Static and Dynamic Pricing of Excess Capacity in a Make-to-Order Environment
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.
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Citations
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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))....
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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....
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References
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)....
[...]
3,059 citations
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)....
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1,520 citations
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....
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...(b) For linear demand, the first-order conditions are sufficient for optimality....
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Frequently Asked Questions (12)
Q2. What is the arrival rate for fill-in customers?
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.
Q3. What is the way to avoid the state of the factory?
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.
Q4. How much is the expected revenue due to the fill-in jobs?
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.
Q5. How much does the expected performance of this pricing scheme exceed that of state-dependent pricing?
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.
Q6. What is the optimal pricing for fill-in jobs?
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.
Q7. What is the reason for the effect of admitting jobs only when idle?
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.
Q8. What is the way to determine the optimal price for fill-in jobs?
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.
Q9. What is the parallel picture in the grocery industry?
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.
Q10. what is the expected waiting time in the system for core arrivals?
The expected waiting time in the system for core arrivals is given by:( ) ( ) +∏ ++= ∑ ∞=− =1100 11 iifjcij c iW µλλµ π.
Q11. How many fill-in jobs can a company complete per month?
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.
Q12. What is the expected waiting time in the system for core customers?
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 .