scispace - formally typeset
Search or ask a question
Journal ArticleDOI

Static and Dynamic Pricing of Excess Capacity in a Make-to-Order Environment

TL;DR: In this paper, the authors examine four pricing policies that span a range of complexity and required knowledge about the status of the production system at the manufacturer, including the optimal policy of setting a different price for each possible state of the queue, and demonstrate numerically the financial gains a firm can achieve by following this policy vs. simpler pricing policies.
Abstract: Recent years have seen advances in research and management practice in the area of pricing, and particularly in dynamic pricing and revenue management. At the same time, researchers and managers have made dramatic improvements in production and supply chain management. The interactions between pricing and production/supply chain performance, however, are not as well understood. Can a firm benefit from knowing the status of the supply chain or production facility when making pricing decisions? How much can be gained if pricing decisions explicitly and optimally account for this status? This paper addresses these questions by examining a make-to-order manufacturer that serves two customer classes - core customers who pay a fixed negotiated price and are guaranteed job acceptance, and "fill-in" customers who make job submittal decisions based on the instantaneous price set by the firm for such orders. We examine four pricing policies that span a range of complexity and required knowledge about the status of the production system at the manufacturer, including the optimal policy of setting a different price for each possible state of the queue. We demonstrate properties of the optimal policy, and we illustrate numerically the financial gains a firm can achieve by following this policy vs. simpler pricing policies. The four policies we consider are (1) state-independent (static) pricing, (2) allowing fill-in orders only when the system is idle, (3) setting a uniform price up to a cut-off state, and (4) general state-dependent pricing. Although general state-dependent pricing is optimal in this setting, we find that charging a uniform price up to a cut-off state performs quite well in many settings and presents an attractive trade-off between ease of implementation and profitability. Thus, a fairly simple heuristic policy may actually out-perform the optimal policy when costs of design and implementation are taken into account.

Summary (5 min read)

1. INTRODUCTION

  • MetalFab, Inc. 1 produces fabricated metal parts mostly for use in the power generation industry.
  • More often, however, GE will change the order quantity and due date several times while the order is outstanding.
  • An alternative approach is to proactively seek fill-in orders when capacity utilization is running low, charging a low price to attract those customers; and when the capacity utilization is high, charging a high price and accepting only limited fill-in orders.
  • Before reviewing the relevant literature, it is important to note that this problem is quite general and is generating much interest beyond high precision job shops like MetalFab.
  • This research is designed to generate insight for managers about the benefits of accounting for the supply chain when making pricing decisions.

2. LITERATURE REVIEW

  • The last two decades have seen significant research progress on the interaction of pricing and operations.
  • This literature falls into two fundamental categories: pricing/inventory models and pricing/queuing models.
  • Federgruen & Heching (1999) consider a infinite horizon, order-up-to model that has a stationary base stock policy as the optimal policy structure in the case of zero leadtimes.
  • When a customer arrives, he receives information on the state of the system and then decides whether to join the queue or not.
  • Users also decide to enter the system based on steady state queue lengths rather than on the current queue status, and the price is simply the opportunity cost of servicing that request.

3.1. Modeling Framework

  • The authors assume that a firm operating a production system faces two types of demand.
  • Given the long-term, contractual agreements between a firm and its core customers, the authors consider the arrival rates of core customers as given.
  • The authors model the above production system as single-server queueing system.
  • General state-dependent pricing, in which no constraints are placed on the pricing policy.
  • These policies were chosen because they represent natural ways in which limited information might be used in practice.

3.2. Model 1: State-Independent Pricing

  • Here the authors study the case of a firm that sets a uniform price for fill-in jobs independent of the current state of the factory.
  • The authors do so for benchmarking purposes because such a static policy is simple and requires no real-time information about factory status.
  • It does, however, require information about time-average system behavior, as might be available from historical data.
  • State-independent pricing is also consistent with the behavior of core customers whose arrival rate is independent of the system state.
  • This is based on the nature of long-term contracts that often specify a fixed price and a service level commitment, where the supplier manages its production system to satisfy those commitments.

Note that c

  • W and f W (where f W is the expected throughput time for fill-in customer orders) are equal for this policy.
  • Problem (1) can be solved via Lagrangean methods (where β denotes the Langrage multiplier).
  • The solution consists of two possible cases depending on whether the waiting time constraint is binding.
  • Here, the value of a job is simply * p , but the authors will demonstrate that this factor takes on more complex forms which yield interpretations that can help managers understand both the benefits and costs of taking on additional work.
  • Jobs from core customers arrive at an average rate of 8 per month and the production system can complete 10 jobs per month on average if continuously busy.

3.3. Model 2: State-Dependent Pricing -Admitting Jobs When Idle

  • As a first-step towards developing more complex dynamic pricing policies, a simple form of dynamic pricing where fill-in arrivals are allowed only when the system is otherwise idle.the authors.
  • This policy captures the idea that a factory manager may wish to only accept fill-in work when the factory is relatively non-congested.
  • Lemma 1 below summarizes some basic properties of such a system.
  • The above factor measures the change in the relative odds that the system is in the set of states in which fill-in arrivals are not allowed.
  • For the case where the waiting time constraint is binding, a second opportunity cost,.

