Lot-Size Planning with Non-Linear Cost Functions Supporting Environmental Sustainability

2. Models Considering Eco-Factors

Ecological impacting factors – eco-factors – both contribute to global environmental friendliness and ecology and can be quantified and charged with costs in order to be included into mathematical models for production planning. Ecological friendliness means in this case that the carbon footprint (Velte et al., 2008) can be reduced or rather the environmental impact can be improved with the optimization of the eco-factors. Quantification means that the eco-factors can be assigned a monetary value based on the lot-size that is allocated to a specific production period.

The following eco-factors are identified as relevant within production planning:

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  Power Usage: Depends on production’s degree of utilization and has to be purchased. Power generation affects the environment based on the respective power plants.

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  Carbon Dioxide Emission: Allows cap and trade on emission markets. Therefore carbon dioxide reduction implies financial opportunities.

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  Water Consumption: Depends on production’s degree of utilization and has to be purchased. Water has the potential to catch up or even replace fossil fuels as core resource in the future.

It is important to note that these cost factors behave in a non-linear way. This is based on the assumption that with an increase in the degree of utilization of a production machine the related eco-factors increase above-average (e.g. power consumption, carbon dioxide emission, etc.).

The existing models and approaches for solving LSP are enhanced with eco-factors. The Wagner-Whitin algorithm (WWA) and the Part-Period Balancing heuristic (PPB) (Tempelmeier, 2005) have been selected for generating lot-sizing schedules. The three identified eco-factors power usage, carbon dioxide emission, and water consumption are dependent on the lot size q of a production period t and affect the total costs of a planning horizon T. The term e(q_t) is introduced in order to enhance the LSP models with an environmental impacting cost factor. The eco-term e(q_t) determines besides inventory costs, setup costs and variable production costs the total costs of a LSP. The term e(q_t) is illustrated in the following: