Upstream boundary tier entities

We let / = 1, , 7, , Wm denote the set of sites in tier m. The entities in the upstream boundary tier collect recycled items from the source, and the source supplies the upstream boundary tier site on the basis of a fee paid by the upstream boundary tier site. We let S denote the collection amount in upstream boundary tier site i e /i and be the collection fee per unit of the recycled item paid by site i e Ii. We characterize the collection amount for the upstream boundary tier site i e /i by a linear function 5, = a, bi, where a,- and fe, are parameters and Ui, bi > 0, The collection fee, pf ° of site i e /i is without sign restriction. The use of a linear function allows the analysis of the problem to be simplified. It captures a qualitative market behavior of increased flow with either an increased payment or a decreased collection fee (charged by the collector). In other words, the upstream boundary tier site may pay or charge for collecting recycled items if p is positive or negative, respectively. 

The principle of the decentralized set-up is that the network system is composed of several independent entities individually operated by self-interested parties. Each independent entity has its own profit function subject to its own processing or transportation constraints, and is not willing to reveal its own information to other entities or the public. Often the decision variables for each entity in a decentralized system are also influenced by other entities decisions. The foundations of the decentralized RPS models are derived from our recent work in a multitiered RPS network. Using this decentralized RPS network framework, we obtain the equilibrium collection fee paid by the upstream boundary tier site and the resulting material flow allocation, as well as the internal transaction prices within the network. 

Acquisition prices are set by the downstream tier and passed back to the upstream tier sequentially from the downstream to upstream boundary tier, as shown in the lower part of Figure 4. In this chapter, we focus on a single horizon of the transaction problem where entities do not collect items in excess of the amount they need to supply to the next tier. Due to this flow conservation rule, the resulting flows can be determined as the acquisition prices are reaUzed. Finally the sites in the first tier decide the collection fees to acquire recycled items from sources. 

---