6 0.9 1.8 0.4 0.3 0.6 2.5 0.1 0.9 0.6 1.4 1.1

2.0 0.8 1.5 0.9 2.0 0.7 0.7 0.6 2.5 0.5 0.4 0.5 0.2 1.8 0.8 0.9 0.5

1.1 2.5 0.4 0.0 1.3 1.3 4.0 0.2 0.9 0.2 1.5 0.5 1.1 0.6 1.0 0.4 0.6

0.0 0.4 0.9 1.3 0.9 0.8 0.5 0.9 2.0 0.4 0.5 1.1 0.3 0.7 0.6 0.6 1.4

0.0 0.4 0.0 0.4 2.0

0.5 0.6 0.2 1.5 0.4 0.7 0.5 0.4 2.0 1.0 0.2 0.4 0.6 1.3 1.5 0.9 0.8 2.5 0.4

1.0 2.5 1.5 1.0 0.3 0.6 0.8 1.8 1.5 1.3 0.8 1.8 1.5 0.5 2.0 0.4 1.3 0.0 0.9

0.2 0.5 0.8 1.8 2.0 0.5 0.3 0.6 2.0 1.0 2.0 0.8 0.4 0.6 0.5 0.7 2.0

0.8 0.2 0.7 0.6 1.0 0.7 0.4 1.5 0.6 2.5 1.1 0.0 0.6 0.4 0.9 0.7 0.8

0.7 0.6 0.0 1.3 2.0 0.5 1.1 0.6 0.9 1.5 0.9 1.8 0.5 0.8 2.0 0.5 0.7

0.5 0.0 2.0 0.6 0.7 2.0 0.5 0.5 1.3 0.6 0.2 0.8 0.5 1.3 1.3 1.3 0.3

1.8 1.1 0.6 1.5 0.2 0.9 1.5 0.7 1.8 0.8 2.0 0.6 0.0 1.0 0.9 0.7 0.5

0.0 2.5

2.5 0.8 0.5 1.0 0.3 1.1 0.9 0.9 0.7 0.8 1.3 0.6 2.5 1.1 1.3 0.5 0.7 1.3 0.8

0.3 2.5 1.3 0.5 0.9 0.5 1.5 1.1 0.8 0.6 0.5 1.0 1.5 1.0 0.2 1.1 2.0 -

a

0.2 -

(-) Impossible subsequence.

Figure 5. Case I schedules. Table 4. Case Study I. Models Characteristicsa precedence schedule IP USIP GP USGP


5a 5a 5b 5a

OF 482.93 482.93 476.30 482.93

gap cont. bin. (%) eqns vars vars 0.01 1324 546 538 2450

40 40 20 976

nodes

482 117 017 861 170 829 201 44 715 820 5 373

CPU (s) 900.00 503.09 41.12 44.37

a

Solved in GAMS 22.8, CPLEX 11, in a Dell Inspiron 1520 2.0 GHz, 2GB RAM.

procedure of monitoring the consecutiveness of two orders is clearly outlined in Figure 3. Constraints 6 and 7 correspond to the mathematical formulation of the aforementioned disjunctive programming expression. Constraint 6 states that the auxiliary variable Posii′j will be set to zero only if batch i is processed before batch i′ and they are allocated to the same equipment unit j, that is, Xii′j ) 1, and simultaneously ∑i′′ *[i,i′] Xii′′j - Xi′i′′j ) 0 (see Figure 3). Posii'j )

∑

i''*[i,i']

(Xii''j - Xi'i''j) + M(1 - Xii'j) ∀i, i' * i, j ∈ (JIi ∩ JIi')

(6)

If the position difference variable Posii′j is equal to zero, that is, when order i has been processed exactly before order i′, eq 7

Figure 6. Case I. Order earliness (in days).

activates the binary variable Seqii′j, that is, Seqii′j ) 1. Therefore, the consecutiveness of the orders is assessed and sequencedependent setup time and/or cost issues can be effectively treated. Posii'j + Seqii'j g 1

∀i, i' * i, j ∈ (JIi ∩ JIi') (7)

Ind. Eng. Chem. Res., Vol. 48, No. 13, 2009 Table 5. Case Study I. Optimal Schedule Results batch

unit

Tsi

Ci

Ei

batch

unit

Tsi

Ci

Ei

i1 i2 i3 i4 i5 i6 i7 i8 i9 i10

J4 J4 J4 J3 J4 J4 J3 J3 J3 J2

13.81 28.19 19.96 22.94 18.71 24.75 17.98 24.5 28.13 27.11

15 28.98 20.78 24.5 19.72 27.95 21 26 30 28.4

0 1.02 1.22 0.5 0.28 2.05 0 0 0 0.6

i11 i12 i13 i14 i15 i16 i17 i18 i19 i20

J2 J2 J1 J1 J4 J4 J1 J2 J2 J4

23.19 14.2 10.94 22.37 21.02 29.22 25.53 28.57 9.87 16.87

26.94 21 22.19 25 24 30 30 30 13 18.47

3.06 0 7.81 0 0 0 0 0 0 0.53

a

max

∀i

(8)

i'*i j∈(JIi∩JIi')

