Robust pricing for airlines with partial information

Abstract

In the spot market for air cargo, airlines typically adopt dynamic pricing to tackle demand uncertainty, for which it is difficult to accurately estimate the distribution. This study addresses the problem where a dominant airline dynamically sets prices to sell its capacities within a two-phase sales period with only partial information. That partial information may show as the moments (upper and lower bounds and mean) and the median of the demand distribution. We model the problem of dynamic pricing as a distributional robust stochastic programming, which minimizes the expected regret value under the worst-case distribution in the presence of partial information. We further reformulate the proposed non-convex model to show that the closed-form formulae of the second-stage maximal expected regret are well-structured. We also design an efficient algorithm to characterize robust pricing strategies in a polynomial-sized running time. Using numerical analysis, we present several useful managerial insights for airline managers to strategically collect demand information and make prices for their capacities in different market situations. Moreover, we verify that additional information will not compromise the viability of the pricing strategies being implemented. Therefore, the method we present in this paper is easier for airlines to use.

Appendices

Appendix 1

See Tables 1, 2 and 3.

Table 1 Closed-form formulae for Proposition 2

Table 2 Closed-form formulae for Proposition 3

Table 3 Closed-form formulae for Proposition 4

Appendix 2

Proof of Proposition 2

The optimal value of problem (9) can be formulated as:

$$ \max_{l \le z \le u} \{ q\min \{ c_{q} ,d(q)z\} - p\min \{ c_{p} ,d(p)z\} . $$

(A-1)

To derive the closed-form formula, we make a classified discussion to the following scenarios:

Case 1

If $d(q)u \le c_{q} , \, d(p)u \le c_{p}$, then (A-1) can be simplified to:

$$ \max_{l \le z \le u} [qd(q) - pd(p)]z. $$

The maximum value of (A-1) is

$$ \left\{ {\begin{array}{*{20}l} {[qd(q) - pd(p)]u,} \hfill & {{\text{if}}\;qd(q) - pd(p) \ge 0,} \hfill \\ {[qd(q) - pd(p)]l,} \hfill & {{\text{otherwise}}{.}} \hfill \\ \end{array} } \right. $$

Case 2

If $d(q)u \le c_{q} , \, d(p)l \le c_{p} \le d(p)u$, then (A-1) can be simplified to:

$$ \max_{l \le z \le u} \left\{ \begin{array}{ll} [qd(q) - pd(p)]z,&\quad {\text{for}}\,\,l \le z < c_{q} /d(q), \hfill \\ qc_{q} - pd(p)z,&\quad {\text{for}}\,\,c_{q} /d(q) \le z \le u. \hfill \\ \end{array} \right. $$

The maximum value of (A-1) is

$$ \left\{ {\begin{array}{*{20}l} {qd(q)u - pc_{p} ,} \hfill & {{\text{if}}\;qd(q) - pd(p) \ge 0,} \hfill \\ {\max \{ [qd(q) - pd(p)]l,qd(q) - pc_{p} \} ,} \hfill & {{\text{otherwise}}{.}} \hfill \\ \end{array} } \right. $$

Case 3

If $d(q)u \le c_{q} , \, c_{p} \le d(p)l$, then (A-1) can be simplified as:

$$ \max_{l \le z \le u} qd(q)z - pc_{p} . $$

The maximum value of (A-1) is

$$ qd(q)u - pc_{p} . $$

Case 4

If $d(q)l \le c_{q} \le d(q)u, \, d(p)u \le c_{p}$, then (A-1) can be simplified to:

$$ \max_{l \le z \le u} \left\{ {\begin{array}{*{20}l} {[qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;l \le z < c_{q} /d(q),} \hfill \\ {qc_{q} - pd(p)z,} \hfill & {{\text{for}}\;c_{q} /d(q) \le z \le u.} \hfill \\ \end{array} } \right. $$

The maximum value of (A-1) is

$$ \left\{ {\begin{array}{*{20}l} {[q - pd(p)/d(q)]c_{q} ,} \hfill & {{\text{if}}\;qd(q) - pd(p) \ge 0,} \hfill \\ {[qd(q) - pd(p)]l,} \hfill & {{\text{otherwise}}{.}} \hfill \\ \end{array} } \right. $$

Case 5

If $d(q)l \le c_{q} \le d(q)u, \, d(p)l \le c_{p} \le d(p)u$, the structure of (A-1) depends on the relative sizes of $c_{q} /d(q)$ and $c_{p} /d(p)$, thus we consider the following two subcases:

Subcase 1 Suppose $c_{q} /d(q) \le c_{p} /d(p)$, then (A-1) can be simplified to:

$$ \max_{l \le z \le u} \left\{ {\begin{array}{*{20}l} {[qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;l \le z < c_{q} /d(q),} \hfill \\ {qc_{q} - pd(p)z,} \hfill & {{\text{for}}\;c_{q} /d(q) \le z < c_{p} /d(p).} \hfill \\ {qc_{q} - pc_{p} ,} \hfill & {{\text{for}}\;c_{p} /d(p) \le z \le u.} \hfill \\ \end{array} } \right. $$

The maximum value of (A-1) is

$$ \left\{ {\begin{array}{*{20}l} {[q - pd(p)/d(q)]c_{q} ,} \hfill & {{\text{if}}\;qd(q) - pd(p) \ge 0,} \hfill \\ {[qd(q) - pd(p)]l,} \hfill & {{\text{otherwise}}{.}} \hfill \\ \end{array} } \right. $$

Subcase 2 Suppose $c_{q} /d(q) > c_{p} /d(p)$, then (A-1) can be simplified to:

$$ \max_{l \le z \le u} \left\{ {\begin{array}{*{20}l} {[qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;l \le z < c_{p} /d(p),} \hfill \\ {qd(q)z - pc_{p} ,} \hfill & {{\text{for}}\;c_{p} /d(p) \le z < c_{q} /d(q),} \hfill \\ {qc_{q} - pc_{p} ,} \hfill & {{\text{for}}\;c_{q} /d(q) \le z \le u.} \hfill \\ \end{array} } \right. $$

