Model: Continuous Newsvendor Problem (CNTNV)

Description:

A vendor orders a fixed quantity of liquid at the beginning of a day to be sold to customers throughout the day. The vendor pays a per-unit order cost \(c\) for the initial inventory and sells it the product to customers at a per-unit price \(s\). At the end of the day, any unsold liquid can be salvaged at a per-unit price, \(w\).

Sources of Randomness:

Each day’s random demand for liquid product follows Burr Type XII distribution and is denoted by \(D\). The parameters of the Burr Type XII distribution are \(α\) and \(β\) so that its cumulative distribution function is given by \(F(x) = 1 - (1+x^α)^{-β}\) where \(x, α,\) and \(β\) are all positive.

Model Factors:

• Cost (\(c\)): The price at which the newsvendor purchases one unit volume of liquid.

  • Default: 5

• Price (\(s\)): The price at which the newsvendor sells one unit volume of liquid.

  • Default: 9

• Salvage Price (\(w\)): The price at which any unsold liquid is sold for salvage.

  • Default: 1

• Alpha (\(α\)): Parameter for the demand distribution.

  • Default: 2

• Beta (\(β\)): Parameter for the demand distribution.

  • Default: 20

• Quantity of Liquid (\(x\)): Amount (volume) of liquid ordered at the beginning of the day.

  • Default: 0.5

Responses:

• Profit: The daily profit; can be negative if a loss is incurred.

References:

Evan L. Porteus. Stochastic inventory theory. In D. P. Heyman and M. J. Sobel, editors, Stochastic Models, volume 2 of Handbooks in Operations Research and Management Science, chapter 12, pages 605–652. Elsevier, New York, 1990.

Optimization Problem: Maximize Profit

Decision Variables:

• Quantity of Liquid (\(x\)): Amount (volume) of liquid ordered at the beginning of the day.

Objectives:

Maximizes the vendor’s expected profit.

Constraints:

Quantity of Liquid must be non-negative: \(x > 0\)

Problem Factors:

• Budget: Max # of replications for a solver to take.

  • Default: 1000

Fixed Model Factors:

• N/A

Starting Solution:

• \(x = 0\)

Random Solutions:

If random solutions are needed, generate \(x\) from an Exponential distribution with mean 1.

Optimal Solution:

Global minimum at \(x^* = (1/((1-r)^{1/β})-1)^{1/α}\). For the default factors, the optimal solution is \(x^*\) = 0.1878.

Optimal Objective Function Value:

For the default factors, the maximum expected profit is 0.4635.