Linear Transportation Costs Research Articles

DEVELOPING SUPPLY CHAIN NETWORK WITH PIECEWISE LINEAR TRANSPORTATION COST FOR A SMALL-AND-MEDIUM ENTERPRISE (SME) IN CILEGON

This study proposed a supply chain network for determining suppliers’ location in which the transportation costs are a piecewise linear function. The supply chain network consists of a production facility, suppliers, and customers. These types of costs are found in the fields of transportation, logistics, and purchasing discount. First, the supply chain network is formulated as the mixed-integer non-linear programming (MINLP) because piecewise linear transportation cost makes the model non-linear. Then, the model is transformed into a mixed-integer programming (MIP) model using the convex-combination method to overcome this nonlinearity. The model was used for solving the problem faced by a small and medium enterprise (SME) in Cilegon. The MIP was solved using the CPLEX software. Sensitivity analysis was carried to provide the SME with several alternatives in handling the suppliers’ location problem

Inbound replenishment and outbound dispatch decisions under hybrid shipment consolidation policies: An analytical model and comparison

We investigate operational decisions in a distribution warehouse facing stochastic demand and incurring fixed and linear transportation costs for the inbound inventory replenishment and outbound dispatch decisions. In order to realize scale economies associated with transportation both on the outbound and inbound sides, dispatch schedules and replenishment decisions at the warehouse must be synchronized over time. Immediate delivery policies on the outbound side are not financially viable because outbound dispatch operations will benefit from temporal shipment consolidation. Our focus in this setting is the analytical modeling of hybrid shipment consolidation policies and their comparison to the time-based and quantity-based counterparts. To this end, we propose analytical (exact and approximate) methods to compute and compare the cost under hybrid policies relative to its alternatives. Since shipment consolidation impacts customer waiting and inventory holding, we also investigate the average delay per order and average inventory per time unit as two important metrics of the distribution operation’s performance, along with the annual cost. We compare these metrics among the alternative under hybrid, time-based, and quantity-based policies. The comparison then allows us to offer an explicit analytical comparison of long-run average cost under these three policies without needing to solve the corresponding optimization problems. Notably, our results offer an analytical characterization of relative cost performance (vis-a-vis the numerical comparison available in the literature) and demonstrate the implications of alternative shipment consolidation policies regardless of the values of model parameters. The results are of practical value in the context of the design and operation of an integrated framework for inventory-transportation systems.

Sparse Regularized Optimal Transport with Deformed q-Entropy.

Optimal transport is a mathematical tool that has been a widely used to measure the distance between two probability distributions. To mitigate the cubic computational complexity of the vanilla formulation of the optimal transport problem, regularized optimal transport has received attention in recent years, which is a convex program to minimize the linear transport cost with an added convex regularizer. Sinkhorn optimal transport is the most prominent one regularized with negative Shannon entropy, leading to densely supported solutions, which are often undesirable in light of the interpretability of transport plans. In this paper, we report that a deformed entropy designed by q-algebra, a popular generalization of the standard algebra studied in Tsallis statistical mechanics, makes optimal transport solutions supported sparsely. This entropy with a deformation parameter q interpolates the negative Shannon entropy (q=1) and the squared 2-norm (q=0), and the solution becomes more sparse as q tends to zero. Our theoretical analysis reveals that a larger q leads to a faster convergence when optimized with the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. In summary, the deformation induces a trade-off between the sparsity and convergence speed.