Incorporating it into the model avoids the creation of binary variables Seqii′j that do not satisfy the order-unit availability specifications of the scheduling problem (see j ∈(JIi∩JIi′)). Objective Function. We now present the different objective functions that have been used in the current work in order to cope with sequence-dependent setup times and/or costs. The earliness and the tardiness for every product order i are given by eqs 9 and 10, respectively. ∀i

(9)

Ti g Ci - Di

∀i

(10)

∑ (R E

i i

+ βiTi)

(13)

First, the revenue maximization, given by eq 14, is considered. Revenue is calculated by adding the revenue contribution, revconi, of each produced batch. max

∑ ∑ revcon Y

(14)

i ij

j∈JIi

The profit maximization constitutes the next economic objective function. Profit is defined as the summation of every batch revenue contribution, revconi, minus the sequencedependent setup costs csdii′j, as given in eq 15.

The minimization of a combined function of earliness and tardiness, as given in eq 11, is one of the most widely used objective functions in the scheduling literature. It is also known as weighted lateness. The weighting coefficients Ri and βi are used to specify the significance of order earliness or tardiness, respectively. min

∀i

Ci e Hor

i

Ei g Di - Ci

(12)

i

Note that if sequence-dependent setup costs are proportional to the sequence-dependent setup times, then the minimization of earliness will correspond to minimization of sequencedependent costs. We also use two economic objective functions in order to highlight the potential economic benefits of implementing the proposed USGP framework. The case studies of cost issues do not include due dates like the case studies of time objective functions; rather they introduce a production time horizon. Thus, the completion time Ci for every batch should be lower than the production horizon Hor. Constraint 13 states this additional restriction. Total demand satisfaction is not imposed.

Finally, the implementation of eq 8 may enhance the computational speed of the proposed model. The number of constraints is increased, but the computational time may be reduced. Seqii'j e 1

∑C i

Time in days.

∑ ∑

6351

If tardiness is not permitted (Ti ) 0) for any order i, the aforementioned objective function can be substituted by the maximization of the order completion time, Ci, as eq 12 states. This objective function is identical to the minimization of earliness.

a

max

∑ ∑ revcon Y

i ij

i

j∈JIi

-

∑∑ ∑ i

Seqii'jcsdii'j

i'*i j∈(JIi∩JIi')

(15) Note that general precedence formulation cannot optimize the sequence-dependent setup costs, since batch consecutiveness is not assessed explicitly. Since scheduling constitutes a part of the supply

(11)

i

Table 6. Case II. Sequence-Dependent Setup Times sdii′j (days)a Order

i1

i2

i3

i4

i5

i6

i7

i8

i9

i10

i11

i12

i13

i14

i15

i16

i17

i18

i19

i20

i21

i22

i23

i24

i25

i1 i2 i3 i4 i5 i6 i7 i8 i9 i10 i11 i12 i13 i14 i15 i16 i17 i18 i19 i20 i21 i22 i23 i24 i25

0.2 0.5 1.1 0.5 0.2 0.9 1.5 2.5 0.8 0.2 0.9 1.8 1.5 1.3 0.7 0.6 0.7 0.0 0.0 0.0

0.3 0.9 0.7 1.0 0.0 0.5 2.0 0.6 1.0 0.7 0.8 1.5 0.9 2.0 0.7 0.5 0.5 0.0 0.0 0.0

0.0 1.3 0.2 0.0 1.3 0.0 0.4 0.5 1.3 0.6 1.3 2.0 1.3 1.5 0.9 1.1 2.0 0.0 0.0 0.0

1.5 0.9 0.5 1.3 1.0 0.0 1.3 0.8 0.8 0.3 1.1 1.5 0.9 0.5 0.8 0.5 0.0 0.0 0.0 0.0 0.0 0.0

0.6 2.5 0.7 0.8 1.0 1.4 0.5 0.6 1.1 0.9 1.3 0.4 0.6 0.4 1.4 0.4 0.0 0.0 0.0 0.0 0.0 0.0

0.5 0.2 0.4 2.0 0.5 0.6 0.0 1.8 0.4 0.3 0.6 2.5 0.1 0.9 0.6 1.4 1.1 0.0 0.0 0.0

0.0 0.8 1.5 0.9 2.0 0.7 0.7 0.6 2.5 0.5 0.4 0.5 0.2 1.8 0.8 0.9 0.5 0.0 0.0 0.0 0.0 0.0

1.1 2.5 0.4 0.0 1.3 1.3 4.0 0.2 0.9 0.2 1.5 0.0 0.0 0.6 1.0 0.4 0.6 0.0 0.0 0.0 0.0 0.0

0.0 0.4 0.9 1.3 0.9 0.8 0.5 0.9 2.0 0.4 0.5 1.1 0.3 0.7 0.6 0.6 1.4 0.0 0.0 0.0 0.0 0.0

0.0 0.4 0.0 0.4 2.0 -

0.5 0.6 0.2 1.5 0.4 0.7 0.5 0.4 2.0 1.0 0.0 0.4 0.6 1.3 1.5 0.9 0.8 2.5 0.4 0.0 0.0