The maximum value of (A-1) is

$$ \left\{ {\begin{array}{*{20}l} {qc_{q} - pc_{p} ,} \hfill & {{\text{if}}\;qd(q) - pd(p) \ge 0,} \hfill \\ {\max \{ [qd(q) - pd(p)]l,qc_{q} - pc_{p} \} ,} \hfill & {{\text{otherwise}}{.}} \hfill \\ \end{array} } \right. $$

Case 6

If $d(q)l \le c_{q} \le d(q)u, \, c_{p} \le d(p)l$, then (A-1) can be simplified to:

$$ \max_{l \le z \le u} \left\{ {\begin{array}{*{20}l} {qd(q)z - pc_{p} ,} \hfill & {{\text{for}}\;l \le z < c_{q} /d(q),} \hfill \\ {qc_{q} - pc_{p} ,} \hfill & {{\text{for}}\;c_{q} /d(q) \le z \le u.} \hfill \\ \end{array} } \right. $$

The maximum value of (A-1) is

$$ qc_{q} - pc_{p} . $$

Case 7

If $c_{q} \le d(q)l, \, d(p)u \le c_{p}$, then (A-1) can be simplified as:

$$ \max_{l \le z \le u} qc_{q} - pd(p)z. $$

The maximum value of (A-1) is

$$ qc_{q} - pd(p)l. $$

Case 8

If $c_{q} \le d(q)l, \, d(p)l \le c_{p} \le d(p)u$, then (A-1) can be simplified to:

$$ \max_{l \le z \le u} \left\{ {\begin{array}{*{20}l} {qc_{q} - pd(p)z,} \hfill & {{\text{for}}\;l \le z < c_{q} /d(q),} \hfill \\ {qc_{q} - pc_{p} ,} \hfill & {{\text{for}}\;c_{q} /d(q) \le z \le u.} \hfill \\ \end{array} } \right. $$

The maximum value of (A-1) is

$$ qc_{q} - pd(p)l. $$

Case 9

If $c_{q} \le d(q)l, \, c_{p} \le d(p)l$, then (A-1) can be simplified to:

$$ \max_{l \le z \le u} qc_{q} - pc_{p} . $$

The maximum value of (A-1) is

$$ qc_{q} - pc_{p} . $$

In summary, the maximum value of (A-1) under any scenario is linear or piecewise linear function of $c_{p}$ and $c_{q}$.□

Proof of Proposition 3

Under the conditions of Lemma 2, model (8) can be written as:

$$ \begin{array}{*{20}l} {\max_{F} } \hfill & {\{ E_{F} [q_{2} \min \{ c_{q} ,d(q_{2} )z\} - p_{2} \min \{ c_{p} ,d(p_{2} )z\} ]} \hfill \\ {\text{subject to}} \hfill & { \, \int_{l}^{u} {dF = 1} ,} \hfill \\ {} \hfill & {\int_{m}^{u} {dF = \frac{1}{2}} ,} \hfill \\ {} \hfill & {dF \ge 0.} \hfill \\ \end{array} $$

(A-2)

By the strong duality theorem, (A-2) can be reformulated as:

$$ \begin{array}{*{20}l} {\min_{{y_{0,} y_{1} }} } \hfill & {\frac{1}{2}(y_{0} + y_{1} ) + \frac{1}{2}y_{0} } \hfill \\ {\text{subject to}} \hfill & {y_{0} + y_{1} 1\{ z \ge m\} \ge q\min \{ d(q)z,c_{q} \} - p\min \{ d(p)z,c_{p} \} , \,\,\, z \in [l,u],} \hfill \\ \end{array} $$

(A-3)

where $1\{ z \ge m\}$ is the indicator function, which is equal to one if $z \ge m$$;$ otherwise, 0.

To derive the closed-form formula, we discuss the following scenarios:

Case 1

If $d(q)u \le c_{q} , \, d(p)u \le c_{p}$, then constraint of (A-3) can be written as:

$$ y_{0} + y_{1} 1\{ z \ge m\} \ge [qd(q) - pd(p)]z,\quad z \in [l,u]. $$

Because $y_{0} + y_{1} 1\{ z \ge m\}$ is a step-shaped piecewise linear function, the above equation can also be further represented as:

$$ \left\{ {\begin{array}{*{20}l} {y_{0} \ge [qd(q) - pd(p)]z,} \hfill & {z \in [l,m),} \hfill \\ {y_{0} + y_{1} \ge [qd(q) - pd(p)]z,} \hfill & {z \in [m,u).} \hfill \\ \end{array} } \right. $$

Then we obtain

$$ \min_{{y_{0,} y_{1} }} \frac{1}{2}(y_{0} + y_{1} ) + \frac{1}{2}y_{0} = \left\{ {\begin{array}{*{20}l} {[qd(q) - pd(p)](m + u)/2,} \hfill & {{\text{if}}\;qd(q) - pd(p) \ge 0,} \hfill \\ {[qd(q) - pd(p)](m + l)/2,} \hfill & {{\text{if}}\;qd(q) - pd(p) < 0.} \hfill \\ \end{array} } \right. $$

Case 2

If $d(q)u \le c_{q} , \, d(p)l \le c_{p} \le d(p)u$, the constraint of (A-3) can be written as:

$$ y_{0} + y_{1} 1\{ z \ge m\} \ge \left\{ {\begin{array}{*{20}l} {[qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;l \le z < c_{p} /d(p),} \hfill \\ {qd(q)z - pc_{p} ,} \hfill & {{\text{for}}\;c_{p} /d(p) \le z \le u.} \hfill \\ \end{array} } \right. $$

Similar to the previous scenario, we consider the following two subcases:

Subcase 1 Suppose that $qd(q) - pd(p) \ge 0$$,$

Here, the step-shaped structure of the constraint is reliant on the relative sizes of $m$ and $c_{p} /d(p)$.

