A revenue management approach to address a truck rental problem

Abstract

This paper describes an application of revenue management techniques and policies in the field of logistics and distribution. In particular, the problem of transportation operators, who offer products for hire, is considered. A product is a truck of a given capacity, which can be rented for one or several time periods, throughout a multi-period planning horizon. The logistic operator can satisfy the demand of a given product with trucks with a capacity greater than that initially required, that is an ‘upgrade’ can take place. In this context, the logistic operator has to decide whether to accept or reject a request and which type of truck should be used to address it. For this purpose, a dynamic programming (DP) formulation of the problem under consideration is devised. The ‘course of dimensionality’ leads to the necessity of introducing different mathematical programming models to represent the problem. The mathematical models we presented are an extension of the well-known approximations for the DP of traditional network capacity management analysis. Based on these models and exploiting revenue management concepts, primal and dual acceptance policies are developed and compared in a computational study.

Appendix

In what follows, we prove that the constraint matrix of the problem R ^ILP introduced in Section 4 is totally unimodular. All the general results about totally unimodular matrices used in the proof can be found in Nemhauser and Wolsey (1998). In particular, we use the definition and the theoretical result reported below.

Definition A.1

• An interval matrix is a 0–1 matrix where the ones appear consecutively in each column.

Theorem A.1

• Interval matrices are totally unimodular.

In what follows, we show that can be transformed into an interval matrix, by performing linear operations on its rows.

A graphical representation of the constraint matrix for the case r>2 is given in Figure A1, where, for the sake of convenience, constraints (3b) and (3c) are reported in reverse order respect to the presentation of the model R ^ILP.

Figure A1

The constraint matrix for r>2.

It is evident from Figure A1 that is a block matrix, in which the number of blocks depends on the number of trucks r and the value of T. In particular, each black block represents a matrix of dimension (T−1) × ∑ _l=0 ^T−1(T−1−l), whose structure is represented in Table A1, whereas each grey block corresponds to an identity matrix of size ∑ _l=0 ^T−1(T−1−l) × ∑ _l=0 ^T−1(T−1−l). Thus, is a Boolean block matrix.

Table A1 A black block of

In order to see concretely the form of the constraint matrix , Table A2 gives a graphical representation of matrix , in the case T=4 and r=2.

Table A2 Matrix , in the case T=4 and r=2

Denote the rows of each block B ^k, k=1,…, r by b ₁ ^k, b ₂ ^k,…, b _γ ^k, where γ=∑ _l=0 ^T−1(T−1−l) and the rows of each block M ^k, k=1,…, r by m ₁ ^k, m ₂ ^k,…, m _T−1 ^k.

We show that, by performing the steps below, the matrix in Figure A1, can be transformed into an interval matrix.

Step 1.:

For the block B ¹, add to each row b _p ¹, p=2,…, γ the previous p−1 rows (ie, the rows b ₁ ¹, b ₂ ¹,…, b _l ¹, for l=1,…, p−1).

Matrices I ₁ ¹, I ₁ ²,…, I ₁ ^r become lower triangular matrices with all entries equal to one. In the case T=4 and r=2, we obtain the constraint matrix in Table A3.

Table A3 Matrix , in the case T=4 and r=2, after Step 1

Step 2.:

For each block B ^k, k=2,…, r add to each row b _p ^k, p=1,…, γ−1, the next γ−p rows (ie, the rows b _l ^k, for l=p+1,…, γ).

Matrices I ₂ ^r+1,…, I ₂ ^r+(r−1),…, I _r ^{(r+(r−1)+⋯+1)} become upper triangular matrices with all entries equal to one. The graphical representation of the constraint matrix, in the case T=4 and r=2, after the execution of Steps 1 and 2 is in Table A4.

Table A4 Matrix , in the case T=4 and r=2, after Steps 1 and 2

Step 3.:

For each block B ^k, k=2,…, r−1, add to the p−th row of the k−th block the p−th row of the (k+1)th block (ie, add to rows b _p ^k, p=1,…, γ the rows b _p ^k+1, for p=1,…, γ).

Matrices 0₁ ^k,…, 0_(r−k) ^k) become matrices of ones. It is important to observe that for the considered numerical example (ie, T=4 and r=2), no operations are executed during Step 3.

Step 4.:

For each block M ^k, k=1,…, r, execute the operations reported in Figure A2. The aim is to modify the zero entries of the M ^k blocks, k=1,…, r, in such a way that the resulting blocks are matrix of all ones. This is achieved by exploiting the specific structure of the blocks themselves.

Figure A2

Operations executed for the block M ^k, k =1, …, r−1.

By executing Steps 1, 2, 3 and 4 on the matrix reported in Table A2, we obtain the matrix of Table A5.

Table A5 Matrix , in the case T=4 and r=2, after steps 1, 2, 3 and 4

Step 5.:

Make a permutation of the rows of the transformed matrix, so that the following sequence of the blocks is obtained: B ¹, M ¹,…, M ^r, B ²,…, B ^r.

The final matrix (see Table A6) is an interval matrix and then it is totally unimodular.

Table A6 The interval matrix