Mixed Integer Linear Programming in Process Scheduling: Modeling ...

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138 FLOUDAS AND LIN where I j is the set of tasks that can be performed in unit ( j). Furthermore, if a task starts in a unit at a time interval, no other task can start in the same unit until this task is finished. This requirement is enforced as follows: � i ′ ∈I j t+αij−1 � t ′ =t Wi ′ jt ′ − 1 ≤ M(1 − Wijt), ∀ j ∈ J, i ∈ I j, t ∈ T, (2) where αij is the fixed processing time of task (i)inunit ( j) and M is a sufficiently large positive number. In addition to the binary variables, the following two sets of continuous variables are also important. Bijt represents the amount of material which starts undergoing task (i) in unit ( j)attime interval (t) (i.e., batch-size), and Sst determines the amount of material state (s) during time interval (t). The batch-size of a task is related to the corresponding assignment variable through the capacity constraints as follows: WijtV Min ij i∈I p s j∈Ji ≤ Bijt ≤ WijtV Max ij , ∀i ∈ I, j ∈ Ji, t ∈ T, (3) where V Min ij and V Max ij are the minimum and maximum capacity of unit ( j) for task (i), respectively. The mass balance can be expressed effectively by establishing the relationships between the inventory levels at two consecutive time intervals through the following constraint: Sst = Ss,t−1 + � ρ p � � is Bi, j,t−αis − ρ c � is Bi, j,t + Rst − Dst, ∀s ∈ S, t ∈ T, (4) where I p s and I c s i∈I c s j∈Ji are the set of tasks that produce and consume state (s), respectively; ρ p is and ρ c is are the fractions of state (s) produced and consumed by task (i), respectively; Ji is the set of units suitable for task (i); αis is the processing time for state (s)bytask (i). Rst is the amount of state (s) received from external sources at time interval (t) and variable Dst represents the amount of state (s) delivered at time interval (t). The restriction on storage of a material state is then represented simply as upper bounds on the variables Sst: 0 ≤ Sst ≤ Cs, ∀s ∈ S, t ∈ T, (5) where Cs is the storage capacity limit for state (s). Other considerations, such as change-overs and limited utility resources, can also be incorporated readily in the discrete-time model (Kondili, Pantelides, and Sargent, 1988, 1993). There has been subsequent progresses following this discretization approach and discrete-time formulation and solution techniques have been developed for a variety of problems. Examples of related work include those presented by Pantelides (1993), Dedopoulos and Shah (1995), Pekny and Zentner (1993), Zentner et al. (1994), Bassett, Pekny, and Reklaitis (1996b) and Elkamel et al. (1997).
MIXED INTEGER LINEAR PROGRAMMING IN PROCESS SCHEDULING 139 The main advantage of the discrete-time representation is that it provides a reference grid of time for all operations competing for shared resources, which renders the possibility of formulating the various constraints in the scheduling problem in a relatively simple manner and usually leads to well-structured mathematical programming problems. However, the discrete-time approach has two main limitations: the discrete approximation of time and the large size of resulting MILP problems. Due to the continuous nature of time and the concept of discretization, the discrete-time formulations are by definition only approximations of the actual problem. Furthermore, one of the key issues in this approach is the selection of the time interval duration, which always presents a tradeoff between the solution quality and the computational requirement. If a coarse discretization scheme is used, the problem size may be tractable, but there is an inevitable loss of model accuracy and it results by definition in suboptimal solutions. Furthermore, for operations with variable processing times such as continuous processes, which consume feeds and produce products continuously and in general can potentially run for a time period of any duration, the discrete-time approach only provides an approximate description of the actual process and the resulting schedule may deviate substantially from the true optimal solution. On the other hand, if a small time interval is used to achieve desirable degree of accuracy, the discrete-time approach inevitably leads to very large combinatorial problems that are difficult or even impossible to solve, especially for medium or large practical applications. 1.3. Continuous-time models To address the aforementioned drawbacks of the discrete-time models, research efforts have been spent in the past decade on the development of more effective and efficient continuous-time approaches for process scheduling problems. For general networkrepresented processes, we classify the continuous-time models that have been presented in the literature into two groups. The first group of models define a set of events (or variable time slots, time points) that are used for all tasks and all units. We denote such formulations as “global event based models.” The second group of models introduce event points on a unit basis, allowing tasks in different units associated with the same event point to take place at different times. We denote such models as “unit-specific event based models.” 1.3.1. Global event based models The time representation scheme employed in the global event based continuous-time scheduling models relies on the introduction of events or variable time slots that are universal for all units and the association of continuous variables to these events or time slots to determine their timings. The earliest work following this direction were presented by Zhang (1995), Zhang and Sargent (1996, 1998), Mockus and Reklaitis (1997a, 1999b), Mockus et al. (1997) and Schilling and Pantelides (1996). Recent developments include those presented by Castro, Barbosa-Póvoa, and Matos (2001), Majozi and Zhu (2001),
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