1.0 2.5 1.5 1.0 0.3 0.6 0.8 1.8 1.5 1.3 0.8 1.8 1.5 0.5 2.0 0.4 1.3 0.0 0.9 0.0 0.0 0.0 0.0 0.0

0.2 0.5 0.8 1.8 2.0 0.5 0.3 0.6 2.0 1.0 2.0 0.8 0.4 0.6 0.5 0.7 2.0 0.0 0.0 0.0

0.8 0.2 0.0 0.6 1.0 0.7 2.0 1.5 0.6 2.5 1.1 0.0 0.6 0.4 0.9 0.7 0.8 0.0 0.0 0.0

0.7 0.6 0.0 1.3 2.0 0.5 1.1 0.6 0.9 1.5 0.9 0.0 0.5 0.8 2.0 0.5 0.7 0.0 0.0 0.0

0.5 0.0 2.0 0.6 0.7 2.0 0.5 0.5 1.3 0.6 0.2 0.8 0.5 1.3 0.0 1.3 0.3 0.0 0.0 0.0

1.8 1.1 0.6 1.5 0.2 0.9 1.5 0.7 1.8 0.8 2.0 0.6 0.0 0.0 0.9 0.7 0.5 0.0 0.0 0.0

0.0 2.0

2.5 0.8 0.5 1.0 0.3 1.1 0.9 0.9 0.7 0.8 1.3 0.6 2.5 1.1 1.3 0.5 0.7 1.3 0.8 0.0 0.0

0.3 2.5 1.3 0.5 0.9 0.5 1.5 1.1 0.8 0.6 0.5 1.0 1.5 1.0 0.2 1.1 2.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -

a


0.2 -

6352


Figure 7. Case II schedules. Table 7. Case Study II. Models Characteristicsa precedence

schedule

IP USIP GP USGP a


7a 7a 7b 7a

OF

gap (%)

eqns

cont. vars

bin. vars

nodes

CPU (s)

575.88 578.36 577.50 579.57

4.45 4.02 1.88 -

2004 804 763 3515

50 50 25 1401

725 1300 286 738

62 243 235 880 1 427 992 213 704

900.00 900.00 900.00 603.45

Solved in GAMS 22.8, CPLEX 11, in a Dell Inspiron 1520 2.0 GHz, 2GB RAM.

Table 8. Case Study II. Optimal Schedule Resultsa batch

unit

Tsi

Ci

Ei

batch

unit

Tsi

Ci

Ei

i1 i2 i3 i4 i5 i6 i7 i8 i9 i10 i11 i12 i13

J4 J4 J4 J3 J3 J4 J4 J4 J3 J2 J2 J2 J1

13.81 29.21 21.18 23.44 19.09 25.77 17.73 24.1 28.13 27.11 23.19 14.2 7.49

15 30 22 25 19.82 28.97 20.94 25.54 30 28.4 26.94 21 18.74

0 0 0 0 0.18 1.03 0.06 0.46 0 0.6 3.06 0 11.26

i14 i15 i16 i17 i18 i19 i20 i21 i22 i23 i24 i25

J4 J1 J1 J1 J2 J2 J3 J3 J3 J3 J3 J3

22.33 18.92 28.75 24.1 28.57 9.87 17.15 19.82 18.22 8.38 25.46 13.55

23.86 23.92 30 28.57 30 13 18.22 23.44 19.09 12 28.13 17

1.14 0.08 0 1.43 0 0 0.78 6.56 0.91 0 1.87 0

a

Time in days.

Table 9. Case Study III-IV Data

Product

production time (hours) economic contribution unit unit unit unit demand (mu/batch) 1 2 3 4 (batches)

cream 1

2

10

5

cream 2 conditioner shampoo lotion

3 1 3.5 1.5

12

10

10

8 6

5

5

9

3 3 3 2

7 12 13

orders notation i1, i6, i7, i14, i15 i2, i8, i9 i3, i10, i16 i4, i11, i13 i5, i12

chain network optimization problem which ought to meet financial goals, it must also be examined and optimized considering financial and economic issues. In view of this industrial strategy in the contemporary highly competitive market environment, the significant advantages of adopting the proposed scheduling framework are clear. Developing an efficient monolithic scheduling/planning

MILP model that considers financial issues greatly facilitates the connection between operational (scheduling) and planning decisionmaking levels. Motivating Example A modified version of a plastic compounding plant, first introduced by Pinto and Grossmann,19 is used as a simple academic example. Table 1 shows the entire input data set. For the sake of simplicity, only the first 15 orders are considered. The optimization goal is to minimize earliness and tardiness (Ri ) 1 and βi ) 5). First, the addressed scheduling problem is solved without considering sequence-dependent setup times, that is, sdii′j ) 0. The objective function value is equal to 6.681 mu; the schedule obtained is shown in Figure 4a. Afterward, the scheduling problem is solved by considering sequence-dependent setup times only between order i12 and order i10 (sdi12i10 j ) 9). As the schedule in Figure 4a shows, order i12 and order i10 are not consecutive; thus, someone will expect to obtain the same schedule even if a sequence-dependent setup time is assigned between these orders. However, this is not the case when the general precedence model is applied. Figure 4b illustrates the schedule obtained, which is, perhaps unsurprisingly, different from the previous one. Its objective function value is equal to 8.472 mu (26% worse). The cause of this fault is that the general precedence sequencing constraints take into account all the sequencedependent setup times (and not only between consecutive orders) of the orders assigned to the same unit j. This point can be seen in eqs 34 and 35 included in the Appendix. It can be clearly seen that sequence-dependent setup times, sdii′j, are taken into account whenever the general precedence sequencing variable GP is active, that is, equals 1. Xii′


6353

a

Table 10. Case Study III-IV. Sequence-Dependent Setup Times sdii′j (h) order

i1

i2

i3

i4

i5

i6

i7

i8

i9

i10

i11

i12

i13

i14

i15

i16

i1 i2 i3 i4 i5 i6 i7 i8 i9 i10 i11 i12 i13 i14 i15 i16

4.5 12.0 3.0 0.0 0.0 4.5 4.5 12.0 3.0 12.0 0.0 0.0 -

6.0 4.5 12.0 3.0 6.0 6.0 0.0 0.0 4.5 12.0 3.0 12.0 0.0 0.0 0.0

9.0 10.5 9.0 4.5 9.0 9.0 10.5 10.5 0.0 9.0 4.5 9.0 0.0 0.0 0.0

10.5 9.0 6.0 1.5 10.5 10.5 9.0 9.0 6.0 0.0 1.5 0.0 0.0 0.0 0.0

9.0 6.0 10.5 9.0 9.0 6.0 6.0 10.5 0.0 10.5 0.0 0.0 -

0.0 4.5 12.0 3.0 0.0 4.5 4.5 12.0 3.0 12.0 0.0 0.0 -

0.0 4.5 12.0 3.0 0.0 4.5 4.5 12.0 3.0 12.0 0.0 0.0 -

6.0 0.0 4.5 12.0 3.0 6.0 6.0 0.0 4.5 12.0 3.0 12.0 0.0 0.0 0.0

6.0 0.0 4.5 12.0 3.0 6.0 6.0 0.0 4.5 12.0 3.0 12.0 0.0 0.0 0.0

9.0 10.5 0.0 9.0 4.5 9.0 9.0 10.5 10.5 9.0 4.5 9.0 0.0 0.0 0.0

10.5 9.0 6.0 0.0 1.5 10.5 10.5 9.0 9.0 6.0 1.5 0.0 0.0 0.0 0.0

9.0 6.0 10.5 0.0 9.0 9.0 6.0 6.0 10.5 10.5 0.0 0.0 -

10.5 9.0 6.0 0.0 1.5 10.5 10.5 9.0 9.0 6.0 0.0 1.5 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -

a


Table 11. Case Study III. Models Characteristicsa precedence

OF

eqns

cont. vars

bin. vars

nodes

CPU (s)

IP USIP GP USGP

32.0 34.5 34.5 35.5

965 387 409 2225

33 33 17 962

334 663 138 399

1 080 605 1 678 431 7 529 622 1 372

600.00 600.00 600.00 1.91

a Solved in GAMS 22.4, CPLEX 10, in a Sun Ultra 4.0 Workstation, 8GB RAM.

Therefore, coming back to the motivating example, because Xi12i10 ) 1 (in the schedule in Figure 4a order i12 is processed before order i10, although they are not consecutive orders) the sequence-dependent setup time sdi12i10 j ) 9 (of two no-consecutive orders) is incorrectly taken into account. In other words, the general precedence model will try to make an order completion time plus the sequence-dependent setup time less than or equal to the starting time of any other following order, and consecutiveness is not explicitly considered. The scheduling problem with sdi12i10 j ) 9 has also been solved using the remaining precedence-based formulations. Table 2 presents a comparative table for all the precedence-based formulations. The USGP formulation reached the optimal schedule in just 0.3 CPU seconds, 16 times faster than the calculation time of the USIP formulation. All in all, in cases where there exist some sequence-dependent setup times higher than some batch processing times, as in this example, GP may result to a myopic solution. If all sequencedependent setup times are lower than all the orders processing times then GP is valid and can be implemented. Nevertheless, note that sequence-dependent cost issues still cannot be addressed explicitly by GP models. Case Studies Case Study I. In this part, a complex case study involving the production scheduling of the first 20 orders for the Pinto

Figure 8. Case III. Optimal USGP schedule.

and Grossmann19 case study is considered. The processing input data are stated in Table 1. Sequence-dependent setup times are also taken into account. Table 3 contains this data set for every pair of orders. The case study’s optimization goal is to maximize order completion times, which is identical to minimizing earliness. The under study scheduling problem has been solved for all precedence-based formulations by setting a computation time limit of 15 min. Figure 5 shows the Gantt charts for the optimal schedules given by each approach. A comparative study of all precedence-based formulations can be found in Table 4. Figure 5a presents the optimal schedule (Etotal ) 17.07 days) of IP, USIP, and USGP frameworks, while the myopic “optimal” GP schedule (Etotal ) 23.7 days) is shown in Figure 5b. A comparative plot for order earliness of the two schedules is illustrated in Figure 6. The GP proposed solution is 39% worse than the optimal solution. Therefore, the inability of the GP model to tackle sequence-dependent setup times is clearly demonstrated once again. This case study was solved in less than a minute (44.37 s) by using the USGP model, while approximately 8.5 min (503.09 s) were needed for USIP to reach the optimal solution. The USGP formulation is 11.5 times faster than the USIP. The IP approach gave the optimal solution after the imposed time limit. Table 5 shows the optimal schedule results. Case Study II. In this section, a more complicated case study is solved. This example contains the scheduling problem of 25 orders when sequence-dependent setup times are considered. Orders processing data and sequence-dependent setup times are shown in Table 1 and Table 6, respectively. Maximizing order completion times is the objective function used here. A computation time limit of 15 min was set. Figure 7a shows the optimal schedule (Etotal ) 27.42 days) for the USGP model. IP and USIP did not reach this optimal

