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Economists express profits as areas representing producer's surplus or consumer's surplus corresponding to a pair of supply and demand functions. A similar representation can be employed in a supply chain network where there can be several producers/suppliers and several consumers/retailers in various trading situations. We aim to graphically represent the profits corresponding to a combination of the decisions of all participants in a supply chain, simultaneously, as areas of non-overlapping regions on the same graph. This way, the shares of the participants in the total can be visually observed and the interdependencies and the inefficiencies in the chain can be detected where the decisions and the corresponding profits are generally interdependent. Such a visual tool can be used in evaluating as well as in designing a supply chain.

Economists express profits as areas representing producer's surplus or consumer's surplus corresponding to a pair of supply and demand functions. A similar representation can be employed in a supply chain network where there can be several producers/suppliers and several consumers/retailers in various trading situations over a single period. A retailer and all suppliers contributing to the production of the item sold by that retailer share the profit created by the final demand of that retailer. We use this fact to graphically represent the profits corresponding to a combination of the decisions of the participants in a supply chain, simultaneously, as areas of non-overlapping regions on the same graph. This methodology is the main contribution of the present study. It also provides geometric conditions for equilibrium. Using this methodology, the profits can be visually observed; the interdependencies and the inefficiencies in the chain can be detected by inspection. This representation can be used in evaluating as well as in designing a supply chain as a complementary visual tool to mathematical analysis. It can be used in teaching courses related to supply chain as well as in situations for which visual impact is more suitable. The tasks performed on a party of a product can be considered as a network. Hence, the suggested representation can also be used for visualizing the value/cost stream generated by the tasks within a firm. The supply chain configuration is important for this representation. Although it is straight forward to represent these profits as areas in two-dimensional plane, our approach is general enough to be applied in different chain configurations: a vertical chain starting with a supplier ending with a retailer; a retailer and his suppliers; a lateral chain consisting of a supplier and his retailers; and finally a supply chain network. In each case, we try to characterize the equilibrium decisions of all parties geometrically, define the areas of non-overlapping regions corresponding to the participants' profits. In the present work, it is not our purpose to find the equilibrium decisions in different supply chain settings; for most of the network configurations considered here, equilibrium decisions have been studied already. The single retailer multiple capacitated supplier chain equilibrium is worked in relatively more detail since its setting is different than those in the literature. A comprehensive review of the supply chain models can be found in Simchi-Levi et al. (2004). Among many others, Lariviere and Porteus (2001) consider equilibrium with single retailer and single supplier and the factors effecting it; Li (2008) and Shang and Song (2007) work on the equilibrium decisions in a vertical supply chain; Chen et al. (2001), Geng and Mallik (2007), Wu et al. (2012) and Chan and Lee (2012) consider single supplier and multiple retailers; Seshadri et al. (1991), Chen et al. (2001), Iyengar and Anuj (2008), Qi (2007) and Glock (2012) consider single buyer and multiple suppliers; Ganeshan (1999) and Seshadri et al. (1991) consider a set of suppliers selling to a set of retailers. They analyze profit maximizing decisions. Cachon and Lariviere (2005) use graphs on which the profits of a single retailer and a single supplier are areas of non-overlapping regions. The present work generalizes the graph to the profits in a supply chain network. We aim (i) to give a representation for any combination of decisions, (ii) to characterize the equilibrium decisions geometrically if possible and (iii) to compare them in terms of this representation. Profits under different settings are usually compared using their magnitudes that depend on participants' decisions as well as the uncontrollable parameters. For example, if a retailer profits more than another, this may be due to his higher selling price rather than his better decisions. In the proposed representation, the profits are necessarily scaled by appropriate market price. As a result, “the market price effect” is reduced in comparisons. Supply chain performance is measured in many dimensions. Gunasekaran et al. (2007) give a general framework detailed in these dimensions including cost and flexibility considerations. We propose a simple efficiency measure in terms of the “scaled” profits in a decentralized chain with respect to the centralized chain to capture the losses due to decentralization, chain configuration, inefficient costs as well as loses due to arbitrary decisions. It shows the percentage of the total available profit that can be created in the market captured by the chain. This measure is straight forward for single retailer chains because it is simply the ratio of the profit of decentralized chain to the profit of centralized chain. When there are several retailers with different market prices in a supply chain it is not equal to that profit ratio; scaling the profits with the corresponding prices it provides a means to aggregate chain profits and to measure efficiency eliminating the market price effect. This measure improves as the chain becomes more “compact”/centralized. We assumed random market demand of the chains however the analysis follows with any demand that is a function of price

Profits of the participants in a supply chain are created by the combinations of their decisions. In the present study, we propose a methodology to represent these profits simultaneously as the areas of non-overlapping regions in (x,y) plane. This methodology can be applied to supply chain networks with mixed configurations over a period. For each configuration, we also give some geometric conditions to visually check if a given set of decisions constitute an equilibrium. So, it can be used for evaluating current distribution of the profits, observing the effects of a change in an existing supply chain as well as designing a new one to eliminate decisions disturbing equilibrium or parameter values causing inefficiencies. Clearly, it is not a substitute for the mathematical analysis; rather it is a complementary tool especially when visual evaluation is necessary. The proposed efficiency measure compares the decentralized and centralized chains. It measures how much the area below the demand curve is converted to profit. It eliminates the effect of market prices so that a participant's dominating profit due to high selling price is subject to a more “fair” comparison. It is very simple for single retailer chains but especially useful when there are several retailers with different market prices and different demand curves since it aggregates profits scaling with the corresponding market price. Using the same methodology, the value/cost stream generated by all the tasks within a firm can also be represented on two dimensional plane.