Hump-Shaped Cross-Price Effects and the Extensive Margin in Cross-Border Shopping

This paper examines the effect of cross-border shopping on grocery demand in Norway using monthly store × category sales data from Norway’s largest grocery chain 2012–2016. The sensitivity of demand to the foreign price is hump-shaped and greatest 30–60 minutes’ driving distance from the closest foreign store. Combining continuous demand, fixed costs of cross-border shopping, and linear transport costs a la Hotelling, we show how this hump shape can arise through a combination of intensive and extensive margins of cross-border shopping. Our conclusions are further supported by novel survey evidence and cross-border traffic data. (JEL D12, F31, L11, L81)

Two-stage risk-averse stochastic programming approach for multi-item single source ordering problem: CVaR minimisation with transportation cost

Integrating inventory and transportation decisions is vital in supply chain management and can enable decision-makers to achieve competitive advantages. This study considers a multi-item replenishment problem (MIRP) with a piece-wise linear transportation cost under demand uncertainty, which usually occurs both in retail and production environment when several items must be ordered from a single supplier. Conventionally, two-stage stochastic programming formulation is risk-neutral, and it lacks robustness in the presence of high data variability. Hence, we introduce the Conditional Value at Risk (CVaR) approach for MIRP. Additionally, we deploy both single and multi-cut L-shaped and the sample average approximation method to circumvent the computational complexity to solve large-scale instances. The data-driven simulation study is used to benchmark the results from deterministic, risk-neutral, and risk-averse stochastic models. The results indicate that under higher data variations, the risk-averse model provides better perspectives for a decision-maker. The results show a 40–50% reduction in lost sales with marginal growth in total cost while considering CVaR instead of a risk-neutral approach.

Two-Player Location-Price Game in a Spoke Market with Linear Transportation Cost

This paper investigates the location game of two players in a spoke market with linear transportation cost. A spoke market model has been proposed, which is inspired by the Hotelling model and develops two-player games in price competition. Using two-stage (position and price) patterns and the backward guidance method, the existence of price and location equilibrium results for the position games is proved.

Equilibrium existence in the Hotelling model with convex production costs

We revisit the spatial duopoly model à la Hotelling (1929), to show that, provided the parameter scaling marginal cost is sufficiently high, quadratic production costs guarantee equilibrium existence in presence of linear transportation costs and a uniform distribution, with minimum product differentiation and no undercutting. We also discuss the conditions under which partial coverage arises. Finally, we extend the duopoly game to generalise the condition for the existence of the pure-strategy equilibrium to a class of convex production cost functions.

Location Problem of Production and Sales under Carbon Tax Policy

This paper considers a location problem of production and sales of a production-oriented enterprise under carbon tax policy. Consumers are distributed on a straight line with linear transportation costs. In addition to paying the transportation costs of its own products, the enterprise is also responsibile for the transportation costs of consumers. A profit maximization model is established, properties of the profit function are analyzed, and a solving method is given. Numerical experiments show that even if the distribution of consumers is center-symmetric, the optimal location is not necessarily center-symmetric, which corrects the illusion of people. It is concluded that different carbon tax policies lead to different cost ratios of different modes of transportation, which lead to different locations and profits.

Mixed duopoly in a Hotelling framework with cubic transportation costs

We examine a two-stage location-price model of a mixed duopoly where a private profit-maximizing firm competes with a public welfare-maximizing firm in a Hotelling-type framework. A noteworthy result in this model is that, with quadratic transportation costs, which has become the usual assumption in the literature, the socially optimal locations are obtained in equilibrium. We show here that under the alternative assumption of cubic transportation costs this result no longer holds: equilibrium locations are socially suboptimal. We find that just as in the case of linear transportation costs, previously studied in the literature, for some locations there is price equilibrium in the second stage of the game and for other locations there is not. But, in contrast with such a case, there is a location pair for which there is price equilibrium in the second stage of the game and neither firm has an incentive to marginally change its location. We also find that, in contrast with the case of quadratic transportation costs, this location pair is not socially optimal.

Spatial market equilibrium in the case of linear transportation costs

In this article, we study the spatial market equilibrium in the case of fixed demands and supply values, the requirement of equality in regard to overall supply and overall demand, and linear transportation costs. The problem is formulated as a nonlinear optimization program with dual variables reflecting supply and demand prices. It is shown that the unique equilibrium commodity assignment pattern is obtained explicitly via equilibrium prices. Moreover, it is proved that in order to obtain absolute values of equilibrium prices, it is necessary to establish a certain base market price. Therefore, once the base market price is given, then other prices are adjusted according to spatial market equilibrium.

