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Keywords:

  • auctions;
  • pricing;
  • scheduling;
  • capacity allocation

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The Model
  5. 3. Profit Maximization When All Information Is Public
  6. 4. Private Information Problem
  7. 5. Computational Analysis
  8. 6. Concluding Remarks
  9. References
  10. Supporting Information

We consider a pricing and short-term capacity allocation problem in the presence of buyers with orders for bundles of products. The supplier's objective is to maximize her net profit, computed as the difference between the revenue generated through sales of products and the production and inventory holding costs. The objective of each buyer is similarly profit maximization, where a buyer's profit is computed as the difference between the time-dependent utility of the product bundle he plans to buy, expressed in monetary terms, and the price of the bundle. We assume that bundles' utilities are buyers' private information and address the problem of allocating the facility's output. We directly consider the products that constitute the supplier's output as market goods. We study the case where the supplier follows an anonymous and linear pricing strategy, with extensions that include quantity discounts and time-dependent product and delivery prices. In this setting, the winner determination problem integrates the capacity allocation and scheduling decisions. We propose an iterative auction mechanism with non-decreasing prices to solve this complex problem, and present a computational analysis to investigate the efficiency of the proposed method under supplier's different pricing strategies. Our analysis shows that the problem with private information can be effectively solved with the proposed auction mechanism. Furthermore, the results indicate that the auction mechanism achieves more than 80% of the system's profit, and the supplier receives a higher percentage of profit especially when the ratio of demand to available capacity is high.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The Model
  5. 3. Profit Maximization When All Information Is Public
  6. 4. Private Information Problem
  7. 5. Computational Analysis
  8. 6. Concluding Remarks
  9. References
  10. Supporting Information

The need for an integrated approach for product pricing and capacity management decisions can be observed in revenue management or equipment procurement considerations, reflecting both short- and long-term issues in the management of production facilities (Fleischmann et al. 2004). In a typical long-term capacity planning and pricing problem, capacity provisions and pricing decisions are usually integrated with models that consider customers' aggregate response, such as price-dependent total demand. In short-term planning, on the other hand, the capacity is usually assumed to be fixed, and the capacity management problem becomes an allocation problem, and, in finding a solution, each customer's individual preferences and response to prices are explicitly taken into consideration. Moreover, when customers' preferences are their private information, centralized decision making is no longer possible, and other decentralized mechanisms, such as auctions (McAfee and McMillan 1987), are needed to solve the allocation problem.

In this study, we consider the pricing and capacity allocation problem of a supplier (or, interchangeably, manufacturer) facing a number of orders from potential buyers. We focus on short-term capacity allocation decisions and assume that the capacity of the manufacturing facility cannot be adjusted during the planning horizon. Integration of pricing decisions with capacity allocation considerations is a challenging task when buyers do have privately held information. In this study, we specifically consider the case where buyers are utility maximizing buyers with privately held information on certain attributes of their orders.

The integrated pricing and capacity allocation problem is typically observed in the process industries such as paper and steel production. In these industries, the final products are in some standard measures—generally some prespecified widths of rolls or sheets—and the demand and supply are indivisible. When we consider the paper industry, we see that the paper suppliers provide paper in various grades, standard widths, and roll sizes. When these grade/width/roll size combinations are treated as “market goods,” the framework we present in this study can be used to model the problem of allocating a paper mill's output. Although there are many B2B electronic exchange websites such as go2paper.com (Go2paper, Inc. 2012) providing a medium to bring the paper buyers and sellers together, their tools lack the capability to capture capacity allocation considerations and deal only with the pricing aspect of the problem.

The capacity allocation dimension of our problem can be viewed as an indirect resource allocation problem, because the solution of the problem involves the allocation of the production facility's capacity among potential buyers while maximizing the net revenue of the manufacturer. In an earlier study, Hylland and Zeckhauser (1979) consider the problem of allocating individuals to positions with limited capacities where the discrete resource allocation problem is solved without any transfer of money. Hall and Liu (2010) consider a capacity and order allocation problem, and investigate the benefits of distributors if they share their allocated capacities. Furthermore, they discuss the benefits of coordination between the manufacturer and the distributors in the multiple product supply chain setting where a manufacturer serves to several distributors. Cho and Tang (2011) consider a supply chain with multiple retailers and one supplier with limited capacity. They allocate supplier capacity when it is exceeded by the retailer orders using the constructed competitive allocation mechanism. This mechanism eliminates the gaming effect and reduces the inefficiencies of the decentralized supply chain.

In the environment we study, it is assumed that the manufacturer, as the owner of the capacity, communicates with the potential buyers through asking prices for the products that constitute the output assortment of the facility. In turn, taking the prices set by the manufacturer into consideration, buyers place orders for bundles of products. In this regard, our problem setting diverges from the single-object auctions (Milgrom and Weber 1982) and has similarities with multi-unit auctions (Demange et al. 1986, Dobzinski and Nisan 2007) and combinatorial auctions (Blumrosen and Nisan 2007, Cramton et al. 2007, de Vries and Vohra 2003), where a fixed number of units of each product type is allocated to buyers. Demange et al. (1986) propose two auction mechanisms for the multi-item case to achieve an incentive-compatible minimum price equilibrium.

