## 2 Mathematical Formulation
### Theoretical Background  
To correctly formulate the objectives of the problem, a series of theorems supported by the existing literature can be used. Out of the three objectives, modelling the energy consumption is the most complex component. Each type of storage location has a different model for the energy consumption, as it varies in magnitude and dynamics. The energy consumed in the cold storage is the more nuanced component. Theorem 1 and its corresponding proof define the energy consumption dynamics of a refrigerator unit, with changing degrees of fullness and hysteresis bands. Theorem 1: _The energy consumption of a refrigerator varies depending on its degree of fullness and the allowed hysteresis band._  
Proof of Theorem 1.: According to [PERSON] [34] and [PERSON] [35], a refrigerator filled with products has a higher specific heat capacity than a empty one. The authors present experimental data showing that an empty refrigerator increases its energy consumption with the increase in the hysteresis band, while one at 40% useful capacity reduces its energy consumption with the increase in the hysteresis band. Consider Theorems 2 and 3. These theorems regard the energy consumption in the warehouse. Theorem 2.2: _The total energy consumption of a warehouse is calculated as the sum of the energy consumptions for the transportation of equipment, building heating and cooling, ventilation, lighting, IT networks, and other energy consumptions._  
Proof of Theorem 2.2.: According to the practical case study by [PERSON] et al. (2018), the total energy consumption is the combination of the stated components, regardless of the warehouse technology and level of automation. Theorem 3.3: _When subjected to heating at the same power output, an empty room increases its temperature faster than a furnished one but also cools down faster._  
Proof of Theorem 3.3.: The emptier a warehouse is, the larger the ratio of air in it; the fuller a warehouse is, the more solid materials there are, and the ratio of air to contents decreases. The mechanisms of heat transfer to the air and to solid objects are different, and their capacity to retain and emit heat also differs. The thermal conductivity of air is smaller than most solid and liquid materials, meaning that it has a lower capacity to exchange heat from and to the environment. According to the experimental work by [PERSON] et al. (2018), an empty room subjected to the same heating power of a furnished one increases the temperature faster than a furnished one. The authors performed a test on an empty and furnished room, where the room was heated for 4 h (starting at 21 \({}^{\circ}\)C), and then allowed to cool for 4 more hours. The empty room increased to 23.3 \({}^{\circ}\)C and then reduced to 20.4 \({}^{\circ}\)C; the furnished room increased to 22.7 \({}^{\circ}\)C and then reduced to 20.7 \({}^{\circ}\)C. This experiment proves Theorem 3. Finally, a theorem regarding how the holding costs are modelled according to the storage degree of fullness can be seen in Theorem 4. Theorem 4.4: _The holding costs of a warehouse are proportional to its degree of fullness._  
Proof of Theorem 4.5.: As stated by [PERSON] et al. (2018), _\"holding costs are continuously incurred at a rate proportional to the storage level\"_. To mathematically formulate each of the objectives, additional notation must be introduced and defined. Table 2 presents the variables required for this problem.