3.4.1 Point-to-Point Distribution Mode with no Vehicle Constraints
Intuitively, PTP mode with no vehicle constraints would result in the best delivery efficiency, because each demand node obtains the replenishment directly. If all emer- gency resources are distributed by PTP mode, then the problem is simple and can be formulated as a linear programming model as follows.
Suppose the emergency distribution network can be constructed as a digraph G(O, V, E, ω), where O = {1,2,· · ·,m} represents the emergency stock- pile depots, and V = {1,2,· · · ,n} stands for the emergency demand nodes.
E = {ei j|i ∈ O,j ∈V}represents the distribution arcs, whereei j is the arc from the stockpile depot ito the emergency demand nodej.ωi j is characterized as the Euclidean distance from stockpile depotito emergency demand nodej.E Qi{i ∈O}
means the original inventory of emergency resources in stockpile depoti. Ij and Qj{j ∈V}stand for the number of infected people and the number of quarantined people in the demand nodej, respectively.dj{j ∈V}is the demand for emergency resources in nodejand can be calculated by Eq. (3.2). And last, the decision vari- ablezi j =1 if the demand nodejwill be serviced by stockpile depoti; otherwise, zi j =0. Suppose that capacity of the vehicle is large enough for each demand node to be satisfied in one trip. Thus, it is easy to formulate the linear programming model for the PTP mode with no vehicle constraints as follows (M1):
(M1) min
i∈O
j∈V
2ωi jzi j (3.4)
s.t.
n j=1
djzi j ≤ E Qi, ∀i ∈O (3.5)
m i=1
E Qizi j ≥dj, ∀j∈V (3.6) m
i=1
zi j ≥1, ∀j ∈V (3.7)
ωi j =
(xi−xj)2+(yi−yj)2, ∀i∈ O,j∈V (3.8)
dj =<k>Ij+Qj, ∀j ∈V (3.9) zi j =0or1, ∀i∈ O, j ∈V. (3.10) Here, Eq. (3.4) is the objective function, searching for the minimum total delivery distance. Equations (3.5) and (3.6) are constraints for flow conservation and guarantee that there are enough emergency resources. Equation (3.7) ensures that all demand nodes would be supplied. Equation (3.8) is the Euclidean distance from stockpile depot i to emergency demand node j, where xand yrepresent the X-coordinate and Y-coordinate, respectively. Equation (3.9) denotes the demand for emergency resources in demand nodej(as introduced in the Sect.3.3.2). And last, Eq. (3.10) is the decision variable. The above model is a simple 0–1 integer programming model, and can be solved by some programming tools (e.g. Matlab, Lingo) directly.
Due to various constraints (such as the number of vehicles is limited, etc.), such a pure point-to-point distribution mode may be infeasible in an actual emergency rescue situation. In reality the delivery mode of the emergency resources is much more complex and difficult. For making a comparison between different emergency distribution modes, we propose a relative evaluation function as follows.
Supposing that the speed of the vehicle is a constantv, we can obtain the minimum waiting time setTwP T Pai t = {t1P T P,t2P T P,· · ·,tnP T P}for all demand nodes in the pure PTP mode based on the optimal solution of (M1). We define the relative timeliness of all demand nodes in this mode as equal to the standard value 1. Meanwhile, we define φelsej as representing the timeliness of demand nodej in other distribution modes as follows:
φelsej = tjP T P
telsej , ∀j ∈V. (3.11)
Herein,telsej is the minimum waiting time of the demand nodejin other shipment systems before it gets the emergency replenishment. Therefore, the relative evaluation function of the timeliness in other distribution modes can be formulated as follows:
else= 1 n
j∈V
φelsej , ∀j ∈V. (3.12)
3.4.2 The Multi-depot, Multiple Traveling Salesmen Distribution Mode with Vehicle Constraints
In this section, we analyze the problem from another point of view. We assume that both the capacity and the number of vehicles are limited. Thus, an inevitable problem for distribution in such a situation is scheduling for all the demand nodes. Under the assumption that each demand point would be satisfied after one replenishment, the problem can be transformed to a multi-depot, multiple traveling salesmen distribution problem, which is characterized as a NP-hard problem.
Suppose that the shipment system can be constructed as a digraph G(O ∪ V,E, ω), where O = {1,2,· · ·,m} represents the emergency stockpile depots, and V = {m+1,m+2,· · · ,m+n}stands for the emergency demand nodes.
E = {ei j|i,j∈ O∪V,i= j}represents the distribution arcs, whereei j is the arc from pointi tojin the network (ifi ∈ O, j ∈ V, that means from the stockpile depotito the demand nodej; ifi ∈V, j ∈ O, that means from the demand nodei back to the stockpile depotj; ifi, j∈V, i= j, that means from the demand node ito the demand nodej; And last, ifi, j ∈ O, i = j, that means from the stockpile depotito the stockpile depotj).ωi j is the Euclidean distance from pointito pointj.
Specially, in order to depict that there is no distribution arc between any two stock- pile depots,ωi j is defined as a large number whilei, j ∈ O, i = j.Rrepresents the feasible path set, andrl stands for the feasible path inR. E Qk{k ∈ O}means the original inventory of the emergency resource in stockpile depotk.Sk(k ∈ O) means the demand nodes set that is supplied by stockpile depotk, and it meets the constraint
k∈OSk = V. Ij andQj{j ∈ V}stand for the number of people who are infected, and the number of people who are quarantined, respectively.dj{j ∈V} is the demand for emergency resources in nodej. Nk(k ∈ O)is the least number of vehicles we needed in depotk, and Qcap represents the capacity of the vehicle.
