Efficient Insertion Heuristic Algorithms for Multi-Trip Inventory Routing Problem with Time Windows, Shift Time Limits and Variable Delivery Time

Abstract

Efficient insertion heuristic algorithms allowing multi trips per vehicle (EIH-MT) and allowing a single trip per vehicle with post-processing greedy heuristic (EIH-ST-GH) are proposed to solve the multi-trip inventory routing problem with time windows, shift time limit and variable delivery time (MTIRPTW-STL-VDT) with short planning horizon. The proposed algorithms are developed based on an original algorithm with two enhancements. First, the delivery volumes, the associated beginning delivery times and the exact profits are calculated and maintained. Second, the process to finalize a best-objective and feasible solution is developed. These algorithms are shown to have the complexity of O(n⁴). These heuristics maximize the profit function, which is the weighted summation of total delivery volume and negative total travel time. EIH-MT and EIH-ST-GH are performed on 280 instances based on Solomon’s test problems with three weight sets. Best-objective solutions are examined to illustrate the feasibility of various constraints. The trade-offs between total delivery volume and total travel time are observed when varying weight values. There is not a single winner heuristic based on the number-of-vehicles, profit and CPU criteria across the three customer configuration types. On average performance, EIH-ST-GH is preferred over EIH-MT for cluster configuration type with the following average improvement percentages: 1.03% for profit, 2.93% for number-of-vehicles and 38.68% for CPU. For random and random-cluster configuration types, EIH-ST-GH should be preferred because of better profit (0.27% for random and 0.22% for random-cluster) and CPU (46.96% for random and 44.06% for random-cluster) improvements. In the comparison of the multi-trip algorithms against the single-trip algorithm, the benefits in reducing the number of vehicles on-average are shown across all customer configuration types.

Appendices

Appendix 1 Function ${f}_i^{r_k^m}(t)$ in Campbell and Savelsbergh (2004a)

1.1 Additional Notations

MDVB_i(t)maximum delivery volume at customer i when the delivery at customer i begins at time t for t ∈ [E_i, L_i]

MDVE_i(t)maximum delivery volume at customer i when the delivery at customer i ends at time t for t ∈ [E_i + P_i ⋅ MDVB_i(E_i), L_i + P_i ⋅ MDVB_i(L_i)]

${\tilde{U}}_i$increase rate of maximum delivery volume at customer iwhen considering the completed delivery time

${f}_i^{r_k^m}(t)$total maximal delivery volume to all customers up to and including i when the delivery at i is completed at time t in route k of tour m

${\tilde{e}}_i^{r_k^m},{\tilde{l}}_i^{r_k^m}$earliest and latest beginning delivery time at customer i on route k of tour m used in the development of ${f}_i^{r_k^m}(t)$

${\tilde{t}}_{i,v}^{r_k^m}$beginning delivery time associated with the v^th inflection point in the function ${f}_i^{r_k^m}(t)$at customer i in route k of tour m

MDVB_i(t), MDVE_i(t) and ${\tilde{U}}_i$ are determined by eqs. (42)–(44). The determination of ${f}_i^{r_k^m}(t)$is explained by the small example in Fig. 3, which shows the route k of tour m. This route starts from the depot node 0, customer 1, customer 2 and depot node n + 1. The earliest beginning delivery times (E_i), latest beginning delivery times (L_i), delivery rates (P_i), fixed stop times (S_i), minimum delivery volume (D_i), demand consumption rates (U_i) are also shown in Fig. 3. Figure 4 portrays the functions MDVB_i(t) and MDVE_i(t) of the two customers. The slope of MDVB_i(t) is U_i, whereas the slope of MDVE_i(t) is ${\tilde{U}}_i$

Fig. 3

A small example

Fig. 4

MDVB_i(t) and MDVE_i(t) of customers 1 and 2 in the small example

.

$$ {MDVB}_i(t)={C}_i-{I}_i+{U}_i\cdot t $$

(42)

$$ {MDVE}_i(t)={MDVB}_i\left({E}_i\right)+{\tilde{U}}_i\cdot \left(t-\left({E}_i+{P}_i\cdot {MDVB}_i\left({E}_i\right)\right)\right) $$

(43)

$$ {\tilde{U}}_i=\frac{U_i\cdot \left({L}_i-{E}_i\right)}{L_i+{P}_i\cdot {MDVB}_i\left({L}_i\right)-\left({E}_i+{P}_i\cdot {MDVB}_i\left({E}_i\right)\right)} $$

(44)

