A multi-period bi-level model for a competitive food supply chain with sustainability considerations

The use of food additives and greenhouse gas (GHG) emissions are major factors that affect consumer health, which cannot be ensured, therefore, unless the food supply chain is controlled. This paper presents a bi-level optimization approach to address competition between the members of a food supply chain, including two suppliers and one manufacturer. The manufacturer seeks to determine prices, production, and investment in reducing GHG emissions to maximize profits at the upper level. A multi-period model is presented, in which the manufacturer seeks to reduce production costs by maintaining inventory and, of course, accepting the risk of waste. At the lower level, the suppliers compete to determine the price of raw materials and the amount of additives to maximize their profits. Two linear approximation methods are proposed to linearize the nonlinear model. According to the results, the lowest relative gaps obtained from solving the model for two numerical examples using the second linear approximation method are lower than those obtained by the first method. A sensitivity analysis of the effective parameters demonstrates that an increase in the dependence of demand on price leads to a 30% decrease in the profit of the entire supply chain. The manufacturer faces an average 3% drop in demand with an increase of 4%-7% in the rate of production waste. An 18% increase in the cost of GHG emission reduction leads to a 5% decrease in total supply chain profit. Hence, the increase in the cost of reducing GHG emissions is not economical and environment-friendly for the supply chain members. A budget increase is appropriate in terms of the economic and environmental goals, but its impact on the manufacturer's decision variables varies by period.