3.4. Model 3: State-Dependent Pricing -Constant Price Up To Cutoff State

  • Here the authors study a pricing scheme in which fill-in job arrivals are allowed only if the system is in a relatively uncongested set of states and all fill-in arrivals are charged a uniform price.
  • Lemma 2 below presents the properties of such a queueing system.
  • Note that, for the above example, this policy yields a significant improvement in expected profitability over the policy of admitting fill-in jobs only when idle -the gain is approximately 65%!.
  • The typical request from Marketing is to take every job, whereas the response from Manufacturing is often "the authors are too busy.".
  • This example helps determine what "busy" means, and it illustrates the potential gains from optimally choosing the cut-off state and the price jointly.

3.5. Model 4: General State-Dependent Pricing

  • Of course, such a scheme entails tracking the status of the production process carefully so that accurate prices can be specified.
  • Theorem 5 below demonstrates that the set of optimal prices for problem (8) are monotone in the state.
  • The basic setting is the same as prior examples.
  • The progression of prices and arrival rates shown in Table 1 confirms their intuition about these relationships, namely that as the system becomes more congested, a higher price is charged for fill-in arrivals and demand is curtailed.
  • It is thus encouraging that the incremental gain is quite small.

4. POLICY COMPARISON

  • Figure 1 provides a comparison of the optimal fill-in job arrival rates under each of the four examples discussed in Section 3, plotted with respect to system state, i.e., the number of jobs in queue plus any being served.
  • Clearly the state-independent or static policy yields a conservative approach to fill-in customers; a relatively low arrival rate is maintained to compensate for lack of state information.
  • The policy of admitting fill-in work only when idle yields a contrasting approach.
  • While the "uniform up to a cut-off" and "general state dependent pricing" policies look quite different from this perspective, it is notable that their relative financial performance is much less divergent.
  • This result was discussed in Section 3 for this particular numerical example and is explored in more depth in this section.

INSERT FIGURE 1 ABOUT HERE

  • Expanding on Examples 1-4, the authors vary two basic model characteristics: (1) the system utilization due to core (long-term) customers, denoted by to this as a "large" market demand function.
  • As mentioned above, when core job utilization is 0.9, the waiting time constraint is binding and fill-in arrivals are not feasible, as revealed in Figure 2 .
  • Figure 2 also shows that general state-dependent pricing sets a performance frontier that encompasses the other policies.
  • Of course, the feasible set of general state-dependent pricing solutions is a superset of the feasible solutions under each of the other policies, and therefore the solution must be at least as good as any of the others.
  • The explanation for this effect is that admitting jobs only when idle gets costly for a lightly loaded system; such a policy does not allow any queueing of fill-in jobs, which can result in unnecessary idle time.

INSERT FIGURE 2 ABOUT HERE

  • Figure 2 also reveals that the relative value of using internal system state information to make job pricing and acceptance decisions increases as c ρ increases.
  • The large relative gain results because the performance gap between the policies remains significant even as the absolute gains decline.
  • While the relative gains from full state-dependent pricing can be tremendous, the costs of using a policy of uniform pricing up to a cut-off state instead of full state-dependent pricing are much more moderate.
  • The curve is very similar to that of Figure 3 for general statedependent pricing.
  • These gains are quite modest when compared to the value of using internal state information, i.e., the gains from implementing a state-dependent pricing policy vs. a state-independent policy.

INSERT FIGURE 3 ABOUT HERE INSERT FIGURE 4 ABOUT HERE

  • For these cases, the waiting time constraint is non-binding and the optimal solution for the policies of uniform pricing up to a cut-off state and general state-dependent pricing both collapse to the state-independent pricing solution, as described in Theorems 3 and 4.
  • As c ρ increases, waiting time constraint becomes binding and these two state-dependent policies begin to differ from and outperform state-independent pricing.

INSERT FIGURE 5 ABOUT HERE

  • Figure 6 presents the relative gains from general state dependent pricing vs. stateindependent pricing, and Figure 7 presents the relative gains from uniform pricing up to a cut-off state vs. state-independent pricing.
  • These plots are analogues of Figures 3 and 4 for the small market case.
  • Figure 6 reveals that the relative value of state information increases dramatically as the system becomes more congested.
  • Figure 7 again supports the idea that the relative gains from full-state dependent pricing vs. uniform pricing up to a cutoff are limited -here the maximum gain is 6.9% at 88 .

INSERT FIGURE 6 ABOUT HERE INSERT FIGURE 7 ABOUT HERE

  • An analytic result that lends some support to the numerical results of this section is provided via the concept of entropy from the field of information theory.
  • It is well known that the steady-state state distribution for the M/M/1 queue follows a geometric distribution (see, for example, Gross & Harris (1985) ).
  • Therefore, more information is revealed from observing the state of a highly utilized system than would be revealed by observing a less highly utilized production system.

INSERT FIGURE 8 ABOUT HERE

  • The interpretation of entropy as a measure of the complexity of the system required to generate a random variable can also provide some insights into the performance of the four pricing policies considered in this paper (see, for example, Cover and Thomas (1991) for an explanation of Kolmogorov complexity).
  • Each of these policies (except static pricing as in Model 1) requires a signal from the factory floor in order to establish the current price for fill-in work.
  • The authors can revisit the examples of Section 3 and use this notion of complexity to evaluate the performance/complexity tradeoff for the policies they consider.
  • Here again, the authors see evidence that the policy of uniform pricing up to a cutoff state deserves attention: it far outperforms the other policies on this metric.