6354


Figure 9. Case IV. Optimal USGP schedule. Table 12. Case Study IV. Models Characteristicsa precedence

OF

eqns

cont. vars

bin. vars

nodes

CPU (s)

IP USIP USGP

34.5 34.5 35.5

965 387 2225

33 33 962

334 663 399

1 407 686 1 046 543 16 605

600.00 600.00 24.99

a Solved in GAMS 22.4, CPLEX 10, in a Sun Ultra 4.0 Workstation, 8GB RAM.

Table 13. Case Study V Data processing times (days) Order

release time (day)

due date (day)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0.0 5.0 0.0 6.0 0.0 2.0 3.0 0.0 2.0 6.0 0.0 1.5 0.0 0.0 5.5 0.0 2.0 1.5 0.0 1.0

10 22 25 20 28 30 17 23 30 21 30 28 15 29 12 19 30 27 30 24

unit ready time (days)

unit 1

unit 2

unit 3

unit 4

6.80 1.80 3.30

5.00 4.80

5.10 5.60 3.60

2.55 2.10

3.20 5.20

3.30 12.60

3.80 1.10 3.40 2.00

3.50 8.00 4.44

2.10 4.17

2.60 2.60 3.60

-

3.15 4.35 5.55 3.60

2.95

3

2

3.90 3

solution before the imposed time limit. The myopic “optimal” GP schedule (Etotal ) 31.7 days) is presented in Figure 7b. Once more, GP yields to a shortsighted solution, as it proposes a schedule that is 7.8% worse than the optimal solution. The computational characteristics for each model are shown in Table 7. It can be seen that only the proposed USGP model can reach the optimal solution in approximately 10 min (603.45 s). The IP solution has a 4.45% integrality gap and the USIP solution has a 4.02% integrality gap. The USGP optimal schedule results are shown in Table 8. Case Study III. This case study, first introduced by Majozi et al.,20 considers a pharmaceutical facility that produces lotions, conditioners, shampoos, and two types of creams. Table 9 shows the problem’s input data. Sequence-dependent setup times, which are stated in Table 10, and product demands have been introduced in addition to the original problem data. A 36-day production time horizon is considered. The problem’s objective function is to maximize revenue. A time limit of 10 min (600 s) was imposed. Only the USGP model was able to reach a solution (in just 2 s) before the time limit. Table 11 presents the

characteristics of the models. The USGP optimal schedule is shown in Figure 8. A total revenue of 35.5 mu is achieved and all orders are produced. Case Study IV. The previous problem is addressed again but now the problem’s objective function is to maximize profit. Profit is calculated as the sum of the orders’ revenue minus sequence-dependent setup costs included in this part. It is assumed that the sequence-dependent setup cost per hour is 0.1 mu. The production time horizon is 36 days. This problem was solved by all precedence-based formulations apart from GP, as it cannot assess consecutiveness and therefore cannot deal with sequence-dependent costs. A time limit of 10 min (600 s) was imposed. Only USGP was capable of obtaining the optimal solution before the time limit; it took almost half a minute (25.99 s) to solve the problem. The optimal schedule, with a profit equal to 35.5 mu, is presented in Figure 9. The characteristics of the models are shown in Table 12. Case Study V. In this part, we consider an industrial case study first introduced by Cerda´ et al.21 and then studied by Meńdez et al.22 Order processing data and sequencedependent setup times can be found in Table 13 and Table 14, respectively. The problem’s objective function is to minimize the earliness and tardiness (Ri ) 1 and βi ) 21). A special characteristic of this case study is the existence of forbidden processing sequences (see Table 14), which the GP formulation cannot deal with. Order release times and unit ready times are also taken into account. Figure 10 depicts the optimal schedule for the IP, USIP, and USGP models. The objective function value is equal to 2.064 mu, which results in an Etotal ) 43.35 days. Table 15 shows the characteristics of the models. Once again, the proposed USGP formulation reached the optimal solution faster than the IP and USIP models. The optimal schedule results are included in Table 16. Remarks Several instances of three different case studies found in the literature have been solved. Different objective functions have also been considered. In all cases our proposed USGP scheduling framework reached the optimal solution in a reasonable computational time, even in cases in which the other precedence-based frameworks did not find the optimal solution inside the imposed time limit. The main reason is that the USGP branch-and-bound exploring tree is much shorter than those of the other formulations (see the tables of the model characteristics for each case study). Further study is needed to ensure that this will be the case in any problem instance. The authors would like to clarify that they do not claim that the USGP formulation will be the fastest for every case study. For the cases solved, USGP is the fastest, but there