The single supplier–multiple buyers with private information problem, along with its variants, has been extensively discussed in the game theory literature (see the references cited above), and market-based mechanisms, such as iterative auctions, have been proposed. In our problem, however, because the number of units of each product type to be sold is dependent on the production or scheduling (or capacity allocation) decisions during the planning horizon, the number of units to be auctioned is not fixed a priori. More specifically, the problem we study is along the lines of decentralized scheduling problems addressed in Hall and Liu (2008a, 2008b), Kutanoğlu and Wu (1999), Liu (2007), Wellman et al. (2001), Reeves et al. (2005), where the capacity of the facility is directly allocated to the buyers. In one of the earlier studies, Kutanoğlu and Wu (1999) consider a job-shop setting and discuss the links between Lagrangean-based decomposition and combinatorial auctions. They present an allocation mechanism that auctions time slots without taking incentive compatibility issues into consideration. Wellman et al. (2001) consider a problem with multiple buyers each having a single job that can be preemptively processed on a common facility and present ascending auction mechanisms for individual or bundles of time slots. Reeves et al. (2005) study buyers' bidding strategies when the auction mechanism involves simultaneous markets for individual time slots. Hall and Liu (2008a, 2008b), Liu (2007) study the non-preemptive single machine case with buyers each having a single job and discuss the properties of the auction mechanism, along with the complexity of the winner determination problem, when time slots, fixed time blocks, and flexible time blocks are auctioned as market goods. A computational study of the proposed auctions reveals that auctions with flexible time-blocks deliver better performance.

In this study, we consider a different problem by considering buyers with desired bundles that may consist of multi-unit demands for multiple products. In regard to the auction mechanism design, we also follow a different path by treating products themselves as the market goods and simultaneously allocating capacity to products and products to buyers. In addition to designing an auction mechanism, which is the primary objective of the study, another objective is to develop alternative pricing strategies and study their impact on the allocation efficiency of the proposed mechanism. To achieve these ends, the study is structured as follows: in section 'The Model', we present a detailed description of our problem setting. In section 'Profit Maximization When All Information Is Public', for mainly benchmarking purposes, we study variants of the problem under the assumption that all relevant information is public. In section 'Profit Maximization When All Information Is Public', we first consider the problem of maximizing a system's net profit (section 3. 'The System's Net Profit without Pricing'). We then study the manufacturer's pricing problem (section 'Manufacturer's Net Profit with Pricing') when all information is public and the incentive compatibility issues are taken into consideration. We conclude section 'Profit Maximization When All Information Is Public' with a discussion of the variants of the manufacturer's pricing problem with constant and time-dependent product prices, quantity discounts, and constant and time-dependent delivery prices. In section 'Private Information Problem', we present an iterative auction mechanism for the case where buyers do not share their private information. The main components of the auction mechanism, winner determination and pricing problems, are discussed in sections 'Winner Determination Problem' and 'Pricing Problem', respectively. In section 'Computational Analysis', we present a computational analysis of the proposed auction mechanism's performance in terms of various efficiency criteria. We conclude in section 'Concluding Remarks' with an overview of the main findings of the study and a discussion of future research directions.

2. The Model

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The Model
  5. 3. Profit Maximization When All Information Is Public
  6. 4. Private Information Problem
  7. 5. Computational Analysis
  8. 6. Concluding Remarks
  9. References
  10. Supporting Information

The manufacturer's planning horizon spans time periods t = 1,2,…,T. The product assortment offered by the facility includes J product types. The manufacturing process of the facility is modeled as a single machine that can process one unit of a product type at any point in time. Each product type j, j = 1,2,…,J, requires a processing time of inline image, which is assumed to be the manufacturer's private information. The manufacturer's operations are subject to two types of costs: inline image, the per unit time processing cost for product type j, and inline image, the per unit time holding cost for product type j. We assume that there are no sequence-dependent setup times between the product types.

The manufacturer has N potential buyers, each with an order for bundle inline image, where inline image, is the number of units of product type j buyer i wants to purchase. We also assume that inline image. The earliest delivery time buyer i is willing to accept is inline image and the time-dependent utility of bundle inline image when delivered at time inline image is inline image We do not make any assumptions about the functional form inline image. In a typical setting, however, inline image can be expressed as

  • display math

where inline image is the desired delivery time, inline image (inline image) is the bundle inline image's early (late) delivery cost per unit time. We note that the maximum possible utility of buyer i is equal to a constant, inline image, and it is achieved only when inline image expressed in monetary terms. Hence, the type of a buyer is determined by the earliest delivery time, inline image, the desired bundle delivery time, inline image, the maximum utility that a buyer could have, inline image, and the time-dependent net utility function, inline image.

Let inline image be the unit price of product type j when delivered at time t, set by the manufacturer. In addition, we may further extend the manufacturer's price menu to include a time-dependent delivery price inline image The delivery price can also be modeled as a special type of a product by introducing an additional unit of this product type to every bundle. However, to investigate the impact of the delivery price with this approach, we need to create additional bundle data sets for the cases in which the delivery price is included in the pricing scheme. We choose to model the delivery price as a separate variable to study the corresponding cases. We note that the manufacturer's pricing scheme is anonymous, i.e., the manufacturer does not employ a price-based discrimination among the buyers. Then, the net utility of buyer i, when his bundle inline image is delivered at time inline image with prices inline image and inline image is given by inline image For a given matrix inline image of product prices and a vector inline image of delivery prices, each buyer's decision problem is equivalent to a search on the time line: inline image

We assume that, except the product quantities each buyer wants to purchase, (i.e., inline image), information of a buyer is private. In this private information setting, the manufacturer's objective is to find the product and delivery prices (i.e., inline image and inline image) that will maximize her net profit in the planning horizon. However, because buyers' cost parameters are their private information, the manufacturer cannot solve her problem centrally. In section 'Profit Maximization When All Information Is Public', we will discuss the problem where all information is considered to be public before we present an iterative auction mechanism to solve the manufacturer's problem in section 'Private Information Problem' The solution values obtained for the problem where all information is public will constitute the benchmark profit levels in evaluating performance of the auction mechanism.

3. Profit Maximization When All Information Is Public

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The Model
  5. 3. Profit Maximization When All Information Is Public
  6. 4. Private Information Problem
  7. 5. Computational Analysis
  8. 6. Concluding Remarks
  9. References
  10. Supporting Information

In this section, we assume buyers' information is public. First, we ignore incentive compatibility and pricing issues, and model the system's net profit maximization problem. We then consider the manufacturer's and system's net profit maximization problems under different pricing schemes and in the presence of incentive compatibility considerations. We note that the problems analyzed in this section are relevant only for benchmarking the performance of the private information approach, which will be subsequently discussed.

