3. Mathematical Modelling
3.1. Mathematical Model
Index:
i =age of vehicle with i ∈ ={0,1,2,3,…I}
j = year, with j ∈ ={0,1,2,3,…,J}
k = type of vehicles, with k ∈ ={1,2} ; k=1 is diesel truck, and k=2 is an electric truck Decision variables:
𝐵𝐵𝑖𝑖𝑖𝑖𝑖𝑖 = numbers of vehicle type-k, with age-i, that are salvaged at the end of year j
𝑃𝑃𝑖𝑖𝑖𝑖 = numbers of vehicle type-k, that are purchased at the beginning of year j 𝐴𝐴𝑖𝑖 = numbers of battery bought at the beginning of year-j
𝐾𝐾𝑖𝑖𝑖𝑖 = numbers of battery salvaged at the end of year-j, with age-i 𝐵𝐵𝑖𝑖 = numbers of battery swapping station needed at year-j 𝑉𝑉𝑖𝑖𝑖𝑖𝑖𝑖 = numbers of vehicle type-k, that are used at year j, with age-i 𝐷𝐷𝑖𝑖𝑖𝑖 = numbers of battery owned at age-i at year-j
𝑁𝑁𝑖𝑖 = numbers of diesel trucks needed to deliver all of demand at year-j without using electric truck
22 Variables:
𝐷𝐷𝑖𝑖𝑖𝑖𝑖𝑖 = total annual distance travelled for all of vehicle type-k, with age-i, at year-j 𝑀𝑀𝑖𝑖𝑖𝑖 = maintenance cost per mile for vehicle type-k with age-i
𝑜𝑜𝑖𝑖𝑖𝑖 = salvage value of vehicle type-k, with age-i 𝐷𝐷𝑖𝑖 = minimum battery needed in the system at year j 𝑓𝑓𝑖𝑖 = frequency to visit BSS at year j
𝐶𝐶𝐶𝐶 = cycle time or total waiting time to get a fully charged battery at BSS that start from electric truck enter BSS until get battery swapped done
𝐶𝐶𝐶𝐶𝐹𝐹 = waiting time in queue until get swapped battery process 𝑤𝑤𝐷𝐷 = arrival rate of electric truck to BSS