6355

a

Table 14. Case V. Sequence-Dependent Setup Times sdii’j (days) order

i1

i2

i3

i4

i5

i6

i7

i8

i9

i10

i11

i12

i13

i14

i15

i16

i17

i18

i19

i20

i1 i2 i3 i4 i5 i6 i7 i8 i9 i10 i11 i12 i13 i14 i15 i16 i17 i18 i19 i20

1.00 1.40 2.10 1.50 0.30 F 0.40 -

0.15 1.80 0.95 F 0.55 0.25 0.70 -

F 1.10 0.30 F 1.25 0.60 F 0.80 1.30 1.05 0.80 F F -

0.30 0.70 0.75 1.45 0.20 0.50 F 0.15

0.05 F 0.85 F 0.50 0.40 0.80 F 0.30 0.45 0.65 F

0.65 F F 0.70 0.80 F 1.30 0.50 0.40 0.90 F 0.85 F

F 0.30 0.90 1.65 F 1.00 1.55 1.20 F 1.10 0.80 0.55

0.60 0.45 0.20 0.30 0.50 0.45

0.85 1.60 1.20 0.70 0.25 F 0.35 -

0.40 0.20 0.50 F F 0.65 1.15 0.35 0.80 0.75 0.60 F 0.40

0.25 0.50 1.00 0.60 1.40 F 0.70 -

F 0.75 0.90 1.10 1.05 0.15 0.40 0.30 0.85 F 0.90 F

0.35 0.70 F 1.20 F 0.85 1.15 0.15 F 0.20 0.45 F F -

0.70 0.80 0.55 1.30 1.05 0.65 0.50 0.40

0.45 F 0.20 F 0.10 0.95 1.30 0.75 0.20 0.55 1.05 F

0.25 0.90 0.80 0.35 1.00 0.50 0.75 -

F 0.60 F 0.35 0.50 F 0.15 0.40 0.80 0.25 0.60 0.30 F

0.65 F 0.50 0.70 F 1.20 1.00 F 0.55 0.30 0.15 0.45 F

0.30 F F 1.30 0.90 0.80 F 0.20 0.95 0.35 0.50 F 0.15 0.60 F

0.75 F 0.80 0.25 0.30 1.25 F 0.65 F 0.30 0.45 F -

a

(F) forbidden processing sequences; (-) impossible subsequence.

Figure 10. Case V. Optimal schedule. Table 15. Case Study V. Models Characteristicsa precedence

OF

eqns

cont. vars

bin. vars

nodes

CPU (s)

IP USIP USGP

2.064 2.064 2.064

1184 568 1202

303 426 902

241 326 250

50 373 14 503 3 565

71.45 15.52 13.18

a Solved in GAMS 22.8, CPLEX 11, in a Dell Inspiron 1520 2.0 GHz, 2GB RAM.

Table 16. Case Study V. Optimal Schedule Resultsa batch

unit

Tsi

Ci

Ei

batch

unit

Tsi

Ci

Ei

i1 i2 i3 i4 i5 i6 i7 i8 i9 i10

J1 J3 J3 J2 J4 J2 J3 J4 J1 J1

2.7 18.8 21.7 12.1 25.45 26.4 13.05 9.3 26.8 14.5

9.5 20.6 25 17.2 28 30 15.15 21.9 30 19.7

0.5 1.4 0 2.8 0 0 1.85 1.1 0 1.3

i11 i12 i13 i14 i15 i16 i17 i18 i19 i20

J3 J4 J1 J2 J2 J3 J1 J1 J3 J4

25.75 22.95 9.85 17.9 7.46 15.95 20.1 22.85 27.05 4.95

26.85 25.05 13.35 25.9 11.9 18.55 22.7 26.45 30 8.85

3.15 2.95 1.65 3.1 0.1 0.45 7.3 0.55 0 15.15

a

Times in days.

may be cases in which another precedence-based formulation is faster. However, we did not observe such a case. It is also worth mentioning that while computational effort is significant, memory requirements are also important. One apparent drawback of the USGP scheduling formulation seems to be that it needs more memory (bigger model size) than the other formulations, which may potentially limit the size of the problems that can be solved. However, the current advances in computer systems moderate this effect. Note that, despite its

bigger model size, USGP outperformed the existing smallersized models in all cases and with medium-size scheduling problems. A large number of authors developed various compact (small model size) scheduling formulations in an attempt to reduce computational time. However, according to Nemhauser and Wolsey, “It is instinctive to believe that computation time increases and computational feasibility decreases as the number of constraints increases. But, trying to find a formulation with a small number of constraints is often a bad strategy...There are problems in which the quality of the formulation depends on the choice of variables...”.23 It is important to keep in mind that large-scale scheduling problems cannot be solved by pure mathematical scheduling models; none of the precedence-based models can solve to optimality a highly sophisticated scheduling problem. A normal tactic is to decompose the scheduling problem with different decomposition methods, such as Bender’s decomposition or Lagrange relaxation, with a mathematical scheduling model as the core. Therefore, the USGP model can constitute the core of a decomposition scheme. This seems a promising research line because USGP has been found to perform well in mediumsized scheduling problems. Summary and Final Considerations In the current work, the new unit-specific general precedence formulation, based on a continuous time-domain representation, is proposed as a means of tackling scheduling problems with sequence-dependent setup time and/or cost issues. The proposed model takes into account sequence-

