In many cases where an STPU is used to represent a plan or schedule, the ability to optimise a function of STPU time bounds subject to the constraint that the network is dynamically controllable make it far more useful than being able only to test if dynamic controllability holds. For example, we showed how it enables optimisa- tion of a chance-constrained pSTN under dynamic controllability, leading to better solutions than possible if strong controllability is imposed. It also enabled us to mea- sure schedule robustness in new ways, and thereby test the hypothesis that greater
§3.5 Conclusions 55 <=0.2 <=0.4 <=0.6 <=0.8 <=1 Feasible Improvement from SC to DC % Prob lems 0 10 20 30 40
Figure 3.9: Reduction in makespan achieved with dynamic as opposed to strong controllability. Instances in the last column are infeasible under strong controllability
but have a valid dynamic execution strategy.
flexibility leads to more robust schedules.
Dynamic controllability is a disjunctive linear constraint. Comparing solution approaches, we found that dealing with disjunctions explicitly, as in the CDRU algo- rithm [Yu et al., 2014], is the most efficient. However, the MIP and NLP formulations are very flexible, allowing controllability to be combined with other constraints. Our constraint model follows closely the reduction rules of the algorithm by Morris et al. [2001].
Chapter4
Dynamic Controllability of CCTPU
Yu et al. [2014] extended the STPU to the Controllable Conditional Temporal Problem with Uncertainty (CCTPU) by considering controllable choices as discrete variables. In CCTPU, the links with a label, which is a conjunction of assignments of discrete variables, are activated by those assignments. Yu et al. [2014] introduced a relaxation method to solve over-constrained CCTPU, by making a static assignment of choice variables and relaxing time constraints to produce a dynamically controllable STPU. In this chapter, to implement the original intent of dynamic control, we introduce the definition of dynamic controllability of a CCTPU that considers making assign- ments of both time points and discrete choices dynamically by extending it to the discrete variables of the CCTPU. From Yu et al. [2014], we borrow the idea of using dynamic controllability checking algorithms of STPU (See Section 2.2) to find con- flicts, which represent the reason why an STPU is not dynamically controllable but extend it to extract a complete set of conflicts.
When implementing the dynamic controllability checking process, some assump- tions are needed: (1) each discrete choice is made at one time point, which isno later than any other time points related to the choice; (2) only the uncontrollable events definitely completed before the time point to make a choice can be treated as an ob- servable condition; and (3) each discrete choice is made following the observation of one, or a sequence of, uncertain events. These assumptions are more conserva- tive than the original concept of dynamic controllability. Our dynamic controllability checking algorithm for a CCTPU is sound but only complete with assumptions. It guarantees to find dynamic strategies under these assumptions.
Last but not least, a similar problem – the dynamic controllability of Conditional Simple Temporal Networks with Uncertainty (CSTNU) – has been studied by Huns- berger et al. [2012]; Combi et al. [2014], which consider conditions as uncontrollable and observable propositions. But we only discuss controllable discrete variables that do not depend on observations.
The work in this chapter extends the conference publication [Cui and Haslum, 2017] by completing the theoretical definitions and validation of the algorithms.
We start by illustrating an example in Section 4.1. In Section 4.2, we introduce the problem statement that includes the definitions CCTPU, dynamic assignment of discrete variables with assumptions and dynamic controllability of CCTPU. We
briefly review conflict resolutions of STPU in Section 4.3.1, which is the basis of our algorithm that extracts a complete set of conflicts (Section 4.3.2 ) in order to get the DC envelope of an STPU (Section 4.3.3). In Section 4.4, we illustrate how to check dynamic controllability of a CCTPU by aggregating and checking its dynamically controllable envelopes. Section 4.5 shows the correctness and completeness of the algorithms. Section 4.6 and 4.7 are experiments and conclusions, respectively.
4.1
An Illustrative Example
To illustrate the motivation for dynamic controllability of a CCTPU, we take the example of Mr. P’s travel plan after work. After leaving work at 5pm, Mr. P is going grocery shopping before having dinner, then catching a bus home. Shopping may take 30 to 50 minutes, depending on how crowded it is. For dinner, Mr. P has two options: He can have a quick dinner at KFC, which only takes 20-30 minutes, or go for his favourite steak. This takes longer, 40-60 minutes, but the restaurant is closer to the bus stop. The bus leaves at 6.50pm. Mr. P neither wants to miss this bus, which will make him wait an hour for the next bus nor arrive at the bus stop before than 6.40pm, to avoid staying out in the cold weather. Mr. P needs to decide his schedule for these two hours.
The CCTPU in Figure 4.1 models Mr. P’s problem. The discrete variablec1models
the choice of dinner (c1=Kfor KFC andc1 =Sfor steak). Contingent links (dashed
lines: E1→ E10, E2 → E20 andE3 → E30) represent uncertainty, such as the time it takes to shop and have dinner. Those durations are not decided by Mr. P but depend on factors outside his control. The other links express constraints on the solution, in the form of bounds on the difference between two time points. Some links have labels, which are the assignments of the discrete choice variable that active those links. For example, constraints E10 → E3, E3 → E30 and E30 → E only need to be met ifc1 =K. S E1 E10 E2 E3 E20 E30 E [0, 5] [15, 20] c1= K [5, 15 ] c1 = S [15, 20] c1 = K [5, 15] c1= S [30, 50] [20, 30] c1 =K [40, 60] c1 =S [100, 110]
Figure 4.1: The CCTPU of Mr. P’s travel problem.
Unfortunately, because of the uncertainty, neither of the two dinner options, if chosen in advance, lead to a schedule that is guaranteed to satisfy Mr. P’s require- ments. If he goes to KFC and the uncontrollable links E1 and E2 take their lower