𝑤𝑤𝐷𝐷 = rate of effective battery swapping processing time These variables calculation is explained in the previous chapter.
Parameters:
𝐶𝐶𝑖𝑖𝑖𝑖 = purchase cost for vehicle type-k at year-j 𝑑𝑑 = salvage value of battery
𝑂𝑂𝑖𝑖 = fuel price at year j ($/km) 𝐷𝐷𝑖𝑖 = electricity price ($/kWh)
𝑑𝑑 = power needed from electricity to move electric vehicle or electricity rate (kWh/km) r = yearly discounted interest rate
𝐸𝐸 = levelized annual investment cost for switching station 𝑞𝑞𝑖𝑖 = maintenance cost for battery switching station at year j 𝑤𝑤𝑖𝑖𝑖𝑖 = emission rate of vehicle type-k at year j(kg/km)
𝐷𝐷𝑖𝑖 = emission cost at year j ($/kg)
𝑀𝑀 = waiting time cost at battery switching station to get a fully charged battery 𝑀𝑀𝑖𝑖 = battery cost at year-j
𝑃𝑃𝑖𝑖 = allocated budget or cost at year j
𝑅𝑅𝑖𝑖 = total range of travelled with fully charged battery at year j
𝑎𝑎𝑖𝑖𝑖𝑖 = initial vehicle (at the beginning of year j=0) with age-i that the company already has and are not part of budget allocated to buy new vehicle
𝑊𝑊𝑖𝑖 = payload or weight capacity of vehicle type k 𝑤𝑤𝑖𝑖 = average weight of customer demand at year j 𝐺𝐺𝑖𝑖 = annually reduction of emission target at year j (%)
𝐺𝐺 = total emission reduction at the end of planning horizon (%) 𝑜𝑜 = station utilization
𝐷𝐷 = effective swapping time process needed to swap used battery with a fully charged battery
𝐷𝐷𝐷𝐷 = coefficient of variation for arrival of vehicle
𝐷𝐷𝐷𝐷 = coefficient of variation for processing time or swapping service time 𝛼𝛼 = service level
