# (285e) Pharmaceutical Research and Development Pipeline Planning Using Multi-Stage Stochastic Programming

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Introduction

The development of new pharmaceutical products contains significant risks and costs. For every drug that reaches the market, thousands of compounds were tested with five to ten reaching the clinical trials stage. Furthermore, the development of a single drug often requires more than \$1 billion in development costs. Given the amount of money and risk involved, optimization of pipeline development decisions can provide immediate returns. An overview of the approaches that have been proposed to address this problem can be found in Shah (2004). In this work, we build upon the multi-stage stochastic programming (MSSP) formulation developed in Colvin and Maravelias (2008) to provide a more general framework for the optimization of the pharmaceutical pipeline.

Stochastic Programming

In a MSSP formulation, all possible uncertainty realizations are represented by a finite set of scenarios (s in S). Each of these scenarios is then treated as a multi-stage deterministic problem, with a set of constraints to connect decisions between indistinguishable scenarios. In the problem addressed in this paper, stages correspond to time periods (t in T) of fixed length into which the planning horizon is divided. The constraints connecting scenarios are referred to as non-anticipativity constraints (NACs) and ensure that the decision-maker cannot anticipate future outcomes (i.e. cannot use information about future uncertainty realizations improperly). For a basic formulation, these constraints represent more than 99% of all constraints in the problem.

Problem Statement

The formal statement of the problem we examine in this work is the follwoing:

A pharmaceutical company has a set of candidate drugs, i in I, in the clinical trials portion of its research and development (R&D) pipeline. Each of these drugs must complete a set of tasks, j in J, including a subset of sequential clinical trials {PI, PII, PIII} before it can be marketed and any profit realized. The tasks require the use of a shared set of resources, r in R, and the following information is assumed to be known deterministically: i) the income of drugs successfully completing all clinical trials; ii) the income received if rights to a drug were sold prior to the completion of all tasks; iii) the cost and duration of task (i,j); iv) the probability of any uncertain outcome; v) the cost and availability of resource r (as discussed later a resource can be expanded, extracted, outsourced and can be either renewable or non-renewable); vi) resource requirements.

The goal is to maximize the expected net present value (NPV) of the research and development pipeline. We assume that uncertainty occurs only in the outcome of clinical trials. Further, we assume that the revenue decreases with delays in development and that there is a time value of money. Because revenue for successful development is assigned only at the completion of PIII clinical trials and the time horizon is finite, we assign a weighted expected revenue to drugs under development at the end of the time horizon.

Mathematical Formulation

Within each scenario, the basic model contains resource constraints, task precedence constraints and an objective function. In addition, there are NACs to connect decisions across scenarios and constraints that weight the NPV of each scenario according to its probability of occurrence. We focus on four specific elements that allow for the improved modeling of the pharmaceutical R&D pipeline: a generalized task list, the modeling of resources, the inclusion of risk constraints and the interdependence of outcomes between drugs.

First, we expand the scope of our framework by considering additional types of tasks. Specifically, development tasks with no uncertainty (e.g. process development, marketing) are incorporated. These tasks can be carried out in parallel with clinical trials or before/after clinical trials. Furthermore, we consider drugs that have already completed up to two of the clinical trials at the beginning of the planning horizon. We also allow drugs that have successfully completed at least PI clinical trials to be outlicensed rather than fully developed. It is important to note that as additional tasks are incorporated and the precedence tree is generalized, our formulation becomes similar to the resource constrained project scheduling problem (RCPSP), (Herroelen and Leus 2005). The main differences include the stochastic outcome of tasks, that not all tasks need to be completed, and that the objective function is significantly different.

Second, we consider different types of resources. In addition to renewable resources (e.g. lab space or trained workers), we consider non-renewable resources such as active ingredients and funds for research, as well as resources that are expanded or contracted via optimization decisions or fixed events. Furthermore, we account for outsourcing of tasks if all resources are utilized.

Third, we discuss methods for risk management. We present methods to constrain risk either by limiting the probability of having negative NPV or limiting the loss from scenarios with a negative NPV. We also discuss how risk management changes the optimal solutions.

Fourth, we present methods to account for drug interdependencies. Drugs containing similar compounds can have correlated probabilities of successfully passing clinical trials. In the MSSP framework, each scenario corresponds to a specific outcome. This means that if the correlation is known, the probabilities of the scenarios can be calculated offline and input as parameters with no additional work necessary.

Finally, we present a series of examples to illustrate various aspects of our framework and their effect on the quality of solution. In addition, we show how different solution techniques, such as a rolling horizon and a branch and cut algorithm can be employed for the solution of instances containing up to eight drugs and thousands of scenarios.

Colvin, M., Maravelias, C. T. (2009). Scheduling of Testing Tasks and Resource Planning in New Product Development Using Stochastic Programming. Computers & Chemical Engineering, 33 (5), 964-976.

Herroelen, W., Leus, R. (2005) Project scheduling under uncertainty: Survey and research potentials. European Journal of Operational Research. 165, 289-306.

Shah, N. (2004). Pharmaceutical Supply Chains: Key Issues and Strategies for Optimization. Computers & Chemical Engineering, 28(6/7), 971-983