The development of ${f}_i^{r_k^m}(t)$is performed starting from the first customer to the last customer in the route as follows. Suppose ${e}_1^{r_k^m}={E}_1=6$ and ${l}_2^{r_k^m}={L}_2=17$. Then, ${e}_2^{r_k^m}$ and ${l}_1^{r_k^m}$are calculated based on minimum delivery volume:

$$ {e}_2^{r_k^m}=\max \left\{{e}_1^{r_k^m}+{P}_1\cdot {D}_1+{T}_{12}+{S}_2,{E}_2\right\}=\max \left\{6+0.5\times 2+3+1,9\right\}=11 $$

$$ {l}_1^{r_k^m}=\min \left\{{l}_2^{r_k^m}-{S}_2-{T}_{12}-{P}_1\cdot {D}_1,{L}_1\right\}=\min \left\{17-1-3-0.5\times 2,10\right\}=10 $$

The function ${f}_i^{r_k^m}(t)$ of the first customer in the route is simply equal to its MDVE_i(t) function:

${f}_1^{r_k^m}(t)$=MDVE₁(t) for $t\in \left[{e}_1^{r_k^m}+{P}_1\cdot {MDVB}_1\left({e}_1^{r_k^m}\right),{l}_1^{r_k^m}+{P}_1\cdot {MDVB}_1\left({l}_1^{r_k^m}\right)\right]$.

The function ${f}_i^{r_k^m}(t)$ of the i^th customer in the route is a piecewise linear function with at most i pieces. In the determination of ${f}_2^{r_k^m}(t)$, we first set ${\tilde{e}}_1^{r_k^m}={e}_1^{r_k^m}$and ${\tilde{l}}_1^{r_k^m}={l}_1^{r_k^m}$, and determine ${\tilde{e}}_2^{r_k^m},$${\tilde{t}}_{2,1}^{r_k^m}$, ${\tilde{l}}_2^{r_k^m}$and the associated MDVB₂ values as follows:

$$ {\tilde{e}}_2^{r_k^m}={\tilde{e}}_1^{r_k^m}+{P}_1\cdot {MDVB}_1\left({\tilde{e}}_1^{r_k^m}\right)+{T}_{12}+{S}_2=6+0.5\times 2+3+1=11 $$

$$ {\tilde{t}}_{2,1}^{r_k^m}={\tilde{l}}_1^{r_k^m}+{P}_1\cdot {MDVB}_1\left({\tilde{l}}_1^{r_k^m}\right)+{T}_{12}+{S}_2=10+0.5\times 4+3+1=16 $$

$$ {\tilde{l}}_2^{r_k^m}={l}_2^{r_k^m}=17 $$

$$ {MDVB}_2\left({\tilde{e}}_2^{r_k^m}\right)=4+0.25\times \left(11-9\right)=4.5 $$

$$ {MDVB}_2\left({\tilde{t}}_{2,1}^{r_k^m}\right)=4+0.25\times \left(16-9\right)=5.75 $$

$$ {MDVB}_2\left({\tilde{l}}_2^{r_k^m}\right)=4+0.25\times \left(17-9\right)=6 $$

The ${f}_2^{r_k^m}(t)$values are then determined:

$$ {\tilde{f}}_2\left({\tilde{e}}_2^{r_k^m}+{P}_2\cdot {MDVB}_2\left({\tilde{e}}_2^{r_k^m}\right)\right)={\tilde{f}}_1\left({\tilde{e}}_1^{r_k^m}+{P}_1\cdot {MDVB}_1\left({\tilde{e}}_1^{r_k^m}\right)\right)+{MDVB}_2\left({\tilde{e}}_2^{r_k^m}\right)=2+4.5=6.5 $$

$$ {\tilde{f}}_2\left({\tilde{t}}_{2,1}^{r_k^m}+{P}_2\cdot {MDVB}_2\left({\tilde{t}}_{2,1}^{r_k^m}\right)\right)={\tilde{f}}_1\left({\tilde{l}}_1^{r_k^m}+{P}_1\cdot {MDVB}_1\left({\tilde{l}}_1^{r_k^m}\right)\right)+{MDVB}_2\left({\tilde{t}}_{2,1}^{r_k^m}\right)=4+5.75=9.75 $$

$$ {\tilde{f}}_2\left({\tilde{l}}_2^{r_k^m}+{P}_2\cdot {MDVB}_2\left({\tilde{l}}_2^{r_k^m}\right)\right)={\tilde{f}}_1\left({\tilde{l}}_1^{r_k^m}+{P}_1\cdot {MDVB}_1\left({\tilde{l}}_1^{r_k^m}\right)\right)+{MDVB}_2\left({\tilde{l}}_2^{r_k^m}\right)=4+6=10 $$

The associated completed delivery times at customer 2 are the following:

$$ {\tilde{e}}_2^{r_k^m}+{P}_2\cdot {MDVB}_2\left({\tilde{e}}_2^{r_k^m}\right)=13.25 $$

$$ {\tilde{t}}_{2,1}^{r_k^m}+{P}_2\cdot {MDVB}_2\left({\tilde{t}}_{2,1}^{r_k^m}\right)=18.875 $$

$$ {\tilde{l}}_2^{r_k^m}+{P}_2\cdot {MDVB}_2\left({\tilde{l}}_2^{r_k^m}\right)=20 $$

Figures 5 and 6 show the graph of ${f}_1^{r_k^m}(t)$ and ${f}_2^{r_k^m}(t)$, respectively.

