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Ind. Eng. Chem. Res. 1996, 35, 338-342
An Alternate MILP Model for Short-Term Scheduling of Batch Plants with Preordering Constraints Jose M. Pinto† and Ignacio E. Grossmann* Department of Chemical Engineering, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213
An alternate model is presented when preordering constraints are imposed in the short term scheduling of batch plants. The idea is to include the assumption of preordering of orders to units in the underlying representation for the model, as opposed to including preordering as a separate set of constraints. The resulting model not only is conceptually simple but also results in shorter solution times. Introduction Pinto and Grossmann (1995) presented an optimization model and a solution technique for the short-term scheduling of multistage batch plants with equipment in parallel. For the case of large systems a solution method was presented which relies on the use of preordering constraints. Moreover, a decomposition scheme was developed which determines initially feasible assignments that minimize in-process time and further determines a schedule that minimizes earliness and eliminates unnecessary setups. This research note presents an alternate model in which the preordering is embedded in the underlying representation. Specifically, instead of using preordering as a separate set of constraints, a different representation of the time slots for the units is used which explicitly incorporates the preordering. As a result, smaller models arise that lead to a significant reduction in solution times, as will be shown in this note. Figure 1. Definition and time-matching of slots.
Model Description The problem considered is the short-term scheduling for a set of orders with deadlines in a multiproduct batch plant with multiple stages and units in parallel. The problem will be formulated as an MILP model in which the objective is the minimization of earliness. Assumptions A1-A7 described by Pinto and Grossmann (1995) also hold for this model. An additional assumption is: (A8) Orders are preordered in the units. Assumption A8 could be a practical constraint or a heuristic to simplify the problem by applying preordering algorithms, such as single machine scheduling problems. The underlying idea is to exploit a priori the information concerning due dates and processing times. A simple algorithm for preordering based on the bounds for start times of the orders is shown later in the paper. One major aspect of the model is that time is not discretized. Two parallel time coordinates are used for handling the assignment of orders to machines. Orders are represented in one coordinate, while the other coordinate handles units, as shown in Figure 1. When stage l of order i is assigned to unit j, matching takes place by enforcing equality of the start and end times of these coordinates through mixed integer constraints (see eq 5). Binary variables Xijl are used to model the potential assignment of stage l of order i to unit j. When * Author to whom correspondence should be addressed. E-mail:
[email protected]. † Present address: Department of Chemical Engineering, University of Sa˜o Paulo, Sa˜o Paulo, Brazil. E-mail: jompinto@ usp.br.
0888-5885/96/2635-0338$12.00/0
Figure 2. Alternative representations for enforcing preordering constraints.
order i is assigned to unit j (Xijl ) 1), the start times in both coordinates are enforced to be the same. Otherwise, the constraints are relaxed. In this formulation, time slots are defined according to the preordering in the units. Figure 2 illustrates the difference between the previous model and the proposed © 1996 American Chemical Society
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one. Consider that orders 1-6 can be assigned to unit j and that a certain preordering algorithm determined the sequence 1-3-5-2-4-6. Moreover, the assignment determined that only orders 1, 2, and 5 are to assigned to that unit (the remaining orders are assigned to other machines in parallel). In the previous model by Pinto and Grossmann (1995), preordering constraints were specified that would enforce order 1 to occupy the first time slot, order 5 the second one, and 2 the third one (Figure 2a). The remaining time slots would be empty. In the alternate model, there is a one-to-one correspondence between the sequence of orders and the time slots. Therefore, since order 3, which is second in the sequence, is not assigned to the unit, time slot 2 will be empty (Figure 2b). The MILP model for short-term scheduling of batch plants with preordering constraints includes the following objective function and equations:
Min
∑
∑i hi[di - Teiilh ]
(1)
i
Xijl ) 1
∀i, l ∈ Li
(2)
j∈(Ji∩Jl)
∑ ∑ i∈I j
Teij ) Tsij +
∀j
Xijl e |Kj|
(3)
l∈(Li∩Lj)
∑
Xijl(Tij + SUj)
∀i, j ∈ Ji (4a)
∑
Xijl(Tij + SUj)
∀i, l ∈ Li (4b)
l∈(Li∩Lj)
Teiil ) Tsiil +
j∈(Ji∩Jl)
-U(1 - Xijl) e Tsiil - Tsij
∀i, j ∈ (Ji ∩ Jl), l∈Li (5a)
U(1 - Xijl) g Tsiil - Tsij
∀i, j ∈ (Ji ∩ Jl), l∈Li (5b)
∀i, j ∈ Ji, Poi′j ) Poij + 1
Teij e Tsi′j
∀i, l ∈ Li - {lhi}
Teiil e Tsiil+1
∀i
Teiilhi e di Xijl ) 0 or 1
(6a) (6b) (6c)
∀i, j ∈ (Ji ∩ Jl), l∈Li
Tsij, Teij g 0 Tsiil, Teiil g 0
∀i, j ∈ Ji ∀i, l ∈ Li
Objective function (1) represents the total earliness of the orders, i.e., the difference between the completion time of the item and its due date. An alternate objective function is the one that minimizes the total in-process time. The derivation of (7) is detailed in Pinto and Grossmann (1995):
zj )
∑i l∈L ∑ hil[di - ∑ i
j∈(Ji∩Jhl i)
Solution Method If preordering is not specified from practical manufacturing constraints, determining the sequence of time slots in the units can be accomplished with the use of a single machine scheduling algorithm that makes use of due dates and processing times. For instance, a simple method of preordering orders is according to the upper bounds on their start times. Given for each order i the demands di and the processing times Tij in each unit j yield: hl i
