2.2.1. Cost Function Composition

#### (1) Vehicle Fixed Cost

The fixed cost of a vehicle is the cost of purchase, maintenance, labor paid to the driver, etc. It only depends on the number of vehicles performing the task. The function was as follows:

$$Z\_1 = \mathbb{C}\_0 \sum\_{k=1}^m \sum\_{i=1}^u X\_{0i}^k \tag{10}$$

### (2) Transportation Cost

In general, the shorter the distance traveled, the lower the cost. Thus, the transportation cost of the vehicle is proportional to the distance traveled. *Cp* represents the transportation cost within a unit distance. The total transportation cost *Z*<sup>2</sup> could be expressed as:

$$Z\_2 = \sum\_{k=1}^{m} \sum\_{i=0}^{n} \sum\_{j=0}^{n} d\_{ij} \mathbb{C}\_p X\_{ij}^k \tag{11}$$

#### (3) Loss Cost

Both the transit period and the load of goods incurred a loss cost. The continuous lifetime function decayed according to an exponential rate. The decay function of fresh products was: *<sup>E</sup>*(*t*) = *<sup>E</sup>*<sup>0</sup> · *<sup>e</sup>*−*θ<sup>t</sup>* [40]. This formula expresses the quality of the product at different times, where *E*<sup>0</sup> is the product's initial quality and *θ* is the sensitivity factor of the product to time. The loss cost of transit without opening the door was as follows:

$$\mathcal{K}\_{\mathcal{S}} = \sum\_{i=0}^{n} \sum\_{k=1}^{m} Y\_{i\mathbf{k}} q\_i [1 - K\_1 e^{-\theta(t\_i - t\_0)}] \mathcal{R}\_1 \tag{12}$$

Temperature changes caused by opening the door during loading could also affect the freshness of the product. The door opening time was determined using the customer service time *si*. The loss cost caused by loading goods while opening the door was:

$$\mathcal{C}\_{I} = \sum\_{i=0}^{\mathcal{U}} \sum\_{k=1}^{m} Y\_{ik} Q\_{i0} (1 - K\_{2} \mathcal{e}^{-\theta S\_{i}}) R\_{1} \tag{13}$$

We could obtain the total loss cost in the whole distribution process as follows:

$$Z\_3 = \sum\_{i=0}^{u} \sum\_{k=1}^{m} Y\_{ik} R\_1[q\_i(1 - K\_1 e^{-\theta(t\_i - t\_0)}) + Q\_{i0}(1 - K\_2 e^{-\theta S\_i})] \tag{14}$$

#### (4) Penalty Cost

In order to maximize the freshness of our products, we needed to arrive within the time frames set by our customers individually. Otherwise, there would be penalty costs. Suppose that the time window required by the customer was (*ETi*, *LTi*). We could use *ti* to represent the time when the delivery vehicle arrives at customer *i*. If *ETi* ≤ *ti* ≤ *LTi*, no additional penalty cost would be incurred. If the goods were not delivered within the time window, the penalty cost *Z*<sup>4</sup> would be paid. Arriving earlier than the time requested would incur a waiting cost, and arriving later would pay a delayed service fee. The relation was stated as follows:

$$Z\_4 = \gamma\_1 \sum\_{i=1}^{u} \max\{ET\_i - t\_{i\prime}0\} + \gamma\_2 \sum\_{i=1}^{u} \max\{t\_i - LT\_i, 0\} \tag{15}$$