Ɵ𝑖𝑖 = percent depreciation rate for vehicle type-k 𝜂𝜂 = per-km depreciation rate for truck
23
24
𝐵𝐵𝑖𝑖−1≤ 𝐵𝐵𝑖𝑖 ∀𝑗𝑗 (30)
𝐷𝐷𝑖𝑖= �∑𝐼𝐼 𝑉𝑉𝑖𝑖𝑖𝑖2
𝑖𝑖−0 ∀𝑗𝑗 if 𝐵𝐵𝑖𝑖= 0
∑𝐼𝐼𝑖𝑖−0𝑉𝑉𝑖𝑖𝑖𝑖2+ ∑𝐼𝐼𝑖𝑖−0𝛼𝛼. 𝑉𝑉𝑖𝑖𝑖𝑖2+ ∑ 0.83𝑉𝑉𝐼𝐼𝑖𝑖 𝑖𝑖𝑖𝑖2 ∀𝑗𝑗 if 𝐵𝐵𝑖𝑖 > 0 (31)
�𝐷𝐷𝑖𝑖− 𝐷𝐷𝑖𝑖−1�+≤ 𝐴𝐴𝑖𝑖 ∀𝑗𝑗 (32)
𝐼𝐼𝑖𝑖𝑖𝑖 = �𝐴𝐴𝑖𝑖 ∀𝑗𝑗 𝑓𝑓𝑤𝑤𝑤𝑤 𝐷𝐷 = 0
𝐷𝐷(𝑖𝑖−1)(𝑖𝑖−1)− 𝐾𝐾(𝑖𝑖−1)(𝑖𝑖−1) ∀𝑗𝑗 𝑓𝑓𝑤𝑤𝑤𝑤 𝐷𝐷 > 0 (33)
∑𝐼𝐼−1𝑖𝑖=0𝐷𝐷𝑖𝑖𝑖𝑖 ≥ 𝐷𝐷𝑖𝑖 ∀𝑗𝑗 (34)
∑𝐼𝐼−1𝑖𝑖=8𝐷𝐷𝑖𝑖𝑖𝑖 = 0 ∀𝑗𝑗 (35)
𝐾𝐾𝑖𝑖𝑖𝑖 ≤ 𝐼𝐼𝑖𝑖𝑖𝑖 ∀𝐷𝐷, 𝑗𝑗 (36)
𝐴𝐴0𝑖𝑖 = 0 ∀𝑗𝑗 (37)
𝐶𝐶𝐶𝐶 = �0.11 𝑓𝑓𝑤𝑤𝑤𝑤 𝑅𝑅𝑖𝑖 < 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝐷𝐷𝑤𝑤𝑤𝑤 ℎ𝑤𝑤𝑜𝑜𝑤𝑤𝑜𝑜 𝑀𝑀 𝑎𝑎𝐷𝐷ℎ𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑜𝑜𝑠𝑠𝐷𝐷𝐷𝐷𝑑𝑑
0 𝑓𝑓𝑤𝑤𝑤𝑤 𝑅𝑅𝑖𝑖 ≥ 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝐷𝐷𝑤𝑤𝑤𝑤 ℎ𝑤𝑤𝑜𝑜𝑤𝑤𝑜𝑜 𝑀𝑀 𝑎𝑎𝐷𝐷ℎ𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑜𝑜𝑠𝑠𝐷𝐷𝐷𝐷𝑑𝑑 (38)
∑𝐼𝐼𝑖𝑖=0∑𝐾𝐾𝑖𝑖=1𝐷𝐷𝑖𝑖𝑖𝑖𝑖𝑖𝑉𝑉𝑖𝑖𝑖𝑖𝑖𝑖𝑤𝑤𝑖𝑖𝑖𝑖≤ �1 − 𝐺𝐺𝑖𝑖�. 𝑁𝑁𝑖𝑖. 𝑤𝑤𝑖𝑖1. (𝐷𝐷𝑎𝑎𝐷𝐷𝑤𝑤𝐷𝐷𝑤𝑤𝐷𝐷 𝑜𝑜𝑠𝑠𝐷𝐷𝐷𝐷𝑑𝑑. 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝐷𝐷𝑤𝑤𝑤𝑤 ℎ𝑤𝑤𝑜𝑜𝑤𝑤𝑜𝑜 −
𝐷𝐷𝑎𝑎𝐷𝐷𝑤𝑤𝐷𝐷𝑤𝑤𝐷𝐷 𝑜𝑜𝑠𝑠𝐷𝐷𝐷𝐷𝑑𝑑 𝑀𝑀 𝐷𝐷𝐷𝐷𝐷𝐷𝑤𝑤𝑤𝑤𝐷𝐷𝑃𝑃𝐷𝐷𝐷𝐷 𝐷𝐷𝐷𝐷𝑀𝑀𝐷𝐷 𝐷𝐷𝑤𝑤 𝑤𝑤𝑤𝑤 𝑤𝑤𝑤𝑤𝑜𝑜𝑤𝑤𝑑𝑑 𝐷𝐷𝑤𝑤𝐷𝐷𝑠𝑠 𝐷𝐷𝑤𝑤 𝐵𝐵𝐵𝐵𝐵𝐵). 260𝑑𝑑𝐷𝐷𝐷𝐷𝑜𝑜 ∀𝑗𝑗 (39a)
∑𝐼𝐼𝑖𝑖=0∑𝑇𝑇𝑖𝑖=0∑𝐾𝐾𝑖𝑖=1𝐷𝐷𝑖𝑖𝑖𝑖𝑖𝑖𝑉𝑉𝑖𝑖𝑖𝑖𝑖𝑖𝑤𝑤𝑖𝑖𝑖𝑖 ≤ (1 − 𝐺𝐺). 𝐷𝐷𝑤𝑤𝐷𝐷𝐷𝐷𝐷𝐷 𝐷𝐷𝑀𝑀𝐷𝐷𝑜𝑜𝑜𝑜𝐷𝐷𝑤𝑤𝑤𝑤 𝑤𝑤𝑓𝑓 16 𝐷𝐷𝐷𝐷𝐷𝐷𝑤𝑤𝑜𝑜 𝑃𝑃𝐷𝐷𝑓𝑓𝑤𝑤𝑤𝑤𝐷𝐷 𝑠𝑠𝐷𝐷𝐷𝐷𝑤𝑤𝑤𝑤𝐷𝐷𝑤𝑤𝑤𝑤 𝐷𝐷𝐷𝐷𝑀𝑀𝐷𝐷 (39b)
∑𝐼𝐼𝑖𝑖=0∑𝐾𝐾𝑖𝑖=1𝐷𝐷𝑖𝑖𝑖𝑖𝑖𝑖𝑉𝑉𝑖𝑖𝑖𝑖𝑖𝑖𝑤𝑤𝑖𝑖𝑖𝑖≤ 𝐷𝐷𝐷𝐷𝑤𝑤𝑤𝑤𝐷𝐷𝐷𝐷 𝐷𝐷𝑀𝑀𝐷𝐷𝑜𝑜𝑜𝑜𝐷𝐷𝑤𝑤𝑤𝑤𝑖𝑖 ∀𝑗𝑗 (39c)
With:
𝐷𝐷𝐷𝐷𝑤𝑤𝑤𝑤𝐷𝐷𝐷𝐷 𝐷𝐷𝑀𝑀𝐷𝐷𝑜𝑜𝑜𝑜𝐷𝐷𝑤𝑤𝑤𝑤𝑖𝑖 = ��1 − 𝐺𝐺𝑖𝑖�. 𝐷𝐷𝑤𝑤𝑤𝑤𝑜𝑜𝐷𝐷𝐷𝐷 𝐷𝐷𝑀𝑀𝐷𝐷𝑜𝑜𝑜𝑜𝐷𝐷𝑤𝑤𝑤𝑤 𝑃𝑃𝐷𝐷𝑓𝑓𝑤𝑤𝑤𝑤𝐷𝐷 𝑠𝑠𝐷𝐷𝐷𝐷𝑤𝑤𝑤𝑤𝐷𝐷𝑤𝑤𝑤𝑤ℎ𝑤𝑤𝑤𝑤𝐷𝐷𝑜𝑜𝑤𝑤𝑤𝑤 𝑓𝑓𝑤𝑤𝑤𝑤 𝑗𝑗 = 0