• (1) When $m \ge c_{p} /d(p)$$,$ the constraint translates to:

  $$ \left\{ {\begin{array}{*{20}l} {y_{0} \ge [qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;l \le z < c_{p} /d(p),} \hfill \\ {y_{0} \ge qd(q)z - pc_{p} ,} \hfill & {{\text{for}}\;c_{p} /d(p) \le z < m,} \hfill \\ {y_{0} + y_{1} \ge qd(q)z - pc,} \hfill & {{\text{for}}\;m \le z \le u.} \hfill \\ \end{array} } \right. $$

  Then we obtain

  $$ \min_{{y_{0} ,y_{1} }} \frac{1}{2}(y_{0} + y_{1} ) + \frac{1}{2}y_{0} = qd(q)\frac{(m + u)}{2} - pc_{p} . $$

• (2) When $m < c_{p} /d(p)$$,$ the constraint translates to:

  $$ \left\{ {\begin{array}{*{20}l} {y_{0} \ge [qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;l \le z < m,} \hfill \\ {y_{0} + y_{1} \ge [qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;m \le z < c_{p} /d(p),} \hfill \\ {y_{0} + y_{1} \ge qd(q)z - pc_{p} ,} \hfill & {{\text{for}}\;c_{p} /d(p) \le z \le u.} \hfill \\ \end{array} } \right. $$

  Then we obtain

  $$ \min_{{y_{0} ,y_{1} }} \frac{1}{2}(y_{0} + y_{1} ) + \frac{1}{2}y_{0} = [qd(q) - pd(p)]\frac{m}{2} + \frac{{qd(q)u - pc_{p} }}{2}. $$

Subcase 2 Suppose $qd(q) - pd(p) < 0$,

• (1) When $m \ge c_{p} /d(p)$$,$ the constraint translates to:

  $$ \left\{ {\begin{array}{*{20}l} {y_{0} \ge [qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;l \le z \le c_{p} /d(p),} \hfill \\ {y_{0} \ge qd(q)z - pc_{p} ,} \hfill & {{\text{for}}\;c_{p} /d(p) \le z \le m,} \hfill \\ {y_{0} + y_{1} \ge qd(q)z - pc,} \hfill & {{\text{for}}\;m \le z \le u.} \hfill \\ \end{array} } \right. $$

  Then we obtain

  $$ \min_{{y_{0} ,y_{1} }} \frac{1}{2}(y_{0} + y_{1} ) + \frac{1}{2}y_{0} = \max \left\{ {\frac{[qd(q) - pd(p)]l}{2} + \frac{{qd(q)u - pc_{p} }}{2},qd(q)\frac{m + u}{2} - pc_{p} } \right\}. $$

• (2) When $m < c_{p} /d(p)$$,$ the constraint translates to:

  $$ \left\{ {\begin{array}{*{20}l} {y_{0} \ge [qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;l \le z < m,} \hfill \\ {y_{0} + y_{1} \ge [qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;m \le z < c_{p} /d(p),} \hfill \\ {y_{0} + y_{1} \ge qd(q)z - pc_{p} ,} \hfill & {{\text{for}}\;c_{p} /d(p) \le z \le u.} \hfill \\ \end{array} } \right. $$

  Then we obtain

  $$ \min_{{y_{0} ,y_{1} }} \frac{1}{2}(y_{0} + y_{1} ) + \frac{1}{2}y_{0} = \max \left\{ {\frac{[qd(q) - pd(p)]l}{2} + \frac{{qd(q)u - pc_{p} }}{2},\;[qd(q) - pd(p)]\frac{m + l}{2}} \right\}. $$

Case 3

If $d(q)u \le c_{q} , \, c_{p} \le d(p)l$, constraint of (A-3) can be written as:

$$ y_{0} + y_{1} 1\{ z \ge m\} \ge qd(q)z - pc_{p} ,\quad \forall z \in [l,u]. $$

Since $y_{0} + y_{1} 1\{ z \ge m\}$ is a step-shaped piecewise linear function, the above equation can be represented further as:

$$ \left\{ {\begin{array}{*{20}l} {y_{0} \ge qd(q)z - pc_{p} ,} \hfill & {{\text{for}}\;z \in [l,m),} \hfill \\ {y_{0} + y_{1} \ge qd(q)z - pc_{p} ,} \hfill & {{\text{for}}\;z \in [m,u].} \hfill \\ \end{array} } \right. $$

Then we obtain

$$ \min_{{y_{0} ,y_{1} }} \frac{1}{2}(y_{0} + y_{1} ) + \frac{1}{2}y_{0} = qd(q)\frac{m + u}{2} - pc_{p} . $$

Case 4

If $d(q)l \le c_{q} \le d(q)u, \, d(p)u \le c_{p}$, constraint of (A-3) can be written as:

$$ y_{0} + y_{1} 1\{ z \ge m\} \ge \left\{ {\begin{array}{*{20}l} {[qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;l \le z \le c_{q} /d(q),} \hfill \\ {qc_{q} - pd(p)z,} \hfill & {{\text{for}}\;c_{q} /d(q) \le z \le u. \, } \hfill \\ \end{array} } \right. $$

Consider the following two subcases:

Subcase 1 Suppose that $qd(q) - pd(p) \ge 0$,

Under the subcase, the step-shaped structure of the constraint further relies on the relative sizes of $m$ and $c_{q} /d(q)$.

• (1) When $m \ge c_{q} /d(q)$$,$ the constraint translates to:

  $$ \left\{ {\begin{array}{*{20}l} {y_{0} \ge [qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;l \le z \le c_{q} /d(q),} \hfill \\ {y_{0} \ge qc_{q} - pd(p)z,} \hfill & {{\text{for}}\;c_{q} /d(q) \le z \le m,} \hfill \\ {y_{0} + y_{1} \ge qc_{q} - pd(p)z,} \hfill & {{\text{for}}\;m \le z \le u.} \hfill \\ \end{array} } \right. $$

  Then we obtain

  $$ \min_{{y_{0} ,y_{1} }} \frac{1}{2}(y_{0} + y_{1} ) + \frac{1}{2}y_{0} = qc_{q} - pd(p)\frac{{(m + c_{q} /d(q))}}{2}. $$

• (2) When $m < c_{q} /d(q)$$,$ the constraint translates to:

  $$ \left\{ {\begin{array}{*{20}l} {y_{0} \ge [qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;l \le z \le m,} \hfill \\ {y_{0} + y_{1} \ge [qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;m \le z \le c_{q} /d(q),} \hfill \\ {y_{0} + y_{1} \ge qc_{q} - pd(p)z,} \hfill & {{\text{for}}\;c_{q} /d(q) \le z \le u.} \hfill \\ \end{array} } \right. $$