3.1. The System's Net Profit without Pricing

In this section, we assume that the manufacturer and buyers operate within the same business unit, and incentive compatibility issues, therefore prices, are not relevant. In this setting, the objective is to maximize the difference between the net utility, or profit, buyers receive with their accepted bundles and the cost of processing and delivering these bundles.

Let inline image be the bundle (or, interchangeably, order) acceptance decisions where inline image if buyer i's order is accepted to be delivered at time t and 0 otherwise. The objective of the full information problem (CP) can now be expressed as

  • display math(1)

where inline image and inline image are the holding and processing costs of the accepted orders.

For a given set of accepted orders, the production planning problem resembles a lot sizing problem with multiple products. Let inline image be the production planning decisions where inline image if one unit of product type j is scheduled to be completed at time t and 0 otherwise. Similarly, let inline image be the inventory levels where inline image denotes the on-hand inventory position of product type j at time t including the product completed at time t. Assuming inline image the changes in the inventory level of each product type can be written as

  • display math(2)

and the capacity of the production facility can be modeled as

  • display math(3)

and

  • display math(4)

Constraints (3) simply force inline image to be equal to zero in the inline image time interval if the facility is scheduled to complete product j at time t, and constraints (4) eliminate infeasible production completion assignments in the beginning of the planning horizon. In addition to constraints (2), (3), and (4), it is assumed that a bundle can be delivered at most once, i.e.,

  • display math(5)

and at most one product can be completed at time t, i.e.,

  • display math(6)

and decision variables are such that

  • display math(7)

In this problem, we assume that the buyers' utilities are time dependent and may take negative values for certain time points in the planning horizon. Moreover, the capacity of the facility may be inadequate to process all the bundles within this horizon. Therefore, in an optimal solution of the full information problem where the decision maker allocates the capacity of the facility among the buyers, a bundle would not be delivered either because its utility is negative at the delivery time or it is impossible to process that bundle with the others in the planning horizon.

With the production plan inline image and inventory levels inline image that satisfy the delivery requirements of order acceptance decisions inline image the processing and inventory holding costs of the facility can be expressed as inline image CP can now be written as

  • display math
  • display math

We now present a result on the computational complexity of CP:

THEOREM 1. CP is NP-hard in the strong sense.

PROOF. A proof of Theorem 1 is provided in Appendix S1.

In the proof of Theorem 1, we use the reduction technique by considering a very restricted case of the CP: we set the early delivery cost inline image, holding cost inline image, and unit production cost inline image to all be equal to 0 for i, j = 1,…,N. In addition, we take a sufficiently long planning horizon T (i.e., inline image) so that every order can be processed within the planning horizon. Also, we let inline image such that every buyer's utility is positive for all delivery times:

  • display math

Even in this very restrictive setting, the CP is proved to be NP-hard in the strong sense.

3.2. Manufacturer's Net Profit with Pricing

In this section, we assume buyers' information is public and first consider the manufacturer's net profit maximization problem as a Stackelberg game with the manufacturer as a leader. We then consider the problem of the system's net profit maximization.

The manufacturer, as the market leader in the Stackelberg game, determines an anonymous price schedule (i.e., inline image and inline image, and each buyer, as a follower, determines the delivery time that maximizes his net profit under the price schedule announced by the manufacturer. If the maximum net profit of a buyer is negative, the buyer does not submit his order. The manufacturer, therefore, needs to choose a price schedule that would ensure participation of buyers whose orders will be manufactured and delivered in the planning horizon. Recalling that inline image if buyer i decides to submit his order to be completed at time inline image, we should have

  • display math(8)

Similarly, if buyer i decides not to submit his order in the (1,T) time interval, we should have

  • display math(9)

To simplify the search process, if no buyer submits an order to be delivered at time inline image, the prices for this particular period are assumed to be large:

  • display math(10)

where M is large constant. The manufacturer's problem MP can now be written as follows:

  • display math
  • display math
  • display math

The last three constraints are expressed in conditional form to simplify the exposition; they can be transformed into regular constraints with the introduction of additional binary variables.

We now present a result on the computational complexity of MP.

THEOREM 2. MP is NP-hard in the strong sense.

PROOF. A proof of Theorem 2 is provided in Appendix S1.

We use the reduction technique by considering a very restricted case of the MP in the proof of Theorem 2. Even in this very restrictive setting, the MP is proved to be NP-hard in the strong sense.

The problem MP can be readily extended to maximize the system's net profit (SP) by changing the objective function as follows:

  • display math

We note that SP, too, is NP-hard in the strong sense.

3.3. Alternative Pricing Schemes

The pricing strategies studied in this study will mainly be the variations of linear and anonymous, i.e., non-discriminatory, product prices. The variations we consider throughout this study are constant and time-dependent product prices and product prices with quantity discounts. In the linear, non-discriminatory price setting, the total bundle price is simply the summation of the individual product prices multiplied by the required quantities. For the cases where we assume constant product prices, the prices are the same for every customer independent of time point t in the planning horizon. The product prices are identical for every customer for a specific delivery time t for the time-dependent non-discriminatory product price assumption. The delivery price is defined as the fixed price charged per bundle without taking the contents of the bundle into consideration. We note that when delivery prices are positive, the pricing scheme no longer qualifies as a linear pricing strategy.

The formulations of MP and SP, as presented in section 'Manufacturer's Net Profit with Pricing', assume that product and delivery prices are time dependent. When the product prices are constant, we include the following constraint in MP and SP:

  • display math(11)

where inline image is a new set of decision variables denoting constant product prices. Similarly, when delivery prices are assumed to be constant, we include the following constraint in MP and SP:

  • display math(12)

where L is the constant delivery price.

Finally, with quantity discounts, we expand decision variables inline image to account for quantities ordered by different buyers by adding a third index. Let inline image be the maximum demand for product type j: inline image. Then, to allow for quantity discounts in pricing, we include the following constraint in MP and SP:

  • display math(13)

and compute the total price of bundle inline image to be delivered at time t as inline image (in lieu of inline image). We note that the constraints (13) allow for time-varying quantity discounts, and to have a quantity discount scheme that is not time dependent, an additional set of constraints can be included in MP and SP:

  • display math(14)

where inline image are additional decision variables.

