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Continuously increasing price in a gradual usage inventory cycle: an optimal strategy for coordinating production with pricing for a supply chain.


The inventory literature has received attention in manufacturing industry almost a century ago. In 1913, Ford Harris suggested an idea of how many parts to make in a batch. The Harris order batching model has then been forgotten until the idea was later published by Wilson (1934) in Harvard Business Review. The batching rule then became known as the Wilson Economic Order Quantity (EOQ) as it applies to inventory management.

The model assumes that a retailer of a product buys the product at a constant unit cost, incurs a fixed cost per order, stores the product at a constant carrying cost per unit of inventory per year, and faces a deterministic and constant demand rate over an infinite horizon, the retailer's optimal strategy is to buy a fixed quantity every time he or she replenishes the inventory. Ignoring inventory related costs, classical price theory tells us that when a product's demand is price sensitive but the demand curve is known and stationary, the retailer's optimal strategy is to charge a single price throughout the year. Although Whitin (1955) was the first one to integrate the concepts of inventory theory with the concepts of price theory to investigate the simultaneous determination of price and order quantity decisions of a retailer, he never stated so explicitly. Whitin's (1955) model would adhere to all the assumptions of the EOQ model stated above, except that demand is price sensitive, with a known and stationary demand curve, a retailer's optimal strategy would be, once again, to buy a fixed quantity for every inventory cycle and to sell it at a single price.

Kunreuther and Richard(1971) then showed that when demand is price elastic, centralized decision-making (using simultaneous determination of optimal price and order quantity) was superior to the common practice of decentralized decision-making whereby the pricing decisions were made by the marketing department while the order quantity decisions were made by the purchasing department independently. Although Kunreuther and Richard (1971) were perhaps unaware of Whitin's (1955) paper, their model was very similar to Whitin's (1955) model. Assuming a known and stationary demand curve along with the remaining conditions of the EOQ model, Arcelus et al (1987, page 173) asserted: "given constant marginal costs of holding and purchasing the goods, the firm will want to maintain the same price throughout the year". Again, they assumed a fixed single selling price throughout each inventory cycle. What they did not realize is that, even though marginal holding costs are constant per unit, a firm's holding costs at any particular time within an inventory cycle are a function of inventory on hand, which itself is a function of the time from the beginning of the inventory cycle.

Since Whitin's (1955) work, numerous authors (Tersine & Price,1981; Arcelus & Srinivasan,1987; Ardalan, 1991; Hall, 1992; Martin, 1994; Arcelus & Srinivasan, 1998; Abad, 2003) have used Whitin's (1955) and Kunreuther and Richard's(1971) models as foundations to their own models. But none of these authors have ever questioned Whitin's (1955) and Kunreuther and Richard's (1971) assumption that the retailer's optimal strategy would be to sell the product at a fixed price throughout the inventory cycle. The fact that Whitin's (1955) and Kunreuther and Richard's (1971) assumption of a single price throughout an inventory cycle leads to suboptimal profits for the retailer is due to declining carrying costs as a function of time. However, any optimization model allowing a retailer with a price-insensitive demand to set the selling price arbitrarily would push the price to infinity. In other words, in that situation, price is not seen as a decision variable for any mathematical model. Given an arbitrary price (and corresponding demand), the retailer's only strategy is to minimize his inventory ordering and holding costs by using the EOQ model.

Considering a situation of price sensitive demand, Abad (1997; 2003) found that, in the case of a temporary sale with a forward buying opportunity, a retailer's optimal strategy is to charge two different prices during the last inventory cycle of the quantity bought on sale--a low price at the beginning of the inventory cycle and a higher price starting somewhere in the middle of the cycle. Yet, Abad (1997; 2003) did not consider a similar strategy in every regular inventory cycle of a product with price sensitive demand. Inspired by Abad's (1997; 2003), Joglekar et al (2008) showed that a continuously increasing price strategy: charging a relatively low selling price at the beginning of an inventory cycle when the on-hand inventory is large, would lead to higher profit.

With the widespread use of revenue management or yield management techniques (Feng & Xiao, 2000; McGill &. van Ryzin,1999; Smith, Leimkuhler & Darrow, 1992; Talluri & van Ryzin,2000; Weatherford &. Bodily, 1992. in the airline, car rental, and hotel industries today, a time-dependent (or dynamic) pricing strategy has become commonly adopted. Revenue management techniques are typically applied in situations of fixed, perishable capacity and a possibility of market segmentation (Talluri & van Ryzin, 2000). In recent years, retail and other industries have begun to use dynamic pricing policies in view of their inventory considerations. The recent recession has brought forth dynamic pricing to a new light. As retail sales dropped, businesses were facing unusual built-ups of inventory that would lead to order cancellations affecting all parties of the supply chain.

