Finding nadir points in multi-objective integer programs

Abstract

We address the problem of finding the nadir point in a multi-objective integer programming problem. Finding the nadir point is not straightforward, especially when there are more than three objectives. The difficulty further increases for integer programming problems. We develop an exact algorithm to find the nadir point in multi-objective integer programs with integer-valued parameters. We also develop a variation that finds bounds for each component of the nadir point with a desired level of accuracy. We demonstrate on several instances of multi-objective assignment, knapsack, and shortest path problems that the algorithms work well.

Appendix: an example

We demonstrate the algorithm on the following four-objective knapsack problem with 10 items:

$$\begin{aligned}&\text {``}\hbox {Max}\text {''}{}\left\{ {\mathbf{Vx}} \right\} \\&\hbox {subject}\;\hbox {to} \\&\mathbf{wx}\le C \\&\mathbf{x}\in \left\{ {0,1} \right\} ^{10} \end{aligned}$$

where

$$\begin{aligned} \begin{array}{ll} \mathbf{V}=\left[ {\begin{array}{llllllllll} 77 &{} 26 &{} 57 &{} 22 &{} 10 &{} 32 &{} 98 &{} 77 &{} 43 &{} 19 \\ 56 &{} 22 &{} 28 &{} 37 &{} 89 &{} 17 &{} 65 &{} 85 &{} 10 &{} 41 \\ 89 &{} 33 &{} 47 &{} 63 &{} 31 &{} 39 &{} 27 &{} 66 &{} 49 &{} 13 \\ 71 &{} 41 &{} 23 &{} 44 &{} 61 &{} 97 &{} 27 &{} 25 &{} 89 &{} 59 \\ \end{array}} \right]&\begin{array}{l} \mathbf{w}=\left[ \begin{array}{llllllllll}91 &{} 95 &{} 87 &{} 42 &{} 12 &{} 55 &{} 24 &{} 17 &{} 96 &{} 77\end{array} \right] \quad \\ {C=\left[ {298} \right] } \\ \mathbf{x}=\left[ {{\begin{array}{llllllllll} {\hbox {x}_{1} }&{} {\hbox {x}_{2} }&{} {\hbox {x}_{3} }&{} {\hbox {x}_{4} }&{} {\hbox {x}_{5} }&{} {\hbox {x}_{6} }&{} {\hbox {x}_{7} }&{} {\hbox {x}_{8} }&{} {\hbox {x}_{9} }&{} {\hbox {x}_{10} } \\ \end{array} }} \right] ^\mathrm{T}\\ \end{array}\\ \end{array} \end{aligned}$$

To find the exact nadir for criterion 1, set $g^{*}=0,\,n = 1$. Table  5 shows the payoff table.

Table 5 The payoff table for the example problem

1.1 Iteration 0

Step 0 (Initialization) $t=0$. From the payoff table, $z_1^{ IP (2)} =\left( {\hbox {3}0\hbox {3, 373, 289, 287}} \right) , \, z_1^{ IP (3)} =\left( {\hbox {327, 342, 325, 317}} \right) ,$ and $z_1^{ IP (4)} =\left( {\hbox {337, 322, 3}0\hbox {1, 37}0} \right)$. Since $z_1^{ IP (2)} =\mathop {\min }\limits _{i\ne 1} z_1^{ IP (i)}, \, m=2$. Then, $lz_3^1 =289+1,\,lz_4^1 =287+1$ and $uz_1 =303$. $lz_1 =96$ due to Corollary 1.

Step 1 Since $\frac{\left( {uz_n -lz_n } \right) }{\left( {z_n^{ IP \left( n \right) } -lz_n } \right) }=\frac{\left( {303-96} \right) }{\left( {351-96} \right) }>g^{*}=0,\,uz_1^*=uz_1 -1=302$ and go to Step 2.

1.2 Iteration 1

Step 2 $t= 1$. Partition $P_{1(2)}^1$ into two submodels using the procedure of Lokman and Köksalan [6]:

$$\begin{aligned}&P_{1(2)}^1 \\&\hbox {Max}\;z_2 \left( \mathbf{x} \right) +\varepsilon _1 z_1 \left( \mathbf{x} \right) +\varepsilon z_3 \left( \mathbf{x} \right) +\varepsilon z_4 \left( \mathbf{x} \right) \\&\hbox {subject}\;\hbox {to} \\&z_3 \left( \mathbf{x} \right) \ge 290-M+My_{31} \\&z_4 \left( \mathbf{x} \right) \ge 288-M+M\left( {1-y_{31} } \right) \\&z_1 \left( \mathbf{x} \right) \le 302\\&z_1 \left( \mathbf{x} \right) \ge 96\\&\mathbf{x}\in \mathbf{X} \end{aligned}$$

