Optimal trading and shipping of agricultural commodities

Abstract

We develop and implement a model for a profit maximizing firm that provides an intermediation service between commodity producers and commodity end-users. We are motivated by the grain intermediation business at Los Grobo—one of the largest commodity-trading firms in South America. Producers and end-users are distributed over a realistic spatial network, and trade with the firm through contracts for delivery of grain during the marketing season. The firm owns spatially distributed storage facilities, and begins the marketing season with a portfolio of prearranged purchase and sale contracts with upstream and downstream counterparts. The firm aims to maximize profits while satisfying all previous commitments, possibly through the execution of new transactions. Under realistic constraints for capacities, network structure and shipping costs, we identify the optimal trading, storing and shipping policy for the firm as the solution of a profit-maximizing optimization problem, encoded as a minimum cost flow problem in a time-expanded network that captures both geography and time. We perform extensive numerical examples and show significant efficiency gains derived from the joint planning of logistics and trading.

Appendix

We outline some additional details of the model that were not included in Section 3.

1. Quality of grain

We also consider a model with two levels of quality: dry and humid grain. We split the flow variables that go from origins to plants into two classes, using super-indices d and h, respectively.

• Dry flow from origin to plant (tonnes): x _op ^d(o, p, t).

• Humid flow from origin to plant (tonnes): x _op ^h(o, p, t).

To link back to the original variables, we use the constraint x _op ^d(o, p, t)+x _op ^h(o, p, t)=x _op (o, p, t). Other flow variables consist of dry grain exclusively since the only locations where humid grain can be transformed into dry one are in the plants, and humid grain cannot be delivered to destinations. We assume that each origin provides a given percentage of humid and dry grain, depending on the location and period, as follows.

• Percentage of humid grain (%): PercHum(o, t).

• Total dry supply from prearranged contracts (tonnes):

  

• Total humid supply from prearranged contracts (tonnes):

  

These constraints encode the balance of mass for grain of both qualities, dry and humid:

• Dry grain can be shipped to plants or to destinations: DryGrains(o, t)+x _purchFwd (o, t)=∑ _p x _op ^d(o, p, t)+∑ _d x _od (o, d, t). In agreement with market practices, forward contracts are written on dry grain.

• Humid grain must go to plants for processing: HumidGrains(o, t)=∑ _p x _op ^h(o, p, t). Notice that this grain cannot be stored in the origin.

• Aggregating the humid grain, we compute the total cost of drying (ARS) as ProcessingCost(t)=∑ _{o, p} UnitDryingCost(p)x _op ^h(o, p, t).

2. Bounds for new transactions

The model allows new transactions up to the minimal amount required to compensate imbalances per basin, with no restriction regarding the locations associated to those new contracts. We aggregate the transaction volume in each basin (purchases−sales) computing the net inventory for each period. When the net inventory is negative (purchases−sales<0)—meaning that in that particular basin and period new purchases are needed to fulfil prearranged sales—new transactions are allowed by an amount up to minus the net inventory.

• Total prearranged contracts for the basin corresponing to destination d (tonnes):

  

• Inventory in basin corresponding to destination d and period t>t ₁:

  

• Inventory in basin corresponding to destination d and period t=t ₁:

  

• Constraint that limits new contracts: ∑ _o x _purchFwd (o, t) +∑ _d x _purchSpot (d, t)⩽∑ _{d, t} Inventory(d, t)⁻, where Inventory(d, t)⁻=−min(Inventory(d, t), 0) is the negative part of the inventory.