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Mathematical Problems in Engineering
Volume 2009, Article ID 932676, 15 pages
http://dx.doi.org/10.1155/2009/932676
Research Article

Optimal Pricing and Production Master Planning in a Multiperiod Horizon Considering Capacity and Inventory Constraints

1School of Engineering, Center for Quality and Manufacturing, Monterrey Institute of Technology, Monterrey, Mexico
2Industrial and Systems Engineering Department, School of Engineering, Monterrey Institute of Technology, Monterrey, Mexico
3Management Department, School of Business, Monterrey Institute of Technology, Monterrey, Mexico

Received 24 January 2009; Revised 7 May 2009; Accepted 23 June 2009

Academic Editor: Wei-Chiang Hong

Copyright © 2009 Neale R. Smith et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We formulate and solve a single-item joint pricing and master planning optimization problem with capacity and inventory constrains. The objective is to maximize profits over a discrete-time multiperiod horizon. The solution process consists of two steps. First, we solve the single-period problem exactly. Second, using the exact solution of the single-period problem, we solve the multiperiod problem using a dynamic programming approach. The solution process and the importance of considering both capacity and inventory constraints are illustrated with numerical examples.

1. Introduction

In the recent years, interest has increased in problems involving the joint optimization of pricing and production decisions. Such problems may not have been very practical some years ago since manufacturing firms have not traditionally been in close contact with the final consumers, making it hard to predict demand as a function of price. However, nowadays new sales channels such as the internet allow direct sales, making easy to observe end customer behavior [1]. Under these conditions, joint price and production planning problems arise in the manufacturing sector. According to Farnham [2], direct sales have grown 1% per year faster than traditional retail sales for the last ten years and that direct sales reached $30 billion in 2003, making it a significant business sector. Manufacturing firms can now, due to direct sales and the internet, maintain close contact with their customers as is currently done by the airlines, and change prices more quickly and at a much lower cost. In addition, firms that sell to a large number of small retailers can apply pricing models considering the numerous retailers to be their final customers.

Joint pricing and production decisions problems are treated in literature; see, for example, Elmaghraby and Keskinocak [3]. However, problems with both capacity and inventory constraints are not common. A literature review is provided below. According to our experience as consultants, the master planning linear programming models currently in use at several large companies reveal that both capacity and inventory constraints are commonly used in practice. The upper bounds on inventory result from either storage and/or budget limitations or company policies. The inventory constraints are of two types, absolute limits, which are considered in this paper, and days-of-cover requirements, which will be addressed in a future paper.

Food and beverage producers may be able to apply the proposed model. Within the Mexican market, these firms supply a very large number of small stores in addition to large supermarkets. The large number of stores is the set of customers served directly by the producers. The producer and many of the retailers are owned by the same parent corporation, making the application of dynamic pricing to the end consumer feasible. The food and beverage producers’ market exhibits a fairly stable demand with seasonal variation and identifiable long-term trends. Due to specialized storage requirements and expiration dates, inventory storage capacity cannot be easily expanded, making upper limits on inventory especially useful. The model we study in this paper is based on the planning models currently in use at such companies (cost minimizing linear programming models) with the addition of the pricing decision, profit maximization, and a focus on a single-item model in a multi period horizon. Furthermore, we do not require the demand pattern to be seasonal, only that the behavior of demand as a function of the price to be known in each period. We consider that a single-item model with multiple time periods is a reasonable starting point that can be used as a base for further work involving more realistic multiitem formulations.

In general, capacity constraints are uncommon in literature and inventory constraints even more so. One notable paper that contains inventory limits but no capacity limits is Cheng [4] who examines an EOQ model with pricing considerations and an inventory constraint. Additionally, pricing problems at the aggregate, master planning level with multiple discrete time periods were not considered before. Extending in this direction, we formulate and solve a multi period horizon with a single-item problem in a joint price and production master panning optimization subject to a capacity and inventory constrains.

Although we consider a deterministic model, it is to be noted that revenue optimization (revenue management) with stochastic elements in the service sector has received considerable attention by many researchers. For an overview see Bitran and Caldentey [5] and Boyd and Bilegan [1]. The most notable examples are the applications in the airline industry [6, 7]. Related techniques have also been applied in hotel [8], restaurant [9, 10], and retail [11, 12] areas. Gallego and Van Ryzin [13] develop a model that applies in the airline, hotel and retail settings. The inclusion of pricing decisions in revenue management models is fairly recent. See Gallego and van Ryzin [14], Feng and Gallego [15], and Bitran and Mondschein [16] for examples of the early work in pricing within a revenue management context. More recently Chan et al. [17] study delayed production and delayed pricing strategies for a multiperiod model.

