Introduction
I: DC J: Depot
1.6 Prospects for Further Applications .1 Variants of the Modified Savings Method
18 Yoshiaki shimizu
points (σ = {0.1, 0.2, 0.3, 0.4, 0.5}) and four different ranges of demand variation (ρ = {0.2, 0.3, 0.4, 0.5}). In all, 600 optimization problems were solved under the same conditions as before. The results are shown in Figures 1.13 and 1.14.
Figure 1.13 shows the total cost for ranges of demand deviation and ordering point. Due to the nondeterministic parameter setting, a complicated profile is found. However, the overall shape is plausible since the region of minimum cost moves to a higher ordering point as the deviation increases. This suggests that, in terms of cost manage- ment, it is important to control the ordering point or inventory level according to the demand variation. When we separate the inventory cost from the total cost, its changes are rather simple, as shown in Figure 1.14. Since a higher stock level incurs a greater holding cost, the cost increases proportionally with the ordering point regardless of the variability of demand.
Finally, from these parametric studies, we claim that the applied model behind the mathematical formulation is adequate. The plau- sibility of the results supports the viability of the approach if it were used in a real-world optimization with actual parameters.
1.6 Prospects for Further Applications
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0.500 0.440 0.380 0.320
Demand deviation range r (–)
ma De de nd tio via an n r r ge ) (–
Ordering point s(–)
Ordering point s(–)
Total cost unit (–)
0.260 0.200
0.500 0.440 0.380 0.320 0.260 0.200 0.1000.1800.2600.3400.4200.500
0.100
2.401e+6 2.394e+6 2.388e+6 2.382e+6 2.376e+6 2.370e+6 0.180 0.260 0.340 0.420 0.500 Figure 1.13 Total cost for various demand deviations and ordering points.
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s
c q D d q d q D d
q D D d
ij v
v i i v i v j j
v i j i
( ) ( ) ( )
( )
γ
α β α β α β
α β
= + + + +
− + +
0 0 0
0 −−(qv +Dj)αdijβ (1.17) where
α and β denote the elastic coefficients for weight and distance, respectively
γ is a constant
When α = β = γ = 1, this expression refers to the ordinary Weber basis.
Moreover, it is common to consider pickup problems instead of delivery ones in reverse logistics. Here, every vehicle visits the pickup points and returns to the depot directly. Letting Pd be pickup demand, we can derive the savings value as follows (refer to Figure 1.15):
s
c q d q Pd d q Pd d
q Pd d
ij v
v j v i i v j j
v i ij
( ) ( ) ( )
( )
γ
α β α β α β
α β
= + + + +
− +
0 0 0
−−(qv+Pdi+Pdj)αdβj
0 (1.18) The previous idea can be extended to the case where vehicles stop at an intermediate destination before returning to the depot. This is the case, for example, when a vehicle visits a disposal site to dump waste
Inventory cost unit (–)
Demand deviation range r (–) Ordering point s (–) 2.709e+5
2.564e+5 2.419e+5
2.274e+5 2.129e+5 1.984e+5
0.500 0.440 0.380 0.320 0.260 0.200 0.1000.1800.2600.3400.4200.500
Figure 1.14 Inventory cost for various demand deviations and ordering points.
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or a remanufacturing facility to deliver used products. In this case, we can modify this equation as follows:
s
c q Pd d q d q d q Pd d
q P
ij v
v i iR v R v j v j jR
v
( ) ( ) ( )
( γ
α β α β α β α β
= + + + + +
− +
0 0
ddi)αdijβ−(qv +Pdi +Pdj)αdβjR (1.19) where the subscript R denotes the intermediate destination.
Similarly, we have the following expression in the case of VRPSPD (refer to Figure 1.16):
s
c D q d Pd q d D q d
Pd q
ij v
i v i i v i j v j
j v
( ) ( ) ( ) ( )
( )
γ
α β α β α β
α
= + + + + +
+ +
0 0 0
dd D D q d
Pd D q d Pd Pd q d
j i j v i
i j v ij i j v j
0 0
0
β α β
α β α β
− + +
− + + − + +
( )
( ) ( ) (1.20)
q
i j
q d0,iiq
Initial: round trip0 Intermediate Final: circular trip
+ + – –
–
– –
(a)
i j
d0,j q d0,i
diR djR
Initial Final
q q 0
q q
q
Drop-by point Intermediate (b)
dR0
(Pdi+q)
(Pdi+q)
(Pdi+Pdj+q) (Pdj+q)
(Pdi+Pdj+q) (Pdi+q)
(Pdj+q) (Pdi+q)
dj,0
di,0 d0,j
Figure 1.15 Scheme to derive savings value for pickup VRP for either (a) direct or (b) drop-by pickup.