The combinations of the above discussed product and delivery price options result in 12 different pricing strategies, which will be analyzed in detail in section 'Computational Analysis'

4. Private Information Problem

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The Model
  5. 3. Profit Maximization When All Information Is Public
  6. 4. Private Information Problem
  7. 5. Computational Analysis
  8. 6. Concluding Remarks
  9. References
  10. Supporting Information

In this section, we consider the problem in case the buyers keep all the information regarding their bundles, except the desired product types and quantities, private. We propose an auction mechanism through which the manufacturer can allocate its capacity when the buyers do not share their private information on their preferred delivery times, the maximum utilities they could gain from receiving the bundles, and the structure of their utility functions. This section is structured as follows: in section 'Iterative Auction Mechanism', we describe the auction mechanism and its bidding structure, in section 'Winner Determination Problem' we study the manufacturer's winner determination problem, and in section 'Pricing Problem', we study the pricing problem of the manufacturer.

4.1. Iterative Auction Mechanism

The literature on the allocation and scheduling of capacity using a market-based mechanism focuses mainly on the case where market good is an individual time slot or combination of time slots (Hall and Liu 2008b, Liu 2007, Reeves et al. 2005, Wellman et al. 2001).

The allocation of the time slots via iterative auction mechanisms has been shown to be useful in job scheduling. Dewan and Joshi (2002) use an iterative auction mechanism to schedule the jobs within a distributed decision-making environment utilizing global information. Their proposed iterative price adjustment method is found to be improving the system performance, although the job and machine agents solve myopic problems.

When the market good is defined as a single slot or a combination of time slots in an auction, the buyers bid for the time slots, and this requires them to know about the processing requirements of their orders. In this study, we take a different path and assume that the finished products of the manufacturing facility are the market goods, keeping the processing requirements of each product type the private information of the manufacturer. We also assume that the buyers bid for the bundles of products and are single-minded (Blumrosen and Nisan 2007), i.e., they are interested in a single bundle of products and receive zero value for any other bundle of products. Although the value a buyer gains from the bundle is time variant, he is still interested in the identical bundle of products and hence is deemed as single-minded. Recalling that inline image is the bundle of buyer i, i = 1,2,…,N, we define the triplet inline image as the bid of buyer i for bundle inline image to be delivered at time t with a bundle price of inline image. With this single-minded bidding policy, a buyer bids for multiple market goods that constitute his/her bundle. As the prices are announced for every time point, a buyer considers the time where he/she maximizes his/her net utility, and if this net utility is positive, he/she extends a single bid for this point in time.

We propose an iterative auction mechanism to solve the private information case. In continuous iterative auctions, a bidder can update his bid at any time point, whereas in discrete or round-based auctions, the bidders are allowed to update their bids after the manufacturer sets the prices for the current round. If the prices in successive rounds of the auction are non-decreasing, then the auction is referred to as a non-decreasing price auction. We consider a round-based price-setting auction with non-decreasing prices throughout this study. The market goods we consider in the auction are the product types the manufacturer can deliver. The manufacturer sets the prices for each product type, and each buyer announces the time period when he wants the products that are in his bundle to be delivered at the posted prices (de Vries and Vohra 2003).

In each iteration of the auction, the manufacturer solves two problems: (1) the Winner Determination Problem (WDP) to maximize her net profit with the submitted bids and (2) the Pricing Problem (PP) to adjust the prices in a direction that could increase his net profit in the next iteration of the auction. The steps of the iterative auction can be summarized as follows:

  • Step 0:
    Let r be the iteration index and inline image be an indicator function that denotes if buyer i's bid is a winning bid or not in the rth iteration of the auction. Also let inline image be the per unit bidding increment in the rth iteration of the auction, and inline image be the maximum number of rounds. Let the initial value of r be equal to zero, the initial bid of buyer i be equal to inline image, and inline image. Let also inline image and inline image be the initial product and delivery prices, respectively, the manufacturer chooses.
  • Step 1:
    Let r = r + 1. If inline image go to Step 4.
  • For all i = 1,2,…,N, such that inline image let inline image and inline image

For all i = 1,2,…,N, such that inline image:

  • Let inline image be the time where the buyer i's net utility is maximized with the current ask prices of the auction, that is, inline image where inline image
  • If inline image buyer i submits bid inline image where inline image otherwise we let inline image and inline image
  • Step 2:
    If inline image and inline image for all i = 1,2,…,N, i.e., there are no new bids, go to Step 4; otherwise WDP is solved with the current bids (section 'Winner Determination Problem') to determine the winning bids in the rth iteration of the auction, i.e., inline image
  • If there is a new winning bid, i.e., there exists a buyer i for which inline image and inline image invite all winning buyers to re-submit their bids with the prices inline image, inline image, and the bid increment inline image, and re-solve the Winner Determination Problem with the adjusted bids. Proceed with Step 3 when there are no new winning bids.
  • Step 3:
    Given inline image compute the prices inline image and inline image for the next iteration of the auction (section 'Pricing Problem'), and go to Step 1.
  • Step 4:
    The auction is terminated with the allocation inline image.

We note that, in Step 2, the re-submission of bids is required to maintain uniformity in prices. As we will shortly discuss within the context of the Pricing Problem, the re-submitted bids would be larger than their original values at most by the bid increment. Some of the bids may be withdrawn in the process, however, as we will discuss in the computational results section, this does not negatively affect the overall effectiveness of the auction mechanism.

4.2. Winner Determination Problem

Given a set of K ≤ N bids (i.e., triplets inline image where i(k) is the index of the buyer who submitted bid k), the WDP involves maximization of the manufacturer's net profit by selecting the bids that would be accepted and generating a production plan, or a schedule, to deliver the products demanded by the accepted bids.