TsiU il ) di -
∑
min {Tij}
l′g1 j∈(Ji∩Jl′)
l-1
∑ min {Tij} l′)1 j∈(J ∩J ) i
hl i-1
Xijl′(Tij + SUj)] (7)
j∈(Ji∩Jl′)
Equations 2 and 3 represent the assignment of orders to units. In (2), stage l of order i is assigned to exactly one unit j. The number of orders assigned to unit j
∀i, l∈Li
For each unit j, select among all the orders allowed to be processed on it, given by the set Ij. Sequence the orders according to increasing Tsiil and construct the matrix Poij which contains the sequence of products for each of the units. Break ties with release dates, which for the case of multistage plants can be defined simply as lower bounds for the start times:
TsiLil )
XijlhiTij -
∑ ∑ l′)1
cannot exceed the total amount of available time slots, as enforced in (3). Nevertheless, eq 3 is important to reduce the combinatorial aspect of the problem. The idea is to make use of the parallel structure of the plant and eliminate clear nonoptimal solutions. In (4) the completion and start times are related in terms of the assignment variables. Equations (5) ensure that, when the assignment of stage l of order i is made to unit j, there is a match between the start times Tsij and Tsiil. Constraint (6a) requires that two successive time slots cannot overlap in unit j. Note that the preordering determines the sequence of orders in the time slots. As for the completion and start times for two successive stages, it is imposed in (6b) that the start time in stage l + 1 can be performed any time after the completion of stage l. The completion of the last stage of order i has to satisfy its due date in (6c). Objective function (1) subject to constraints (2)-(6) defines model AM1. Model AM2 involves (7) subject to the same set of constraints. Compared to model M1 of Pinto and Grossmann (1995), model AM1 involves similar end-time, timematching, timing, and due date constraints. One major difference occurs in the assignment of orders. Since M1 does not consider preordering, the binary 0-1 variables Wijkl denote the assignment of stage l of order i to slot k of unit j, while in AM1 the allocation of stage l of order i to unit j defines its relative position. When the preordering constraints (23) and (24) or Padberg and Alevras’ constraints (25) are added to model M1 of Pinto and Grossmann (1995), it becomes equivalent to AM1 in terms of solution space. Similarly, model AM2 has the same solution space and objective function of M2.
∀i, l∈Li
l′
Other single machine sequencing algorithms are described in Appendix I of Pinto and Grossmann (1995) and can also be found in French (1982). Model AM1 can be solved with the strategy S1 which consists of two steps: (1) solve the MILP model AM1 in which setup times may be overestimated (i.e., setups assigned regardless of whether the same product is
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Table 1. Computational Results for Example 1 with 5 Orders binary vars cont. vars. rows
model reformulated M1
161 365 449 105 669 1092 161 365 492 105 266 511
reformulated M1 and preordering constraintsa reformulated M1 and Padberg and Alevras’ ordering constraintsa model AM1
CPU timeb
nodes
LP relaxation solution
optimal integer solutionc
58.87
851
6154
6151
9.53
50
6151
6102
2.9
12
6151
6102
45558
6276
6102
3069
a Preordering constraints correspond to eqs 23 and 24; Padberg and Alevras’ order preserving constraints are represented in eq 25 in Pinto and Grossmann (1995). b Seconds on HP 9000-730 with GAMS/OSL. cObjective function corresponds to:
Max[
∑∑ i l∈Li
hilTeil -
max(di)
∑
hνiSEi] where hil ) 0.2
i
processed consecutively) and (2) solve an LP to eliminate unnecessary setups for fixed assignment determined in step 1. The solution strategy S2 adopted for large problems is as follows: (1) solve the problem AM2 in which the total in-process is minimized and assignments of orders to units are determined with overestimated setup times; (2) solve problem AM1, minimizing earliness for fixed assignments (fixed Xijl) and eliminating unnecessary setups. It is important to note that feasibility in step 2 is guaranteed since both AM1 and AM2 consist of the same set of constraints. In fact, the solution strategies are the same as the ones presented in Pinto and Grossmann (1995). The solution methods are tested in the problem of example 1 in which a larger number of orders are considered. Computational Results The modeling system GAMS version 2.25 (Brooke et al., 1992) was used to implement the model and the solution method. The models were solved with OSL (IBM, 1991) on an HP 9000-730. The numerical tests were performed with data from example 1, which consists of a plant with 5 stages and 25 units, and described in Pinto and Grossmann (1995). It is interesting to note, that if model AM1 is used with strategy S1, the performance of the model is poor. The optimal results obtained with 5 orders are shown in Table 1 for the alternate model AM1 and compared to Pinto and Grossmann’s (1995) reformulated model M1 with no preordering, M1 with preordering constraints, and M1 with Padberg and Alevras’ orderpreserving constraints. Note that the first entry in this table has a higher objective function value because it is not constrained to the preordering. From Table 1, it is evident that the alternate model has poor performance: the solution time, when the models are solved to global optimality, increases from less than 1 min and 851 nodes enumerated (the worst case which corresponds to the reformulated model) to 51 min and 45 558 nodes. Also, although in all cases the integrality gaps are not high, model AM1 presents the highest one at 2.85%, compared to 0.05% and 0.80% for the reformulated model M1 and the models with preordering constraints.