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Just by applying these formulas to derive the initial solution of the respective VRP, we can simply follow the overall procedure described earlier to obtain the respective final solutions.
1.6.2 Application of Parallel Computing Techniques
Unlike conventional studies, our method can cope with large-scale problems in a practical and flexible manner. However, there still exists a great need for solving problems more quickly and efficiently in order to make responsive decisions in global markets. For such require- ments, it is natural to develop a parallel implementation for logistics optimization. In a special issue of Parallel Computing, Laporte and Musmanno (2003) emphasized the importance of parallel computing in logistics not only due to the large scale of these problems but also because of real-time applications arising in the delivery of emergency services and in courier or dial-a-ride services.
To make our hybrid approach suitable for parallel optimization, we developed a binary particle swarm optimizer (PSO) and sub- stituted it for the modified tabu search used in the previous proce- dure. Compared with an individual search such as tabu search, the population-based PSO algorithm is well suited to implementing a parallel computation (Shimizu and Ikeda, 2010). The first application considered a rather simplified formulation for strategic problems using master–worker parallelism, as illustrated in Figure 1.17. Then, more complicated configurations were examined. For these, a novel paral- lel procedure similar to the island model used in genetic algorithms was developed, employing multithreading techniques so that the idle time for the parallel computation becomes very small. Moreover, the effect of the topology of subpopulations (Figure 1.18) and the manner
i j +
– –
0 (Di+q) (Dj+q)
(Pdi+q) (Pdj+q)
(Pdi+Dj+q)
(Di+Dj+q) (Pdi+Pdj+q)
Figure 1.16 Scheme to derive savings value for VRPSPD.
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Master Determine optimal
location of DC
Task assignment Reporting
DC location Routes between customers and DC
Master–worker cluster
Slave1 Slave2 SlaveM
Hybrid procedure Determine optimal transportation routes
Figure 1.17 Scheme under master–worker parallelism.
Subpopulation 1 (a)
(b)
Subpopulation 5 Subpopulation 2
Subpopulation 3 Subpopulation 4
Migration
Migration Migration
Migration Migration
Individual
Migration
Migration Subpopulation
Figure 1.18 Topology for island model parallelism: (a) random ring (RR) or (b) two-node torus ring (2nR).
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of information exchange between subpopulations were analyzed.
Finally, we showed that such an approach can solve huge problems with more than 20,000 customers while maintaining high efficiency of the parallelization, as shown in Figure 1.19.
1.6.3 Enhancement for Practical Use
To realize the planning system illustrated in Figure 1.1, it is essential to provide a user-friendly interface to manage the system. In the planning section on the production side, this goal is closely related to data han- dling and visualization of the circumstances at hand. For this, we can effectively utilize some software developed for the Google Maps appli- cation programming interface (API). We have developed the following stepwise procedure by using JavaScript and appropriate free software:
Step 1: Collect the addresses of locations in an Excel spread sheet or text file.
Step 2: Add longitude and latitude information for every loca- tion in the sheet.
Step 3: Calculate the distance between every pair of locations by using the Google geocoding API.
Step 4: Solve the optimization problem by the proposed method.
Step 5: Display the routes obtained from Step 4 in Google Maps.
Figure 1.20 shows some results for an illustrative problem, in which every depot has a single route, |I| = 1, |J| = 3, and |K| = 17. In this figure,
0.9 0.85
0.8 1 3 5 7 9 11 13 15
0.95 1
Number of PC (–)
Parallelization efficiency (–)
RR 2nR
Figure 1.19 Efficiency of parallelization against the number of processors.
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for simplicity, the routing paths from only depot 1 are shown. Marks for locations (A–B–…–K–A) and dotted arrows are superimposed to help visualize the actual circular route. We can see that this kind of visual information is very helpful for some tasks at an operational level. However, there still remain many possibilities to add more and valuable service information from geographical information system (GIS) applications and the Google Maps API.