WDP can be cast as a shortest path problem as follows: Let G be a network with a source node, a sink node, and T + K disjoint sets of nodes. A given set of nodes corresponds either to the completion of a time period (there are T such node sets) or to the acceptance–rejection decision about bid inline image. We note that K ≤ N. Each node in the network characterizes the state of the inventory levels and the production system and has J + 2 elements. Element j, j = 1, 2, …, J, corresponds to the inventory levels of product type j, J + 1st element corresponds to the type of the product that is being processed by the facility, and finally the J + 2nd element denotes the remaining processing time of the product being processed.

In Figure 1, we illustrate the nodes and related arcs of the first four node sets in the network representation of a two-product problem. The processing times of Products 1 and 2 are 2 and 3, respectively. We assume that there is a bid as (2,(1,0),5). The set of nodes grouped in the column denoted by t = 1 represents the end of the first time period, and at the end of the first time period the system can be in one of the following states: the inventory levels of Products 1 and 2 are zero, and Product 1 is in the production process with one time unit of processing left (top node); the inventory levels of Products 1 and 2 are zero, and Product 2 is in the production process with two time units of processing left (middle node); the inventory levels of Products 1 and 2 are zero, and the facility is idle (bottom node). The arc cost between the initial node of the network and the top node at level t = 1 of the network is equal to inline image and, similarly, the arc cost between the initial node of the network and the middle node is equal to inline image The facility is idle in the first time period when the system is in the state represented by the bottom node; therefore the arc cost between the initial node and the bottom node is equal to zero.

image

Figure 1. Network Representation of the Winner Determination Problem

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As there are no bids to be evaluated at the end of the first time period, we proceed with the set of nodes (grouped in the column denoted by t = 2) that represent the system's possible states at the end of the second time period. The bottom node in this set of nodes represents the case where the processing of Product 1 is finished and the completed unit is in the inventory. The cost of the arc that enters this node is equal to inline image, the unit time processing cost of Product 1. Similarly, the middle node represents the state where the processing of Product 2 has continued in the second time period, with one time unit of processing left. Similarly, the cost of entering arc is equal to inline image. The other three nodes represent the possible decisions that could be taken in the beginning of the second time period and are identical to the node set denoted by t = 1.

The next node set, grouped in the column denoted by k = 1, represents the acceptance–rejection decision of the first bid ((2,(1,0),5)). Note that this bid can be accepted only when the inventory position of Product 1 is at least one. The arc from the bottom node of node set t = 2 to the fourth node of node set k = 1 represents the acceptance decision, and its cost is equal to inline image All other arc costs are equal to zero, because they correspond to rejection of the bid.

The node set and relevant arcs for t = 3 are created in a similar manner. The arcs that emanate from the bottom node of node set k = 1 must now account for the inventory holding cost of Product 1 along with the corresponding production costs. For example, the arc cost between the bottom node of node set k = 1 and bottom node of node set t = 3 is equal to inline image Finding the shortest path in this network will give the optimal solution of WDP.

The above outlined network will have at most T + N stages. The number of nodes in each stage will depend on the individual processing times of products. For example, under the assumption that inline image and including the state where the inventory level is equal to 0, the inventory level of Product j, j = 1, 2, …, J, can have at most inline image different states. Similarly, in terms of production decisions, we have J + 1 options for the J + 1st element, and at most T stages for the J + 2nd element of a given node. Therefore, the number of nodes at any stage is bounded above by inline image and the total number of nodes in the network is bounded above by inline image which is inline image when we assume that T > N. Similarly, because a node can be entered from at most J + 1 nodes, the number arcs in the network is inline image. When J, the number of products, is bounded above by a constant, the shortest path problem can be solved in polynomial time.

4.3. Pricing Problem

In order to execute the iterative auction mechanism presented in section 4. 'Iterative Auction Mechanism', we have to solve the pricing problem in Step 3. The solution procedure we propose for this problem is built on the concept of “pseudo-dual prices.” In multi-round combinatorial auctions, the bidders submit bids on the bundles and the auctioneer designates a temporary allocation of her capacity to the bidders at each round. In this case, the bidders can potentially use the approximate dual information of the winner determination problem to bid more effectively in the subsequent rounds. However, when the linear programming relaxation of the winner determination problem does not carry the integrality property, the bidders cannot use the dual information directly. Then, a pseudo-dual pricing approach can be employed to construct the guiding prices relying on the solution of the winner determination problem in some sense. The derivation of the pseudo-dual prices, in general, involves the solution of the dual of the linear relaxation of WDP (Kwasnica et al. 2005, Rassenti et al. 1982). The pseudo-dual pricing approach updates prices in a direction that might lead to a competitive equilibrium where the system's total payoff is maximized (Bichler et al. 2009). However, as discussed in Pikovsky and Bichler (2005), in combinatorial auctions with indivisible goods, a competitive equilibrium may not exist with linear anonymous prices. Hall and Liu (2011) further show that an optimal allocation that is in equilibrium may not exist for the combinatorial market goods defined by Wellman et al. (2001). Therefore, with linear anonymous prices, the proposed pseudo-dual pricing scheme is simply an approximate solution procedure.

The objective in the pseudo-dual pricing approach is to update the prices such that (1) the updated prices support the allocation obtained with the current prices as closely as possible, and (2) price increases between successive updates are as small as possible. When these two requirements are met, the buyers' participation in the auction is sustained for higher number of iterations.

We present the pricing problem with the most general pricing strategy that includes time-dependent product prices with quantity discounts and time-dependent delivery prices. Given prices inline image, and winning bids inline image, in the rth iteration of the auction, the prices for the r + 1st iteration can be set by consecutively solving the Pricing Problems inline image and inline image.