Our work differs from most of the works in literature review in that we consider both capacity and inventory limits. We also address the problem at a master planning level rather than at the faster paced lot scheduling level where rapid and frequent price changes may not be feasible. It is important to point out that Chan et al. [17] study delayed production and delayed pricing strategies rather than optimal simultaneous determination of both pricing and production, placing their work in a separate category.

The remainder of the paper is organized as follows. Section 2 describes the multiperiod price-optimizing model. Section 3 shows how the multiperiod problem can be simplified in the single-period case with known initial and ending inventories. Section 4 provides a closed form solution of the single-period problem of Section 3, assuming a demand function of the exponential form. Section 5 shows how the result of Section 4 can be used to solve the multiperiod model using a dynamic programming approach. Numerical examples are presented in Section 6 and conclusions are provided in Section 7.

2. Model Description

We formulate a single-item price-optimizing master planning problem. The problem is to determine for each period of a discrete time, finite planning horizon, the optimal sales price, production quantity, and sales amount for a single-item. In each period, a production capacity, a variable cost of production, a fixed cost, a safety stock requirement, and a demand function that returns demand as a function of price are considered. The production capacity, variable cost, fixed cost, and safety stock requirement are allowed to vary in each time period. The demand function is allowed to vary over time but is always of the same parametric form. The following notation is defined:

𝑝𝑡: sales price in period 𝑡,𝑛𝑡: production quantity in period 𝑡,𝑠𝑡: sales quantity in period 𝑡,𝐼𝑡: inventory in period 𝑡,𝑉𝑡: variable cost per unit produced in period 𝑡,𝐹: fixed cost per time period,𝐶𝑡: production capacity in period 𝑡,𝐼𝑡min: safety stock requirement in period 𝑡,𝐼𝑡max: maximum inventory limit in period 𝑡,𝐼0: given value of initial inventory,𝐷𝑡(𝑝𝑡): demand function in period 𝑡.

The demand function is assumed to be continuous, nonincreasing and asymptotically equal to zero. The general multiperiod model for 𝑇 periods is the following:

max𝑍=𝑇𝑡=1𝑝𝑡𝑠𝑡𝑉𝑡𝑛𝑡𝑇𝐹,(2.1) s.t.

𝑠𝑡𝐷𝑡𝑝𝑡𝑛𝑡,(2.2)𝑡𝐶𝑡𝐼𝑡,(2.3)𝑡=𝐼𝑡1+𝑛𝑡𝑠𝑡𝐼𝑡,(2.4)𝑡𝐼𝑡max𝐼𝑡,(2.5)𝑡𝐼𝑡min𝑡.(2.6) In addition, all variables are assumed to be nonnegative. For simplicity the nonnegativity constraints are not explicitly expressed. The objective function (2.1) is to maximize profit. Notice that the fixed cost does not play a role in the optimization, it has been included only to clarify that profit is to be maximized. Constraint (2.2) limits the sales amount to the demand. Constraint (2.3) ensures that production will not exceed the available capacity. Constraint (2.4) is an inventory balance equation. Constraints (2.5) and (2.6) keep the inventory between specified maximum and minimum limits. In order to show how to solve the multiperiod problem, we first provide a solution to a simplified problem with a single-period, assuming that the initial and ending inventories, 𝐼0 and 𝐼1, are known and feasible with respect to (2.5) and (2.6).

Notice that although an inventory holding cost parameter is not included in the objective function (2.1), it is possible to model the financial opportunity costs of holding inventory by multiplying the terms of (2.1) by the appropriate discount factors. The resulting objective is then to maximize the present value (NPV) of the future cash flows. For examples of this approach please see Hadley [18], Park and Sharp-Bette [19], Sun and Queyranne [20], and Smith and Martínez-Flores [21]. In Smith and Martínez-Flores [21] it is shown that the traditional approach and net present value (NPV) approach can yield different optimal costs and inventory policies. It is important to mention that the papers listed in Table 1 do not consider the NPV approach. The NPV model, assuming that all cash flows occur at the end of a period and eliminating the constant terms in (2.1), is the following:

tab1
Table 1: Inventory models with price and production decisions.

