Figure 3. Bulldozer supply chain conguration

(a)

(b)

Figure 4. Pareto fronts (a) and Pareto normalized fronts (b) withdifferent combinations of n, m and e

(a)

(b)

Figure 5. Pareto fronts (a) and Pareto normalized fronts (b) withdifferent combinations of neb and nsb

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Accordingly, this step can only be carried out if thematching procedure was already performed for the firsterror image. Therefore, only areas that were not removedduring the first matching procedure are extended bycorresponding areas of the subsequent error images.Otherwise, the noise (falsely detected areas) would causean enlargement of incorrectly detected areas. The red shortdashed rectangles in Figure 8 mark 2 examples of suchcorresponding areas. Resulting areas that are too largeare removed from the error images In and In+1. This isindicated by the areas in the right lower corner of errorimage In in Figure 8. As can be seen, the resulting errorimage In from Figure 8 is used as input (error image In) inFigure 7. Without the extension of the areas, the midmostcandidate in Figure 7 would have been rejected.

As some real moving objects are sometimes not detectedin an error image as a result of an inaccurate optical flowcalculation or (radial) distortion, the temporal matchingwould fail. This could already be the case if only onearea in one error image is missing. Thus, candidates thatwere detected once in 3 temporal succeeding error imagesand 4 greyscale images (original images), respectively, arestored for a sequence of 3 error images subsequent to theimage where the matching was successful, cf. Figure 9(a).Their coordinates are updated for the succeeding errorimages by using the optical flow data. As a consequence,they can be seen as candidates for moving objects inthe succeeding images, but they are not used within thematching procedure as input. If within this sequenceof images a corresponding area is found again, it is

stored for a larger sequence of images (more than 3) andits coordinates are updated for every succeeding errorimage. The number of sequences depends on the followingcondition:

=

{c+ccc | c = c2c | c = c,

(13)

where c is the number of found corresponding areas andc is the number of missing corresponding areas for onecandidate starting with the image where the candidatewas found again. If < 0 > 10, the candidate isrejected. Moreover, the candidate is no longer stored if itwas detected again in 3 temporal succeeding images. Inthis case, it is detected during the matching procedure.An example concerning to this procedure is shown inFigure 9(b). As one can imagine, error image In inFigure 9(a) is equivalent (except area-extension) to In+1in Figure 7, whereas error image In in Figure 9(b) isequivalent to In+2 in Figure 9(a).

For a further processing of the data, only the position(shown as small black crosses in the left lower corners ofthe rectangles in Figures 7 and 9) and size of the rectanglesmarking the candidates are of relevance. Thus, for everyerror image the afore mentioned information is storedfor candidates that were detected during the matchingprocedure, for candidates that were detected up to 3 errorimages before and for candidates that were found again(see above). On the basis of this data, candidates that arevery close to each other are combined and candidates thatare too large are rejected.

(a)

(b)

Figure 9. Preventing rejection of candidates for moving objects that were detected only in a few sequences. (a) Storage of candidatesfor which a further matching fails. These candidates are marked by a blue dotdashed rectangle. The green dashed rectangle marks acandidate for which a corresponding area was found again and the red short-dashed rectangle marks a candidate with successful matching.(b) Storage of candidates for which a corresponding area was found again. The 2 areas drawn with transparency in error image In indicatethe position of the candidates, but they are not part of the error image.

5. Experimental tests

Several experimental tests have been conducted in orderto show the performance of the multi-objective beesalgorithm parameters on the solutions, and to find thecombination of them that gives the best Pareto front.All the experiments are carried out with the numberof iterations fixed to 1000 and a patch size ngh of 1.In all tests the algorithm is run 100 times and resultsare added after each run in order to get more robustsolutions. The weighted sum approach has been used toevaluate the quality of the Pareto fronts. For each coupleof normalized solutions of all combinations of the beesalgorithm parameters, the Z (main objective) function iscalculated, expressed by Equation 8:

Z = w1TCn + w2TLTn (8)

where TCn and TLTn are the normalized solutions and w1and w2 are the weights where the summation is equal to 1.To minimize both the cost and lead-time simultaneously, itneeds to find the minimum value of Z function accordingto the weight values which regulate the importance of eachobjective function as stipulated by the decision-maker. Forthis reason, in each test the Pareto front which containsthe minimum of Z has been considered as the fittest.

During experiments, the parameters of the bees algorithmhave been tuned by trial and error empirically to find thebest combination of the parameters set.

In the first experimental attempt, the four differentcombination of n, m and e are tested given in Table 2,while neb and nsb are fixed to 10 and 6 respectively, andthe equal weight values (0.5, 0.5) have been utilized forboth cost and lead-time.

In the second experimental attempt, n, m and e areset to 50, 15 and 5 respectively and the weight values areselected equal (0.5, 0.5), then the best parameter sets forthe neb and nsb are searched; the combination sets utilizedin this study are given in Table 3.