In problem inline image, the objective is to minimize the maximum difference between the bid amount of a non-winning bid and the price of that bid with the adjusted prices in the rth iteration of the auction:

  • display math
  • display math(15)
  • display math(16)
  • display math(17)
  • display math(18)
  • display math(19)
  • display math(20)

In the model of inline image, constraints (15) and (16) try to carry the results of WDP obtained in the rth iteration of the auction to the r + 1st iteration. Because the ideal prices where the WDP solution is perfectly supported may not exist, the constraints (16) include non-negative decision variables inline image, which represent the required bid adjustment to make the non-winning status of buyer i's bid compatible with the prices that will be offered in the r + 1st iteration. In inline image, we aim to minimize the maximum value of such adjustments. The ideal value of inline image, inline image's optimal solution value, is zero; however, when inline image, there exist multiple solutions where some constraints may be unnecessarily adjusted. Therefore, in inline image, we aim to minimize the sum of all adjustments, given each adjustment is less than or equal to the maximum adjustment obtained in inline image. We present the mathematical model of pricing problem inline image in Appendix S1.

Finally, in inline image and inline image we minimize the maximum price and the sum of prices, respectively, to maintain price increases at the lowest possible level. We present the models for pricing problems inline image and inline image in Appendix S1.

5. Computational Analysis

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The Model
  5. 3. Profit Maximization When All Information Is Public
  6. 4. Private Information Problem
  7. 5. Computational Analysis
  8. 6. Concluding Remarks
  9. References
  10. Supporting Information

In this section, we present a computational analysis of the full information and private information problems. All problems are solved with IBM ILOG (2012) Optimization Studio v.12 on an Intel® 2.10 GHz Coreinline image2 Duo processor with 3 GB of RAM.

We first consider six sets of problems with N = 12,J = 2, T = 100. For each set, we consider 50 randomly created problem instances. In all problem sets, we let per unit time processing costs and holding costs for product types j = 1, 2 be equal, i.e., inline image and inline image We generate buyers' utilities in Problem Sets 1–3 as follows: inline image. In Problem Sets 4–6, buyers' utilities are generated as follows: inline image. We present the remaining problem parameters in Table 1.

Table 1. Parameter Values
Set no. inline image inline image inline image inline image inline image inline image
  1. inline imageinline image is an indicator function with a value of 1 when inline image, 0 otherwise.

1(6,4)(DU(1,3),DU(1,3))DU(40,50)0U(1.0,2.5)U(1.0,2.5)
2(5,3) inline image DU(30,70)0U(0.5,1.5)U(0.5,1.5)
3(3,1) inline image DU(30,70)0U(0.5,2.5)U(0.5,2.5)
4(6,4)(DU(1,3),DU(1,3))DU(40,50) inline image U(0.5,1.5)
5(3,2)(DU(1,3),DU(1,3))DU(40,50) inline image U(0.5,1.5)
6(3,1)(DU(1,3),DU(1,3))DU(30,70) inline image U(0.5,1.5)

We consider 12 different pricing strategies, as enumerated in the first three columns of Table 2 (see section 'Alternative Pricing Schemes' for a discussion of pricing strategies). We will refer to the pricing strategies using a three field notation; for example, pricing strategy C/Z/N will refer to the strategy with Constant product prices/Zero delivery price/No quantity discount. All studied pricing strategies are uniform, i.e., all product types are sold and bundles are delivered at the same, possibly time-dependent, prices for all buyers. In pricing strategies with quantity discounts, however, there exists an indirect discrimination, because buyers with bundles that contain more units of a product type may receive lower prices (Stole 2007). We also note that, within the context of our pricing framework and when the delivery prices are equal to zero, the pricing strategies we present fall into the category of linear pricing schemes.

5.1. Analysis of Benchmark Solutions in the Presence of Full Information

In Table 2, we first report the results of the full information problems in 300 randomly generated instances (50 instances for each problem set listed in Table 1). The average solution times for CP, SP, and MP have been 1.07, 5.95, and 28.90 CPU seconds, respectively. For each problem instance-pricing strategy combination, we first solve CP, without taking pricing issues into consideration, to determine the maximum profit of the system. We then solve the full information problem, with the pricing considerations, first to maximize the system's net profit (SP), then to maximize the manufacturer's net profit (MP). We report the results as a fraction of the system's maximum profit without pricing considerations. In other words, we report the allocation efficiency obtained in SP and MP in relation to the maximum possible profit for the system.

Table 2. Performance with the Full Information Approach in Problems with J = 2 and N = 12
Pricing strategyObjective
Product pricesDelivery pricesQuantity discountsSystem's profitManufacturer's profit
Total (%)Manufacturer (%)Buyers (%)Total (%)
ConstantZeroNo 97.79 81.13 8.84 89.98
ConstantZeroYes97.9990.123.2593.36
ConstantConstantNo97.8688.134.2992.42
ConstantConstantYes97.9991.292.8194.10
ConstantVariableNo99.8598.060.8298.88
ConstantVariableYes99.8698.700.5199.20
VariableZeroNo99.9597.781.3199.09
VariableZeroYes99.9599.010.4299.44
VariableConstantNo99.9598.880.5199.39
VariableConstantYes99.9599.150.3899.52
VariableVariableNo99.9599.050.4399.48
VariableVariableYes99.9599.170.3599.52

The results reported in Table 2 indicate that pricing strategies we consider in this study are not guaranteed to achieve the maximum system profit. In SP, pricing strategies with variable product prices or variable delivery prices, on the other hand, can achieve maximum system profit in most of the problem instances. The simplest pricing scheme C/Z/N, results in an approximately 2.21% (10.02%) average loss of efficiency in SP (MP), whereas the more flexible strategies C/V/Y and V/·/· result in less than 1% average efficiency loss in both SP and MP.

Another interesting observation is the existence of linear prices that would transfer almost all of the system's maximum profit to the manufacturer. In maximizing the manufacturer's profit, pricing strategy V/Z/Y, for example, achieves an average total efficiency of 99.44%, and less than 0.5% of this total efficiency is transferred to buyers to meet their incentive compatibility requirements. In a sense, when all information is public, the buyers do have very little power, and the manufacturer can extract almost all of the system's profit with specific pricing strategies that include variable product prices.

