Production, Manufacturing and LogisticsCompact bidding languages and supplier selection for markets with economies of scale and scope
Highlights
► We introduce a bidding language for markets with economies of scale and scope. ► We propose a mixed integer program to solve the supplier selection problem. ► We discuss the impact of language features on computational effort and total spend. ► We conclude that practically relevant problem sizes can be solved to optimality.
Introduction
Economies of scale and scope describe key characteristics of a supplier’s production function that influence allocations and prices on procurement markets. This paper is motivated by real world procurement negotiations of an industry partner, where procurement managers need to purchase large volumes of multiple items. This might be the yearly demand for different types of tires in the car industry or for different types of memory chips or hard disks in the PC industry. Often in these circumstances, economies of scale are jointly present with economies of scope. On the one hand suppliers that set up a finishing line for a certain product have high setup costs, but low marginal costs leading to a unit price degression. On the other hand economies of scale arise in shipping and handling a larger number of items to a customer, and in the joint procurement of raw materials. Unit prices can not only decrease, however. A supplier might increase unit prices, if the demand exceeds his capacity and he has to work shifts or purchase items from third-party suppliers.
Split-award auctions are regularly used for multi-item, multi-unit negotiations, where the best bidder gets the larger share of the volume for a particular quantity and the second best bidder gets a smaller share (e.g., a 70/30% split) (Anton and Yao, 1992, Perry and Sakovics, 2003, Anton et al., 2009). This is done to assure supply if one supplier is unable to fulfill his contract and a minimal supplier pool in the long run. With significant economies of scale, suppliers face a strategic problem in simple split-award auctions with only a single unit price. Since there is uncertainty about which quantity they will get awarded, they might speculate and bid less aggressively based on the unit cost for the smaller share. In other words, simple split award auctions do not allow suppliers to adequately express economies of scale.
In the recent years, driven by the new possibilities of the Internet, a growing literature is devoted to the design of optimization-based markets (aka. smart markets) (Gallien and Wein, 2005), and in particular to combinatorial auctions, where bidders are allowed to submit bids on packages of discrete items (Cramton et al., 2006). The promise of these mechanisms is that by allowing market participants to reveal more comprehensive information about cost structures or utility functions, this can drastically increase allocative efficiency and lead to higher economic welfare. Unfortunately, the matching of complex preference profiles typically leads to hard optimization problems.
The literature in this field is typically focused on multi-item but single-unit negotiations and respective auction formats do not easily extend to multi-unit markets with economies of scale. While preference elicitation is already a fundamental problem for bidders in single-unit combinatorial auctions, it becomes prohibitive in multi-unit combinatorial auctions. Markets with significant economies of scale and scope require a fundamentally different bidding language that allows to specify discount rules rather than a huge number of multi-unit package bids.
In this paper, we will introduce a compact bidding language and the respective allocation problem for procurement markets with economies of scale and scope. This language combines logical spend conditions defined on volume and quantity with different types of discounts. Two central types of volume discounts discussed in the literature are incremental discounts and total quantity discounts. Total quantity discounts have been described as a discount policy, where the supplier has specified a number of quantity intervals (aka. discount intervals), and the price per unit for the entire quantity depends on the discount interval in which the total amount ordered lies (Goossens et al., 2007). In contrast, incremental volume discounts describe a discount policy, where the discounts apply only to the additional units above the threshold of the quantity interval. In business practice, such discount policies are often also defined on spend or on spend and quantity for one or more items. In addition, we will also allow for lump sum discounts, defining a one time reverse payment on overall spend or quantity.
So far, optimization formulations only exist for incremental or for total quantity discount bids, defined on quantity purchased. We focus on a bidding language, which provides considerably more flexibility in the discount policies used. Often increased expressiveness of a formal language comes at the cost of increased computational complexity (Papadimitriou, 1993, Dantsin et al., 2001). It is important to understand, if increased expressiveness of this bidding language also leads to a higher computational burden during the supplier selection.
Procurement managers are not only interested in the cost-minimal solution to this optimization problem. They typically apply a number of side constraints in an interactive manner to find a good solution that considers a number of strategic and operational goals. Examples are upper and lower bounds on the number of suppliers or the volume awarded to one or a group of suppliers. These side constraints are not always known in advance exactly and whether they should be considered or not also depends on their impact on the total cost. For example, a purchasing manager wants to have no more than five suppliers, but he would also be willing to accept six or seven, if it reduces total cost significantly. This interactive exploration of different award scenarios based on allocation constraints and different types of volume discounts is also referred to as scenario analysis, and a number of decision support tools are provided by e-sourcing vendors in this area (Gartner, 2008). Although, this type of decision support is vital for many companies and leading to multi-million dollar procurement decisions, there is surprisingly little academic literature in this field.
Scenario analysis poses tight time constraints to allow for an interactive exploration of different award scenarios. As we will show, the winner determination problem is an -complete problem. A main research question in this paper is, for which problem sizes (number of bidders, bids, items, and discount intervals) a procurement manager can hope to solve respective instances optimally in acceptable time. Optimality of the solutions is desirable, as the allocation problem tries to minimize the cost of the buyer, and even a small decrease in procurement cost will directly impact company gain. In addition, we will analyze the impact of different discount policies on total cost. Some discount policies make it easier to approximate the underlying cost functions closely.