𝑇𝑡=1(1+𝑟)𝑡𝑝𝑡𝑠𝑡𝑉𝑡𝑛𝑡,(2.7) where 𝑟 models the financial opportunity cost. Although an operational inventory holding cost (cash cost) cannot be modeled in this way, in practice, the financial opportunity cost tends to be by far the largest portion of the inventory holding cost [22], making this modeling technique adequate for a wide range of applications.

3. The Single-Period Problem with Known Initial and Ending Inventories

The multiperiod model given by (2.1) to (2.6) can be simplified in the single-period case. The single-period model without assuming 𝐼1 is given as

max𝑍=𝑝1𝑠1𝑉1𝑛1𝐹(3.1) s.t. 𝑠1𝐷1𝑝1𝑛,(3.2)1𝐶1𝐼,(3.3)1=𝐼0+𝑛1𝑠1𝐼,(3.4)1𝐼1max𝐼,(3.5)1𝐼1min.(3.6) As before, all variables are assumed to be nonnegative and the initial inventory (𝐼0) is assumed to be feasible with respect to (2.5) and (2.6). Now, the problem with given initial and ending inventories, 𝐼0 and 𝐼1, respectively, that are feasible with respect to (2.5) and (2.6) can be formulated as

max𝑍=𝑝1𝑠1𝑉1𝑛1𝐹(3.7) s.t. 𝑠1=𝐷1𝑝1𝑛,(3.8)1𝐶1𝐼,(3.9)1=𝐼0+𝑛1𝑠1𝑛,(3.10)10.(3.11) Constraints (3.5) and (3.6) can be eliminated because 𝐼0 and 𝐼1 are assumed to be feasible. A proof to justify the equality in (3.8) can be found in the appendix. Using (3.8) and (3.10), formulations (3.7)–(3.11) can be simplified to

𝑝max𝑍=1𝑉1𝑝𝐷1+𝑉1𝐼0𝐼1𝐹(3.12) s.t. 𝐷1𝑝1𝐶1+𝐼0𝐼1𝐷,(3.13)1𝑝1𝐼0𝐼1.(3.14) In the next section a closed form solution to (3.7)–(3.11), assuming a specific parametric form of the demand function, is derived. We drop the subscripts for simplicity.

4. Closed Form Solution with an Exponential Demand Function

We now present an analytic solution assuming an exponential demand function given by

𝑝𝐷(𝑝)=𝑀exp𝑘,(4.1) where 𝑀 is the 𝑦-intercept (demand) with a price equal to zero and 𝑘>0 is a price scaling constant. See Ladany [34] and Smith and Achabal [35] for previous examples of the use of this function to model demand as a function of price. Notice that with the exponential demand function given above, problem (3.12)–(3.14) is infeasible when 𝐼0𝐼1>𝑀, since 𝑀 is the absolute upper bound on demand and when 𝐼1𝐼0>𝐶, since 𝐶 is the absolute upper bound on production. The following proposition gives the optimal closed form solution.

Proposition 4.1. The optimal values of sales price, sales quantity, and production quantity for problem (3.7)–(3.11) with 𝐷(𝑝)=𝑀exp(𝑝/𝑘) with 𝐼0𝐼1𝑀 and 𝐼1𝐼0𝐶 are given by 𝑝=minmax𝑉+𝑘,𝑘ln(𝑀)ln𝐶+𝐼0𝐼1,𝑘𝐼ln(𝑀)ln0𝐼1,if𝐼1𝐼0<0,max𝑉+𝑘,𝑘ln(𝑀)ln𝐶+𝐼0𝐼1,if0𝐼1𝐼0<𝐶,if𝐼1𝐼0𝑠=𝐶,(4.2)𝑝=𝐷𝑛,(4.3)=𝑠𝐼0+𝐼1.(4.4)

