A robust optimization approach to enhancing reliability in production planning under non-compliance risks

Certain regulated industries are monitored by inspections that ensure adherence (compliance) to regulations. These inspections can often be with very short notice and can focus on particular aspects of the business. Failing such inspections can bring great losses to a company; thus, evaluating the risks of failure against various inspection strategies can help it ensure a robust operation. In this paper, we investigate a game-theoretic setup of a production planning problem under uncertainty in which a company is exposed to the risk of failing authoritative inspections due to non-compliance with enforced regulations. In the proposed decision model, the inspection agency is considered an adversary to the company whose production sites are subject to inspections. The outcome of an inspection is uncertain and is modeled as a Bernoulli-distributed random variable whose parameter is the mean of non-compliance probabilities of products produced at the inspected site and, therefore, is a function of production decisions. If a site fails an inspection, then all its products are deemed adulterated and cannot be used, jeopardizing the reliability of the company in satisfying customers’ demand. In the proposed framework, we address two sources of uncertainty facing the company. First, through the adversarial setting, we address the uncertainty arising from the inspection process as the company does not know a priori which sites the agency will choose to inspect. Second, we address data uncertainty via robust optimization. We model products’ non-compliance probabilities as uncertain parameters belonging to polyhedral uncertainty sets and maximize the worst-case expected profit over these sets. We derive tractable and compact formulations in the form of a mixed integer program that can be solved efficiently via readily available standard software. Furthermore, we give theoretical insights into the structure of optimal solutions and worst-case uncertainties. The proposed approach offers the flexibility of matching solutions to the level of conservatism of the decision maker via two intuitive parameters: the anticipated number of sites to be inspected, and the number of products at each site that are anticipated to be at their worst-case non-compliance level. Varying these parameters when solving for the optimal products allocation provides different risk-return tradeoffs and thus selecting them is an essential part of decision makers’ strategy. We believe that the robust approach holds much potential in enhancing reliability in production planning and other similar frameworks in which the probability of random events depends on decision variables and in which the uncertainty of parameters is prevalent and difficult to handle.