5. CONCLUSIONS AND FUTURE RESEARCH

  • The objectives of this paper are twofold.
  • First, although there has been much research in the area of dynamic pricing, very few papers have integrated supply chain issues with pricing policies.
  • This result suggests that while managers can improve profit markedly by using information about the status of the factory, a fairly simple pricing policy that requires limited information gathering can perform extremely well.
  • The authors argue that existence of such fully state-dependent pricing policies may actually motivate some fill-in customers to become core customers.
  • This research presents many opportunities for future work.

Did you find this useful? Give us your feedback

Content maybe subject to copyright    Report

Static and Dynamic Pricing Of Excess Capacity in a Make-
To-Order Environment
Joseph M. Hall
Praveen K. Kopalle
David F. Pyke
Tuck School of Business at Dartmouth
100 Tuck Hall, HB 9000
Hanover, NH 03755
Phone: 603-646-0778
Fax: 603-646-1308
Joseph.M.Hall@Dartmouth.EDU
Praveen.K.Kopalle@Dartmouth.EDU
David.F.Pyke@Dartmouth.EDU

1
Static and Dynamic Pricing Of Excess Capacity in a Make-
To-Order Environment
ABSTRACT
Recent years have seen advances in research and management practice in the area of
pricing, and particularly in dynamic pricing and revenue management. At the same time,
researchers and managers have made dramatic improvements in production and supply chain
management. The interactions between pricing and production/supply chain performance,
however, are not as well understood. Can a firm benefit from knowing the status of the supply
chain or production facility when making pricing decisions? How much can be gained if pricing
decisions explicitly and optimally account for this status? This paper addresses these questions
by examining a make-to-order manufacturer that serves two customer classes – core customers
who pay a fixed negotiated price and are guaranteed job acceptance, and “fill-in” customers who
make job submittal decisions based on the instantaneous price set by the firm for such orders.
We examine four pricing policies that span a range of complexity and required knowledge about
the status of the production system at the manufacturer, including the optimal policy of setting a
different price for each possible state of the queue. We demonstrate properties of the optimal
policy, and we illustrate numerically the financial gains a firm can achieve by following this
policy vs. simpler pricing policies. The four policies we consider are (1) state-independent
(static) pricing, (2) allowing fill-in orders only when the system is idle, (3) setting a uniform
price up to a cut-off state, and (4) general state-dependent pricing. Although general state-
dependent pricing is optimal in this setting, we find that charging a uniform price up to a cut-off
state performs quite well in many settings and presents an attractive trade-off between ease of
implementation and profitability. Thus, a fairly simple heuristic policy may actually out-perform
the optimal policy when costs of design and implementation are taken into account.

2
1. INTRODUCTION
MetalFab, Inc.
1
produces fabricated metal parts mostly for use in the power generation
industry. The parts are made from expensive materials – some 4x8-foot sheets of material cost
$20,000 – and, not surprisingly, the fabricated parts have very tight tolerances. MetalFab is a
large job shop with about 60 highly skilled shop floor employees who operate metal bending and
metal cutting machines, as well as a variety of welding equipment. At this writing,
approximately 80% of MetalFab’s output is sold directly to General Electric, or to first tier GE
suppliers. In keeping with MetalFab’s policy, we will refer to this output as belonging to GE.
MetalFab can forecast orders from GE and GE’s suppliers, but the forecast error can be
quite high. Sometimes MetalFab production planners will have firm forecasts – and these
forecasts remain firm until the order is delivered. More often, however, GE will change the
order quantity and due date several times while the order is outstanding. In fact, GE’s systems
will occasionally produce a purchase order that is already past due when the order is placed.
(The authors observed a case in which an order placed in September had a due date of the
previous May!) Taken together, the blend of firm forecasts, changes, and emergency orders
create a situation that is well captured by a mean forecast with a fairly high variance around that
mean. Similar situations could arise in cases when a single large customer aggregates demand
forecasts from many different locations and provides an aggregate request to its supplier.
From MetalFab’s perspective, GE orders form the core of its business, while the other
orders that may take up the remaining 20% of its capacity are treated as “fill-in” orders. From a
marketing standpoint, one approach to accepting and pricing fill-in orders is to take as many as
possible, charge the same price as the core orders, and let the production planners and factory
workers try to keep up. The danger with this approach, of course, is that it may not be a long-run
profit-maximizing strategy; and service performance, for both GE and fill-ins, could suffer. An
alternative approach is to proactively seek fill-in orders when capacity utilization is running low,
charging a low price to attract those customers; and when the capacity utilization is high,
charging a high price and accepting only limited fill-in orders. This alternative raises the issue of
how to (i) price dynamically over time depending on the state of the production system, (ii)
endogenously determine “low” versus “high” capacity utilization, and (iii) incorporate core