  Then we obtain

  $$ \min_{{y_{0} ,y_{1} }} \frac{1}{2}(y_{0} + y_{1} ) + \frac{1}{2}y_{0} = [qd(q) - pd(p)]\frac{{m + c_{q} /d(q)}}{2}. $$

Subcase 2 Suppose that $qd(q) - pd(p) < 0$,

• (1) When $m \ge c_{q} /d(q)$$,$ the constraint translates to:

  $$ \left\{ {\begin{array}{*{20}l} {y_{0} \ge [qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;l \le z \le c_{q} /d(q),} \hfill \\ {y_{0} \ge qc_{q} - pd(p)z,} \hfill & {{\text{for}}\;c_{q} /d(q) \le z \le m,} \hfill \\ {y_{0} + y_{1} \ge qc_{q} - pd(p)z,} \hfill & {{\text{for}}\;m \le z \le u.} \hfill \\ \end{array} } \right. $$

  Then we obtain

  $$ \min_{{y_{0} ,y_{1} }} \frac{1}{2}(y_{0} + y_{1} ) + \frac{1}{2}y_{0} = q\frac{{c_{q} + d(q)l}}{2} - pd(p)\frac{m + l}{2}. $$

• (2) When $m < c_{q} /d(q)$$,$ the constraint translates to:

  $$ \left\{ {\begin{array}{*{20}l} {y_{0} \ge [qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;l \le z \le m,} \hfill \\ {y_{0} + y_{1} \ge [qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;m \le z < c_{q} /d(q),} \hfill \\ {y_{0} + y_{1} \ge qc_{q} - pd(p)z,} \hfill & {{\text{for}}\;c_{q} /d(q) \le z \le u.} \hfill \\ \end{array} } \right. $$

  Then we obtain

  $$ \min_{{y_{0} ,y_{1} }} \frac{1}{2}(y_{0} + y_{1} ) + \frac{1}{2}y_{0} = [qd(q) - pd(p)]\frac{m + l}{2}. $$

Case 5

If $d(q)l \le c_{q} \le d(q)u, \, d(p)l \le c_{p} \le d(p)u$, to recognize the structure of the constraint, again, we consider the two subcases, namely $qd(q) - pd(p) \ge 0$ and $qd(q) - pd(p) < 0$. The challenge lies in further consideration of full permutation of three values $m, \, c_{q} /d(q)$ and $c_{p} /d(p)$ under each subcase. Accordingly, we have $2 \times A_{3}^{3} = 12$ scenarios.

Subcase 1 Suppose that $qd(q) - pd(p) \ge 0$$,$

• (1) When $c_{q} /d(q) \le m < c_{p} /d(p)$, the constraint translates to:

  $$ \left\{ {\begin{array}{*{20}l} {y_{0} \ge [qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;l \le z < c_{q} /d(q),} \hfill \\ {y_{0} \ge qc_{q} - pd(p)z,} \hfill & {{\text{for}}\;c_{q} /d(q) \le z < m,} \hfill \\ {y_{0} + y_{1} \ge qc_{q} - pd(p)z,} \hfill & {{\text{for}}\;m \le z < c_{p} /d(p),} \hfill \\ {y_{0} + y_{1} \ge qc_{q} - pc_{p} ,} \hfill & {{\text{for}}\;c_{p} /d(p) \le z \le u.} \hfill \\ \end{array} } \right. $$

  Then we obtain

  $$ \min_{{y_{0} ,y_{1} }} \frac{1}{2}(y_{0} + y_{1} ) + \frac{1}{2}y_{0} = qc_{q} - pd(p)\frac{{c_{q} /d(q) + m}}{2}. $$

• (2) When $c_{q} /d(q) \le c_{p} /d(p) < m$, the constraint tranlates to:

  $$ \left\{ {\begin{array}{*{20}l} {y_{0} \ge [qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;l \le z < c_{q} /d(q),} \hfill \\ {y_{0} \ge qc_{q} - pd(p)z,} \hfill & {{\text{for}}\;c_{q} /d(q) \le z < c_{p} /d(p),} \hfill \\ {y_{0} \ge qc_{q} - pc_{p} ,} \hfill & {{\text{for}}\;c_{p} /d(p) \le z < m,} \hfill \\ {y_{0} + y_{1} \ge qc_{q} - pc_{p} ,} \hfill & {{\text{for}}\;m \le z < u.} \hfill \\ \end{array} } \right. $$

  Then we obtain

  $$ \min_{{y_{0} ,y_{1} }} \frac{1}{2}(y_{0} + y_{1} ) + \frac{1}{2}y_{0} = [qd(q) - pd(p)]\frac{{c_{q} }}{2d(q)} + \frac{{qc_{q} - pc_{p} }}{2}. $$

• (3) When $c_{p} /d(p) \le m < c_{q} /d(q)$, the constraint tranlates to:

  $$ \left\{ {\begin{array}{*{20}l} {y_{0} \ge [qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;l \le z < c_{p} /d(p),} \hfill \\ {y_{0} \ge qd(q)z - pc_{p} ,} \hfill & {{\text{for}}\;c_{p} /d(p) \le z < m,} \hfill \\ {y_{0} \ge qd(q)z - pc_{p} ,} \hfill & {{\text{for}}\;m \le z < c_{q} /d(q),} \hfill \\ {y_{0} + y_{1} \ge qc_{q} - pc_{p} ,} \hfill & {{\text{for}}\;c_{q} /d(q) \le z < u.} \hfill \\ \end{array} } \right. $$

  Then we obtain

  $$ \min_{{y_{0} ,y_{1} }} \frac{1}{2}(y_{0} + y_{1} ) + \frac{1}{2}y_{0} = q\frac{{c_{q} + d(q)m}}{2} - pc_{p} . $$

