An optimal variable pricing model for container line revenue management systems

Abstract

This paper discusses a variable pricing method, which is a special technique of revenue management, in the context of the competitive market of the liner shipping industry. Most papers on competitive variable pricing are based on two fundamental assumptions: (i) the length of a price-adjusting period is short enough so that in each period at most one customer can arrive; and (ii) the real-time inventory levels of all firms constitute public information. This paper relaxes both assumptions so that each interval between two consecutive freight rate changes allows more than one shipper to arrive and that a line knows only its competitors’ initial carrying capacity at the beginning of the sales process. A multi-iteration of genetic algorithm is proposed and numerically tested.

Appendices

Appendix 1

According to Kinderlehrer and Stampacchia (1980) and Nagurney (1999), there is at least one solution to variational inequity: < F(X^b), X − X^b > ≥ 0, ∀X^b ∊ Φ_b, if Φ_b is closed, bounded, and convex, and F(X) is continuous.

For line i, the Lagrange function for optimization is

$$L_{i} = \sum\limits_{t = 1}^{D} {p_{i}^{(t)} \cdot \rho_{i}^{(t)} \cdot \lambda^{(t)} } + \gamma_{i} \left( {S_{i} - \sum\limits_{t = 1}^{D} {\rho_{i}^{(t)} \cdot \lambda^{(t)} } } \right).$$

(8)

Its partial derivatives with respect to p ^{( t)} _i and γ_i are, respectively,

$$\frac{{\partial L_{i} }}{{\partial p_{i}^{(t)} }} = \rho_{i}^{(t)} \cdot \lambda^{(t)} - \lambda^{(t)} \cdot \frac{{\partial \rho_{i}^{(t)} }}{{\partial p_{i}^{(t)} }} \cdot \left( {\gamma_{i} - p_{i}^{(t)} } \right),$$

(9)

$${\text{and}}\,\frac{{\partial L_{i} }}{{\partial \gamma_{i} }} = S_{i} - \sum\limits_{t = 1}^{D} {\rho_{i}^{(t)} \cdot \lambda^{(t)} } .$$

(10)

In (2),

$$\frac{{\partial \rho_{i}^{(t)} }}{{\partial p_{i}^{(t)} }} = - \beta \cdot \rho_{i}^{(t)} \cdot (1 - \rho_{i}^{(t)} ).$$

The corresponding variational inequity can be expressed as

$$\sum\limits_{i = 1}^{N} {\sum\limits_{t = 1}^{D} {\left\{ {[ - \rho_{i}^{(t)} \cdot \lambda^{(t)} + \lambda^{(t)} \cdot \frac{{\partial \rho_{i}^{(t)} }}{{\partial p_{i}^{(t)} }} \cdot (\gamma_{i} - p_{i}^{(t)} )] \cdot [p_{i}^{(t)} - p_{i}^{(t)*} ]} \right\}} } + \sum\limits_{i = 1}^{N} {\left\{ {[S_{i} - \sum\limits_{t = 1}^{D} {\rho_{i}^{(t)} \cdot \lambda^{(t)} ]} \cdot [\gamma_{i} - \gamma_{i}^{*} ]} \right\}}.$$

(11)

Therefore, to the game of this section, Φ_b = {p ^{( t)}₁ , …, p ^{( t)} _N ; γ₁, …, γ_N}, and p ^{( t)} _i ∈[0, ζ_i] and γ_i∈[0, ε_i], where ζ_i ≥ 0 and ε_i ≥ 0, without a loss of generality.