�1 − 𝐺𝐺𝑖𝑖�. 𝐷𝐷𝐷𝐷𝑤𝑤𝑤𝑤𝐷𝐷𝐷𝐷 𝐷𝐷𝑀𝑀𝐷𝐷𝑜𝑜𝑜𝑜𝐷𝐷𝑤𝑤𝑤𝑤𝑖𝑖 𝑓𝑓𝑤𝑤𝑤𝑤 𝑗𝑗 > 0
∑𝐼𝐼𝑖𝑖=0∑𝐾𝐾𝑖𝑖=1𝐷𝐷𝑖𝑖15𝑖𝑖𝑉𝑉𝑖𝑖15𝑖𝑖𝑤𝑤15𝑖𝑖 ≤ (1 − 𝐺𝐺15). 𝑁𝑁15. 𝑤𝑤151. (𝐷𝐷𝑎𝑎𝐷𝐷𝑤𝑤𝐷𝐷𝑤𝑤𝐷𝐷 𝑜𝑜𝑠𝑠𝐷𝐷𝐷𝐷𝑑𝑑. 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝐷𝐷𝑤𝑤𝑤𝑤 ℎ𝑤𝑤𝑜𝑜𝑤𝑤𝑜𝑜 −
𝐷𝐷𝑎𝑎𝐷𝐷𝑤𝑤𝐷𝐷𝑤𝑤𝐷𝐷 𝑜𝑜𝑠𝑠𝐷𝐷𝐷𝐷𝑑𝑑 𝑀𝑀 𝐷𝐷𝐷𝐷𝐷𝐷𝑤𝑤𝑤𝑤𝐷𝐷𝑃𝑃𝐷𝐷𝐷𝐷 𝐷𝐷𝐷𝐷𝑀𝑀𝐷𝐷 𝐷𝐷𝑤𝑤 𝑤𝑤𝑤𝑤 𝑤𝑤𝑤𝑤𝑜𝑜𝑤𝑤𝑑𝑑 𝐷𝐷𝑤𝑤𝐷𝐷𝑠𝑠 𝐷𝐷𝑤𝑤 𝐵𝐵𝐵𝐵𝐵𝐵). 260𝑑𝑑𝐷𝐷𝐷𝐷𝑜𝑜 (39d)
∑𝐼𝐼𝑖𝑖=0∑𝐾𝐾𝑖𝑖=1𝐷𝐷𝑖𝑖𝑖𝑖𝑖𝑖𝑉𝑉𝑖𝑖𝑖𝑖𝑖𝑖𝑤𝑤𝑖𝑖𝑖𝑖≤ 𝑀𝑀𝐷𝐷𝑀𝑀𝐷𝐷𝑀𝑀𝑜𝑜𝑀𝑀 𝐴𝐴𝐷𝐷𝐷𝐷𝑤𝑤𝑤𝑤𝐷𝐷𝑃𝑃𝐷𝐷𝐷𝐷 𝐸𝐸𝑀𝑀𝐷𝐷𝑜𝑜𝑜𝑜𝐷𝐷𝑤𝑤𝑤𝑤𝑖𝑖 ∀𝑗𝑗 (39e)
∑𝐼𝐼𝑖𝑖=0∑𝑇𝑇𝑖𝑖=1∑𝐾𝐾𝑖𝑖=1𝐷𝐷𝑖𝑖𝑖𝑖𝑖𝑖𝑉𝑉𝑖𝑖𝑖𝑖𝑖𝑖𝑤𝑤𝑖𝑖𝑖𝑖 ≤ 𝑀𝑀𝐷𝐷𝑀𝑀𝐷𝐷𝑀𝑀𝑜𝑜𝑀𝑀 𝐶𝐶𝑤𝑤𝐷𝐷𝐷𝐷𝐷𝐷 𝐴𝐴𝐷𝐷𝐷𝐷𝑤𝑤𝑤𝑤𝐷𝐷𝑃𝑃𝐷𝐷𝐷𝐷 𝐸𝐸𝑀𝑀𝐷𝐷𝑜𝑜𝑜𝑜𝐷𝐷𝑤𝑤𝑤𝑤 𝑓𝑓𝑤𝑤𝑤𝑤 16 𝐷𝐷𝐷𝐷𝐷𝐷𝑤𝑤𝑜𝑜 (39f)
∑𝐼𝐼𝑖𝑖=0∑𝐾𝐾𝑖𝑖=1𝐷𝐷𝑖𝑖𝑖𝑖𝑖𝑖𝑉𝑉𝑖𝑖𝑖𝑖𝑖𝑖𝑤𝑤𝑖𝑖𝑖𝑖≤ 𝑀𝑀𝐷𝐷𝑀𝑀𝐷𝐷𝑀𝑀𝑜𝑜𝑀𝑀 𝑠𝑠𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝐷𝐷𝑜𝑜𝑜𝑜𝐷𝐷𝑎𝑎𝐷𝐷 𝐷𝐷𝐷𝐷𝑤𝑤𝑤𝑤𝐷𝐷𝐷𝐷 𝐷𝐷𝑀𝑀𝐷𝐷𝑜𝑜𝑜𝑜𝐷𝐷𝑤𝑤𝑤𝑤𝑖𝑖 ∀𝑗𝑗 (39g)
∑𝐼𝐼𝑖𝑖=0∑𝐾𝐾𝑖𝑖=1𝐷𝐷𝑖𝑖15𝑖𝑖𝑉𝑉𝑖𝑖15𝑖𝑖𝑤𝑤15𝑖𝑖 ≤ 𝑀𝑀𝐷𝐷𝑀𝑀𝐷𝐷𝑀𝑀𝑜𝑜𝑀𝑀 𝐴𝐴𝐷𝐷𝐷𝐷𝑤𝑤𝑤𝑤𝐷𝐷𝑃𝑃𝐷𝐷𝐷𝐷 𝐸𝐸𝑀𝑀𝐷𝐷𝑜𝑜𝑜𝑜𝐷𝐷𝑤𝑤𝑤𝑤15 (39h)
𝑤𝑤𝑗𝑗
𝑊𝑊1≤ 𝑁𝑁𝑖𝑖 ≤(𝑤𝑤𝑗𝑗𝑊𝑊+𝑊𝑊1)
1 ∀𝑗𝑗 (40)
𝐵𝐵𝑖𝑖𝑖𝑖𝑖𝑖, 𝑃𝑃𝑖𝑖𝑖𝑖, 𝐴𝐴𝑖𝑖, 𝐾𝐾𝑖𝑖𝑖𝑖, 𝐵𝐵𝑖𝑖, 𝑉𝑉𝑖𝑖𝑖𝑖𝑖𝑖, 𝐷𝐷𝑖𝑖𝑖𝑖, 𝑁𝑁𝑖𝑖 ∈ 𝐷𝐷𝑤𝑤𝐷𝐷𝐷𝐷𝑤𝑤𝐷𝐷𝑤𝑤 (41)
25
The equation (23) describes the objective function for this vehicle replacement model. The objective of this model is to find the replacement schedule for a diesel truck to electric truck, the time when to purchase and salvage battery includes its battery quantity, the time to build BSS and the numbers of BSS needed. The objective function for this model is minimizing total cost of ownership of the vehicle, battery switching stations, and electric battery. The cost that included in total cost of ownership are purchasing cost of vehicle, salvage value of vehicle, maintenance and energy cost of vehicle, emission cost, battery switching station investment cost, battery switching station maintenance cost, purchasing cost of battery, salvage value of battery, maintenance cost of battery and waiting cost for electric truck to get a fully charged battery at BSS.
Equation (24) explains that purchase cost for the vehicle, investment cost to build BSS and purchasing cost for the battery should not exceed yearly budget. Equation (25) explains the value of an electric vehicle owned. The first part of equation describes that newly purchased vehicle will be operated and is considered to have an age of 0.The second part of the equation (25) describes that the number of the vehicle used this year are consist of the number of the vehicle used last year plus the new vehicle purchased this year minus the vehicle salvaged this year. The last part of equation explains that the vehicles that company has before the planning period can be used as distribution vehicle
Equation (26) explains that newly purchased vehicles should not be salvaged immediately before they are used. Equation (27) explains that the numbers of salvaged vehicle should not be more than the number of vehicles owned in the same year. Equation (28) explains that the average weight of daily demand each year that needs to be loaded by total vehicles should not exceed the maximum payload of vehicles.