• (4) When $c_{p} /d(p) \le c_{q} /d(q) < m$, the constraint tranlates to:

  $$ \left\{ {\begin{array}{*{20}l} {y_{0} \ge [qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;l \le z < c_{p} /d(p),} \hfill \\ {y_{0} \ge qd(q)z - pc_{p} ,} \hfill & {{\text{for}}\;c_{p} /d(p) \le z < c_{q} /d(q),} \hfill \\ {y_{0} \ge qc_{q} - pc_{p} ,} \hfill & {{\text{for}}\;c_{q} /d(q) \le z < m,} \hfill \\ {y_{0} + y_{1} \ge qc_{q} - pc_{p} ,} \hfill & {{\text{for}}\;m \le z \le u.} \hfill \\ \end{array} } \right. $$

  Then we obtain

  $$ \min_{{y_{0} ,y_{1} }} \frac{1}{2}(y_{0} + y_{1} ) + \frac{1}{2}y_{0} = qc_{q} - pc_{p} . $$

• (5) When $m \le c_{q} /d(q) < c_{p} /d(p)$, the constraint tranlates to:

  $$ \left\{ {\begin{array}{*{20}l} {y_{0} \ge [qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;l \le z < m,} \hfill \\ {y_{0} + y_{1} \ge [qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;m \le z < c_{q} /d(q),} \hfill \\ {y_{0} + y_{1} \ge qc_{q} - pd(p)z,} \hfill & {{\text{for}}\;c_{q} /d(q) \le z < c_{p} /d(p),} \hfill \\ {y_{0} + y_{1} \ge qc_{q} - pc_{p} ,} \hfill & {{\text{for}}\;c_{p} /d(p) \le z \le u.} \hfill \\ \end{array} } \right. $$

  Then we obtain

  $$ \min_{{y_{0} ,y_{1} }} \frac{1}{2}(y_{0} + y_{1} ) + \frac{1}{2}y_{0} = [qd(q) - pd(p)]\frac{{m + c_{q} /d(q)}}{2}. $$

• (6) When $m \le c_{p} /d(p) < c_{q} /d(q)$, the constraint tranlates to:

  $$ \left\{ {\begin{array}{*{20}l} {y_{0} \ge [qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;l \le z < m,} \hfill \\ {y_{0} + y_{1} \ge [qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;m \le z < c_{p} /d(p),} \hfill \\ {y_{0} + y_{1} \ge qd(q)z - pc_{p} ,} \hfill & {{\text{for}}\;c_{p} /d(p) \le z < c_{q} /d(q),} \hfill \\ {y_{0} + y_{1} \ge qc_{q} - pc_{p} ,} \hfill & {{\text{for}}\;c_{q} /d(q) \le z \le u.} \hfill \\ \end{array} } \right. $$

  Then we obtain

  $$ \min_{{y_{0} ,y_{1} }} \frac{1}{2}(y_{0} + y_{1} ) + \frac{1}{2}y_{0} = [qd(q) - pd(p)]\frac{{m + c_{q} /d(q)}}{2}. $$

Subcase 2 Suppose that $qd(q) - pd(p) < 0$,

• (1) When $c_{q} /d(q) \le m < c_{p} /d(p)$, the constraint translates to:

  $$ \left\{ {\begin{array}{*{20}l} {y_{0} \ge [qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;l \le z < c_{q} /d(q),} \hfill \\ {y_{0} \ge qc_{q} - pd(p)z,} \hfill & {{\text{for}}\;c_{q} /d(q) \le z < m,} \hfill \\ {y_{0} + y_{1} \ge qc_{q} - pd(p)z,} \hfill & {{\text{for}}\;m \le z < c_{p} /d(p),} \hfill \\ {y_{0} + y_{1} \ge qc_{q} - pc_{p} ,} \hfill & {{\text{for}}\;c_{p} /d(p) \le z \le u.} \hfill \\ \end{array} } \right. $$

  Then we obtain

  $$ \min_{{y_{0} ,y_{1} }} \frac{1}{2}(y_{0} + y_{1} ) + \frac{1}{2}y_{0} = [qd(q) - pd(p)]\frac{l}{2} + \frac{{qc_{q} - pd(p)m}}{2}. $$

• (2) When $c_{q} /d(q) \le c_{p} /d(p) < m$, the constraint tranlates to:

  $$ \left\{ {\begin{array}{*{20}l} {y_{0} \ge [qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;l \le z < c_{q} /d(q),} \hfill \\ {y_{0} \ge qc_{q} - pd(p)z,} \hfill & {{\text{for}}\;c_{q} /d(q) \le z < c_{p} /d(p),} \hfill \\ {y_{0} \ge qc_{q} - pc_{p} ,} \hfill & {{\text{for}}\;c_{p} /d(p) \le z < m,} \hfill \\ {y_{0} + y_{1} \ge qc_{q} - pc_{p} ,} \hfill & {{\text{for}}\;m \le z \le u.} \hfill \\ \end{array} } \right. $$

  Then we obtain

  $$ \min_{{y_{0} ,y_{1} }} \frac{1}{2}(y_{0} + y_{1} ) + \frac{1}{2}y_{0} = [qd(q) - pd(p)]\frac{l}{2} + \frac{{qc_{q} - pc_{p} }}{2}. $$

• (3) When $c_{p} /d(p) \le m < c_{q} /d(q)$, the constraint tranlates to:

  $$ \left\{ {\begin{array}{*{20}l} {y_{0} \ge [qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;l \le z < c_{p} /d(p),} \hfill \\ {y_{0} \ge qd(q)z - pc_{p} ,} \hfill & {{\text{for}}\;c_{p} /d(p) \le z < m,} \hfill \\ {y_{0} \ge qd(q)z - pc_{p} ,} \hfill & {{\text{for}}\;m \le z < c_{q} /d(q),} \hfill \\ {y_{0} + y_{1} \ge qc_{q} - pc_{p} ,} \hfill & {{\text{for}}\;c_{q} /d(q) \le z \le u.} \hfill \\ \end{array} } \right. $$

  Then we obtain

  $$ \min_{{y_{0} ,y_{1} }} \frac{1}{2}(y_{0} + y_{1} ) + \frac{1}{2}y_{0} = \max \left\{ {q\frac{{c_{q} + d(q)m}}{2} - pc_{p} ,[qd(q) - pd(p)]\frac{l}{2} + \frac{{qc_{q} - pc_{p} }}{2}} \right\}. $$

