Downstream tier sites

The decision timeline for an A/-tiered problem is shown in Figure 4, where the upper arrows indicate the entity tasks and the lower arrows show the information disclosure timeline. The flow functions are independently designed by the upstream tier sites and communicated to the subsequent downstream tier sites. The algorithm starts at the sites in tier 1 to determine the flow function between itself and the second tier sites, given the source supply functions, which describe the variation of the collected amount with the collection fee between the sites in tier 1 and the sources. The sites in tier 1 communicate flow functions to the sites in intermediate tier 2. Each intermediate tier site independently determines the associated flow functions and communicates them to its next tier sites. This proceeds sequentially until the last tier is reached. The sites in the downstream... 

At the other end of the system, we assume that the amount of raw materials resulting from the decomposition of end-of-life products and used products is relatively small compared to the quantity in the virgin raw material and brand-new product markets. This observation leads to the assumption that the selling prices of raw materials or used products in final demand markets are hxed amounts, not affected by the sales quantities. We let denote the selling price obtained in downstream boundary tier site k e P . In other words, the two main exogenous information streams to the system are the source supply functions of the collected recycled item amount in the upstream boundary tier, represented by Si = ai -I- bifor site i e /i, and the selling price obtained in the down-stream boundary tier, denoted by for site k e Im- Here, a, bi, and... 

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. 

We let x-J denote the material flow from site i e Im to site j e Im+i and denote the aggregate flow shipped from the sites in tier M — 1 to downstream boundary tier site k e Im- The decision variables for the system are the optimal material flow allocation, and within the system and the collection fee, in the upstream boundary tier. We also define the following system parameters ... 

We assume the upstream tier designs the price-flow contract, shown in the equation (1), below which is a mechanism describing the correspondence between the acquisition prices offered by downstream sites and the flow amount supplied by the upstream sites to its subsequent downstream sites. The material... 

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