Proof. The unconstrained version of problem (3.12)–(3.14) using the exponential demand function is given by 𝑝max𝑍=𝑝𝑉𝑀exp𝑘𝐼+𝑉0𝐼1𝐹.(4.5) Problem (4.5) is solved by setting the derivative with respect to 𝑝 equal to zero, 𝑑𝑝𝑑𝑝(𝑝𝑉)𝑀exp𝑘𝑀𝑝=0,exp𝑘1+(𝑝𝑉)𝑘𝑝exp𝑘𝑀(𝑝=0,𝑉𝑝+𝑘)exp𝑘=0.(4.6) One solution is 𝑝=𝑉+𝑘 (the other is at infinity). The solution 𝑝=𝑉+𝑘 can be shown to be a maximum by verifying that the second derivative is negative at that point. The second derivative is given by 𝑑2𝑝𝑑𝑝(𝑝𝑉)𝑀exp𝑘=𝑀𝑘2𝑝(𝑝𝑉)exp𝑘2𝑀𝑘𝑝exp𝑘.(4.7) With 𝑝=𝑉+𝑘, we obtain 𝑀𝑘exp𝑉+𝑘𝑘,(4.8) which can be seen to be negative by inspection. Problem (4.5) thus has a maximum at 𝑝=𝑉+𝑘, is strictly decreasing for 𝑝>𝑉+𝑘 and strictly increasing for 𝑝<𝑉+𝑘. This result will be used later.
For ease of reference, we define 𝐿1=𝐷1𝐶+𝐼0𝐼1,𝐿2=𝐷1𝐼0𝐼1,if𝐼0𝐼1>0,,if𝐼0𝐼10.(4.9) Notice that these quantities are related to the right-hand sides of (3.13) and (3.14). The relationship 𝐿1𝐿2 can be seen to be true by inspection. Constraints (3.13) and (3.14) can be solved for 𝑝 to obtain 𝑝𝐿1,(4.10)𝑝𝐿2,(4.11) respectively. Since 𝐿1𝐿2, three cases are possible.
Case 1. 𝑉+𝑘𝐿1𝐿2.Case 2. 𝐿1𝑉+𝑘𝐿2.Case 3. 𝐿1𝐿2𝑉+𝑘.In Case 1 the prices given by (4.10) and (4.11) at equality are both to the right of the unconstrained maximum at 𝑉+𝑘. Therefore (4.10) is the binding constraint that determines the solution to the problem. In Case 2 the unconstrained solution at 𝑉+𝑘 is between the prices given by (4.10) and (4.11). Neither constraint is binding so the solution is at 𝑉+𝑘. In Case 3 the prices given by (4.10) and (5.1) are both to the left of the unconstrained maximum at 𝑉+𝑘. Therefore (4.11) is the binding constraint that determines the optimal price. It can be easily verified that (4.2) provides the correct answer in each case. Expression (4.3) follows from the proof of (3.8) given in the appendix and (4.4) follows from (3.10).

5. Solving the Multiperiod Problem

To solve the multiperiod model a dynamic programming solution approach employing the result of Proposition 4.1 is developed in this section. In order to simplify the procedure for dynamic programming, we allow only integer inventory quantities. This is well justified because the inventory quantities would be integer values in a real application. Letting t represent the time period, the recursive relationship for backward induction is

𝑓𝑡𝐼𝑡1=max𝐼𝑡=𝐼𝑡min,,𝐼𝑡max𝑓𝑡𝐼𝑡1,𝐼𝑡,(5.1) with

𝑓𝑡𝐼𝑡1,𝐼𝑡=𝑝𝑡𝑠𝑡𝑉𝑡𝑛𝑡+𝑓𝑡+1𝐼𝑡,(5.2) where 𝑓𝑡(𝐼𝑡1,𝐼𝑡) is the contribution of periods from time 𝑡 until the end of the horizon given period 𝑡 begins with 𝐼𝑡1 inventory, ends with 𝐼𝑡 inventory and optimal decisions are made thereafter. The value of 𝑓𝑇+1(𝐼𝑇) is by definition equal to zero and 𝑝𝑡, 𝑠𝑡, and 𝑛𝑡 are defined as follows:

𝑝𝑡=𝑉minmax𝑡+𝑘𝑡,𝑘𝑡𝑀ln𝑡𝐶ln𝑡+𝐼𝑡1𝐼𝑡,𝑘𝑡𝑀ln𝑡𝐼ln𝑡1𝐼𝑡,if𝐼𝑡𝐼𝑡1𝑉<0,max𝑡+𝑘𝑡,𝑘𝑡𝑀ln𝑡𝐶ln𝑡+𝐼𝑡1𝐼𝑡,if0𝐼𝑡𝐼𝑡1<𝐶𝑡,,if𝐼𝑡𝐼𝑡1=𝐶𝑡,𝑠(5.3)𝑡=𝐷𝑡𝑝𝑡𝑛,(5.4)𝑡=𝑠𝑡𝐼𝑡1+𝐼𝑡.(5.5) When a discounted cash flow approach is used, (5.2) becomes

𝑓𝑡𝐼𝑡1,𝐼𝑡=(1+𝑟)𝑡𝑝𝑡𝑠𝑡𝑉𝑡𝑛𝑡+𝑓𝑡+1𝐼𝑡.(5.6) In the implementation of the method, when 𝑠𝑡=0, which can occur when the ending inventory is greater than the initial inventory by an amount exactly equal to the available capacity, the price is not relevant and is set equal to any positive constant in order to correctly evaluate the objective function. When 𝐼𝑡1𝐼𝑡>𝑀𝑡, which makes the problem infeasible, the objective function is set to a negative number, effectively eliminating such a combination from further consideration. The problem is also infeasible when 𝐼𝑡𝐼𝑡1>𝐶𝑡 since 𝐶𝑡 is the absolute upper bound on production. These cases are explicitly excluded from consideration.