In this paper, we extend the dynamic pricing model to incorporate the gradual usage (or gradual production/ replenishing) inventory. (See Chase et al, 2010) for the gradual usage model.) Consider a retailer that is also a manufacturer of a product. Depending on the demand and the ordering quantity, the retailer would produce the inventory gradually until it fulfills the planned order quantity. Since the build-up is gradual, the actual inventory holding cost will be lowered and that the maximum inventory level will not be as high as the basic EOQ. In light of the currently changing demands with economic conditions, this model is appropriate when the retailer/manufacturer would have to adjust his or her inventory policy and production schedule according to the current price elasticity of demand dictated by the market.

Although this model is limited by the fact that it assumes deterministic customer behavior (or deterministic price elasticity) and any lack of competitive reactions to one's action, it does provide the ground work to take on a new direction of inventory management in supply chain. In addition, it provides a way to reset initial pricing to optimize profit, and the marginal rate of price increase during an inventory cycle. The most beneficial of it all is that the solutions are only dependent on the assumed price elasticity and cost data, and can be revised easily should the market conditions change again.

In what follows, we first recapitulate Whitin's (1955) and Kunreuther and Richard's (1971) fixed price strategy model with gradual usage cycle as described in Chase et al (2010). In the next section, we present our own model. The derived model has a polynomial objective function, and hence, does not have a closed form solution. However, we are still able to provide the optimal initial pricing and the marginal rate of price increase during the cycle easily. Using Microsoft Solver[R], we utilize several numerical examples with a linear demand curve and varied values of relevant parameters. The final section provides the conclusions of our analysis along with some directions for future research.


Both papers of Whitin (1955), Kunreuther and Richard (1971) consider a situation where all the other assumptions of the EOQ model are valid but demand is price sensitive, with a known and stationary demand curve. Whitin's (1955) notation is different from Kunreuther and Richard's (1971) notation. There are also some minor differences in the details of the two models. In addition with Chase et al (2010) for gradual usage assumption, the following captures the basics of these models. Although the model is applicable to any form of the demand function, for simplicity, we use a linear demand function. Let the following notations hold,

C = retailer's known and constant unit cost of buying the product,

S = retailer's known and constant ordering cost per order,

I = retailer's carrying costs per dollar of inventory per year,

[P.sub.1] = retailer's selling price per unit in this model,

m = constant production rate per period,

t = time elapsed from the beginning of an inventory cycle,

[Y.sub.1] = retailer's profit per cycle,

[Z.sub.1] = retailer's profit per period

It is assumed that,

[P.sub.1] > C, and [D.sub.1] = retailer's annual demand as a function of the selling price, [P.sub.1], hence, [D.sub.1] = a - b [P.sub.1] where a and b are nonnegative constants, a representing the theoretical maximum annual demand (at the hypothetical price of $0 per unit) and b representing the demand elasticity (i.e., the reduction in annual demand per dollar increase in price). Although P1 would remain constant throughout a cycle, we choose to express it in an affine function form to be consistent with the price increasing model in the next section.

Note that since [D.sub.1] must be positive for the conceivable range of values of [P.sub.1], a > b[P.sub.1] for that range of values of [P.sub.1], and since [P.sub.1] > C, it follows that a > bC.


[T.sub.1] = the duration of retailer's gradual replenishment in an inventory cycle, and

[T.sub.2] = the duration of retailer's inventory cycle (replenishment and consumption).

Note that [T.sub.2] > [T.sub.1]. Further, let

Q = retailer's order quantity per order in this model. Then, Q = [D.sub.1][T.sub.2] = (a-b[P.sub.1])[T.sub.2]. The maximum inventory level would then be m[T.sub.1]-[DT.sub.1] = (m-a+b[P.sub.1])[T.sub.1].

