Integrated Optimization of Procurement, Processing, and Trade of Commodities

Abstract

We consider the integrated optimization problem of procurement, processing, and trade of commodities in a multiperiod setting. Motivated by the operations of a prominent commodity processing firm, we model a firm that procures an input commodity and has processing capacity to convert the input into a processed commodity. The processed commodity is sold using forward contracts, while the input itself can be traded at the end of the horizon. We solve this problem optimally and derive closed-form expressions for the marginal value of input and output inventory. We find that the optimal procurement and processing decisions are governed by price-dependent inventory thresholds. We use commodity markets data for the soybean complex to conduct numerical studies and find that approximating the joint price processes of multiple output commodities using a single, composite output product and using the approximate price process to determine procurement and processing decisions is near optimal. Compared to a myopic spread-option-based heuristic, the optimization-based dynamic programming policy provides significant benefits under conditions of tight processing capacities and high price volatilities. Finally, we propose an approximation procedure to compute heuristic policies and an upper bound to compare the heuristic against, when commodity prices follow multifactor processes.

Dynamic Risk Management of Commodity Operations: Model and Analysis

“…The results in Devalkar et al (2011) are a special case of the results presented here (notice that Υ (k) n in equation (15) reduces to the discounted expected marginal value of input inventory when markets are complete). We see from the proof of Theorem 2 that the optimal procurement and processing quantities are governed by price and horizon dependent thresholds and the optimal policy is as given in Corollary 1.…”

“…With the discrete prices, we use equations (10) and (13) to compute ∆ m n and Θ (k) n and thereby the procurement and processing policy at each node in the tree. The computational time required to calculate the optimal policy parameters (Table 3) are not significantly different from the times reported in Devalkar et al (2011) to compute the optimal risk-neutral policy.…”

“…As a result, the uncertainty can be perfectly hedged by trading in the financial markets and as Lemma 1 below states, the optimal commitment policy and the marginal value of output inventory are the same as those for a firm facing no risk costs (see Devalkar et al 2011). LEMMA 1.…”

“…Both these papers consider the optimal input procurement and inventory management policies for a firm facing stochastic demand and fluctuating commodity prices. Devalkar et al (2011) is closely related to our work but falls in the (NC) category, where the authors consider operations of a risk-neutral firm in the presence of uncertain commodity prices and procurement and processing constraints. The firm procures an input commodity, processes the input to produce output commodities that are then traded.…”

“…Substituting this, we get H n ( ⃗ X, ⃗ α n−1 , I m n ) = (⃗ α n−1 ) T ⃗ M n + CV aR (p) (η n ,n) (X n ) + βE π n Also notice that V N −1 is separable in Q N −1 and e N −1 and furthermore, linear in Q N −1 with the coefficient of Q N −1 dependent only on I m N −1 . Because the revenue from selling a unit of output depends only on market uncertainty, we can use induction arguments similar to those in Devalkar et al (2011) and show that V n (·) is separable in Q n and that the optimal commitment policy and marginal value of output inventory is as given in the Lemma.…”