6. Numerical Examples

In this section, some numerical examples will be presented to illustrate the dynamic programming solution approach on a small three-period problem. Let, 𝑟=0.01, 𝐼0=1, 𝑀1=10, 𝑀2=12, 𝑀3=15, 𝑘1=3, 𝑘2=2, 𝑘3=8, and 𝐶𝑡=4, 𝑉𝑡=2, 𝐼𝑡min=0, 𝐼𝑡max=3 for all 𝑡. Table 2 for 𝑡=3 is populated using (5.1)–(5.5). The value of 𝑓4() is by definition equal to zero.

tab2
Table 2: For t = 3.

Table 3 for 𝑡=2 is populated similarly.

tab3
Table 3: For t = 2.

Table 4 for 𝑡=1 is populated similarly.

tab4
Table 4: For t = 1.

The optimal solution is shown in Table 5. The optimal sales prices, sales quantities, and production quantities can be found using (5.3), (5.4), and (5.5), respectively. Notice that the sales and production quantities are not integer values. In practical master planning applications that are solved using linear programming, noninteger values are acceptable approximations due to the aggregate nature of the products, the medium to long-term time horizons involved, and the typically large quantities planned to be produced.

tab5
Table 5: Optimal solution for example 1.

Three additional illustrative examples will be presented to highlight some of the behavior of the model. The following example illustrates how it is possible to have an optimal solution in which, although it is feasible to produce and sell the optimal quantity in one period when that period is considered in isolation, the sale in that period will be limited to allow greater sales in a later more profitable period. Let 𝑟=0.01, 𝐼0=0, 𝑀1=100, 𝑀2=100, 𝑀3=610, 𝑘1=5.1, 𝑘2=5.1, 𝑘3=8, and 𝐶𝑡=21, 𝑉𝑡=10, 𝐼𝑡min=0, 𝐼𝑡max=40 for all 𝑡. The optimal solution is shown in Table 6. Notice how although it is feasible to sell 5.18 units in periods 1 and 2 (which would be optimal for those periods taken in isolation), the optimal solution is to limit sales in the first two periods to allow greater sales in the last period.

tab6
Table 6: Optimal solution for example 2.

Our next example illustrates what we call horizon decoupling which can help solve problems with many time periods when production costs are constant or decrease over time. It is worth noticing that the horizon decoupling is an example of a regeneration point, which is a fundamental construct of planning horizon theory. See, for instance, Chand et al. [36] for a review of literature on planning horizon theory. This example is identical to the previous one with the exception that 𝐶2=𝐶3=35. The optimal solution is shown in Table 7. Notice that the capacity in the last period is not enough to produce its optimal (when considered in isolation with infinite capacity) sales amount of 64.30 units. It, therefore, remains coupled to previous periods. Further notice that the two last periods do have between them enough capacity to produce their optimal (when each period is considered in isolation with infinite capacity) sales amounts of (64.30+5.18=69.48). They, therefore, decouple from previous periods and can be solved independently of any previous periods. This property of the problem may allow problems with long planning horizons to be split into several smaller problems with shorter planning horizons that can be solved more easily. Notice, however, that if we let 𝑉1=9, the optimal solution calls for inventory to be accumulated at the end of period 1 to take advantage of the lower production cost. The optimal solution with 𝑉1=9 is shown in Table 8. Thus, it cannot be assumed that the planning horizon will decouple when production costs are increasing over time. It is important to note that if the setup costs are included, the planning horizon results would change.

tab7
Table 7: Optimal solution for example 3.
tab8
Table 8: Optimal solution for example 4.

