Closed-loop scheduling: Major considerations, paradoxes and remedies

Process industries employ a complex chain of operations (processes) which compete for limited resources. Hence a resource allocation problem arises wherein decisions with regards to the start time of operations, how much to process, the equipment to utilize and several associated decisions need to be taken on a repetitive basis. In the last two decades, there has been an increasing thrust towards using advanced computational tools to achieve more profitable schedules. Hence, working with scheduling models to optimally plan operations has become an important problem in process systems engineering [1,2]. Although the process systems engineering (PSE) community has worked on building accurate models and better solution methods, the aspect of repetitive scheduling has received limited attention. Rescheduling has been emphasized in some works [3-9], but in most cases scheduling is still thought to be a static open loop problem, wherein, if rescheduling is carried out, the emphasis is only on restoring feasibility or optimality to the current static schedule. Quantifying the quality of actual implemented schedule obtained by rescheduling has not been addressed. In a dynamic environment multiple disturbances, such as task delays, yield losses, unit breakdowns or rush order arrival can render a previously computed schedule, suboptimal or even infeasible. In addition, Subramanian et. al. [10] emphasized, that rescheduling also needs to be performed due to advancement of the scheduling horizon, over which the schedule is computed, even when there are no disturbances. In this work, we first and foremost show that open-loop and closed-loop scheduling are two different problems, even in the deterministic case, when no uncertainty is present. We also show how equally good open-loop schedules can translate to very different closed-loop schedules, so much so, that it could mean a difference between, no production versus production at full capacity. In addition, we stumble upon a paradox, wherein, solving a well-defined open-loop problem to optimality in every iteration, leads to a worse closed-loop schedule, than if this same open-loop problem was to be solved to a suboptimal solution. Thereafter, we investigate various design attributes of the open-loop problem that affect the quality of the resulting implemented closed-loop schedule. The design attributes we study, are scheduling horizon length, rescheduling frequency and optimality gap of each open-loop optimization. We choose illustrative production networks, and extensively study combinations of the aforementioned attributes, over a reasonably exhaustive set of short term demand patterns, production load, and scheduling objectives. From these test cases, we identify trends, and problem characteristics which can facilitate in carefully choosing the three attributes. First, we find that it is important to reschedule periodically, even when there are no "trigger" events, something that is in contrast with the current approaches to rescheduling. Second, we show that finding suboptimal open-loop schedules does not necessarily "accumulate" in the closed-loop. On the contrary, suboptimal solutions are "corrected" through repetitive revisions due to feedback. Third, we observe that there exist lower and upper thresholds for appropriate rescheduling frequency and moving horizon length, operating outside of which leads to substantial deterioration in closed-loop performance. Fourth, we show that the various design attributes work in conjunction with each other, and hence it is important to study them simultaneously rather than individually. The aforementioned observations lead to some interesting questions: 1) Should we reschedule on encountering trigger events (e.g., new order arrival, unit breakdown) or at a predetermined frequency or both? 2) Is it better to reschedule fast, even if we obtain suboptimal solutions, or less frequently but obtain optimal open-loop schedules? 3) Is it better to consider models over short scheduling horizons solved to optimality; or long horizons, which require larger models possibly leading to suboptimal solutions? 4) How can we choose the rescheduling frequency and moving horizon length? Can we trade one (e.g., long horizon) for the other (e.g., higher rescheduling frequency)? If yes, what are the trade-offs? We address the following questions through a series of cases studies and then draw some general conclusions. Lastly, we explore objective function modifications and addition of constraints to the open-loop problem as methods to improve closed-loop performance. Specifically, we show (1) how the former, among others, can help us "choose" among many equivalent open-loop solutions; and (2) how the latter can be used to enforce solution characteristics which can possibly lead to suboptimal open-loop schedules but ultimately result in higher quality closed-loop (implemented) schedules. In this work, we use state-task network representation [11] and the discrete time-grid state-space model proposed by Subramanian et al. [10]. However, the framework and analysis presented, is also applicable to all discrete-time models and can be extended to continuous-time models represented in state-task network as well as resource-task network [12] representations.