Hump-Shaped Cross-Price Effects and the Extensive Margin in Cross-Border Shopping

This paper examines the effect of cross-border shopping on grocery demand in Norway using monthly store*category sales data from Norway's largest grocery chain 2011-2016. The sensitivity of demand to foreign price is hump-shaped and greatest 30-60 minutes' driving distance from the closest foreign store. Combining continuous demand, fixed costs of cross-border shopping and linear transport costs `a la Hotelling we show how this hump-shape can arise through a combination of intensive and extensive margins of cross-border shopping. Our conclusions are further supported by novel survey evidence and cross-border traffic data.

Hump-shaped Cross-price Effects and the Extensive Margin in Cross-border Shopping

This paper examines the effect of cross-border shopping on grocery demand in Norway using monthly store*category sales data from Norway's largest grocery chain 2011-2016. The sensitivity of demand to foreign price is hump-shaped and greatest 30-60 minutes' driving distance from the closest foreign store. Combining continuous demand, fixed costs of cross-border shopping and linear transport costs `a la Hotelling we show how this hump-shape can arise through a combination of intensive and extensive margins of cross-border shopping. Our conclusions are further supported by novel survey evidence and cross-border traffic data.

How transport costs affect the decision to purchase a new or a remanufactured good

We provide the first strategic analysis of the interaction between a continuum of potentially green consumers and two firms in regional science. Firm 1 (2) sells new (remanufactured) toner cartridges. Each firm selects its price and a consumer purchases from the firm that offers her the highest utility. Utility is given by a surplus measure, the price, and by the transport cost incurred in traveling to a firm’s location. We first derive the best response functions of the two firms. Second, we stipulate a numerical value for the surplus measure and show that when the two firms select their “monopoly” prices, the Nash equilibrium is unique. Third, we specify a linear transport cost function with a constant coefficient and show that the costlier it is for consumers to get to the locations of the two firms, the higher is the price charged by these two firms. Finally, our analysis shows that there is a need to study models in which the two toner cartridges are dissimilar, the interaction between consumers and firms is repeated, and behavioral factors are taken into account.

Uniqueness of market equilibria on networks with transport costs

We study the existence and uniqueness of equilibria for perfectly competitive markets in capacitated transport networks. The model under consideration is rather general so that it captures basic aspects of related models in, e.g., gas or electricity networks. We formulate the market equilibrium model as a mixed complementarity problem and show the equivalence to a welfare maximization problem. Using the latter we prove uniqueness of the resulting equilibrium for piecewise linear and symmetric transport costs under additional mild assumptions. Moreover, we show the necessity of these assumptions by illustrating examples that possess multiple solutions if our assumptions are violated.

Green zoning regulation under price discrimination

This study aims to examine urban zoning within a linear city in a Bertrand duopolistic competition framework with price discrimination and linear transportation costs. It analyses the effects of introducing an environmental area where economic and residential activity are not allowed. The welfare function used to determine the optimal size of the green area allows for a possible regulator’s bias in favour of firms/consumers. It is shown that location-price competition can be either reduced or increased depending on the size of the green area. The results indicate when a regulator implements green zoning, under linear transportation costs, influences the optimal location of firms (because these locations depend on the size of the green zone). In consequence, zoning may be used as an effective industrial or urban policy tool.