5.2. Analysis of the Performance of Iterative Auction Algorithm in the Private Information Problem

In Table 4, we report the performance of the iterative auction mechanism presented in section 'Iterative Auction Mechanism' over the same set of problem instances in the case when the buyer information other than the requested bundle components is private. The iterative auction mechanism, which is an approximate solution procedure to solve this private information problem, implicitly addresses the question of how to allocate the profits of the system between the manufacturer and the buyers. The initial prices and bid increment values, which are set in Step 0 of the iterative auction, could have a significant impact on this allocation. Therefore, we have selected the initial prices and bid increment values as given in Table 3 after a number of experiments aiming to maximize the manufacturer's net profit. The maximum number of rounds is selected as 100, and the bid increment value is increased by 0.05 every 10 rounds.

Table 3. Selection of Initial Prices and Bid Increments
 Pricing strategy
  C/V V/V
 C/ZC/CV/ZV/C
Bid increment (inline image)0.50.51.01.0
Initial price of product type j (inline image) inline image 0.0 inline image 0.0

While selecting the initial product prices, we only use the requested product's information (such as the per unit time processing cost and the processing time for product type j, inline image and inline image) which is known to the manufacturer.

Table 4. Iterative Auction's Performance in Problems with J = 2 and N = 12
Pricing strategyProfits with auctionNumber of iterations
Product pricesDelivery pricesQuantity discountsManufacturer (%)Buyers (%)Total (%)
ConstantZeroNo 51.11 31.75 82.86 62.14
ConstantZeroYes54.9233.2188.1252.19
ConstantConstantNo50.9333.9084.8363.64
ConstantConstantYes53.1732.5185.6860.49
ConstantVariableNo52.5534.3886.9354.18
ConstantVariableYes53.2033.7986.9952.62
VariableZeroNo49.9939.1889.1777.08
VariableZeroYes50.2238.3488.5686.35
VariableConstantNo53.7733.8187.5874.29
VariableConstantYes54.2633.8488.1082.06
VariableVariableNo52.5635.2787.8385.98
VariableVariableYes52.1135.4887.5990.94

The results reported in Table 4 indicate that, with different pricing strategies, the iterative auction is able to achieve an average system efficiency in the [82.86,89.17] range with an average value of 87.02% and an average profit for the manufacturer, in terms of system's maximum total profit, in the [49.99, 54.92] range with an average value of 52.40%. In terms of the manufacturer's net profit, the C/Z/Y pricing strategy, although it is not one of the more flexible strategies, delivers a slightly better performance. The observation that the iterative auction is not able to fully exploit the higher levels of pricing flexibility (with the V/V/Y pricing strategy, for example) highlights the approximate nature of pseudo-dual prices used in solving the pricing problem. We also note that, in problems with J = 2, N = 12, and T = 100, the average (maximum) CPU time for one iteration of the auction has been 0.18 (3.07) seconds.

The iterative auction algorithm performs differently for different problem sets as well as for different pricing strategies. As expected, variable product prices make the algorithm run longer on the average. For the strategies for which the product prices are constant, 28% of the problems hit to the iteration limit whereas 35% of the problems with variable product prices do so. When we investigate this proportion for different problem sets, we observe that 66% of the instances in Problem Set 5 hit the iteration limit while only 12% of the instances do so for Problem Sets 2 and 3. This is because, in Problem Set 5, desired delivery times are tighter than they are for different buyers in Problem Sets 2 and 3. Tighter delivery times cause more buyers to bid for the same interval, resulting in more rounds for the auction.

5.3. Analysis of the Effect of Quantity Discounts on the Iterative Auction's Performance

As far as the effect of quantity discounts on the algorithm's efficiency is considered, one can observe that the algorithm performs remarkably well when the delivery prices are zero and the product prices are constant. Efficiency of the algorithm is diminished, but still statistically significant, when the delivery prices exist but are constant. The effect of quantity discounts is not statistically significant at the 5% confidence level when product prices are variable even if the delivery prices are kept constant.

5.4. Analysis of the Effect of Different Levels of Capacity Scarcity on the Iterative Auction's Performance

In an auction setting, the ratio of demand to the available capacity can have a significant impact on the profit the auctioneer, in our case the manufacturer, can achieve. In Problem Sets 1, 2, and 4 (see Table 1) the processing times of product types are larger, and therefore the potential output of the facility is smaller. An output level that is relatively small with respect to buyers' requirements causes prices to increase more rapidly as the auction progresses. To illustrate the impact of ratio of demand to the available capacity on profit performance, we divide the problem sets into two groups: problem sets with high (1, 2, and 4) and low (3, 5, and 6) ratios. We note that, although the processing times are higher in the first group of problem sets, the buyers' requirements are more or less the same in both groups of problem sets.

In Figure 2, we report the performance of the iterative auction for these two groups separately and with the C/Z/Y, V/Z/Y, and V/C/Y pricing strategies. An analysis of the results presented in Figure 2 reveals that, in problems with higher demand to available capacity ratios, all three pricing strategies deliver profit levels that are at least 10% higher than the levels obtained in problems with lower ratios. The difference between two sets is statistically significant at the 5% confidence level. This observation clearly demonstrates that the proposed auction mechanism is able to capture the dynamics of capacity scarcity and translate it into increased profits for the manufacturer.

image

Figure 2. Impact of High and Low Demand to Capacity Ratios on Profit Levels

Download figure to PowerPoint

5.5. Analysis of the Iterative Auction's Performance on Larger-Size Private Information Problems

In Table 5, we report the performance of the iterative auction over a set of 50 problems with three product types (J = 3), 16 buyers (N = 16), and planning horizon of 150 time periods (T = 150). Problem instances are generated using the following parameter values: inline image and, for i = 1,2,…,N, inline image whereinline image. In Step 0 of the iterative auction, inline image is selected as 150, and the initial prices and bid-increment values are set as presented in Table 3.We also consider three pricing strategies: C/Z/Y, V/Z/Y, and V/C/Y.