Equation (29) and (30) describe BSS in the system. The first part of the equation (29) explains that the system will need to build battery switching stations if the electric truck has range distance less than the potential distance truck needs to travel each day. The numbers of battery switching station in the system depend on the electric vehicle owned and BSS capacity. The system needs to ensure that each battery switching station does not serve electric vehicles more than its capacity. For this case, the capacity of a battery switching station is 75 electric vehicle. The second part of equation (29) explains that the number of BSS will also depend on the fraction of demand electric vehicle needs to serve. The equation holds the assumption that all of the demand distributed equally around Netherlands and the BSS, as well as electric truck, will only be operated in the fraction area based on the fraction of demand that electric vehicle distributed. This equation is needed to make sure that electric trucks have enough BSS spread within Netherlands and can reach BSS within allowable time, which is 15 minutes For this case, to ensure this allowable time can be reached, the potential number of BSS in Netherlands needs to be 267. The last part of equation (29) explains that there will be no BSS needed if the electric truck has range distance high enough to deliver demand in one day. Equation (30) explains that the BSS has been built should not be closed
Equation (31) until equation (37) explains about battery constraint in the system. The first part of equation (31) explains that the minimum number of battery needed by the system depends on the number of electric vehicles owned and battery switching station. The first part of the equation describes that if the system does not need battery swapping station, the minimum battery which system needs will be same with the number of electric trucks. The second part of the equation
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(31) describes the minimum number of battery needed if the system needs battery switching station. This second part is derived based on the explanation of minimum battery needed in the system in the previous chapter, which is shown in equation (15). The equation (51) has non-linear function of 𝑉𝑉𝑖𝑖𝑖𝑖2 and 𝐵𝐵𝑖𝑖. This non-linear function needs to be linearized and constraint at equation (31) shows the linearized function of equation (15). This linearization process to make equation (15) becomes constraints at equation (31) is explained at the second part of this chapter that discuss about linearization at sub-chapter 3.2.
Equation (32) explains the purchased battery at year-j should be at least have same numbers of additional battery needed. Equation (33) explains the number of battery in the system. The first part of the equation describes that the newly purchased battery will be operated and is considered to have the age of 0. The second part of the equation describes that the battery owned will be equal to the number of battery owned minus the salvaged battery at the previous year.
Equation (34) describes the battery owned in the system should be at least have same numbers with minimum battery needed. Equation (35) explains that battery should only have a maximum age of seven with assumption battery life time is seven years. Equation (36) explains that a number of salvaged battery should not be more than the numbers of battery owned in the same year. Equation (37) explains that newly purchased battery should not be salvaged
Equation (38) describes the value of waiting time of each electric vehicle in the system to get a fully charged battery at battery switching station. The first part of the equation is the approximation of waiting time for each electric truck if the electric truck needs to visit BSS due to limited driving distance. The second part of the equation describes the waiting time when the electric truck does not need to visit BSS. This constrain derived based on the equation of waiting time that is explained in the previous chapter at equation (9). The equation (9) has non-linear function of 𝑉𝑉𝑖𝑖𝑖𝑖2 if 𝐶𝐶𝐶𝐶 at equation (9) inputed in the objective function. This non-linear function needs to be linearized and constraint at equation (38) is a linear form of it. This linearization process is explained further at the second part of this chapter at sub-chapter 3.2 that discuss about linearization.
Equation (39) contains six parts that can use to explain the emission reduction target to control EV’s adoption. Those six equations do not need to be used together but can be picked one based on the preference to achieve emission reduction based on the percentage emission reduction(%) or based on the numerical emission reduction (kg CO2eq). The equation (39a) explains that the real emission that generated each year by the company to deliver demand by using either electric truck, diesel truck, or electric truck and diesel truck combination should be less than the value of emission allowed to be generated if the company use all of the diesel trucks to deliver demand.
The equation (39b) describes that the total emission generated during planning horizon should be less than allowable emission based on the emission reduction target and total emission generated for 16 years before the planning horizon begins. The equation (39c) explains that the emission generated each year by the company to deliver demand should be less than maximum allowable emission target based on the previous maximum emission target and progressive emission reduction target. The equation (39d) describes that real emission at the end year of planning horizon should be less than the value of emission allowed to be generated if the company use all of the diesel trucks to deliver demand at that year.The equation (39e), (39f), (39g), and (39h) have the same concept with equation (39a),(39b), (39c) and (39d) but instead describe the emission reduction target by percentage, the emission reduction is explained by the numerical
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maximum allowable emission (kg). The second part explains that the emission reduction by using electric vehicle should be not less than numerical emission reduction target (kg). The equation (39h) is similar to the emission reduction target strategy that is used in the Netherlands.
Equation (40) explains the number of diesel needed at year-j to deliver all of the demands without using electric truck should be more than daily demand at year-j devided by diesel truck capacity and should be less than the value of daily demand at year-j added by diesel vehicle capacity and divided by diesel truck capacity. Equation (41) describes that the decision variables are integer numbers.