From [T.sub.1] [greater than or equal to] t [greater than or equal to] [T.sub.2], the retailer would be consuming from the inventory at a rate of [D.sub.1]t until the end of the cycle, [T.sub.2]. That is, we have

(m-a + b[P.sub.1])[T.sub.1] = [D.sub.1]([T.sub.2] - [T.sub.1]), or

[T.sub.1] = [(a - b[P.sub.1])/m][T.sub.2] (1)

The retailer's profit per cycle, Y is given by

[Y.sub.1] = ([P.sub.1] - C)[D.sub.1][T.sub.2] - IC[[T.sub.1](m - a + b[P.sub.1])/2][T.sub.1] - IC[[T.sub.1](m-a + b[P.sub.1])/2]([T.sub.2] - [T.sub.1]) - S (2)

Thus, the retailer's profit per period, [Z.sub.1] is obtained by [Y.sub.1]/[T.sub.2], and by substituting (1) into [Y.sub.1], we have,

[Z.sub.1] = ([P.sub.1] - C)(a - b[P.sub.1]) - IC[(m/2) - (a - b[P.sub.1])/2][(a - b[P.sub.1])/m][T.sub.2] - (S/[T.sub.2]) (3)

Differentiating [Z.sub.1] with respect to [P.sub.1] and [T.sub.2], the first-order conditions for the maximization of this function are:

[P.sub.1] = [(a/b) + C + (IC[T.sub.2]/2) - (aIC[T.sub.2]/m)]/[2 - (bIC[T.sub.2]/m)] (4)

[T.sup.2.sub.2] = 2S/{(IC)(a - b[P.sub.1])[1 - (a - b[P.sub.1])/m]} (5)

Notice that Equation (5) is equivalent of the gradual usage (or production) model as illustrated in

Chase et al (2010) .

Combining Equations (4) and (5) and simplifying, we have an explicit quartic equation in terms of [T.sub.2]:


As we can see here, we can solve Equation (6) for [T.sub.2] which has 4 roots. To obtain these roots, one may use MathCAD to find all the closed form solutions in terms of the coefficients of [T.sub.2], and substituting them into (3) to see which root would optimize [Z.sub.1]. On the other hand, one may use other software to solve (4) and (5) simultaneously. However, we elect to use Excel Solver[R] to solve for [P.sub.1] and [T.sub.2] simultaneously for optimal [Z.sub.1]. To ensure solution quality, we have also implemented the constraints such as [P.sub.1] > C, and so on. We then verify the solution with (4), (5) and (6). Once care is taken to input these conditions and reasonable starting values, in our experimentation, Excel Solver has never failed to return the best real solution (if any). Hence, we believe that practicing managers would be adequately served by the use of Excel Solver. They need not worry about obtaining all the roots of the quartic equation.


We retain all of the assumptions of the foregoing model, except that now we assume that the retailer uses a continuously increasing price strategy within each inventory cycle.

Let us add the following notation:

[P.sub.2](t) = the retailer's selling price at time t, and [P.sub.2](t) = f+gt, where f and g are nonnegative decision variables, and f > C. f represents the retail price at the beginning of each inventory cycle and g represents the annual rate of increase in the retail price throughout an inventory cycle.

[Y.sub.2] = retailer's profit per cycle, and

[Z.sub.2] = retailer's profit per period.

X(t) = the retailer's inventory level at time t, with X(0) = 0, X([T.sub.2]) = 0, and

X([T.sub.1]) = the maximum inventory level in an inventory cycle.

[D.sub.2](t) = a - b[P.sub.2](t) = retailer's demand as a function of the selling price.

Given that price is a function of time, now the retailer's annual demand rate will also be a function of time. Hence, we should redefine demand as

[D.sub.2](t) = a-b[P.sub.2](t) = a-bf-bgt (7)

Since at the beginning of the inventory cycle, the retailer orders a quantity Q to meet the cycle time demand, and with the gradual production rate, m,

Q = m[T.sub.1]

X(t), the inventory level at time t, for 0 < t < [T.sub.1], is

X(t) = [[integral].sup.t.sub.0][m - D(t)]dt = mt-at+bft+1/2[bgt.sup.2]

At [T.sub.1], the maximum inventory level is reached, and

X([T.sub.1]) = m[T.sub.1] - a[T.sub.1] + bf[T.sub.1] + 1/2bg[T.sub.1.sup.2] (8)

At the end of an inventory cycle, X([T.sub.2]) = 0. That is,

X([T.sub.2]) = m[T.sub.1] - a[T.sub.2] + bf[T.sub.2] + 1/2bg[T.sub.2.sup.2] = 0 which implies

[T.sub.1] = (a[T.sub.2] - bf[T.sub.2] - 1/2bg[T.sub.2.sup.2])/m (9)

The retailer's profit per inventory cycle is given by,


Substituting (9) after evaluating (10), we obtain,


The profit per period, [Z.sub.2] = [Y.sub.2]/[T.sub.2] is,


Differentiating [Z.sub.2] with respect to [T.sub.2], f, and g, for the first order optimal conditions by setting them to zero, we have, respectively,




From (14) and (15), we obtain,



g = (3/m)ICf(1-b)+1/2IC(17) (17)

Notice that f represents the initial price set at the beginning of the inventory cycle, and the marginal increase during the cycle is given by g as in Equation (17). Also note that when m approaches infinity as in the case of basic EOQ models with instantaneous replenishment, g would be equal to IC, same as the basic model.