3
customer arrivals (i.e., orders from core customers) in determining the pricing policy for fill-in
customers. For example, when a potential fill-in customer asks for a bid for a given part, what
price should MetalFab quote? Should that price depend on the current level of congestion at the
factory? If so, how? Finally, what benefits are available if the firm wisely uses capacity
information when making pricing decisions? This paper addresses these questions.
We focus attention on four models that take into consideration both core and fill-in
customer arrival rates: (1) state-independent (static) pricing – where MetalFab sets a price, p, for
fill-in customers without regard to the current state of the factory; (2) allowing fill-in jobs at a
chosen price p only when the factory is idle; (3) allowing fill-in jobs at a chosen price p only
when there are s or fewer jobs in the production system, where both s and p are decision
variables; and (4) general state dependent pricing – i.e. potentially setting a different price for
fill-in orders for every possible state of the factory. To ensure satisfactory service, we impose a
constraint on expected waiting time for core customers. We compare the optimal solutions
obtained in the above four cases, report the magnitude of the benefit from utilizing increasing
amounts of information, illustrate interesting properties of the solutions, and examine conditions
under which one solution is superior to another.
Before reviewing the relevant literature, it is important to note that this problem is quite
general and is generating much interest beyond high precision job shops like MetalFab. With the
advent of modern pricing software such as that offered by DemandTec, ProfitLogic, and
KhiMetrics, many companies now are devoting considerable time and energy to getting prices
right. However, recent trade press articles suggest that firms have traditionally been slow to
adopt sophisticated pricing models (Reda 2002), have priced products solely on cost (At What
Price? Guidelines for a Customer-Focused Pricing Strategy 2000), and often simply employ
“what-if” analyses without incorporating the interactions across functional areas (Retail Revenue
Management 2001, Lester 2002). Further, the transition to the Euro has elevated this issue for
companies doing business in Europe, and many are appointing senior “pricing officers” with
direct responsibility over pricing decisions. Furthermore, many firms are beginning to realize
that price changes should be made with a deeper understanding of the supply chain (Cisco
Thought Leadership Summit 2001). If a firm cuts price to stimulate demand, but the factory or
supply chain is currently overloaded, they risk some very unhappy customers. On the other
1
The name of this privately owned company has been disguised at the owner’s request.

4
hand, if the supply chain and factories currently have excess capacity, marketing personnel may
wish to decrease price to consume some of that capacity. In addition, some of the leading
suppliers of supply chain software are developing linkages to pricing software. Manugistics, for
instance, has bought Talus, a revenue management software provider with the expressed intent of
linking these two areas. This research is designed to generate insight for managers about the
benefits of accounting for the supply chain when making pricing decisions.
The rest of this paper is organized as follows. In Section 2 we review the relevant
literature. In Section 3, we present the four models and corresponding analytical results. A
numerical comparison of the policies is presented in Section 4. Section 5 contains a summary
discussion and directions for future research.
2. LITERATURE REVIEW
The last two decades have seen significant research progress on the interaction of pricing
and operations. This literature falls into two fundamental categories: pricing/inventory models
and pricing/queuing models. In the first case, generally speaking, prices are determined jointly
with inventory decisions, or are determined based on current inventory levels. In the second
case, prices are used to control the arrival rate to a queue or queues and may or may not be set
based on the current queue length. For a recent review of this literature, see Fleischmann, Hall &
Pyke (2004).
Pricing/inventory models have a long history, beginning with single period, single price,
single quantity models like those of Whitin (1955), Karlin & Carr (1962), and Lau & Lau (1988).
Multiperiod models typically assume a single, constant price, and deterministic demand.
Examples include Wagner & Whitin (1958a), Wagner & Whitin (1958b), Thomas (1970),
Kunreuther & Richard (1971), Kunreuther & Schrage (1973), Pekelman (1974), Eliashberg &
Steinberg (1987), Eliashberg & Steinberg (1991), Gilbert (2000), Arvind Rajan & Steinberg
(1992), and Sogomonian & Tang (1993).
Models with stochastic demand include Thomas (1974) who addresses an N period
problem and proposes a heuristic policy of the form (s, S, p), where price is a parameter in the
probability distribution of demand. Federgruen & Heching (1999) consider a infinite horizon,
order-up-to model that has a stationary base stock policy as the optimal policy structure in the

Citations
More filters
Journal ArticleDOI
TL;DR: In this paper, a unified framework for classifying various marketing-operations interface models is presented, which may serve as a guide to navigate through the sea of research articles in this important area.
Abstract: Marketing and operations are two key functional areas that contribute to the success of a firm. By acquiring and analyzing information regarding customers and competitors, marketing can be viewed an external-focused functional area that determines “what” kind of products (or services) a company should provide through “which” channel at “what” price. By viewing this marketing plan as the “demand” from an internal customer, operations is by-and-large an internal-focused functional area that examines “how” to deliver this demand by using internal or external resources. Due to their inherent roles and responsibilities, coordination and collaborations between marketing and operations areas can be difficult in practice. As such, the conflict between marketing and operations arises when the operation’s “supply” does not meet the marketing’s “demand.” Over the last two decades, researchers have developed different quantitative models to examine the issue of coordination/collaboration in the context of marketing operations interfaces. The intent of this paper is two-fold. We present a unified framework for classifying various marketing-operations interface models that may serve as a guide to navigate through the sea of research articles in this important area. Also, by examining some missing gaps, we discuss some topics for potential future research.

