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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.

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 [

Joint pricing and production decisions problems are treated in literature; see, for example, Elmaghraby and Keskinocak [

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 [

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 [

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. [

The remainder of the paper is organized as follows. Section

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:

The demand function is assumed to be continuous, nonincreasing and asymptotically equal to zero. The general multiperiod model for

Notice that although an inventory holding cost parameter is not included in the objective function (

Inventory models with price and production decisions.

Author(s) | Year | Inventory model | Capacity constrain | Inventory constrain | Multiperiod horizon |
---|---|---|---|---|---|

Whitin [ | 1955 | It was linked the price policy and inventory theory and determined the combined policy that yield the maximum profit | No | No | No |

Thomas [ | 1970 | Determines simultaneously the price and production decision with a known deterministic demand function | No | No | Yes |

Kunreuther and Richard [ | 1971 | Determines the price and ordering decision considering a stationary demand curve | No | No | No |

Cheng [ | 1990 | An economic order quantity (EOQ) model that integrates the product pricing and order sizing decisions with storage space and inventory investment limitations | No | Yes | No |

Harris and Pinder [ | 1995 | Determines optimal price and capacity decision for a single-period | No | No | No |

Kim and Lee [ | 1998 | Determines the optimal price, lot size and the capacity decision for a firm with constant price-dependent demands | Yes | No | No |

Gilbert [ | 1999 | Determines the optimal price and production schedule for a product with seasonal demand | No | No | Yes |

Bhattacharjee and Ramesh [ | 2000 | Determines the optimal price and lot size for a product with fixed life perish-ability for a certain number of periods | No | No | Yes |

Gilbert [ | 2000 | Determines the optimal price and production schedule for a product with seasonal price dependent demand | Yes | No | Yes |

Zhao and Wang [ | 2002 | Coordination of price and production schedules in a decentralized supply chain | No | No | Yes |

Chen and Simchi-Levi [ | 2003 | Determines the price and productions decisions | No | No | Yes |

Deng and Yano [ | 2006 | Setting prices and choosing production quantities for a single product over a finite horizon for a capacity-constrained manufacturer facing price-sensitive demands | Yes | No | Yes |

Chan et al. [ | 2006 | Study delayed production and delayed pricing strategies for a multiple period horizon under a general, nonstationary stochastic demand function with a discrete menu of prices | Yes | No | Yes |

This paper | This year | Determines the optimal pricing and production master planning in a multi period horizon considering capacity and inventory constraints | Yes | Yes | Yes |

The multiperiod model given by (

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

The optimal values of sales price, sales quantity, and production quantity for problem (

The unconstrained version of problem (

For ease of reference, we define

To solve the multiperiod model a dynamic programming solution approach employing the result of Proposition

In this section, some numerical examples will be presented to illustrate the dynamic programming solution approach on a small three-period problem. Let,

For

0 | 1 | 2 | 3 | |||
---|---|---|---|---|---|---|

0 | 33.288 | 29.726 | 23.526 | 13.263 | 33.288 | 0 |

1 | 35.311 | 33.288 | 29.726 | 23.526 | 35.311 | 0 |

2 | 37.252 | 35.311 | 33.288 | 29.726 | 37.252 | 0 |

3 | 39.193 | 37.252 | 35.311 | 33.288 | 39.193 | 0 |

Table

For

0 | 1 | 2 | 3 | |||
---|---|---|---|---|---|---|

0 | 36.472 | 36.534 | 36.515 | 36.223 | 36.534 | 1 |

1 | 38.433 | 38.495 | 38.475 | 38.456 | 38.495 | 1 |

2 | 40.314 | 40.456 | 40.436 | 40.416 | 40.456 | 1 |

3 | 41.442 | 42.337 | 42.397 | 42.377 | 42.397 | 2 |

Table

For

0 | 1 | 2 | 3 | |||
---|---|---|---|---|---|---|

1 | 44.124 | 44.105 | 44.086 | 44.047 | 44.124 | 0 |

The optimal solution is shown in Table

Optimal solution for example 1.

1 | 0 | 1.89 | 5.0 | 0.89 |

2 | 1 | 1.62 | 4.0 | 2.62 |

3 | 0 | 4.30 | 10.0 | 3.30 |

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

Optimal solution for example 2.

1 | 17 | 4.0 | 16.42 | 21.0 |

2 | 34 | 4.0 | 16.42 | 21.0 |

3 | 0 | 55.0 | 19.25 | 21.0 |

Our next example illustrates what we call

Optimal solution for example 3.

1 | 0 | 5.18 | 15.10 | 5.18 |

2 | 28 | 5.18 | 15.10 | 33.18 |

3 | 0 | 63.00 | 18.16 | 35.00 |

Optimal solution for example 4.

1 | 16 | 5.00 | 15.28 | 21.00 |

2 | 28 | 5.18 | 15.10 | 17.18 |

3 | 0 | 63.00 | 18.16 | 35.00 |

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:

Optimal solution for example 5.

1 | 1 | 4.0 | 16.09 | 5.0 |

2 | 3 | 3.0 | 17.53 | 5.0 |

3 | 0 | 8.0 | 24.37 | 5.0 |

Projected plan assuming aggregate capacity.

1 | 0 | 2.52 | 18.4 | 2.52 |

2 | 0 | 2.52 | 18.4 | 2.52 |

3 | 0 | 9.96 | 22.4 | 9.96 |

Projected plan assuming no constraints on inventory.

1 | 2 | 3.0 | 17.53 | 5.0 |

2 | 5 | 2.0 | 19.56 | 5.0 |

3 | 0 | 10.0 | 22.36 | 5.0 |

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 [

For problem (

Assume that

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.