6356


dependent changeovers and finite release times for units and orders. Forbidden processing sequences can also be explicitly considered. Despite the considerable complexity of the problems addressed, the proposed model is able to reach a solution in a shorter time than the existing precedence-based frameworks, which is evidenced by the case studies. The proposed model can resolve medium-sized scheduling problems in a reasonable computational time, and the development of decomposition techniques based on the current model in order to solve large-scale industrial scheduling problems is a very promising research direction.

specific immediate precedence, and Meńdez and Cerda´15 for the general precedence model. Immediate Precedence (IP).

∑ YF

(16)

∑ (YF

ij

+ Yij) ) 1

∀i

(17)

j∈JIi

Acknowledgment The authors appreciate the financial support received from the Spanish Ministry of Science and Innovation (FPU grant). Financial support from the EU project MRTN-CT-2004-5 12233 and the national projects I-0898 and DPI2006-05673 is also gratefully acknowledged.

∀j

e1

ij

i∈IJj

∑ pt (YF

Ci ) Tsi +

ij

ij

+ Yij)

∀i

(18)

j∈JIi

Tsi' g Ci +

∑ (sd

+ sui'j)Yij - M(1 - XIP ii' )

ii'

∀i, i'

j∈JIi

(19)

Nomenclature Sets i ) batch orders j ) processing units JIi ) available processing units to process order i

Ci g


i

+ suij)(YFij + Yij)

ij

∀i

j∈JIi

(20)

Parameters csdii′j ) sequence-dependent setup cost of order i and order i′ in unit j Di ) due date of order i Hor ) production time horizon M ) a very large number ptij ) processing time of order i in unit j roi ) release time of order i ruj ) ready time of unit j sdii′j ) sequence-dependent setup time of order i and order i′ in unit j suij ) sequence-independent setup time of order i in unit j

YFij + Yij e YFi'j - XIP ii' + 1

YFij + Yij e 1 - XIP ii'

∑ YF

Variables

ij

∀i, i', j ∈ (JIiJIi')

∑X

+

IP i'i

j∈JIi

Ci ) completion time for order i Posii′j ) position difference between orders i and i′ when both are allocated to the same unit j Ei ) earliness for order i Ti ) tardiness for order i Binary Variables

∀i, i', j ∈ (JIi ∩ JIi') (21)

)1

∀i

(22)

(23)

i'

∑X

IP ii'

e1

∀i

(24)

i'∈IJj

Unit-Specific Immediate Precedence (USIP).

Seqii′j ) if order i′ is processed exactly after order i when both are allocated to the same unit j, then Seqii′j ) 1, otherwise Seqii′j ) 0 Xii′j ) if order i is processed before i′, when both are assigned to the same unit j, then Xii′j ) 1, otherwise Xii′j ) 0 Yij ) if order i is allocated to unit j, then Yij ) 1, otherwise Yij ) 0

∑ YF

ij

e1

∀j

(25)

i∈IJj

∑I ∑ X

j∈J

USIP ii'j

)1

∀i

(26)

i'∈IJj

i

Greek Symbols Ri ) weighting coefficient for earliness for order i βi ) weighting coefficient for tardiness for order i

∑X

USIP ii'j

∀i

e1

(27)

i'∈IJj

Appendix In this section, the existing precedence-based mathematical formulations found in the literature are presented in brief. For a more detailed description see Meńdez et al.22 regarding the immediate precedence formulation, Cerda´ et al.21 for the unit-

YFij +

∑X

i'∈IJj

USIP i'ij

+

∑

∑

USIP Xii'j' e1

∀i, j ∈ JIi

j'∈JIi,j*j' i'∈(IJj∩IJj')

(28)


Ci ) Tsi +

∑ (pt

ij

+ suij)(YFij +

j∈JIi

∑X

∀i

USIP i'ij )

i'∈IJj

(29)

Tsi g Ci' +

∑

sdi'ijXUSIP - M(1 i'ij

j∈(JIi∩JIi')

Ci g


∑

j∈(JIi∩JIi')

i

ij

+ suij)(YFij +

j∈JIi

∑X

XUSIP i'ij ) ∀i, i' USIP i'ij )

(30) ∀i

i'∈IJj

(31)

General Precedence (GP).