Fig. 5

Function ${f}_1^{r_k^m}(t)$ in the small example

Fig. 6

Function ${f}_2^{r_k^m}(t)$ in the small example

Appendix 2 Profit Determination Procedure in Campbell and Savelsbergh (2004a)

Given ${f}_i^{r_k^m}(t)$, ${g}_i^{r_k^m}$ and ${tg}_i^{r_k^m}$, the increase in delivery volume on a route associated with an insertion can be determined in O(log n) time. First, compute ${g}_j^{r_k^m}$. There are two possible cases.

Case 1: If ${e}_j^{r_k^m}+{D}_j{P}_j+{T}_{j,i}+{S}_i>{tg}_i^{r_k^m}$, then set

$$ {g}_j^{r_k^m}={D}_j+{g}_i^{r_k^m}-\left({e}_j^{r_k^m}+{D}_j{P}_j+{T}_{j,i}+{S}_i-{tg}_i^{r_k^m}\right)/{P}_i $$

(45)

This is because the volume deliverable to the succeeding customers will decrease at rate 1/P_i after ${tg}_i^{r_k^m}$.

Case 2: If ${e}_j^{r_k^m}+{D}_j{P}_j+{T}_{j,i}+{S}_i\le {tg}_i^{r_k^m}$, then set

$$ {g}_j^{r_k^m}={g}_i^{r_k^m}+\max \_{d}_j^{r_k^m}\left({tg}_i^{r_k^m}-{S}_i-{T}_{j,i}\right) $$

(46)

Where

$$ \max \_{d}_j^{r_k^m}\left({tg}_i^{r_k^m}-{S}_i-{T}_{j,i}\right)=\min \left(\begin{array}{l}{Q}_m-{q}_{r_k^m}^{\mathrm{min}},\\ {}{C}_j-{I}_j+\left(\frac{tg_i^{r_k^m}-{S}_i-{T}_{j,i}-{C}_j{P}_j+{I}_j{P}_j}{1+{U}_j{P}_j}\right){U}_j,\\ {}{C}_j-{I}_j+{U}_j{l}_j^{r_k^m},\\ {}\left({tg}_i^{r_k^m}-{S}_i-{T}_{j,i}-{e}_j^{r_k^m}\right)/{P}_j\end{array}\right) $$

(47)

${tg}_j^{r_k^m}$ associated with the four quantities in $\max \_{d}_j^{r_k^m}\left({tg}_i^{r_k^m}-{S}_i-{T}_{j,i}\right)$ are

• (i) ${tg}_j^{r_k^m}\in \left[\frac{Q_m-{q}_{r_k^m}^{\mathrm{min}}-{C}_j+{I}_j}{U_j},{tg}_i^{r_k^m}-{S}_i-{T}_{j,i}-{P}_j\left({Q}_m-{q}_{r_k^m}^{\mathrm{min}}\right)\right]$,

• (ii) ${tg}_j^{r_k^m}=\frac{tg_i^{r_k^m}-{S}_i-{T}_{j,i}-{C}_j{P}_j+{I}_j{P}_j}{1+{U}_j{P}_j}$, (iii) ${tg}_j^{r_k^m}={l}_j^{r_k^m}$ and (iv) ${tg}_j^{r_k^m}={e}_j^{r_k^m}$.

After determining ${g}_j^{r_k^m}$, set

$$ {\widehat{q}}_{r_k^m}^{\mathrm{max}}=\min \left\{{Q}_m,{g}_j^{r_k^m}+{f}_{i-1}^{r_k^m}\left({t}_j^{g,{r}_k^m}-{S}_j-{T}_{i-1,j}\right)\right\} $$

(48)

The objective function is the combination of the change in delivery volume and the change in travel time associated with an insertion:

$$ \mathrm{Profit}\left(i,j,k,m\right)=\alpha \left({\widehat{q}}_{r_k^m}^{\mathrm{max}}-{q}_{r_k^m}^{\mathrm{max}}\right)-\beta \left({T}_{i-1,j}+{T}_{j,i}-{T}_{i-1,i}\right) $$

(49)