According to the above experiments on parametertuning, the best parameter combination of the beesalgorithm has been utilized for the next section which willbe based on finding the Pareto front lines according to thedifferent combination of weights.

6. Results and discussion

6.1. Effect of n, m and e

The Pareto front line of each combination of n, m and e isgiven in Figure 4(a) and the normalized results are givenin Figure 4(b).

According to the Figure 4, the first combination gives 15points, and second, third and fourth give 14, 13 and 13respectively. On the other hand, Z function values arelower using the last two combinations.

The minimum value of the Z function is 0.29600567which corresponds to a total cost of 127023480 $ and

a total lead-time of 46 days was found with the thirdparameter combination set (n=50, m=15, e=5). For furtheranalysis, this combination set has been utilized.

6.2. Effect of neb and nsb

The Pareto front line of each combination of the neb andnsb parameters are given in Figure 5(a) and the Paretofront line of the normalized results are given in Figure5(b). According to Figure 5, the best combination of neband nsb is (neb=10 and nsb=6). The minimum value of theZ function gives a total cost of 127023480 $ and a totallead-time of 46 days which indicates the Pareto front lineof the red line in the Figure 5(a), i.e., neb and nsb are 10and 6 respectively.

According to the above results, the best combinationof the parameter sets are n=50, m=15, e=5, neb=10 andnsb=6. These values have been utilized to calculate thePareto fronts according to the weights values.

6.3. Effect of weights and comparison to ant colony optimization

In this section, the Pareto front line is calculated withdifferent combinations of weights.

(a)

(b)

Figure 6. Pareto fronts lines (a) and Pareto normalized fronts lines(b) with nine combinations of weights

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Figure 7. Boxplot of Z functions results for two samples obtainedwith the same parameters values of the bees algorithm-basedapproach

In Figure 6(a), nine different combinations of the weightshave been utilized such as (w1, w2) =(w1, 1-w1), wherew1=0.1,0.2...0.9. In Figure 6(b), the normalized resultsare given. For instance using w1=0.1 and w2=0.9, moreimportance is given to the minimization of the totallead-time instead of the total cost, using w1=0.5 andw2=0.5 the two objectives have the same importance, andfinally in the case of w1=0.9 and w2=0.1, the minimizationof the total cost is favoured.

It is clear that each point of each Pareto front line canbe taken as a solution to the problem depending on thedecision-maker. It is also possible to find a representativepoint for each curve according to the minimization of theZ function. Finally, the nine couples of the total cost andthe total lead-time which give the minimum value of theZ function for all the combinations of weights are given inTable 4 with network configuration. The results show that,when w1 < w2 the Z function minimum value is located inthe zone of the lower total lead-time and higher total cost(I), when w1 = w2 the Z function minimum value is locatedin a central zone (II), while when w1 > w2 the Z functionminimum value is located in the zone of the higher totallead-time and lower total cost (III).

All results are compared with those obtained by the antcolony optimization in [23] where the Pareto front lineswere calculated with three values of which regulatesthe importance of two objective functions; = 0.1corresponds to w1=0.9 and w2=0.1, = 0.5 to w1=w2=0.5and = 0.9 to w1=0.1 and w2=0.9. According to theresults in [23], it shows that each value of covers onlysome portion of the Pareto front line. However, the resultscomputed from the bees algorithm shows that each weightvalue covers more portions of the Pareto front line. Foreach couple of the weights, the minimum total lead-timevalue of 30 days has been found, while the minimumcost value depends on the weights. The minimum totalcost found is 126421236 $ for w1=0.8 and w2=0.2. Thus,the bees algorithm found more Pareto non-dominatedsolutions than the ant colony approach and obtainedlower costs. For instance, comparing the results using = 0.5 for the ant colony approach and w1=w2=0.5 forthe bees algorithm approach, both of them found theminimum total lead-time of 30 days, but the first one

found a minimum total cost of 127193700 $ while the beesalgorithm approach found 126595752 $.

The bees algorithm-based approach has been proven togive repeatable results as shown in Figure 7. Comparingthe boxplots of two samples of results in terms of Zfunction obtained with the same weights and parameters,very similar distributions have been achieved, with meansof 0.395 and 0.392, and variances of 0.0030 and 0.0025.

7. Conclusions

In this study, a multi-objective bees algorithm-basedsupply chain design model has been proposed andapplied on a supply chain problem. This problem dealswith the resource options selection for a multi-productand multi-delivery supply chain in order to minimize twoobjective functions simultaneously, namely, total cost andtotal lead-time of the network. Several tests have beenconducted to find the optimum parameters for the beesalgorithm. Subsequently, Pareto front lines have beencomputed with different weight combinations. The resultsshowed the efficiency of the proposed model. The Paretosolutions of the proposed model have been comparedwith those obtained by an ant colony optimization [23].This showed that the bees algorithm is a more powerfultool for finding a better Pareto solution for supply chainproblems.

Future work will consider increasing the complexityof the problem and testing it using the current version ofthe bees algorithm, in addition, an improved version ofthe bees algorithm will be researched.

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