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

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 discretetime multiperiod horizon. The solution process consists of two steps. First, we solve the singleperiod 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.


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 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.

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:

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, I 0 and I 1 , 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: 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.

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 I 1 is given as 3 4 5

3.6
As before, all variables are assumed to be nonnegative and the initial inventory I 0 is assumed to be feasible with respect to 2.5 and 2.6 .Now, the problem with given initial and ending inventories, I 0 and I 1 , respectively, that are feasible with respect to 2.5 and 2.6 can be formulated as 3.9 Constraints 3.5 and 3.6 can be eliminated because I 0 and I 1 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

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.

Closed Form Solution with an Exponential Demand Function
We now present an analytic solution assuming an exponential demand function given by where M is the y-intercept demand with a price equal to zero and k > 0 is a price scaling constant.See Ladany 34 Proof.The unconstrained version of problem 3.12 -3.14 using the exponential demand function is given by Problem 4.5 is solved by setting the derivative with respect to p equal to zero,

Mathematical Problems in Engineering
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 With p V k, we obtain 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 4.9 Notice that these quantities are related to the right-hand sides of 3.13 and 3.14 .The relationship L 1 ≤ L 2 can be seen to be true by inspection.Constraints 3.13 and 3.14 can be solved for p to obtain respectively.Since L 1 ≤ L 2 , three cases are possible.
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 .

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 trepresent the time period, the recursive relationship for backward induction is

5.3
s t D t p t , 5.4 n t s t − I t−1 I t .

5.5
When a discounted cash flow approach is used, 5.2 becomes In the implementation of the method, when s t 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 I t−1 − I t > M t , 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 I t − I t−1 > C t since C t is the absolute upper bound on production.These cases are explicitly excluded from consideration.

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, I 0 1, M 1 10, M 2 12, M 3 15, k 1 3, k 2 2, k 3 8, and C t 4, V t 2, I min t 0, I max t 3 for all t.Table 2 for t 3 is populated using 5.1 -5.5 .The value of f * 4 • is by definition equal to zero.
Table 3 for t 2 is populated similarly.Table 4 for t 1 is populated similarly.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.
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, I 0 0, M 1 100, M 2 100, M 3 610, k 1 5.1, k 2 5.1, k 3 8, and C t 21, V t 10, I min t 0, I max t 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.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 C 2 C 3 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 V 1 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 V 1 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.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, I 0 0, M 1 100, M 2 100, M 3 110, k 1 5, k 2 5, k 3 7, and C t 5, V t 10, I min t 0, I max t 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.

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 s 1 ≤ D p 1 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 s * / D 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 s * p * − V n * ≥ sp − V n 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 p s > p * such that s * D p s .This implies that s * p * − V n * < s * p s − V n * , which contradicts the optimality of p * .
The demand function is assumed to be continuous, nonincreasing and asymptotically equal to zero.The general multiperiod model for T periods is the following: sales price in period t, n t : production quantity in period t, s t : sales quantity in period t, I t : inventory in period t, V t : variable cost per unit produced in period t, F: fixed cost per time period, C t : production capacity in period t, t ≥ I min t ∀t.

Table 1 :
Inventory models with price and production decisions.
and Smith and Achabal 35for 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 I 0 − I 1 > M, since M is the absolute upper bound on demand and when I 1 − I 0 > 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 M exp −p/k with I 0 − I 1 ≤ M and I 1 − I 0 ≤ C are given by

Table 5 :
Optimal solution for example 1.

Table 6 :
Optimal solution for example 2.

Table 7 :
Optimal solution for example 3.

Table 8 :
Optimal solution for example 4.

Table 9 :
Optimal solution for example 5.

Table 10 :
Projected plan assuming aggregate capacity.

Table 11 :
Projected plan assuming no constraints on inventory.