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
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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.
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1. INTRODUCTION
MetalFab, Inc.
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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
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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
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The name of this privately owned company has been disguised at the owner’s request.
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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