Product Differentiation Research

Product differentiation has been an important field in industrial organization economics and has inspired theoretical and empirical research on competition in strategic management and marketing, among other disciplines. To better understand this important field, we review the literature in a critical manner. Here we have two purposes. First, we update some recent developments in the last decade or two. One interesting aspect of recent developments is an active incorporation of consumer heterogeneity, elastic demand in the model, two-side markets with network externalities, among others. Second, we assess the sensitivity of equilibrium outcomes on their assumptions. A critical review of the literature allows us to argue that the assumptions of quadratic transportation costs, elastic demand, and concave consumer distribution are influential in determining the level of differentiation. These assumptions, individually or together, increase product differentiation, possibly to the maximum level, by making consumers quite sensitive to (even small) price changes. And we also argue that when we assume multiple products and asymmetric cost positions, we have results similar to those suggested by resource endowments argument in strategic management. Based on the review, we offer several suggestions for future research. First, given the influential effect of the assumption of quadratic transportation costs, we suggest that researchers should make it clear why they choose to assume quadratic transportation costs over linear transportation costs. We believe this issue could be solved rather easily if researchers are more explicit about the applications of derived results rather than the derivation itself. Second, the impact of elastic demand on the degree of differentiation suggests that researchers should be particularly cautious in empirical research. Any condition that triggers price rigidity should be explicitly represented in the equation. Otherwise the degree of differentiation would not be properly estimated due to the endogeneity caused by the absence of this condition on the right side of the equation. Third, regarding unknown distribution of consumer density, researchers should be advised to include income distribution in their empirical model. Unlike unknown distribution of consumer heterogeneity, income distribution could be available if the target of the application of derived results is known. Lastly, it may prove to be quite fruitful if researchers build upon some of recent developments. Researchers may want to extend vertical differentiation in two-side markets with network externalities that may have interesting implications on on-demand economy or researchers may want to take up the notion of variable elasticity of substitution that could open up new possibilities of expanding research streams on supply-side (e.g., production costs) of product differentiation.

Spatial firm competition in two dimensions with linear transportation costs: simulations and analytical results

Models of spatial firm competition assume that customers are distributed in space and transportation costs are associated with their purchases of products from a small number of firms that are also placed at definite locations. It has been long known that the competition equilibrium is not guaranteed to exist if the most straightforward linear transportation costs are assumed. We show by simulations and also analytically that if periodic boundary conditions in two dimensions are assumed, the equilibrium exists for a pair of firms at any distance. When a larger number of firms is considered, we find that their total equilibrium profit is inversely proportional to the square root of the number of firms. We end with a numerical investigation of the system's behavior for a general transportation cost exponent.

Costly location in Hotelling duopoly

We introduce a cost of location into Hotelling׳s (1929) [9] spatial duopoly model. We derive the general conditions on the cost-of-location function under which a pure strategy price-location Nash equilibrium exists. With linear transportation cost and a suitably specified cost of location that rises toward the center of the Hotelling line, symmetric equilibrium locations are in the outer quartiles of the line, ensuring the existence of pure strategy equilibrium prices. With quadratic transportation cost and a suitably specified cost of location that falls toward the center of the line, symmetric equilibrium locations range from the center to the end of the line.

Local market structure in a Hotelling town

We develop a theoretical framework to study the location-price competition in a Hotelling-type network game, extending the Hotelling model, with linear transportation costs, from a line (city) to a network (town). We show the existence of a pure Nash equilibrium price if, and only if, some explicit conditions on the production costs and on the network structure hold. Furthermore, we prove that the local optimal localization of the firms are at the cross-roads of the town.

Efficient formulation and heuristics for multi-item single source ordering problem with transportation cost

Integrated inventory and transportation decisions are critical in the supply chain, providing significant gains for all parties. In this paper, we present a mathematical formulation for the dynamic demand multi-item single source replenishment problem with a piecewise linear transportation cost. Through an extensive experimental study, we find that the new formulation provides a tighter LP relaxation of the problem, while requiring fewer computational resources to optimally solve the problem when compared with existing model in the literature. We also present a new metaheuristic for this general class of coordinated capacitated replenishment problems. On average, the solutions from heuristics are within 1.23% of the optimal solution for the comprehensive set of test problems.