• (4) When $c_{p} /d(p) \le c_{q} /d(q) < m$, the constraint tranlates to:

  $$ \left\{ {\begin{array}{*{20}l} {y_{0} \ge [qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;l \le z < c_{p} /d(p),} \hfill \\ {y_{0} \ge qd(q)z - pc_{p} ,} \hfill & {{\text{for}}\;c_{p} /d(p) \le z < c_{q} /d(q),} \hfill \\ {y_{0} \ge qc_{q} - pc_{p} ,} \hfill & {{\text{for}}\;c_{q} /d(q) \le z < m,} \hfill \\ {y_{0} + y_{1} \ge qc_{q} - pc_{p} ,} \hfill & {{\text{for}}\;m \le z \le u.} \hfill \\ \end{array} } \right. $$

  Then we obtain

  $$ \min_{{y_{0} ,y_{1} }} \frac{1}{2}(y_{0} + y_{1} ) + \frac{1}{2}y_{0} = \max \left\{ {qc_{q} - pc_{p} ,[qd(q) - pd(p)]\frac{l}{2} + \frac{{qc_{q} - pc_{p} }}{2}} \right\}. $$

• (5) When $m \le c_{q} /d(q) < c_{p} /d(p)$, the constraint tranlates to:

  $$ \left\{ {\begin{array}{*{20}l} {y_{0} \ge [qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;l \le z < m,} \hfill \\ {y_{0} + y_{1} \ge [qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;m \le z < c_{q} /d(q),} \hfill \\ {y_{0} + y_{1} \ge qc_{q} - pd(p)z,} \hfill & {{\text{for}}\;c_{q} /d(q) \le z < c_{p} /d(p),} \hfill \\ {y_{0} + y_{1} \ge qc_{q} - pc_{p} ,} \hfill & {{\text{for}}\;c_{p} /d(p) \le z \le u.} \hfill \\ \end{array} } \right. $$

  Then we obtain

  $$ \min_{{y_{0} ,y_{1} }} \frac{1}{2}(y_{0} + y_{1} ) + \frac{1}{2}y_{0} = [qd(q) - pd(p)]\frac{m + l}{2}. $$

• (6) When $m \le c_{p} /d(p) < c_{q} /d(q)$, the constraint tranlates to:

  $$ \left\{ {\begin{array}{*{20}l} {y_{0} \ge [qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;l \le z < m,} \hfill \\ {y_{0} + y_{1} \ge [qd(q) - pd(p)]z,} \hfill & {{\text{for}}\;m \le z < c_{p} /d(p),} \hfill \\ {y_{0} + y_{1} \ge qd(q)z - pc_{p} ,} \hfill & {{\text{for}}\;c_{p} /d(p) \le z < c_{q} /d(q),} \hfill \\ {y_{0} + y_{1} \ge qc_{q} - pc_{p} ,} \hfill & {{\text{for}}\;c_{q} /d(q) \le z \le u.} \hfill \\ \end{array} } \right. $$

  Then we obtain

  $$ \min_{{y_{0} ,y_{1} }} \frac{1}{2}(y_{0} + y_{1} ) + \frac{1}{2}y_{0} = \max \left\{ {[qd(q) - pd(p)]\frac{m + l}{2},[qd(q) - pd(p)]\frac{l}{2} + \frac{{qc_{q} - pc_{p} }}{2}} \right\}. $$

Case 6

If $d(q)l \le c_{q} \le d(q)u, \, c_{p} \le d(p)l$, constraint of (A-3) can be written as:

$$ y_{0} + y_{1} 1\{ z \ge m\} \ge \left\{ {\begin{array}{*{20}l} {qd(q)z - pc_{p} ,} \hfill & {{\text{for}}\;l \le z < c_{q} /d(q),} \hfill \\ {qc_{q} - pc_{p} ,} \hfill & {{\text{for}}\;c_{q} /d(q) \le z \le u.} \hfill \\ \end{array} } \right. $$

Next consider the relative sizes of $m\,{\text{and}}\,c_{q} /d(q)$:

Subcase 1 If $m \ge c_{q} /d(q)$, the constraint translates to:

$$ \left\{ {\begin{array}{*{20}l} {y_{0} \ge qd(q)z - pc_{p} ,} \hfill & {{\text{for}}\;l \le z < c_{q} /d(q),} \hfill \\ {y_{0} \ge qc_{q} - pc_{p} ,} \hfill & {{\text{for}}\;c_{q} /d(q) \le z < m,} \hfill \\ {y_{0} + y_{1} \ge qc_{q} - pc_{p} ,} \hfill & {{\text{for}}\;m \le z \le u.} \hfill \\ \end{array} } \right. $$

Then we obtain

$$ \min_{{y_{0} ,y_{1} }} \frac{1}{2}(y_{0} + y_{1} ) + \frac{1}{2}y_{0} = qc_{q} - pc_{p} . $$

Subcase 2 If $m < c_{q} /d(q)$, the constraint translates to:

$$ \left\{ {\begin{array}{*{20}l} {y_{0} \ge qd(q)z - pc_{p} ,} \hfill & {{\text{for}}\;l \le z < m,} \hfill \\ {y_{0} + y_{1} \ge qd(q)z - pc_{p} ,} \hfill & {{\text{for}}\;m \le z < c_{q} /d(q),} \hfill \\ {y_{0} + y_{1} \ge qc_{q} - pc_{p} ,} \hfill & {{\text{for}}\;c_{q} /d(q) \le z \le u.} \hfill \\ \end{array} } \right. $$

Then we obtain

$$ \min_{{y_{0} ,y_{1} }} \frac{1}{2}(y_{0} + y_{1} ) + \frac{1}{2}y_{0} = q\frac{{c_{q} + d(q)m}}{2} - pc_{p} . $$

