We analyse a principal-agent contracting model with asymmetric information between a supplier and a retailer. Both the supplier and the retailer have the classical non-linear economic ordering cost functions consisting of ordering and holding costs. We assume that the retailer has the market power to enforce any order quantity. Furthermore, the retailer has private holding costs. The supplier wants to minimise his expected costs by offering a menu of contracts with side payments as an incentive mechanism. We consider a general number of discrete singledimensional retailer types with type-dependent default options.A natural and common model formulation is non-convex, but we present an equivalent convex formulation. Hence, the contracting model can be solved efficiently for a general number of retailer types. We also derive structural properties of the optimal menu of contracts. In particular, we completely characterise the optimum for two retailer types and provide a minimal list of candidate contracts for three types. Finally, we prove a sufficient condition to guarantee unique contracts in the optimal solution for a general number of retailer types.

We analyse a two-echelon discrete lot-sizing problem with a supplier and a retailer under information asymmetry. We assume that all cost parameters are time independent and that the retailer has single-dimensional continuous private information, namely either his setup cost or his holding cost. The supplier uses mechanism design to determine a menu of contracts that minimises his expected costs, where each contract specifies the retailer's procurement plan and a side payment to the retailer. There is no restriction on the number of contracts in the menu.To optimally solve this contracting problem we present a two-stage approach, based on a theoretical analysis. The first stage generates a list of procurement plans that is sufficient to solve the contracting problem to optimality. The second stage optimally assigns these plans to the retailer types and determines all side payments. The result is an optimal menu with finitely many contracts that pools retailer types. We identify cases for which the contracting problem can be solved in polynomial time and provide the corresponding algorithms. Furthermore, our analysis reveals that information asymmetry leads to atypical structures in the plans of the optimal menu, e.g., plans violating the zero-inventory property. Our solution approach and several results are directly applicable to more general problems as well.

a b s t r a c tWe consider the Maximum Weighted Coverage problem (MCP). We can relate the MCP to optimisation problems using submodular functions. Performance guarantees of the Swap Local Search algorithm are known for these problems, but can be improved for the MCP. Our main contribution is a constructive proof of tight performance guarantees for Swap Local Search applied to the MCP, which provides insight into the structure of worst-case MCP instances, and has the potential to be applicable to other optimisation problems.

We consider principal-agent contracting models between a seller and a buyer with singledimensional private information. The buyer's type follows a continuous distribution on a bounded interval. We present a new modelling approach where the seller offers a menu of finitely many contracts to the buyer. The approach distinguishes itself from existing methods by pooling the buyer types using a partition. That is, the seller first chooses the number of contracts offered and then partitions the set of buyer types into subintervals. All types in a subinterval are pooled and offered the same contract by the design of our menu.We call this approach robust pooling and apply it to utility maximisation and cost minimisation problems. In particular, we analyse two concrete problems from the literature. For both problems we are able to express structural results as a function of a single new parameter, which remarkably does not depend on all instance parameters. We determine the optimal partition and the corresponding optimal menu of contracts. This results in new insights into the (sub)optimality of the equidistant partition. For example, the equidistant partition is optimal for a family of instances for one of the problems. Finally, we derive performance guarantees for the equidistant and optimal partitions for a given number of contracts. For the considered problems the robust pooling approach has good performances with only a few contracts.