In summary, the contributions in this paper are the following: We will introduce a compact bidding language for markets with economies of scale and scope, referred to as . Our bidding language allows for two different types of discounts, which have already been discussed in the literature: incremental and total quantity discount bids. Our approach allows to handle both types of volume discounts, and we have seen several applications, where different bidders submit different types of volume discount bids. In addition to previous approaches, allows for lump sum discounts on total spend to model economies of scope and various conditions on spend or quantity for the different discount types. As a result, is considerably more expressive than previous approaches and gives suppliers high flexibility in specifying their offerings. Apart from expressiveness, we introduce description length as an important criterion for bid languages, since bidders cannot be expected to submit arbitrarily many parameters or bids. We will see that there are considerable differences between bundle bids, bids with total quantity or with incremental volume discounts.
In this work, we will investigate the buyer’s problem, who needs to select quantities from suppliers providing bids in such that his costs are minimized and his demand is satisfied. We will refer to this problem as supplier quantity selection (SQS) problem and propose a respective mixed integer program (MIP). Modeling matters and there are considerable differences in the solution time depending on different model formulations. We will also discuss additional allocation constraints as they are typically used for scenario navigation.
Procurement managers need a clear understanding of which problem sizes they can analyze in an interactive manner during the scenario analysis or in dynamic auctions. Therefore, we will show that SQS is -complete, and report on an extensive evaluation of the empirical hardness of the supplier quantity selection problem. Similar analyses have recently been performed for the winner determination problem in combinatorial auctions (Leyton-Brown et al., 2009). It is important that problem instances for the experimental evaluation mirror real-world characteristics. We introduce a cost function for markets with scale and scope economies and generate bids based on this cost function. and the software framework used in this paper have already been used to support a number of high-stakes sourcing decisions with an industry partner. The synthetic bids matched the characteristics of those that we also found in the field. The experimental results show that realistic problem sizes of the SQS problem can be solved to optimality in a matter of minutes with IBM’s CPLEX (version 12) and Gurobi 2.0. (All results reported are based on CPLEX.) We also find that the shape of the underlying cost function and the demand can have a significant impact on the runtime of the problems and empirical evaluations need to be interpreted with care.
Previous work has only focused on the computational complexity of the winner determination problem. The cost curves in our experimental evaluation allow us to compare the total cost achieved with bids and different types of volume discounts and bids for split-award auctions. While we do not discuss mechanism design questions in our analysis, we assume a direct revelation mechanism where bidders submit bids that best reflect their cost curves. Even if bidding behavior in the lab or in the field is different, this result suggests that a richer bidding language can lead to considerably lower cost and more efficient results in markets with economies of scale and scope with . However, we also find that total quantity discounts with only a few intervals can lead to higher spend than simple split-award auctions.
In Section 2, we provide an overview of related literature. Section 3 introduces , a bid language for markets with economies of scale and scope, and discusses relevant features of the language. In Section 4 we formulate the winner determination problem as a mixed integer program, and propose various extensions in Section 5. Section 6 describes the experimental design, while Section 7 summarizes the main results of our experimental analysis. Finally, Section 8 provides conclusions and an outlook on future research.
Section snippets
Related literature
The literature on supplier selection and volume discounts includes studies of various discounting schemes, such as unit discounts (Silverson and Peterson, 1979), inventory models with demand uncertainty and incremental quantity discounts and carload quantity discounts (Jucker and Rosenblatt, 1985, Lee and Rosenblatt, 1986). Munson and Rosenblatt (1998) provide a perspective on discounts used in practice, while Chaudhry et al. (1993) discuss a vendor selection model in the presence of price
Bidding language
In this section, we will introduce a bidding language allowing to describe a supplier’s cost function. We focus on markets with economies of scale and scope, where bidders typically express discounts in order to reflect these economic characteristics. A language in computer science and logic assigns a semantic to a syntax. Bidding languages have been studied in the context of combinatorial auctions (Boutilier and Hoos, 2001, Abrache et al., 2004, Nisan, 2006). However, this research typically
The supplier quantity selection problem
In the following, we will investigate a buyer’s problem, who needs to select quantities from each supplier providing bids in such that his costs are minimized and his demand is satisfied. We will refer to this problem as supplier quantity selection (SQS) problem and introduce a respective mixed integer program (MIP) in the following.
We will first introduce some necessary notation. We will use uppercase letters for parameters, lowercase letters for decision variables, and calligraphic fonts
Scenario analysis
We will focus on scenario analysis as a typical use case. During scenario analysis, procurement managers typically use additional side constraints to explore different award scenarios. For example, a purchasing manager might be interested in an optimal allocation with a maximum of 5 winners, or the optimal allocation, where the spend on an individual supplier is limited to 1 million dollars due to certain risk considerations. An ex-post analysis based on already submitted bids also allows to
Experimental setup
Initial experimental analyses on the winner determination problem in combinatorial auctions have used simple value distributions, which have been criticized mainly for lacking economic justification (Andersson et al., 2000, de Vries and Vohra, 2003). Leyton-Brown et al. (2009) put emphasis on the selection of realistic instance distributions for the analysis of a computational problem. Note, that the cost function imposes some structure on the bids generated, which has an impact on the
Experimental results
We will now summarize the results of our experiments with respect to computation time to solve the SQS problem and total spend.
Summary and conclusions
We have suggested a bidding language for markets with economies of scale and scope and a respective mixed integer program to solve the resulting supplier quantity selection (SQS) problem. The bidding language is considerably more expressive than what has been discussed in the literature so far and includes incremental volume discounts, total quantity discounts, lump sum discounts, and a variety of conditions defined on spend and quantity of selected items. While both, incremental volume
Acknowledgement
This project is supported by the Deutsche Forschungsgemeinschaft (DFG) (BI 1057/1-1).
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