Of course, when we replace f and g in Equation (13), we would obtain an explicit polynomial function of [T.sub.2] , that is, of fifth order, or quintic function. There is, unfortunately, no close form solution for any polynomial function higher than quartic. For practitioners, we would, once again, rely on Excel Solver which has demonstrated to be a reliable tool for obtaining an optimal solution.


Consider a situation where the retailer's cost of a product is $5 per unit. The theoretical maximum periodic demand is 20 units and periodic demand declines at the rate of 1 unit for each dollar's increase in the price. The ordering costs are $100 per order and the carrying costs are $0.05 per dollar of inventory per period. The production rate is 40 units per period. We will refer to this set of assumptions as the base case.

Table 1 summarizes the base case assumptions, the optimal decisions and the consequences under the two models. As can be seen there, in the fixed price model, the optimal retail price is $15.64 per unit and the optimal inventory cycle time is 10.35 periods. This means that the retailer would order 45.088 units per order and would obtain per period profit of $31.08.

In the continuously increasing price production model, the starting retail price is $12.75 per unit at the beginning of the cycle and that price increases at the rate of $0.125 per period. The optimal cycle time is 12.118 periods. This means that the retail price at the end of the cycle is $14.26 per unit, the retailer would order 69.548 units per order and would achieve a periodic profit of $39.88.

In addition to reporting these numbers, Table 1 also presents the percent differences between the two models for each relevant decision and consequence. Observe that, in percent terms, the differences are rather substantial. In comparison with the fixed price strategy, the continuously increasing price strategy results in a longer cycle time of 17.08 percent. At the beginning of an inventory cycle, under the continuously increasing price model, the retail price is smaller than what it is under the fixed price model. However, by the end of the cycle, the retail price under the continuously increasing price model is still lower than what it is under the fixed price model. As a result, the periodic demand is larger under the continuously increasing price model. The average cycle profit shows a substantial improvement of 28.31 percent in the continuously increasing price model compared to the per cycle profit under the fixed price model. This shows that continuously pricing model performs substantially better than a fixed pricing model by taking advantage of the gradual usage/replenishment, and the pricing elasticity.

The specific numerical results we have obtained are a function of the numerical assumptions we have made. Hence, in order to identify the circumstances under which the continuously increasing price strategy would be particularly desirable, we carried out a sensitivity analysis, as described in the following section.


Table 2 summarizes the results of an analysis where we increased the value of each one of our parameters (except the last row which indicates a decrease in production level), one at a time, while maintaining the values of the other parameters constant. In each case, Table 2 shows the consequences of these changes on the retailer's annual profits under the two models and the percentage increase in the periodic profit that the retailer obtains by using the continuously increasing price gradual usage strategy as against using the fixed price strategy. For comparison purposes, the first row of Table 2 repeats the profit results of the two models in the base case.

While other things remain constant, any increase in ordering cost, elasticity, inventory holding rate, and production/replenishment rate would still favor the continuously increasing price strategy. With the exception of production/replenishment rate, the periodic profits can increase rather significantly over those of the fixed price model, although the differences are getting narrower. In particular, when the production/replenishment rate is doubled, the profit difference is only 1.69%. This, we believe is due to the fact the demand rate, m, is due to the fact that when increasing production/replenishment rate would be equivalent to switching gradual production to instantaneous delivery of the basic EOQ model. As Joglekar et al (2008) has pointed out that the percentage changes would be small under the basic EOQ assumptions.

The additional analysis of decreasing the production/replenishment rate, m, is shown in the last row of Table 2. Notice that the gradual usage model discussed in Chase et al (2010) exhibits lower inventory cost if m approaches the periodic demand. In the continuously increasing price strategy, the profit increase is even more pronounced given the fact that we would now take advantage of the slower replenishment, as well as the price elasticity leading to a 23.96% increase with m = 20 over the fixed price model.

Of course, our sensitivity analysis focused on changes in one parameter at a time. When several parameters are favorable to the continuously increasing price strategy, the gains offered by this strategy may be even more significant.


