Minimize bus fleet cost

Model Formulation

Indexes

Age of a bus in years: i \in \mathbb{Z}, i = \{0,1,2,...,M\}

Time periods: j \in \mathbb{Z}, j = \{0,1,2,...,N\}

Decision Variables

$X_{ij}$ = the number of $i$-year old buses used in year $j$,

$Y_{ij}$ = the number of $i$-year old buses salvaged at the end of year $j$,

$P_j$ = number of buses purchased at the beginning of year $j$.

Parameters

$u_i$ = utilization (annual miles traveled by an $i$-year old bus),

$d_j$ = demand (miles traveled by all buses) in year $j$,

$b_j$ = budget (available for purchasing new buses) constraint in year $j$,

$v$ = purchase cost of a bus,

$f_i$ = fuel economy (mpg) for an $i$-year old bus,

$g_j$ = fuel price($/gallon) in year fuel price($/gallon) in year $j$,

$m_i$ = per-mile operation and maintenance costs for an $i$-year old bus,

$s_i$ = salvage revenue from selling an $i$-year old bus,

$e$ = utilization emissions in GHG

$r$ = discount rate

Initial conditions

$h_i$ = the number of $i$-year old buses available at the beginning.

Objective Function

\sum_{j=0}^{99}v_jP_j(1 + r)^{-j}

+\sum_{i=0}^{19}\sum_{j=0}^{99}(\frac{g_ju_i}{f_i})X_{ij}(1+r)^{-j}

+\sum_{i=0}^{19}\sum_{j=0}^{99}m_iu_iX_{ij}(1+r)^{-j}

+\sum_{i=0}^{19}\sum_{j=0}^{99}t_iu_ieX_{ij}(1+r)^{-j}

-\sum_{i=0}^{19}\sum_{j=0}^{99}s_iY_{ij} (1+r)^{-j}

Constraints

1. The total purchase cost for a period ($j$) should not exceed the total budget for

purchasing buses in that period ($j$).

v_jP_j \leq b_j

\qquad \forall j \in \mathbb{Z}, j = \{0,1,2,...,N\}

2. The number of buses purchased in a year ($j$) equals new buses in that year ($j$),

except for the current time.

P_j = X_{0j}

\qquad \forall j \in \mathbb{Z}, j = \{1,2,...,N-1\}

3. The number of ($i$)-year old vehicles in the fleet ($X$) at any time ($j$) should not be

less than the minimum number of buses ($d$) needed to work ($u$).

\sum_{i=0}^{19}u_i \centerdot X_{ij} \geq d_j

\qquad \forall i \in \mathbb{Z}, i = \{0,1,2,...,M-1\}

4. The age ($i$) of any vehicle in use ($X$) increases by one year after each time period

($j$); the vehicle is either used or sold.

X_{(i-1)(j-1)} = X_{ij} + Y_{ij}

\qquad \forall i \in \mathbb{Z}, i = \{1,2,...,M\},\forall j \in \mathbb{Z}, j = \{1,2,...,N\}

5. The sum of new buses purchased ($P$) in a year ($j$) and existing ($i$)-year old buses

already in use ($X$) should equal the number of $i$-year old buses available at the beginning.

P_j + X_{ij} = h_i

6. When a bus reaches its maximum age ($M$), it is sold or salvaged.

X_{Mj} =0

\qquad \forall j \in \mathbb{Z}, j = \{0,1,2,...,N\}

7. A newly purchased bus should not be sold before it is used.

Y_{0j} = 0

\qquad\forall j \in \mathbb{Z}, j = \{0,1,2,...,N\}

8. The decision variables associated with purchasing, utilization, and salvaging decision

s must be integer numbers.

P_j, X_{ij},Y_{ij} \forall i \in \mathbb{Z}, i = \{0,1,2,...,M\},\forall j \in \mathbb{Z}, j = \{0,1,2,...,N\}