Table 5. Iterative Auction's Performance in Problems with J = 3 and N = 16
Pricing strategyProfits with autionNumber of iterations
Product pricesDelivery pricesQuantity discountsManufacturer (%)Buyers (%)Total (%)
ConstantZeroYes 52.83 30.87 83.70 60.33
VariableZeroYes46.1039.6285.72139.70
VariableConstantYes50.4835.0585.53132.73

The figures reported in Table 5 are in line with the observations that have been made earlier: in terms of the manufacturer's objective function, the C/Z/Y pricing strategy delivers a slightly better performance. In this problem set, the average (maximum) CPU time for one iteration of the auction has been 0.53 (1.28) seconds, demonstrating the iterative auction approach's scalability to moderately large problems.

6. Concluding Remarks

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The Model
  5. 3. Profit Maximization When All Information Is Public
  6. 4. Private Information Problem
  7. 5. Computational Analysis
  8. 6. Concluding Remarks
  9. References
  10. Supporting Information

In this study, we consider a pricing and short-term capacity allocation problem in the presence of buyers with potential orders for bundles of products. We specifically consider the case where buyers' preferences are their private information and propose a market-based mechanism. The literature on capacity allocation using market-based mechanisms focuses mainly on the case where the market good is an individual time slot or combination of time slots. In this study, we take a different path and assume that the finished products of the manufacturing facility are market goods, keeping processing requirements of each product type private information of the manufacturer. To our knowledge, our study is unique in the development of a model that first allocates a facility's capacity to products and then products to buyers via such mechanisms in a setting where the buyers' utilities are private information.

We propose a round-based price-setting auction with non-decreasing prices. The winner determination problem integrates the product allocation and scheduling decisions. We consider a variety of anonymous pricing strategies that include linear prices, quantity discounts, and delivery prices.

One of the main contributions of our study is to provide a comprehensive evaluation of efficiency of alternative pricing schemes via an extensive computational analysis. The discussion presented in section 'Computational Analysis' puts forward a number of managerial insights. When all information is public, most of the studied pricing strategies are able to achieve the system's maximum profit, although they are not guaranteed to do so from a theoretical standpoint. When we solve the problem of maximizing the manufacturer's profit, again with complete information about buyers' bundles and their related utilities, most of the pricing strategies achieve very high efficiency levels that are almost completely transferred to the manufacturer. This observation highlights the power the buyers might gain by keeping their information private, because, when all information is public, the manufacturer can easily capture almost all of the system's maximum profit. By keeping their information private, buyers drive the manufacturer to solve her capacity allocation problem with market-based mechanisms, resulting in a transfer of 30% (with some pricing schemes up to 39%) of the system's maximum profit to the buyers. The iterative auction design we propose in this study, with its efficiently solvable Winner Determination Problem for a bounded number of product types, can solve the allocation problem in a very short period of time.

In the computational analysis with randomly created problems, we also demonstrate the impact of degree of capacity scarcity, which is defined as the ratio of demand to capacity, on the allocation of the system's profit between the manufacturer and the buyers. Certain pricing strategies, such as C/Z/Y, V/Z/Y, and V/C/Y, achieve an average system efficiency of 90% in all of the problem sets. In problem sets with higher demand to capacity ratios, however, they allocate a higher percentage of this efficiency to the manufacturer. When this ratio is higher, the manufacturer may feel entitled to a higher portion of the achieved efficiency. In line with this observation, our proposed auction design captures the dynamics of capacity scarcity and allocates a higher portion of system's profit to the manufacturer when the demand to capacity ratio is higher.

Throughout this study, we build a foundation for the pricing and short-term capacity allocation problem in the presence of a single manufacturer and multiple buyers with potential orders for bundles of various products. To conclude this study, we present the limitations of the proposed models and method as well as some future research themes that we believe are important in fully addressing a real-life pricing and short-term capacity allocation problem faced in the process industries.

In this study, we use a pseudo-dual pricing scheme to construct the guiding prices in the proposed iterative auction algorithm. The approximate nature of the pseudo-dual pricing is a limitation on the iterative auction algorithm's capability to fully exploit the pricing flexibility. Hence, it would be very interesting to test the iterative auction algorithm's performance on capturing the benefits of the pricing flexibility when different pricing schemes instead of the pseudo-dual approach are applied.

The iterative auction algorithm we propose in this study is actually quite scalable with respect to the problem size, allowing for an analysis that considers higher number of buyers and product types. However, generating the benchmark results for these larger instances for comparison purposes becomes a significantly difficult task due to the intractability of the full information models presented in section 'Profit Maximization When All Information Is Public'

The bidding strategy of the buyers plays an important role in the progress of the iterative auction algorithms. In this study, we employ the myopic best response bidding strategy. Although alternative bidding strategies, such as the power set bidding, did not provided significantly different auction outcomes in an earlier study (Bichler et al. 2009), it would still be interesting to experiment with other bidding strategies in the context of the problem addressed in this study.

As a major future research direction, there is a need to extend the models we built in this study to the situation where there exist competing manufacturers in the market. In that case, a comparison of the multiple manufacturers' revenues (or profits) and the buyers' extracted utilities compared to the single manufacturer case would provide additional managerial insights to the decision makers.

In an extended model of competing manufacturers' case, it would also be interesting to investigate the various degrees of information sharing between the buyers and the manufacturers and the effects of the new settings on the system's profit and the utilities extracted by each party.

References

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  2. Abstract
  3. 1. Introduction
  4. 2. The Model
  5. 3. Profit Maximization When All Information Is Public
  6. 4. Private Information Problem
  7. 5. Computational Analysis
  8. 6. Concluding Remarks
  9. References
  10. Supporting Information
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Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The Model
  5. 3. Profit Maximization When All Information Is Public
  6. 4. Private Information Problem
  7. 5. Computational Analysis
  8. 6. Concluding Remarks
  9. References
  10. Supporting Information
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