The last example we present illustrates the value of solving a joint pricing and production problem taking into account capacity and inventory constraints. The parameters of the problem are the following: 𝑟=0.01, 𝐼0=0, 𝑀1=100, 𝑀2=100, 𝑀3=110, 𝑘1=5, 𝑘2=5, 𝑘3=7, and 𝐶𝑡=5, 𝑉𝑡=10, 𝐼𝑡min=0, 𝐼𝑡max=2 for all 𝑡. The optimal solution is shown in Table 10. The optimal objective value is $161.96. Now assume the marketing department sets prices using the same data but assuming that the aggregate capacity over the next three periods is equal to 15. That is, an aggregate capacity limit is imposed rather than a period by period limit. The projected plan under these assumptions is shown in Table 9. The projected objective value would be $165.87. Now assuming that the marketing department executed to the planned prices, but production is now constrained by the real inventory and capacity limits, the greatest possible profit is only $97.96, well below both the projected plan and the optimal plan. Now suppose that marketing creates a pricing plan taking into account the period by period capacity limits but fails to consider the inventory limits. The projected plan is shown in Table 11. The projected objective value would be $165.36. Now assuming that marketing executes to the planned prices, but production is now constrained by the real inventory and capacity limits, the greatest possible profit is only $97.44, also well below both the projected plan and the optimal plan. These examples show that neglecting to consider capacity and/or inventory constraints can have very significant detrimental effects on the profitability of the firm.

tab9
Table 9: Optimal solution for example 5.
tab10
Table 10: Projected plan assuming aggregate capacity.
tab11
Table 11: Projected plan assuming no constraints on inventory.

7. Conclusions and Recommendations for Further Research

In conclusion, we derive an exact solution to the single-period price-optimizing master planning problem with deterministic demand and inventory and capacity constraints for the case with known initial and ending inventories. In addition, we show how to solve the multiperiod version of the problem using a dynamic programming approach. Our direct observation of the planning models in use at a variety of industries shows that the types of constraints we consider are commonly used in practice but largely missing in literature. We also show that inventory holding costs can be included in the model by discounting the terms of the objective function. The numerical examples presented serve to highlight the importance of taking into account capacity and inventory constraints when generating a pricing and production plan. The implication for practitioners is that potentially significantly higher profits can be obtained through price optimization, making sure to consider the firms capacity and inventory constraints.

It is worth noting that our proposed model has three main advantages. First, our model considers both capacity and inventory limits. Our consulting experience shows that firms take both types of constraints into account, making their inclusion desirable. Second, we address the problem at a master planning level, where setups are usually not considered. The previously published works address similar problems at a lot scheduling level despite the fact the price changes (with the exception of discounts) are often not feasible in the short term. Third, our model considers the net present value approach instead of the traditional approach.

The research presented in this paper may be extended in several ways. One extension is to solve the model with an upper bound on the price or, alternatively, on the allowable change in price between periods, which is a realistic market scenario. In addition, solution approaches could be developed for multiitem and stochastic versions of the problem. Models with days-of-cover constraints would also be relevant research topics as would be the inclusion of setup costs in a mixed integer formulation. An additional recommendation is to investigate a dynamic control version of the problem, which would recast the problem from a planning level to an operational level. Additional possible extensions are to reformulate the model to include demand learning effects [37], and to model the supply chain with a supplier-buyer relationship as two-player nonzero sum differential game [38].

Appendix

Justification of the Equality in Constraint (3.8)

Claim 1. For problem (3.7)–(3.11), with (3.8) rewritten as 𝑠1𝐷(𝑝1) and assuming that 𝐷(𝑝) is a demand function that is continuous, nonincreasing and asymptotically equal to zero, if 𝑝, 𝑠, and 𝑛 are optimal, then 𝑠=𝐷(𝑝).

Proof. Assume that 𝑠𝐷(𝑝). This yields two cases. The first is that 𝑠>𝐷(𝑝). This case can be ignored since it is impossible for sales to exceed demand. The remaining case is that 𝑠<𝐷(𝑝). For optimality we require that 𝑠𝑝𝑉𝑛𝑠𝑝𝑉𝑛 for any feasible choice of 𝑠,𝑝, and 𝑛 (notice that 𝑝 is not bounded from above by (3.8)–(3.11). However, if 𝑠<𝐷(𝑝), given that 𝐷(𝑝) is continuous, nonincreasing and asymptotically equal to zero, there exists a feasible 𝑝𝑠>𝑝 such that 𝑠=𝐷(𝑝𝑠). This implies that 𝑠𝑝𝑉𝑛<𝑠𝑝𝑠𝑉𝑛, which contradicts the optimality of 𝑝.

Acknowledgments

This research was partially supported by the research fund no. CAT128 and by the School of Business at Tecnológico de Monterrey. The authors would also like to thank the two anonymous referees for their constructive comments and suggestions that enhanced this paper.

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