i
di
l and hνi ) 105
Table 2. Computational Results for Large-Scale Problems (Example 1) with Strategy S2 reformulated model with preordering constraints no. of orders 5 8 10 20 35 50
binary vars. cont. vars. rows
CPU timesa
105 694 1087 168 1582 2207 210 2483 3255 420 8426 9715 735 21421 23015 1050 47917 48843
6.57 0.84 7.41 26.18 2.80 28.98 20.52 6.30 26.82 161.83 57.12 218.95 1567.40 378.06 1945.46 12752.70 2198.00 14950.70
proposed model binary vars. % cont. vars. CPU CPU time a rows times reduction 105 266 511 168 383 817 210 531 1021 420 1061 2041 735 1856 3571 1050 2651 5101
2.34 0.53 2.87 3.29 0.76 4.05 4.94 1.09 6.07 26.49 3.03 29.52 239.85 18.76 258.61 1166.99 161.75 1328.74
61.27 86.02 77.37 86.52 86.71 91.11
a Seconds on HP 9000-730. The first two entries correspond to CPU times for steps 1 and 2, respectively, using the interior point method in OSL. The third entry is the total time.
Nevertheless, when strategy S2 is applied to the alternate model, the computational requirements decrease significantly. Table 2 shows the size and the solution times of the reformulated model M2 and the proposed model. The reformulated model includes the preordering constraints. The binary assignment variables are the same for both models. However, the number of continuous variables and rows is much smaller for model AM2. The main reason is that in M2 the preordering is imposed by constraints and additional variables, while in AM2 it is embedded in the model representation. Computational experience showed that both formulations M2 and AM2 have zero-integrality gap; i.e., the objective values of the linear programming relaxation and of the MILP problem are the same (see Appendix I for sufficient conditions on the single-stage case). Therefore, both formulations will have the same LP relaxation, and due to the fact that AM2 is a much more compact formulation, it is expected to be solved faster.
Ind. Eng. Chem. Res., Vol. 35, No. 1, 1996 341
In fact, the computational times are reduced by an average of 80%, reaching 91% for the case of 50 orders.
Min
∑Xij ) 1
Xij e |Kj| ∑ i∈I
Sets I Ij J Ji Jl Kj L Li Lj
(I.2)
∀j
(I.3)
j
∀i, j∈Ji
Teij ) Tsij + Xij(Tij + SUj)
∀i, j∈Ji
Teij ) Tsi+1,j
Acknowledgment
Indices i, i′ order j unit l, l′ processing stage for an order hli stage at which order i is completed
∀i
j∈Ji
This note has presented a simple MILP model for short-term scheduling of batch plants which assumes preordering of orders. As shown in the results, if the model is solved with a decomposition method, problems up to 50 orders in 25 machines and 5 stages can be solved in approximately 20 min.
Nomenclature
(I.1)
i
Conclusions
The authors acknowledge support received from FAPESP (Fundac¸ a˜o de Amparo a` Pesquisa do Estado de Sa˜o Paulo, Brazil) under Grant 91/1186-0 and from the National Science Foundation under Grant CTS9209671.