Case 7

If $c_{q} \le d(q)l, \, d(p)u \le c_{p}$, constraint of (A-3) can be written as:

$$ y_{0} + y_{1} 1\{ z \ge m\} \ge qc_{q} - pd(p)z. $$

Then we obtain

$$ \min_{{y_{0} ,y_{1} }} \frac{1}{2}(y_{0} + y_{1} ) + \frac{1}{2}y_{0} = qc_{q} - pd(p)\frac{m + l}{2}. $$

Case 8

If $c_{q} \le d(q)l, \, d(p)l \le c_{p} \le d(p)u$, constraint of (A-3) can be written as:

$$ y_{0} + y_{1} 1\{ z \ge m\} \ge \left\{ {\begin{array}{*{20}l} {qc_{q} - pd(p)z,} \hfill & {{\text{for }}l \le z < c_{p} /d(p),} \hfill \\ {qc_{q} - pc_{p} ,} \hfill & {{\text{for }}c_{p} /d(p) \le z \le u.} \hfill \\ \end{array} } \right. $$

Next consider the relative sizes of $m\,{\text{and}}\,c_{p} /d(p)$:

Subcase 1 If $m \ge c_{p} /d(p)$, the constraint translates to:

$$ \left\{ {\begin{array}{*{20}l} {y_{0} \ge qc_{q} - pd(p)z,} \hfill & {{\text{for}}\;l \le z < c_{p} /d(p),} \hfill \\ {y_{0} \ge qc_{q} - pc_{p} ,} \hfill & {{\text{for}}\;c_{p} /d(p) \le z < m,} \hfill \\ {y_{0} + y_{1} \ge qc_{q} - pc_{p} ,} \hfill & {{\text{for}}\;m \le z < u.} \hfill \\ \end{array} } \right. $$

Then we obtain

$$ \min_{{y_{0} ,y_{1} }} \frac{1}{2}(y_{0} + y_{1} ) + \frac{1}{2}y_{0} = qc_{q} - p\frac{{d(p)l + c_{p} }}{2}. $$

Subcase 2 If $m < c_{p} /d(p)$, the constraint translates to:

$$ \left\{ {\begin{array}{*{20}l} {y_{0} \ge qc_{q} - pd(p)z,} \hfill & {{\text{for}}\;l \le z < m,} \hfill \\ {y_{0} + y_{1} \ge qc_{q} - pd(p)z,} \hfill & {{\text{for}}\;m \le z < c_{p} /d(p),} \hfill \\ {y_{0} + y_{1} \ge qc_{q} - pc_{p} ,} \hfill & {{\text{for}}\;c_{p} /d(p) \le z < u.} \hfill \\ \end{array} } \right. $$

Then we obtain

$$ \min_{{y_{0} ,y_{1} }} \frac{1}{2}(y_{0} + y_{1} ) + \frac{1}{2}y_{0} = qc_{q} - pd(p)\frac{m + l}{2}. $$

Case 9

If $c_{q} \le d(q)l, \, c_{p} \le d(p)l$, constraint of (A-3) can be written as:

$$ y_{0} + y_{1} 1\{ z \ge m\} \ge qc_{q} - pc_{p} . $$

Then

$$ \min_{{y_{0} ,y_{1} }} \frac{1}{2}(y_{0} + y_{1} ) + \frac{1}{2}y_{0} = qc_{q} - pc_{p} . $$

Based on the above observation, we can deduce that the closed-form solutions to the problem (9) in each small domain $\Delta t$ are either linear functions or piecewise linear functions of $c_{p}\, {\text{and}}\,c_{q}$.□

Proof of Proposition 4

By the strong duality theorem, the dual form of model (9) is represented as follows:

$$ \begin{array}{*{20}l} {\min_{{y_{0} ,y_{1} }} } \hfill & {y_{0} + y_{1} } \hfill \\ {{\text{subject}}\;{\text{to}}} \hfill & {2y_{0} + y_{1} \ge q\min \{ d(q)(m - z),c_{q} \} } \hfill \\ {} \hfill & {\quad - p\min \{ d(p)(m - z),c_{q} \} } \hfill \\ {} \hfill & {\quad + q\min \{ d(q)(m + z),c_{q} \} } \hfill \\ {} \hfill & {\quad - p\min \{ d(p)(m + z),c_{q} \} ,\;\;\;0 \le z \le m.} \hfill \\ \end{array} $$

(A-4)

The constraint of model (A-4) is equivalent to the linear programming as follows:

$$ \begin{array}{*{20}l} {\frac{1}{2}\max_{0 \le z \le m} } \hfill & {\{ q\min \{ d(q)(m - z),c_{q} \} } \hfill \\ {} \hfill & {\quad - p\min \{ d(p)(m - z),c_{q} \} } \hfill \\ {} \hfill & {\quad + q\min \{ d(q)(m + z),c_{q} \} } \hfill \\ {} \hfill & {\quad - p\min \{ d(p)(m + z),c_{p} \} .} \hfill \\ \end{array} $$

(A-5)

Let

$$ \varphi (z|q,c_{q} ) = \min \{ d(q)(m - z),c_{q} \} + \min \{ d(q)(m + z),c_{q} \} . $$

$$ \varphi (z|p,c_{p} ) = \min \{ d(p)(m - z),c_{p} \} + \min \{ d(p)(m + z),c_{p} \} . $$

Then (A-5) is represented as follows:

$$ \frac{1}{2}\max_{0 \le z \le m} q\varphi (z|q,c_{q} ) - p\varphi (z|p,c_{p} ). $$

(A-6)

Function $\varphi (z|p,c_{p} )$ exhibits different formations depending on the relative sizes of $c_{p} /d(p), \, m,\,{\text{and}}\,2m$.

Case 1

$c_{p} /d(p) \ge m, \, c_{q} /d(q) \ge m$.