129 citations

Journal ArticleDOI
TL;DR: Detective Winnie arrived at the scene of a fight at a restaurant to find everything in disarray, and found the proprietor, Mr. Sully, thinking he had been robbed and the so‐called fight was nothing more than a red herring.
Abstract: DIRECTIONS: Read the story below. Then answer the questions. When Detective Winnie arrived at the scene, everything was in disarray. Tables were overturned, chairs were splintered, broken dishes and food were scattered everywhere. “It looks like there was a fight,” said the restaurant’s proprietor, Mr. Sully. “Heck of a fight, too, from what I can tell.” “Maybe,” said Mr. Winnie. He walked slowly around the floor, looking closely down at the mess. He was deliberating. “Fifteen years I have owned this restaurant,” Mr. Sully said. “Fifteen years I have trusted the same general manager. But it was after hours. Why were so many people here, eating, after hours? It’s like the place was open, when it wasn’t.” “Maybe,” said Mr. Winnie. “And I guess no one even paid their check,” Mr. Sully went on, dismally. Mr. Winnie looked up. “Mr. Sully,” he announced. “This so‐called ‘fight’ is nothing more than a red herring.” Mr. Sully looked at him in confusion. “I suggest you check your safe,” Mr. Winnie went on. I think you have been robbed. And the longer we stand here wondering about this mysterious dinner and ‘fight’, the more time the thief―I suspect your general manager―has to abscond with your hard‐earned money.” “But...how do you know that the fight was a fake?” Mr. Sully demanded. “Because there are no footprints in the food,” Mr. Winnie said, simply. Mr. Sully looked around, flabbergasted. “So there aren’t!” “It’s as if someone tiptoed through here, knocking over tables and breaking chairs, but stepping around the mess carefully, so they wouldn’t get food on their shoes. If this had been a real fight, the food would have all been smeared everywhere!” 1. Which word is a synonym for disarray?

80 citations

Journal ArticleDOI
TL;DR: This paper motivates the consideration of risk-averse and robust revenue management and briefly introduces revenue managements’ two main methods – capacity control and dynamic pricing – in the classical, risk-neutral setting.

53 citations


Cites background from "Static and Dynamic Pricing of Exces..."

  • ...Application areas of dynamic pricing include, for example, retailing (e.g. Zhao and Zheng (2000), Heching et al. (2002)), low cost airlines (e.g. Marcus and Anderson (2008)), hotels (e.g. Schütze (2008)), and make-to-order manufacturing (e.g. Hall et al. (2009))....

    [...]

Journal ArticleDOI
TL;DR: In this article, the authors investigate the performance of the optimal pricing scheme as well as two commonly used pricing schemes (fixed fee and time-based pricing) for such services on important dimensions such as revenue, demand served, and utilization.
Abstract: In many services, for example, website or landscape design, the value or quality derived by a customer depends upon the service time, and this valuation differs across customers. Customers procure the service based on the expected value to be delivered, prices charged, and the timeliness of service. We investigate the performance of the optimal pricing scheme as well as two commonly used pricing schemes (fixed fee and time-based pricing) for such services on important dimensions such as revenue, demand served, and utilization. We propose a novel model that captures the above features and wherein both service rate and demand are endogenous and functions of the pricing scheme. In particular, service time is an outcome of the pricing scheme adopted and the heterogeneous valuations of customers, unlike in the queueing-based pricing literature. We find that the service system may benefit from a greater variance in consumer valuations, and the performance of pricing schemes is impacted by the shape of the distribution of customers' valuation of service time and the responsiveness desired by customers. Both the fixed fee and time-based schemes do well relative to the optimal pricing scheme in terms of revenue in many plausible scenarios, but there are substantial differences between the pricing schemes in some important operational metrics. For instance, the fixed fee scheme serves more customers and has higher utilization than the time-based scheme. We also explore variants of the fixed and time-based schemes that have better revenue performance and show that the two-part tariff which is a combination of fixed and time-based pricing can do as well as the optimal scheme in terms of revenue.

45 citations

Posted Content
TL;DR: In this article, the authors analyze the dynamic strategic interactions between a manufacturer and a retailer in a decentralized distribution channel used to launch an innovative durable product (IDP), and demonstrate that revenue-sharing contracts can coordinate the IDP's supply chain with both farsighted and myopic retailers throughout the entire planning horizon and arbitrarily allocate the channel profit.
Abstract: We analyze the dynamic strategic interactions between a manufacturer and a retailer in a decentralized distribution channel used to launch an innovative durable product (IDP). The underlying retail demand for the IDP is influenced by word-of-mouth from past adopters and follows a Bass-type diffusion process. The word-of-mouth influence creates a trade-off between immediate and future sales and profits, resulting in a multi-period dynamic supply chain coordination problem. Our analysis shows that while in some environments, the manufacturer is better off with a far-sighted retailer, there are also environments in which the manufacturer is better off with a myopic retailer. We characterize equilibrium dynamic pricing strategies and the resulting sales and profit trajectories. We demonstrate that revenue-sharing contracts can coordinate the IDP's supply chain with both far-sighted and myopic retailers throughout the entire planning horizon and arbitrarily allocate the channel profit.

40 citations


Cites methods from "Static and Dynamic Pricing of Exces..."