∑i hi(j∈J ∑XijTij)
(I.5)
∀i, j∈Ji
Teij e di + U(1 - Xij)
(I.4)
(I.6)
where eq I.6 is a looser constraint obtained by combining the time-matching constraints (5a), written for the completion times Tei and Teij, with the due date constraints (6c). Note also that (5b) is not necessary for the single-stage case. Substituting (I.4) in (I.6) yields (I.7) and also in (I.5) yields (I.8):
∀i, j∈Ji (I.7)
Tsij + Xij(Tij + SUj) e di - U(1 - Xij)
∀i, j∈Ji (I.8)
Tsi+1j ) Tsij + Xij(Tij + SUj)
Also, by replacing (I.8) in (I.7) successively: set of orders set of orders which can be processed in unit j (Ij ⊂ I) set of units set of units which can process item i (Ji ⊂ J) set of units which belong to stage l (Jl ⊂ J) set of slots postulated for unit j set of stages stages involved in the production of i (Li ⊂ L) stage corresponding to unit j (Lj ⊂ L)
Parameters due date of order i di weight for earliness of order i hi position of order i in unit j Poij SUj setup time in unit j processing time of order i in unit j Tij U upper bound on start times Variables starting time in unit j during time slot i Tsij finishing time in unit j during time slot i Teij Tsiil starting time for order i in stage l Teiil finishing time for order i in stage l binary variable that assigns order i stage l to unit Xijl j
i
Ts1j +
∑ Xi′j(Ti′j + SUj) e di - U(1 - Xij) i′)1
∀i, j∈Ji (I.9)
Without loss of generality, Ts1j can be set to zero. Model AS2 can then be written only as a function of the integer variables Xij:
∑i hi(j∈J ∑XijTij)
Min
(I.1)
i
∑Xij ) 1
∀i
(I.2)
j∈Ji
Xij e |Kj| ∑ i∈I
∀j
(I.3)
j
i
∑ Xi′j (Ti′j+ SUj) e di - U(1 - Xij)
∀i, j∈Ji (I.9)
i′)1
Consider a relaxation of AS2, denoted RAS2, which is composed of (I.1), (I.2), and (I.3). Clearly, RAS2 can be solved as an LP due to the fact that the coefficient matrix is unimodular. Moreover, if (I.3) is a nonbinding constraint, i.e., a sufficiently large number of slots is allowed for unit j, the solution of RAS2 is given by:
(Xijˆi ) 1, ∀i) where ˆji ) arg min{Tij} j
Appendix I. On the Zero-Integrality Gap of Model AM2 In this appendix sufficient conditions are given for the zero-integrality gap of model AM2 which minimizes inprocess time for the single-stage case. When orders follow the same sequence in all units and the orders are processed immediately after each other, model AM2 reduces to model AS2 composed of constraints:
If constraints (I.9) are satisfied, then the above solution is optimal for AS2 and therefore AS2 has an integer optimum. However, if constraints (I.9) are active for AS2, then a noninteger solution may arise, as illustrated in the counterexample below. Counterexample. Consider two orders with due dates (di ) 4, i ) 1, 2) to be scheduled in two units. The orders are processed in the 1-2 sequence with the
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processing times: T11 ) 1, T12 ) 3, T21 ) 4, T22 ) 5. Dropping constraints (I.3), the model can be written as:
Min z ) X11 + 4X21 + 3X12 + 5X22 X11 + X21 ) 1 X12 + X22 ) 1 X11 + 4X21 e 4 3X12 + 5X22 e 4 The LP solution is given by X11 ) 1, X12 ) 0, X21 ) 0.75, and X22 ) 0.25 and with objective value zLP* ) 5. The corresponding integer solution is given by X11 ) 0, X12 ) 1, X21 ) 1, and X22 ) 0; the integer objective value is zIP* ) 7. This counterexample relies on the fact that the processing times Tij are different for each order in each unit. Assume now that they are the same for all lines; in this case it is no longer possible to construct a counterexample, and, in fact, the total in-process time becomes independent of the assignment. Assume that for order i the processing times are given by Tij ) τi, ∀j. Then, the objective function (I.1) can be rewritten as:
z)
∑i hi(j∈J ∑XijTij) ) ∑i hij∈J ∑Xijτi ) ∑i hiτij∈J ∑Xij i
i
i
Using (I.2), the objective function is constant with value z ) ∑ihiτi. Thus: if there exists a feasible integer solution, (Xi,ji ) 1, ∀i) w MILP has a zero-integrality gap. Literature Cited Brooke, A.; Kendrick, D.; Meeraus, A. GAMSsA user’s guide (release 2.25); The Scientific Press: San Francisco, CA, 1992. French, S. Sequencing and scheduling: an introduction to the mathematics of the job-shop; Ellis Horwood Ltd.: Chichester, England, 1982. IBM. OSL, Guide and Reference (release 2); IBM: Kingston, NY, 1991. Pinto, J. M.; Grossmann, I. E. A continuous time MILP model for short term scheduling of multistage batch plants. Ind. Eng. Chem. Res. 1995, 34, 3037-3051.
Received for review May 23, 1995 Accepted October 23, 1995X IE9503095
X Abstract published in Advance ACS Abstracts, December 1, 1995.