∑Y

ij

)1

∀i

(32)

j∈JIi

∑ (max[ru , ro ] + pt

∀i

(33)

Ci + sdii'j e Ci' - pti'j - sui'j + M(1 - XGP ii' ) + M(2 - Yij - Yi'j) ∀i, i' * i, j ∈ (JIi ∩ JIi')

(34)

Ci' + sdi'ij e Ci - ptij - suij + MXGP ii' + M(2 - Yij - Yi'j) ∀i, i' * i, j ∈ (JIi ∩ JIi')

(35)

Ci g

j

i

ij

+ suij)Yij

j∈JIi

Appendix Notation IJj ) set of orders i that can be processed into unit j Tsi ) starting time of order i YFij ) allocation binary variable that denotes if order i is processed first in the unit j; in this case, YFij ) 1, otherwise is zero IP IP ) if order i′ is processed exactly after order i, then Xii′ )1, Xii′ otherwise is set to zero USIP ) if order i′ is processed exactly after order i and they Xii′j USIP ) 1, are also allocated into the same unit j, then Xii′j otherwise is set to zero GP GP ) it is Xii′ )1 for every order i′ processed after order i Xii′ independent if they are assigned to the same unit or not; otherwise it is set to zero Literature Cited (1) Yang, W.; Wilhelm, W. Scheduling and lot sizing with sequencedependent setup: A literature review. IIE Trans. 2006, 38, 987–1007. (2) Pinedo, M. Scheduling: Theory, Algorithms and Systems; Prentice Hall: Englewood Cliffs, NJ, 2002. (3) Allahverdi, A.; Gupta, J.; Aldowaisan, T. A review of scheduling research involving setup considerations. Omega 1999, 27 (2), 219–239.

6357

(4) Yang, W.; Liao, C. A review of scheduling research involving setup considerations. Int. J. Syst. Sci. 1999, 30 (2), 143–155. (5) Allahverdi, A.; Ng, C.; Cheng, T.; Kovalyov, M. A survey of scheduling problems with setup times or costs. Eur. J. Oper. Res. 2008, 187, 985–1032. (6) Conway, R., Maxwell, W., Miller, L., Eds. Theory of Scheduling; Massachussets: Addison-Wesley: Reading, MA, 1967. (7) Sherali, H.; Sarin, S.; Kodialam, M. Models and algorithms for a two stage production process. Prod. Plan. Control 1990, 1, 27–39. (8) Paul, R. A production scheduling problem in the glass container industry. Oper. Res. 1979, 27, 290–302. (9) Fortemps, P.; Ost, C.; Pirlot, M.; Teghem, J.; Tuyttens, D. Using metaheuristics for solving a production scheduling problem in a chemical firm. Int. J. Prod. Econ. 1996, 46-47, 13–26. (10) Bitran, G. R.; Gilbert, S. M. Sequencing production on parallel machines with two magnitudes of sequence-dependent setup cost. J. Manuf. Oper. Manage. 1990, 3, 24–52. (11) Narasimhan, S.; Panwalkar, S. Scheduling in a two stage manufacturing process. Int. J. Prod. Res. 1984, 22, 555–564. (12) Andre´s, C.; Albarracıń, J. M.; Tormo, G.; Vicens, E.; GarcıáSabater, J. P. Group technology in a hybrid flowshop environment: A case study. Eur. J. Oper. Res. 2005, 167, 272–281. (13) Allahverdi, A.; Soroush, H. M. The significance of reducing setup times/setup costs. Eur. J. Oper. Res. 2008, 187, 978–984. (14) Meńdez, C.; Cerda´, J.; Grossmann, I.; Harjunkoski, I.; Fahl, M. Review: State-of-the-art of optimization methods for short-term scheduling of batch processes. Comput. Chem. Eng. 2006, 30, 913–946. (15) Mende´z, C.; Cerda´, J. Dynamic scheduling in multiproduct batch plants. Comput. Chem. Eng. 2003, 27, 8–9. (16) Noivo, J.; Ramalhinho-Lourenço, H. Solving two production scheduling problems with sequence-dependent set-up times. J. Econ. Lit. 1998, Class. C61, M11, L60. (17) Meńdez, C.; Cerda´, J. An MILP framework for batch reactive scheduling with limited discrete resources. Comput. Chem. Eng. 2004, 28, 1059–1068. (18) Kopanos, G.; Capoń-Garcıá, E.; Espunã, A.; Puigjaner, L. Costs for rescheduling actions: a critical issue for reducing the gap between scheduling theory and practice. Ind. Eng. Chem. Res. 2008, 47, 8785–8795. (19) Pinto, J.; Grossmann, I. A continuous time mixed integer linear programming model for short term scheduling of multistage batch plants. Ind. Eng. Chem. Res. 1995, 34, 3037–3051. (20) Majozi, T.; Friedler, F. Maximization of throughput in a multipurpose batch plant under a fixed time horizon: S-graph approach. Ind. Eng. Chem. Res. 2006, 45, 6713–6720. (21) Cerda´, J.; Henning, G.; Grossmann, I. A mixed-integer linear programming model for short-term scheduling of single-stage multiproduct batch plants with parallel lines. Ind. Eng. Chem. Res. 1997, 36, 1695–1707. (22) Meńdez, C.; Henning, G.; Cerda´, J. Optimal scheduling of batch plants satisfying multiple product orders with different due-dates. Comput. Chem. Eng. 2000, 24, 2223–2245. (23) Nemhauser, G. L.; Wolsey, L. A. Integer and Combinatorial Optimization; Series in discrete mathematics and optimization; Wiley Interscience: New York, 1988.

ReceiVed for reView July 22, 2008 ReVised manuscript receiVed March 20, 2009 Accepted May 7, 2009 IE801127T

An Efficient Mixed-Integer Linear Programming Scheduling

Recommend Documents