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In order to solve the problem above, we propose to estimate the distance between any S/D pair in a WSN by replacing the inaccurate distance with optimum space step distance and then Eq. (1) can been changed into the following equation: d = h × E ( R) (2) where d and E(R) stand for the hop counts between S and D and the optimum space step distance respectively in a three dimensional WSN. So the most important part of our localization scheme is to obtain the optimum space step distance. 2. 2, if a source node S is given, the optimum space step distance toward the destination D at each step is denoted as Ri which is a random variable.

The prices charged by manufacturer m when transacting with retailer n and consumers at demand market o are ρ mnj and ρ mo . The risk functions are denoted by r2lmj , r2 mnj , rmo . The relationship functions, between manufacturer m and supplier l , manufacturer m and retailer n , manufacturer m and consumers at demand market o , are denoted by Let b2lmj , b2 mnj and bmo , respectively. α m , βm express the related weight of the risk function and the relationship function, respectively. Then we can get the total income for manufacturer follows: m as Multi-objective Optimization of Supply Chain Supernetwork with Electronic Commerce N 2 ρ Max n=1 j=1 q +ρmoqmo − fm( Q2,Q3) −c2mnj ( qmnj ,hmnj ) −c2mo ( qmo,hmo ) − O mnj mnj N 2 o=1 O n=1 j=1 o=1 v ( h ) − v ( h ) − v ( h ) −α r L 2 N l =1 j =1 2lmj r ( q N 2 2mnj n=1 j=1 mnj 49 lmj 2 n=1 j =1 O 2mnj mnj o=1 mo mo m N n=1 2 j =1 2mnj (qmnj , hmnj ) + O N 2 O L 2 ,hmnj ) +rmo ( qmo,hmo ) +βmb2lmj ( hlmj ) +b2mnj ( hmnj ) +bmo ( hmo ) o=1 n=1 j=1 o=1 l=1 j=1 ( qmnj ≥ 0, qlmj ≥ 0,0 ≤ hlmj ≤ 1,0 ≤ hmnj ≤ 1,0 ≤ hmo ≤ 1, qmo ≥ 0, ∀l, j, n, o) (10) Hence, the optimality conditions for all manufacturers can be described simultaneously using the following variational inequality: * * ∂r2lmj ( qlmj , hlmj ) × q −q* + M L 2 α ∂r2lmj ( qlmj* , hlmj* ) −β ∂b2lmj ( hlmj* ) α ( ) m m m lmj lmj ∂qlmj ∂hlmj ∂hlmj m=1 l =1 j=1 m=1 l =1 j =1 2* 3* * * * * M N 2 ∂f ∂ ∂ Q , Q c q , h r q , h ( ) ( ) ( ) 2mnj 2mnj mnj mnj mnj mnj * * × ×( hlmj − hlmj + +αm − ρmnj ) + m ∂q ∂qmnj ∂qmnj m=1 n=1 j =1 mnj * * * * * * M N 2 ∂c ( q ,h ) ∂v ( h ) ∂r ( q ,h ) ∂b ( h ) ( qmnj −qmnj* ) + 2mnj ∂hmnj mnj + 2mnj∂h mnj +αm 2mnj ∂hmnj mnj −βm 2mnj∂h mnj × m=1 n=1 j=1 mnj mnj mnj mnj M L 2 M O ∂f Q2*, Q3* ( ) ∂c ( q* , h* ) ∂r ( q* , h* ) * − hmnj ) + m ∂q + 2mo∂qmo mo +αm 2mo ∂qmo mo − ρmo* ×( qmo −qmo* ) + m=1 o=1 mo mo mo (h mnj * * ∂c2mo ( qmo ,hmo ) + ∂vmo ( hmo* ) +α ∂r2mo ( qmo* ,hmo* ) −β ∂bmo ( hmo* ) × h −h* + ( mo mo ) m m ∂hmo ∂hmo ∂hmo ∂hmo m=1 o=1 L M 2 M L 2 * * * ρlm* j +c2lmj ( qlm* j ,hlm* j ) −ρm* ×( qlmj −qlm* j ) + qlm j −dm ( ρ2 ) ×( ρm −ρm) ≥0 l=1 m=1 j=1 m=1 l=1 j=1 M O ( (q mnj ) ≥ 0, qlmj ≥ 0,0 ≤ hlmj ≤ 1,0 ≤ hmnj ≤ 1,0 ≤ hmo ≤ 1, qmo ≥ 0, ∀l, j, n, o ) (11) cn .

6. Results of Anchor Nodes and Mean Localization Error Fig. 7. Results of Communication Radius and Mean Localization Error Fig. 8. 6, the number of anchor nodes changes from 15 to 35 with node communication radius 60m and 250 nodes in all in the space region. From the figure, the mean localization error of OSSDL is lower than that of DV-Hop at any number of anchor nodes. When there is 26 or 33 anchor numbers, differences between them is the most obvious. From the trend line, we can infer that with the increase of the number of anchor, OSSDL performs better and better than DV-Hop propagation.

### 1996 IEEE International Conference on Systems, Man and Cybernetics by IEEE

by Robert

4.4