  • ...Using a Markovian model, Hall et al. (2009) consider a supply chain in which a make-to-order manufacturer sells a product to the core customers at a fixed price and to ‘‘fill-in’’ customers at a current price....

    [...]

References
More filters
Book
01 Jan 1991
TL;DR: The author examines the role of entropy, inequality, and randomness in the design of codes and the construction of codes in the rapidly changing environment.
Abstract: Preface to the Second Edition. Preface to the First Edition. Acknowledgments for the Second Edition. Acknowledgments for the First Edition. 1. Introduction and Preview. 1.1 Preview of the Book. 2. Entropy, Relative Entropy, and Mutual Information. 2.1 Entropy. 2.2 Joint Entropy and Conditional Entropy. 2.3 Relative Entropy and Mutual Information. 2.4 Relationship Between Entropy and Mutual Information. 2.5 Chain Rules for Entropy, Relative Entropy, and Mutual Information. 2.6 Jensen's Inequality and Its Consequences. 2.7 Log Sum Inequality and Its Applications. 2.8 Data-Processing Inequality. 2.9 Sufficient Statistics. 2.10 Fano's Inequality. Summary. Problems. Historical Notes. 3. Asymptotic Equipartition Property. 3.1 Asymptotic Equipartition Property Theorem. 3.2 Consequences of the AEP: Data Compression. 3.3 High-Probability Sets and the Typical Set. Summary. Problems. Historical Notes. 4. Entropy Rates of a Stochastic Process. 4.1 Markov Chains. 4.2 Entropy Rate. 4.3 Example: Entropy Rate of a Random Walk on a Weighted Graph. 4.4 Second Law of Thermodynamics. 4.5 Functions of Markov Chains. Summary. Problems. Historical Notes. 5. Data Compression. 5.1 Examples of Codes. 5.2 Kraft Inequality. 5.3 Optimal Codes. 5.4 Bounds on the Optimal Code Length. 5.5 Kraft Inequality for Uniquely Decodable Codes. 5.6 Huffman Codes. 5.7 Some Comments on Huffman Codes. 5.8 Optimality of Huffman Codes. 5.9 Shannon-Fano-Elias Coding. 5.10 Competitive Optimality of the Shannon Code. 5.11 Generation of Discrete Distributions from Fair Coins. Summary. Problems. Historical Notes. 6. Gambling and Data Compression. 6.1 The Horse Race. 6.2 Gambling and Side Information. 6.3 Dependent Horse Races and Entropy Rate. 6.4 The Entropy of English. 6.5 Data Compression and Gambling. 6.6 Gambling Estimate of the Entropy of English. Summary. Problems. Historical Notes. 7. Channel Capacity. 7.1 Examples of Channel Capacity. 7.2 Symmetric Channels. 7.3 Properties of Channel Capacity. 7.4 Preview of the Channel Coding Theorem. 7.5 Definitions. 7.6 Jointly Typical Sequences. 7.7 Channel Coding Theorem. 7.8 Zero-Error Codes. 7.9 Fano's Inequality and the Converse to the Coding Theorem. 7.10 Equality in the Converse to the Channel Coding Theorem. 7.11 Hamming Codes. 7.12 Feedback Capacity. 7.13 Source-Channel Separation Theorem. Summary. Problems. Historical Notes. 8. Differential Entropy. 8.1 Definitions. 8.2 AEP for Continuous Random Variables. 8.3 Relation of Differential Entropy to Discrete Entropy. 8.4 Joint and Conditional Differential Entropy. 8.5 Relative Entropy and Mutual Information. 8.6 Properties of Differential Entropy, Relative Entropy, and Mutual Information. Summary. Problems. Historical Notes. 9. Gaussian Channel. 9.1 Gaussian Channel: Definitions. 9.2 Converse to the Coding Theorem for Gaussian Channels. 9.3 Bandlimited Channels. 9.4 Parallel Gaussian Channels. 9.5 Channels with Colored Gaussian Noise. 9.6 Gaussian Channels with Feedback. Summary. Problems. Historical Notes. 10. Rate Distortion Theory. 10.1 Quantization. 10.2 Definitions. 10.3 Calculation of the Rate Distortion Function. 10.4 Converse to the Rate Distortion Theorem. 10.5 Achievability of the Rate Distortion Function. 10.6 Strongly Typical Sequences and Rate Distortion. 10.7 Characterization of the Rate Distortion Function. 10.8 Computation of Channel Capacity and the Rate Distortion Function. Summary. Problems. Historical Notes. 11. Information Theory and Statistics. 11.1 Method of Types. 11.2 Law of Large Numbers. 11.3 Universal Source Coding. 11.4 Large Deviation Theory. 11.5 Examples of Sanov's Theorem. 11.6 Conditional Limit Theorem. 11.7 Hypothesis Testing. 11.8 Chernoff-Stein Lemma. 11.9 Chernoff Information. 11.10 Fisher Information and the Cram-er-Rao Inequality. Summary. Problems. Historical Notes. 12. Maximum Entropy. 12.1 Maximum Entropy Distributions. 12.2 Examples. 12.3 Anomalous Maximum Entropy Problem. 12.4 Spectrum Estimation. 12.5 Entropy Rates of a Gaussian Process. 12.6 Burg's Maximum Entropy Theorem. Summary. Problems. Historical Notes. 13. Universal Source Coding. 13.1 Universal Codes and Channel Capacity. 13.2 Universal Coding for Binary Sequences. 13.3 Arithmetic Coding. 13.4 Lempel-Ziv Coding. 13.5 Optimality of Lempel-Ziv Algorithms. Compression. Summary. Problems. Historical Notes. 14. Kolmogorov Complexity. 14.1 Models of Computation. 14.2 Kolmogorov Complexity: Definitions and Examples. 14.3 Kolmogorov Complexity and Entropy. 14.4 Kolmogorov Complexity of Integers. 14.5 Algorithmically Random and Incompressible Sequences. 14.6 Universal Probability. 14.7 Kolmogorov complexity. 14.9 Universal Gambling. 14.10 Occam's Razor. 14.11 Kolmogorov Complexity and Universal Probability. 14.12 Kolmogorov Sufficient Statistic. 14.13 Minimum Description Length Principle. Summary. Problems. Historical Notes. 15. Network Information Theory. 15.1 Gaussian Multiple-User Channels. 15.2 Jointly Typical Sequences. 15.3 Multiple-Access Channel. 15.4 Encoding of Correlated Sources. 15.5 Duality Between Slepian-Wolf Encoding and Multiple-Access Channels. 15.6 Broadcast Channel. 15.7 Relay Channel. 15.8 Source Coding with Side Information. 15.9 Rate Distortion with Side Information. 15.10 General Multiterminal Networks. Summary. Problems. Historical Notes. 16. Information Theory and Portfolio Theory. 16.1 The Stock Market: Some Definitions. 16.2 Kuhn-Tucker Characterization of the Log-Optimal Portfolio. 16.3 Asymptotic Optimality of the Log-Optimal Portfolio. 16.4 Side Information and the Growth Rate. 16.5 Investment in Stationary Markets. 16.6 Competitive Optimality of the Log-Optimal Portfolio. 16.7 Universal Portfolios. 16.8 Shannon-McMillan-Breiman Theorem (General AEP). Summary. Problems. Historical Notes. 17. Inequalities in Information Theory. 17.1 Basic Inequalities of Information Theory. 17.2 Differential Entropy. 17.3 Bounds on Entropy and Relative Entropy. 17.4 Inequalities for Types. 17.5 Combinatorial Bounds on Entropy. 17.6 Entropy Rates of Subsets. 17.7 Entropy and Fisher Information. 17.8 Entropy Power Inequality and Brunn-Minkowski Inequality. 17.9 Inequalities for Determinants. 17.10 Inequalities for Ratios of Determinants. Summary. Problems. Historical Notes. Bibliography. List of Symbols. Index.

