From a theoretical point of view there are two main contributions in this dis- sertation: we introduced the novel concepts of global chance-constraints and of
optimization-oriented global chance-constraints. The first, as stated, let us model
complex interactions that arise in stochastic constraint programs where several chance-constraints appear together. The second let us apply cost based filter- ing in a stochastic environment, by exploiting cost-based reasoning and/or relax- ations involving decision variables, random variables and the constraints defined on these.
From a practical point of view, our contribution consists in the application of both these techniques to known problems in the area of stochastic inventory control.
Global chance-constraints
There are three main contributions related to this novelty:
• Formal background. We have formally introduced global chance-
constraints, defined as constraints that capture a relation among a non-fixed
number of decision and random variables. These constraints not only are more expressive than the respective aggregation of simple chance-constraints, but they can be associated with more powerful filtering algorithms (Chap. 2).
• Application 1. We have applied global chance-constraints to com-
pute optimal replenishment cycle policy parameters under non-stationary stochastic demand and service level constraints. Global chance-constraints allow the assumption on negative orders adopted in previous works [89, 92] to be relaxed and thus they let us compute the real optimal solution for the problem (Chap. 2).
• Application 2. We exploited global chance-constraints to represent mul-
tiple layers of uncertainty, demand uncertainty and delivery uncertainty, and to compute replenishment cycle policy parameters under non-stationary
stochastic demand, service level constraints and stochastic delivery lag (Chap. 3).
Optimization-oriented global chance-constraints
There are two main contributions related to this novelty:
• Formal background. We have formally introduced optimization-oriented
global chance-constraints, defined as global chance-constraints that encap-
sulate suitable relaxations of the constraints considered. This relaxation, in contrast to conventional optimization-oriented global constraints, may in- volve stochastic variables (Chap. 4).
• Application 3. By using optimization-oriented global chance-constraints,
we have augmented the SCP model originally proposed by Tarim and Smith [92] for computing optimal replenishment cycle policy parameters under non-stationary stochastic demand and service level constraints. In Tarim and Smith’s model domain filtering was originally performed only in a proactive way before starting the search process. The cost-based filter- ing dynamically performed during the search by the optimization-oriented global chance-constraints proposed let us now efficiently compute near- optimal replenishment cycle policy parameters under non-stationary stochas- tic demand and service level constraints (Chap. 5). The augmented model produces run times that are orders-of-magnitude lower than those achieved by the state of the art approach in [92].
A global perspective
Finally we have employed both global chance-constraints and optimization-oriented
global chance-constraints to obtain the state of the art approach for computing re-
plenishment cycle policy parameters under non-stationary stochastic demand and a penalty cost scheme:
• Application 4. We have applied global chance-constraints to model the non-linear cost function that is only approximated by the approach in
[90], which employs a piecewise linear approximation for modeling pe- riod holding and back-ordering costs. In addition to this we have applied
optimization-oriented global chance-constraints to the same model in or-
der to perform cost-based reasoning and thus improve the efficiency of the search process (Chap. 6).
1.4.3
Paper I (Chap. 2): A Global Chance-Constraint for Stochas-
tic Inventory Systems under Service Level Constraints
[75]
SCP has been introduced in [98] to model decision problem involving uncertainty and probability. In contrast to conventional approaches in Stochastic Program- ming, SCP features all the key features of CP: constraint propagation, variable and value selection strategies and so forth.
To solve stochastic constraint programs, Tarim et al. in [91] proposed a se- mantics based on scenario trees. This semantics is extremely flexible, especially for the fact that it lets stochastic constraint programs be compiled down into con- ventional constraint programs, so that conventional constraint solvers can be em- ployed to find a solution. Nevertheless, the framework proposed by Tarim et al. still presents limits: in particular, as formulated in [91], it does not specify how a generic relation among a non-predefined number of decision variables and stochastic variables under a given policy of response should be translated into a conventional constraint program. This is obviously not an easy task, as it is prob- lem dependent.
In order to address this issue we propose in this chapter an extension for SCP:
global chance-constraints. Global chance-constraints, similarly to conventional
global constraints, represent relations among a non predefined number of vari- ables and incorporate dedicated filtering algorithms. In contrast to conventional global constraints, global chance-constraints represent relations among decision and stochastic variables and can model any policy of response.
By means of this novelty and using the scenario based semantics proposed by Tarim et al. [91], in this work we were able to relax the original assumption on negative order quantities that had to be adopted in [89, 92] for computing re-
plenishment cycle policy parameters under non-stationary stochastic demand. In contrast to models previously proposed our model provides (i) the exact cost of an optimal solution, and (ii) exact policy parameters, that is replenishment cycle lengths and order-up-to-levels. A comparison among our approach and previous approaches shows that the discussed assumption does not significantly affect the quality of the policy parameters computed by the models in [89, 92], but it does affect the computed cost, which typically differs significantly from the real cost of the solution provided.
1.4.4
Paper II (Chap. 3): Computing Replenishment Cycle
Policy under Non-stationary Stochastic Lead Time [72]
Also in this chapter we rely on the scenario based semantics originally proposed in [91]. The problem here is to compute replenishment cycle policy parameters under non-stationary stochastic demand, delivery lag and service level constraints. Incorporating a delivery lag in inventory control models is a very active research topic, as the literature review presented in this chapter will show. To the best of our knowledge, this is the first work in which a non-stationary stochastic demand and a non-stationary stochastic delivery lag are considered together when computing replenishment cycle policy parameters under service level constraints.
The first part of this work is dedicated to the derivation of a mathematical model for computing feasible buffer stocks under non-stationary stochastic de- mand, delivery lag and service level constraints. The expression obtained repre- sents a non-linear relation among decision variables (replenishment decisions and inventory levels) and stochastic variables (stochastic demands and delivery lags).
Using the expression derived in the first part of this chapter, we developed a
global chance-constraint and the respective filtering procedure able to take into
account both demand and delivery lag uncertainty while computing buffer stocks required to guarantee the given minimum service level in terms of non-stockout probability. The approach was tested against different delivery lag distributions. The experimental results presented show the behavior of the expected total cost of the optimal policy with respect to the expected value and to the variance of the delivery lag.