Due to changing circumstances in the managed-health-care environment, profit margins of US pharmaceutical companies and the productivity of their Research and Development (R&D) pipelines have started to decline; effective patent lives have been shortened, and patents provide lower barriers to entry even while active. Therefore, it is imperative for pharmaceutical companies to manage their R&D pipelines more effectively to reduce the cost of developing new drugs. This is a challenging task due to the highly stochastic nature of the R&D process: if a drug fails a clinical trial, its development stops and all prior investment is wasted; if it passes all trials, it enters the marketplace and profits are significantly larger than development costs.

The modeling of the pharmaceutical R&D is challenging because uncertainty is revealed only when a trial is performed, which can vary based on decisions made throughout the process. This is known as endogenous uncertainty and can lead to computationally intractable formulations. Previous attempts to address this problem include deterministic approaches where resource constraints are relaxed via outsourcing (Jain & Grossmann, 1999; Maravelias & Grossmann, 2001, 2004), dynamic programming approaches over a reduced state space (Choi et al 2004), and simulation-based optimization (Subramanian et al 2001, 2003) where a deterministic problem is solved, the solution is simulated and a new deterministic optimization problem is solved every time uncertainty is revealed.

In this paper, we aim to develop a multi-period stochastic programming (SP) framework for the management of the pharmaceutical R&D that explicitly accounts for all the different outcomes of clinical trials and the recourse action that the decision-maker can take to compensate for the effect of stochastic elements. Given are a) a planning horizon divided into a set of periods (stages), b) a portfolio of potential drugs with revenues from sales given as decreasing functions of development time, c) limited resources, and d) a set of clinical trials for each drug with their associated cost, duration, probability of success and resource requirements. The model determines the trials to be performed in each planning period and possible set of outcomes (scenarios) with the goal of maximizing the expected net present value. Because all possible outcomes of the trials are evaluated, the solution is guaranteed to be optimal.

Due to the endogenous nature of the uncertainty, it is necessary to introduce non-anticipativity constraints to ensure that decisions made while two scenarios are indistinguishable, are identical. Under generic circumstances it is necessary to have non-anticipativity constraints between every possible pair of uncertain realizations across all time periods, for all decisions, making even small problems computationally intractable. By exploiting the specific structure of this problem, however, we are able to reduce the size of the formulation substantially.

First, we show that the uncertainty in the outcome of clinical trials can be represented using a reduced number of scenarios where multiple outcomes are combined into events. The proposed representation reduces the number of scenarios from an exponential growth in the number of tests to a linear growth, without compromising the quality of uncertainty representation.

Second, a number of properties are presented that enable us to reduce the number of non-anticipativity constraints necessary to model indistinguishable scenarios. Properties such as reflexivity and transitivity allow for a reduction of non-anticipativity constraints in the general case, but the structured nature of the drug development process allows for even greater improvements. By definition, for every pair of different scenarios, there exists at least one trial that passes in one scenario but fails in the other. Using a transitivity argument, it is possible to show that only scenarios where this happens in a single drug must be constrained for non-anticipativity to hold. Furthermore, we show that if a given drug fails two different trials in two scenarios, then non-anticipativity constraints are only necessary if the trials failed are consecutive.

Third, we show that for the enforcement of non-anticipativity constraints it is not necessary to have a binary variable for every pair of scenarios. Because the reduced formulation pairs scenarios differing in only a single trial for a single drug, it is possible to use a formulation based on the running of this trial in either scenario. This eliminates the need to have binary variables indexed by pairs of scenarios.

Employing all of these reductions allows problems to be computationally tractable. For example, the non-naticipativity constraints for a three drug example over a planning horizon divided into 12 periods was reduced from 37 million to approximately 20,000 non-anticipativity constraints. The resulting formulation was solved in under 8 seconds. For a five drug example, the number of constraitns was reduced from 1.55*10^11 constraints to under 1,000,000, allowing the problem to be solved in under an hour on a desktop computer. The proposed formulation can be extended to address problems where outsourcing can be used or outlicensing of drugs upon successful completion of phase I or phase II clinical trials is considered.

1. Jain, V; Grossmann, I. E. Resource-constrained Scheduling of Tests in New Product Development. Ind. Eng. Chem. Res,.38, 3013-3026, 1999.

2. Maravelias, C.T.; Grossmann, I.E. Optimal Resource Investment and Scheduling of Tests for New Product Development. Comput. Chem. Eng., 28, 1021-1038, 2004.

3. Choi, J.; Realff, M.J.; Lee, J.H.. Dynamic Programming in a Heuristically Confined State Space: A Stochastic Resource-constrained Project Scheduling Application. Comput. Chem. Eng., 28 (6-7), 1039-1058, 2004.

4. Subramanian, D.; Pekny, J.F.; Reklaitis, G.V. A Simulation-Optimization Framework for Research and Development Pipeline Management. AIChE J., 47 (10), 2226-2242, 2001.

5. Subramanian, D.; Pekny, J. F.; Reklaitis, G.V.; Blau, G.E. Simulation-Optimization Framework for Stochastic Optimization of R&D Pipeline Management. AIChE Journal, 49(1), 96-112, 2003.