And last, the decision variable,zi j =1 if the vehicle travels from pointito pointj;
otherwise,zi j =0. Thus, the multi-depot, multiple traveling salesmen distribution problem can be formulated as follows (M2):
(M2) min
i∈O∪V
j∈O∪V,i=j
ωi jzi j (3.13)
s.t.
i∈O∪V
xi j =1,∀j ∈V,i = j (3.14)
j∈O∪V
xi j =1,∀i∈V,i = j (3.15)
i∈O
j∈V
zi j =
i∈V
j∈O
zi j (3.16)
j∈Sk
dj ≤E Qk,∀k∈O (3.17)
j∈rl
dj ≤ Qcap,∀rl ∈ R (3.18)
i∈/S
j∈S
zi j ≥1,∀S⊆V,|S| ≥2 (3.19)
Nk=
⎡
⎢⎢
⎢⎢
j∈Sk
dj
Qcap
⎤
⎥⎥
⎥⎥,∀k∈ O (3.20)
ωi j =
(xi−xj)2+(yi−yj)2, ∀i ∈O,j ∈V, ori ∈V,j ∈ O, ori,j ∈V,i = j
(3.21) ωi j =M, ∀i,j ∈ O,i = j (3.22)
dj =<k>Ij+Qj, ∀j ∈V (3.23) zi j =0or 1, ∀i,j∈ O∪V,i = j (3.24) Herein, Eq. (3.13) is the objective function, searching for the minimum total dis- tribution distance. Equations (3.14) and (3.15) ensure that each demand node would be supplied once. Equation (3.16) means all vehicles leaving the stockpile depots must return to them afterwards. Equation (3.17) means there are enough emergency resources for the demand nodes. Equation (3.18) is a constraint for the feasible path, which warrants that the total emergency requirement on the feasible path does not exceed the capacity of the vehicle. Equation (3.19) ensures that there is no sub-loop in the optimal solution. Equation (3.20) is used for calculating the number of vehi- cles required in each depot. Equations (3.21) and (3.22) are the distance constraints.
Equation (3.23) is used for forecasting the demand for emergency resources in each demand node. Finally, Eq. (3.24) is the variable constraint. The above model is a typical NP problem, and would thus be difficult to optimally solve, especially for realistically large-scale problems. Therefore, we should design an algorithm to get the approximate optimal solution.
3.4.3 The Mixed-Collaborative Distribution Mode
In the above two Sects. (3.4.1) and (3.4.2), we discussed the problem from two differ- ent directions. In fact, both of these two pure distribution modes may be infeasible in an actual situation. On the one hand, we may have not enough vehicles to implement all the emergency services by PTP mode. On the other hand, if we adopt the MMTS mode, some of the vehicles may be unused, which reduces the emergency timeli-
ness. Thus, a mixed-collaborative mode, which allows both of these two distribution modes to coexist, is proposed in the following, with the objective of equilibrating the total rescue distance and timeliness. Similarly as before, this problem can also be depicted as follows.
atj represents the time when the distribution vehicle arrives at the demand node j{j ∈V}.[ej,lj]represents the time window for demand nodej{j ∈V}, whereej
means the earliest arrival time (result of the PTP mode), andlj is the latest arrival time (result of the MMTS mode). Here, we design a special time window for such a mixed-collaborative mode using the computational results of the former two pure modes, which guarantee that the outcome of the mixed-collaborative mode will be maintained at a high level.Ni,i ∈Omeans the number of vehicles in stockpile depot i. Suppose that the speed of the vehicle is an invariable constant, thenti j =ωi j/v,
∀i,j ∈O∪V would be the time consumed from pointito pointj. Other parameters are specified in the same way as MMTS mode.
Such a mixed system can be understood as a MMTS mode which allows some requirements to be delivered directly when the constraints are much stricter. Thus, combined with Eq. (3.12) in Sect. 3.4.1, the mixed distribution problem can be formulated as follows (M3):
(M3) max
i∈O
i (3.25)
s.t.
i∈O∪V
xi j =1,∀j ∈V,i = j (3.26)
j∈O∪V
xi j =1,∀i∈V,i = j (3.27)
j∈V
zi j =Ni,∀i ∈ O (3.28)
i∈V
zi j =Nj,∀j∈ O (3.29)
j∈Sk
dj ≤E Qk,∀k∈O (3.30)
j∈rl
dj ≤ Qcap,∀rl ∈ R (3.31)
i∈/S
j∈S
xi j ≥1,∀S ⊆V,|S| ≥2 (3.32)
ωi j =
(xi−xj)2+(yi−yj)2, ∀i ∈O,j ∈V or i∈V,j ∈ O or i,j ∈V,i= j
(3.33) ωi j =M, ∀i,j ∈ O,i = j (3.34)
dj =<k>Ij+Qj, ∀j ∈V (3.35)
ej ≤atj ≤lj,∀j ∈V (3.36)
ati+ti j +(1−zi j)T ≤atj,∀i∈ O∪V,j ∈V,i = j (3.37)
ati =0,∀i ∈O (3.38)
atj>0,ej>0,lj >0,∀j ∈V (3.39) ti j >0,∀i∈ O∪V,j ∈V,i = j (3.40) zi j =0or 1, ∀i,j ∈O∪V,i = j. (3.41) Equation (3.25) is the objective function, searching for the maximizing timeli- ness. Equations (3.28) and (3.29) mean that all vehicles leaving from the stockpile depot must finally return to the depot. Equations (3.35)–(3.39) are the time window constraints, T means a large enough number. Other constraints are defined in the same way as in Sect.3.4.2.