The shape of function $q\varphi (z|q,c_{q} ) - p\varphi (z|p,c_{p} )$ further relies on the relative sizes of $2d(p)m\,{\text{and}}\,2d(q)m$. Thus, we consider the following subcases:

• (1) If $c_{p} \le 2d(p)m, \, c_{q} \le 2d(q)m,$

  $$ \begin{aligned} & \frac{1}{2}\max_{0 \le z \le m} \{ q\varphi (z|q,c_{q} ), \\ & \quad = \left\{ {\begin{array}{*{20}l} {\max \left\{ {[qd(q) - pd(p)]m,[qd(q) - pd(p)]m + \frac{{pd(p)c_{q} }}{2d(q)} - \frac{{pc_{p} }}{2}} \right\},} \hfill & {{\text{if}}\;qd(q) - pd(p) \ge 0,} \hfill \\ {[qd(q) - pd(p)]m} \hfill & {{\text{if}}\;qd(q) - pd(p) < 0,\;\frac{{c_{p} }}{d(p)} \ge \frac{{c_{q} }}{d(q)},\;qc_{q} { - }pc_{p} \le 2[qd(q) - pd(p)]m,} \hfill \\ {\frac{{qc_{q} - pc_{p} }}{2},} \hfill & {{\text{otherwise}}.} \hfill \\ \end{array} } \right. \\ \end{aligned} $$

• (2) If $c_{p} \le 2d(p)m, \, c_{q} > 2d(q)m$,

  $$ \frac{1}{2}\max_{0 \le z \le m} \{ q\varphi (z|q,c_{q} ) - p\varphi (z|p,c_{p} )\} = qd(q)m - pc_{p} /2. $$

• (3) If $c_{p} > 2d(p)m, \, c_{q} \le 2d(q)m$,

  $$ \frac{1}{2}\max_{0 \le z \le m} \{ q\varphi (z|q,c_{q} ) - p\varphi (z|p,c_{p} )\} = [qd(q) - pd(p)]m. $$

• (4) If $c_{p} > 2d(p)m, \, c_{q} > 2d(q)m$,

  $$ \frac{1}{2}\max_{0 \le z \le m} \{ q\varphi (z|q,c_{q} ) - p\varphi (z|p,c_{p} )\} = [qd(q) - pd(p)]m. $$

Case 2

$c_{p} /d(p) < m, \, c_{q} /d(q) \ge m$,

• (1) If $c_{q} \le 2d(q)m$

  $$ \begin{aligned} & \frac{1}{2}\max_{0 \le z \le m} \{ q\varphi (z|q,c_{q} ) - p\varphi (z|p,c_{p} )\} \\ & \quad = \left\{ {\begin{array}{*{20}l} {\max \left\{ {[qd(q) - pd(p)]m + \frac{{pd(p)c_{q} }}{2d(q)} - \frac{{pc_{p} }}{2},qd(q)m - pc_{p} } \right\},} \hfill & {{\text{if}}\;qd(q) - pd(p) \ge 0,} \hfill \\ {} \hfill & {{\text{if}}\;qd(q) - pd(p) < 0,} \hfill \\ {qd(q)m - pc_{p} ,} \hfill & {\frac{{c_{p} }}{d(p)} + \frac{{c_{q} }}{d(q)} \le 2m,} \hfill \\ {} \hfill & {qc_{q} + pc_{p} \le 2qd(q)m,} \hfill \\ {\frac{{qc_{q} - pc_{p} }}{2},} \hfill & {{\text{otherwise}}{.}} \hfill \\ \end{array} } \right. \\ \end{aligned} $$

• (2) If $c_{q} > 2d(q)m$

  $$ \frac{1}{2}\max_{0 \le z \le m} \{ q\varphi (z|q,c_{q} ) - p\varphi (z|p,c_{p} )\} = qd(q)m - pc_{p} . $$

Case 3

$c_{p} /d(p) \ge m, \, c_{q} /d(q) < m$,

• (1) If $c_{q} \le 2d(p)m$

  $$ \begin{aligned} & \frac{1}{2}\max_{0 \le z \le m} \{ q\varphi (z|q,c_{q} ) - p\varphi (z|p,c_{p} )\} \\ & \quad = \left\{ {\begin{array}{*{20}l} {\max \left\{ {\Big[qc_{q} - \frac{{pd(p)c_{q} }}{2d(q)}{-\frac{pc_p}{2}},qc_{q} - pd(p)m} \right\},} \hfill & {{\text{if}}\;qd(q) - pd(p) \ge 0,} \hfill \\ {} \hfill & {{\text{if }}\;qd(q) - pd(p) < 0,} \hfill \\ {qc_{q} - pd(p)m,} \hfill & {\frac{{c_{p} }}{d(p)} + \frac{{c_{q} }}{d(q)} \ge 2m,} \hfill \\ {} \hfill & {qc_{q} + pc_{p} \le 2pd(p)m,} \hfill \\ {\frac{{qc_{q} - pc_{p} }}{2},} \hfill & {{\text{otherwise}}{.}} \hfill \\ \end{array} } \right. \\ \end{aligned} $$

• (2) If $c_{p} > 2d(p)m$

  $$ \frac{1}{2}\max_{0 \le z \le m} \{ q\varphi (z|q,c_{q} ) - p\varphi (z|p,c_{p} )\} = qc_{q} - pd(p)m. $$

Case 4

$c_{p} /d(p) < m, \, c_{q} /d(q) < m$,

• (1) If $qd(q) \ge pd(p)$

  $$ \frac{1}{2}\max_{0 \le z \le m} \{ q\varphi (z|q,c_{q} ) - p\varphi (z|p,c_{p} )\} = \max \left\{ {\Big[qc_{q} - pc_{p} ,qc_{q} - \frac{{pd(p)c_{q} }}{2d(q)} - \frac{{pc_{p} }}{2}} \right\}. $$

• (2) If $qd(q) < pd(p)$

  $$ \frac{1}{2}\max_{0 \le z \le m} \{ q\varphi (z|q,c_{q} ) - p\varphi (z|p,c_{p} )\} = \left\{ {\begin{array}{*{20}l} {qc_{q} - pc_{p} ,} \hfill & {{\text{if }}qc_{q} \le pc_{p} , \, \frac{{c_{q} }}{d(q)} \ge \frac{{c_{p} }}{d(p)},} \hfill \\ {\frac{{qc_{q} - pc_{p} }}{2},} \hfill & {{\text{otherwise}}{.}} \hfill \\ \end{array} } \right. $$

□

In conclusion, the closed-form formulae for problem (9) in each small domain $\Delta t$ are either linear functions or piecewise linear functions of $c_{p}$ and $c_{q}$.