Under this assumption, it is evident that Φ_b is closed, bounded, and convex. Based on (4), the vector function F(X) can be expressed as

$$F(X) = \left[ {\begin{array}{*{20}l} {\sum\limits_{{t = 1}}^{D} {[ - \rho _{1}^{{(t)}} \cdot\lambda ^{{(t)}} + \lambda ^{{(t)}} \cdot\frac{{\partial \rho _{1}^{{(t)}} }}{{\partial p_{1}^{{(t)}} }}\cdot(\gamma _{1} - p_{1}^{{(t)}} )} ],\; \ldots ,\sum\limits_{{t = 1}}^{D} {[ - \rho _{N}^{{(t)}} \cdot\lambda ^{{(t)}} + \lambda ^{{(t)}} \cdot\frac{{\partial \rho _{N}^{{(t)}} }}{{\partial p_{N}^{{(t)}} }}\cdot(\gamma _{N} - p_{N}^{{(t)}} )} ],} \hfill \\ {[S_{1} - \sum\limits_{{t = 1}}^{D} {\rho _{1}^{{(t)}} \cdot\lambda ^{{(t)}} ]} ,\; \ldots ,\;[S_{N} - \sum\limits_{{t = 1}}^{D} {\rho _{N}^{{(t)}} \cdot\lambda ^{{(t)}} ]} } \hfill \\ \end{array} } \right].$$

(12)

From (5), it is evident that F(X) is continuous with respect to {p ^{( t)}₁ , …,p ^{( t)} _N ; γ₁, …, γ_N}.

Overall, the variational equity derived from the game of this section complies with the conditions proven by Kinderlehrer and Stampacchia (1980) and Nagurney (1999). Hence, there is at least one pure-strategy Nash equilibrium in the game.

Appendix 2

Step 1:

Let iter = 1 and set M, the maximum rounds of iteration, and/or the criteria of convergence.

Step 2:

Set the initial value of vectors P ^(t)*₂ , P ^(t)*₃ , …, P ^(t)* _N , t = 1,…, D, e.g., (1 1 1 … 1)_1×D.

Step 3:

With P ^{( t)*}₂ , P ^{( t)*}₃ , …, P ^{( t)*} _N , calculate line 1’s optimal revenue Π ^*₁ and the corresponding P ^{( t) *}₁ for t = 1,…, D by the GA described later.

Step 4:

With P ^{( t)*}₁ , P ^{( t)*}₃ , …, P ^{( t)*} _N , calculate line 2’s optimal revenue Π ^*₂ and the corresponding P ^{( t) *}₂ for t = 1,…, D by the GA.

…

Step N + 2:

With P ^{( t)*}₁ , P ^{( t)*}₂ , …, P ^{( t)*} _{N- 1} , calculate line N’s optimal revenue Π ^* _N and the corresponding P ^(t)* _N for t = 1,…, D by the GA.

Step N + 3:

iter = iter + 1.

Step N + 4:

Return to Step 3, unless iter > M, and/or the criteria of convergence are met.

Step N + 5:

Output every line i’s optimal profit Π ^* _i and their corresponding p ^{( t) *} _i , and end the procedure.

Appendix 3

Step 1:

Generate the initial population for P_i = (p ⁽¹⁾ _i , p ⁽²⁾ _i , …, p ^{( D)} _i ), and let gen = 1.

Step 2:

Decode all groups of chromosomes in the current population and calculate their corresponding revenue Π_i.

Step 3:

gen = gen + 1. If gen exceeds the maximum number of generations, then go to Step 4. Otherwise, generate the population for the next generation:

Step 3.1:

Select a pre-specified percentage, normally known as the “generation gap,” of chromosomes from the population of previous generations with the best fitness values, and remove the other chromosomes.

Step 3.2:

Apply the crossover operator to the chromosomes selected in Step 3.1 randomly but at a certain percentage.

Step 3.3:

Apply the mutation operator to the chromosomes obtained in Step 3.2 randomly but at a certain percentage.

Step 3.4:

Select the offspring chromosomes from those obtained in Step 3.3 with the best fitness values at a certain percentage. Replace the worst chromosomes in the population of the previous generation with the newly selected chromosomes and go to Step 2.

Step 4:

Let line i’’s optimal revenue Π ^* _i  = max(Π_i), and its corresponding P ^* _i  = (p ⁽¹⁾ _i , p ⁽²⁾ _i , …, p ^{( D)} _i ). Return this result and stop.