45,034 citations


"Static and Dynamic Pricing of Exces..." refers background in this paper

  • ...…of entropy as a measure of the complexity of the system required to generate a random variable can also provide some insights into the performance of the four pricing policies considered in this paper (see, for example, Cover and Thomas (1991) for an explanation of Kolmogorov complexity)....

    [...]

  • ...(See, for example, Cover and Thomas (1991) for a discussion of entropy)....

    [...]

Book
01 Jan 1974
TL;DR: The Fundamentals of Queueing Theory, Fourth Edition as discussed by the authors provides a comprehensive overview of simple and more advanced queuing models, with a self-contained presentation of key concepts and formulae.
Abstract: Praise for the Third Edition: "This is one of the best books available. Its excellent organizational structure allows quick reference to specific models and its clear presentation . . . solidifies the understanding of the concepts being presented."IIE Transactions on Operations EngineeringThoroughly revised and expanded to reflect the latest developments in the field, Fundamentals of Queueing Theory, Fourth Edition continues to present the basic statistical principles that are necessary to analyze the probabilistic nature of queues. Rather than presenting a narrow focus on the subject, this update illustrates the wide-reaching, fundamental concepts in queueing theory and its applications to diverse areas such as computer science, engineering, business, and operations research.This update takes a numerical approach to understanding and making probable estimations relating to queues, with a comprehensive outline of simple and more advanced queueing models. Newly featured topics of the Fourth Edition include:Retrial queuesApproximations for queueing networksNumerical inversion of transformsDetermining the appropriate number of servers to balance quality and cost of serviceEach chapter provides a self-contained presentation of key concepts and formulae, allowing readers to work with each section independently, while a summary table at the end of the book outlines the types of queues that have been discussed and their results. In addition, two new appendices have been added, discussing transforms and generating functions as well as the fundamentals of differential and difference equations. New examples are now included along with problems that incorporate QtsPlus software, which is freely available via the book's related Web site.With its accessible style and wealth of real-world examples, Fundamentals of Queueing Theory, Fourth Edition is an ideal book for courses on queueing theory at the upper-undergraduate and graduate levels. It is also a valuable resource for researchers and practitioners who analyze congestion in the fields of telecommunications, transportation, aviation, and management science.

3,059 citations

Journal ArticleDOI
TL;DR: Disjoint planning horizons are shown to be possible which eliminate the necessity of having data for the full N periods and desire a minimum total cost inventory management scheme which satisfies known demand in every period.
Abstract: (This article originally appeared in Management Science, October 1958, Volume 5, Number 1, pp. 89-96, published by The Institute of Management Sciences.) A forward algorithm for a solution to the following dynamic version of the economic lot size model is given: allowing the possibility of demands for a single item, inventory holding charges, and setup costs to vary over N periods, we desire a minimum total cost inventory management scheme which satisfies known demand in every period. Disjoint planning horizons are shown to be possible which eliminate the necessity of having data for the full N periods.

