Production planning under yield and demand uncertainty with yield-dependent cost and price

This paper studies production planning with random yield and demand. It is a departure from previous studies of random yield in that it defines the sale price and the purchasing cost as exogenous and increasing with decreasing yield. While this behavior can be observed in various industries (e.g., citrus), the paper focuses on the olive oil industry as its application. Production of olive oil is a challenging business as olives grow every other year; thus, a risky investment is involved. A new practice among olive oil producers involves leasing farm space from farmers to grow olives. When the yield of olives is low (because of weather, disease, etc.), the oil producer gets a second chance to buy olives from other farmers at a unit cost varying with the yield. In this case, the sale price of olive oil increases in the market place because of the reduced supply. When the yield is high, the company uses some of its olives for olive oil production and some are salvaged. After olives are pressed and olive oil is produced, the company experiences an uncertain demand. The paper makes four contributions: First, it is shown that the objective function is concave in the amount of farm space leased, so that the first-order conditions provide the globally optimal solution. Second, it illustrates how the total production of olive oil changes with the yield. Third, it proves that the optimal amount of farm space leased decreases under the presence of a second (and reliable) source of supply. Finally, unlike traditional yield papers, the fourth result shows that increased yield variance does not necessarily increase the optimal amount of farm space to be leased when there is a second chance to obtain supplies.