Patient-based pharmaceutical inventory management: a two-stage inventory and production model for perishable products with Markovian demand

Abstract

Drug shortages have increased over the past decade, tripling since 2006. Pharmacy material managers are challenged with developing inventory policies given changing demand, limited suppliers, and regulations affecting supply. Pharmaceutical inventory management and patient care are inextricably linked; suboptimal control impacts both patient treatment and the cost of care. We study a perishable inventory problem motivated by challenges in pharmaceutical management. Inpatient hospital pharmacies stock medications in two stages, raw material and finished good (e.g. intravenous). While both stages of material are perishable, the finished form is highly perishable. Pharmacy demand depends on the population and patient conditions. We use a stochastic ‘demand state’ as a surrogate for patient condition and develop a Markov decision process to determine optimal, state-dependent two-stage inventory and production policies. We define two ordering and production scenarios, prove the existence of optimal solutions for both scenarios, and apply this framework to the management of Meropenem.

Appendix

Proof of Lemma 1:

• For a fixed (z, R) there is a finite x₁^*(z, R) which exists such that L is convex in x and L _x (z, R, x₁^*(z, R))=0

Using the Dominated Convergence Theorem (DCT) we can observe that the following relationships exist.

For these relationships to hold, b₂ must be greater than the production cost (i.e., expediting must be more costly than production) and the cost w₂ must be greater than or equal to w₁, that is, the finished good is at least as valuable as the raw material. Therefore, for a fixed (z, R) the single period cost function L (z, R, x) is convex in x and an optimal solution to the minimization function v₁(z, R) exists and is defined as x₁^*(z, R).

Proof of Lemma 2:

• Define R(z) as the raw material order quantity when the demand process is initially in state z. For a fixed demand state, z, there is a finite R^*(z) which exists such that

Therefore, for a fixed z and any production quantity, x, the single period cost function L (z, R, x) is convex in R, and an optimal solution to the minimization function v₁(z, R) exists and is defined as R^*(z).

Recall that L is our single period cost function (Eq. 3) when the demand is in state z, the raw material inventory is R, and the production decision is x. Now if we assume that the decision R can be continuous but the decision x is discrete, we must prove that there still exists an optimal solution for x. The following Proposition defines the analog to Lemma 1.

Proposition 1:

• If κ(x) is a real-valued function of x∈X where X is a bounded subset of Z, and if Δ²κ(x)⩾0 for all x∈Z, then every local minimum of κ(x) is also a global minimum of κ(x) on X.

Proof:

• In order to prove that the proposition is true, we must develop the first difference equation. We define the first difference as where Δf(x)=f(c+h)−f(c) where h is defined as the increment. Rewriting Eq. (3.3) as its discrete counterpart below, the first difference in developed as a function of the decision variable x.

The last term can be further simplified as follows:

Defining the second difference as Δ²f(x)≡f(c+2 h)−2f(c+h)+f(c), we next evaluate the resulting second difference for the single period problem in order to determine if an optimal production decision exists.

Thus, Proposition 1 is proven for the single period problem.

The multi-period problem extends the single period total expected cost function by the addition of positive expected cost. It immediately follows that Proposition 1 holds for the n period problem and thus the second difference for the multi-period total expected cost function v has a positive second difference.