Traditionally, operations researchers (Whitin, 1955: Kunreuther & Richard, 1971, and others) have assumed that when a product's demand curve is known and stationary, a retailer of the product would find it optimal to buy a fixed quantity every time he buys and to sell the product at a fixed price throughout the year. Joglekar et al (2008) found that the continuously increasing price strategy would increase the periodic profit abide the small gains. With the gradual usage assumption, Chase et al (2010) shows that by take advantage of the noninstantaneous production/replenishment rate, the total inventory cost could be reduced. However, when employing the continuously increasing price strategy in this situation as proposed by Joglekar et al (2008), we find that the profit per period would increase significantly as indicated in the numerical examples in the previous sections.

A continuously increasing price strategy might be impractical in the past, but with the new technology today, e-tailers can easily update their prices continuously. Elmaghraby and Keskinocak's (2003) review of dynamic price models indicates that a number of industries are already using continuously changing pricing strategies.

With the recent economic downturn, retailers are more conscientious of their inventory and pricing. The ever changing retail environment may warrant a new and more dynamic strategy in order to remain competitive in the market place. Our model provides a practical and systematic approach to coordinate a supply chain's sales and productions. In addition, it provides an easy computational tool of the initial price and the marginal rate of increase. Should the business environments change; a new pricing scheme can be quickly re-configured and implemented just like what we have shown with our formulas in section 3.

Our numerical example suggests that the advantage of a continuously increasing price strategy is significant and the sensitivity analysis shows that this strategy is particularly desirable when demand is highly price sensitive or when an e-tailer's supplier commands great pricing power. While the continuously increasing price strategy may not be practical for a brick-and-mortar retailer, such a retailer could use the dual price strategy model developed by Joglekar (2003) that a retailer who sets two different prices at two different points in an inventory cycle obtains a greater profit than a retailer using a single fixed price throughout the cycle. Although Joglekar et al (2008) model only shows modest gains under the basic EOQ assumptions, the continuously increasing price strategy with gradual usage (or non-instantaneous replenishment) provides significantly higher profits. In addition, it provides an avenue where retailers can adjust their pricing schemes in a hurry when the market conditions call for sudden and significant changes in price elasticity.

There are several directions in which this research can be further investigated. In this paper, we have extended the Joglekar et al (2008) model for a retailer who is also a manufacturer of the product. Some retailers have now adopted the pre-ordering strategy (such as game console, smart phones, and so on), which is similar to back-ordering. This extension would do well for those with different pre-order and initial prices to maximize profit. Finally, a number of recent models coordinating pricing and order quantity decisions across a supply chain [23-26] have assumed a fixed price for each participant of the supply chain. These models also need to be updated by considering a continuously increasing price strategy.


This research is dedicated to Professor Prafulla Joglekar who passed away in November 2009.


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Patrick Lee, Fairfield University

Prafulla Joglekar, La Salle University (deceased)
Table 1: A numerical example.

Assumptions common to both models

C = $5/unit; a = 20 units/period; b =
1 unit/period; S = $100/order; I =
$0.05/$/period; m = 40 units/period

Optimal Decisions and Consequences
Under the Two Models

                                                   Fixed Price

Optimal        Production periods, [T.sub.1]       1.1272

Decisions      Cycle time, [T.sub.2]               10.35047204

               Price at beginning of cycle, f      $15.64/unit

               Price increase rate per period, g   None

Consequences   Order quantity, Q                   45.088/order

               Price at the end of the cycle       $15.64/unit
               (=f +gt)

               Profit per period, Z                $31.08

               Continuously   Percent
               Increasing     Difference
               Price Model    Between
                              the Models

Optimal        1.7387         54%

Decisions      12.11802081    17.08%

               $12.75         -18.48%

               $0.125/unit    NA

Consequences   69.548/order   54.25%

               $14.26/unit    -8.80%

               $39.88         28.31%

Table 2. Sensitivity Analysis

                                A comparison of the periodic
                                profit under the two models

Changed assumption(s)   Fixed price      Continuously     Percent
                        gradual          increasing       Difference
                        usage model      price gradual    Between the
                                         usage model      Two Models

None (Base Case)        $31.08/period    $39.88/period    28.31%

S = $150/order          $33.22/period    $36.16/period    8.85%

b = 1.2 units/period    $22.71/period    $26.17/period    15.24%

I = 0.1/unit/period     $29.78/period    $33.01/period    10.85%

m = 80                  $37.32/period    $37.95/period    1.69%

m = 20                  $37.40/period    $46.36/period    23.96%
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Author:Lee, Patrick; Joglekar, Prafulla
Publication:Academy of Information and Management Sciences Journal
Article Type:Report
Date:Jan 1, 2012
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