2,114 citations


"Static and Dynamic Pricing of Exces..." refers background in this paper

  • ...Examples include Wagner & Whitin (1958a), Wagner & Whitin (1958b), Thomas (1970), Kunreuther & Richard (1971), Kunreuther & Schrage (1973), Pekelman (1974), Eliashberg & Steinberg (1987), Eliashberg & Steinberg (1991), Gilbert (2000), Arvind Rajan & Steinberg (1992), and Sogomonian & Tang (1993)....

    [...]

01 Jan 2016
TL;DR: In this paper, a forward algorithm for a solution to the following dynamic version of the economic lot size model is given: allowing the possibility of demands for a single item,,inventory holding charges, anid setup costs to vary over N periods, we desire a minimum total cost inventory management scheme which satisfies known demand in every period.
Abstract: A forward algorithm for a solution to the following dynamic version of the economic lot size model is given: allowing the possibility of demands for a single item,,inventory holding charges, anid setup costs to vary over N periods, we desire a minimum total cost inventory management scheme which satisfies known demand in every period. Disjoint planning horizons are shown to be possible which eliminate the necessity of having data for the full N periods.

1,520 citations

Book
01 Apr 1987
TL;DR: As one of the part of book categories, dynamic programming deterministic and stochastic models always becomes the most wanted book.
Abstract: If you really want to be smarter, reading can be one of the lots ways to evoke and realize. Many people who like reading will have more knowledge and experiences. Reading can be a way to gain information from economics, politics, science, fiction, literature, religion, and many others. As one of the part of book categories, dynamic programming deterministic and stochastic models always becomes the most wanted book. Many people are absolutely searching for this book. It means that many love to read this kind of book.

1,311 citations


"Static and Dynamic Pricing of Exces..." refers background in this paper

  • ...Using a per-period profit criterion (Bertsekas (1987)), this translates to maximizing average expected revenue collected per unit time....

    [...]

  • ...(b) For linear demand, the first-order conditions are sufficient for optimality....

    [...]

Frequently Asked Questions (12)
Q1. What have the authors contributed in "Static and dynamic pricing of excess capacity in a make- to-order environment" ?

This paper addresses these questions by examining a make-to-order manufacturer that serves two customer classes – core customers who pay a fixed negotiated price and are guaranteed job acceptance, and “ fill-in ” customers who make job submittal decisions based on the instantaneous price set by the firm for such orders. The authors examine four pricing policies that span a range of complexity and required knowledge about the status of the production system at the manufacturer, including the optimal policy of setting a different price for each possible state of the queue. The authors demonstrate properties of the optimal policy, and they illustrate numerically the financial gains a firm can achieve by following this policy vs. simpler pricing policies. The four policies the authors consider are ( 1 ) state-independent ( static ) pricing, ( 2 ) allowing fill-in orders only when the system is idle, ( 3 ) setting a uniform price up to a cut-off state, and ( 4 ) general state-dependent pricing. Although general statedependent pricing is optimal in this setting, the authors find that charging a uniform price up to a cut-off state performs quite well in many settings and presents an attractive trade-off between ease of implementation and profitability. 

the arrival rate for fill-in customers depends on the prices charged via a downward sloping demand function, ( )ff pλ , where fp is the price charged for fill-in customers. 

From a managerial standpoint, if the factory is very lightly loaded or customers are relatively insensitive to delays, it is best to ignore the state of the factory in deciding which jobs to accept – only price should be used to discourage or encourage customers. 

The probability that this system is idle is approximately 0.0603, and thus the expected additional revenue due to the fill-in jobs is approximately $1073 per month. 

It is interesting to note that the expected performance of this relatively simple form of state-dependent pricing exceeds that of state-independent pricing by approximately 8.4% in this example. 

Of note in Theorem 3 is that for a non-binding waiting time constraint, it is optimal to use state-independent pricing for fill-in jobs. 

The explanation for this effect is that admitting jobs only when idle gets costly for a lightly loaded system; such a policy does not allow any queueing of fill-in jobs, which can result in unnecessary idle time. 

In this subsection the authors study a general state-dependent pricing scheme in which price maybe changed dynamically without constraint, i.e., fill-in job arrivals at time t are charged a price that is a function of the congestion levels in time t. 

A parallel picture in the grocery industry is the emergence of loyalty cards where “loyal” (or long-term) customers are promised better deals than “walk-in” customers who have to pay the price they face during that week. 

The expected waiting time in the system for core arrivals is given by:( ) ( ) +∏ ++= ∑ ∞=− =1100 11 iifjcij c iW µλλµ π. 

Example 4. Following Examples 1-3, fill-in customers exhibit demand of the form ( ) ppf 1.0100 −=λ ; core customers arrive at an average rate of 8 per month and the production system can complete 10 jobs per month on average. 

The expected waiting time in the system for core customer arrivals is given by:( ) ( ) − + − + +=µ λ µ λµλλ µ π cc fc cW1111 12 0 .