Submodel 1	Submodel 2
$\hbox {Max}\;z_2 \left( \mathbf{x} \right) +\varepsilon _1 z_1 \left( \mathbf{x} \right) +\varepsilon z_3 \left( \mathbf{x} \right) +\varepsilon z_4 \left( \mathbf{x} \right)$	$\hbox {Max}\;z_2 \left( \mathbf{x} \right) +\varepsilon _1 z_1 \left( \mathbf{x} \right) +\varepsilon z_3 \left( \mathbf{x} \right) +\varepsilon z_4 \left( \mathbf{x} \right)$
$\hbox {subject}\;\hbox {to}$	$\hbox {subject}\;\hbox {to}$
$z_3 \left( \mathbf{x} \right) \ge -M$	$z_3 \left( \mathbf{x} \right) \ge 290$
$z_4 \left( \mathbf{x} \right) \ge 288$	$z_4 \left( \mathbf{x} \right) \ge -M$
$z_1 \left( \mathbf{x} \right) \le 302$	$z_1 \left( \mathbf{x} \right) \le 302$
$z_1 \left( \mathbf{x} \right) \ge 96\quad \;$	$z_1 \left( \mathbf{x} \right) \ge 96\quad \;$
$\mathbf{x}\in \mathbf{X}$	$\mathbf{x}\in \mathbf{X}$
Optimal solution: (258,334*,239,313)	Optimal solution: (237,325,301,357)

$$\begin{aligned} \mathbf{dz}^{1}=\left( {258,334,239,313} \right) \end{aligned}$$

Step 3 Solve $D_{1(2)}^1$. The solution yields $\mathbf{z}^{1}=\left( {\hbox {327, 342, 325, 317}} \right)$ that dominates $\mathbf{dz}^{1}$. Set $lz_3^2 =326$ and $lz_4^2 =318$.

1.3 Iteration 2

Step 2. $t = 2$. Partition $P_{1(2)}^2$ into three submodels using the procedure of Lokman and Köksalan [6]:

$$\begin{aligned}&P_{1(2)}^2 \\&\hbox {Max}\;z_2 \left( \mathbf{x} \right) +\varepsilon _1 z_1 \left( \mathbf{x} \right) +\varepsilon z_3 \left( \mathbf{x} \right) +\varepsilon z_4 \left( \mathbf{x} \right) \quad \\&\hbox {subject}\;\hbox {to} \\&z_3 \left( \mathbf{x} \right) \ge 290-M+My_{31} \\&z_4 \left( \mathbf{x} \right) \ge 288-M+M\left( {1-y_{31} } \right) \\&z_3 \left( \mathbf{x} \right) \ge 326-M+My_{32} \\&z_4 \left( \mathbf{x} \right) \ge 318-M+M\left( {1-y_{32} } \right) \quad \\&z_1 \left( \mathbf{x} \right) \le 302\\&z_1 \left( \mathbf{x} \right) \ge 96\; \\&\mathbf{x}\in \mathbf{X} \end{aligned}$$

Submodel 1	Submodel 2	Submodel 3
$\hbox {Max}\;z_2 \left( \mathbf{x} \right) +\varepsilon _1 z_1 \left( \mathbf{x} \right)$	$\hbox {Max}\;z_2 \left( \mathbf{x} \right) +\varepsilon _1 z_1 \left( \mathbf{x} \right)$	$\hbox {Max}\;z_2 \left( \mathbf{x} \right) +\varepsilon _1 z_1 \left( \mathbf{x} \right)$
$\quad \quad +\varepsilon z_3 \left( \mathbf{x} \right) +\varepsilon z_4 \left( \mathbf{x} \right)$	$\quad \quad +\varepsilon z_3 \left( \mathbf{x} \right) +\varepsilon z_4 \left( \mathbf{x} \right)$	$\quad \quad +\varepsilon z_3 \left( \mathbf{x} \right) +\varepsilon z_4 \left( \mathbf{x} \right)$
$\hbox {subject}\;\hbox {to}$	$\hbox {subject}\;\hbox {to}$	$\hbox {subject}\;\hbox {to}$
$z_3 \left( \mathbf{x} \right) \ge -M$	$z_3 \left( \mathbf{x} \right) \ge 290$	$z_3 \left( \mathbf{x} \right) \ge 326$
$z_4 \left( \mathbf{x} \right) \ge 318$	$z_4 \left( \mathbf{x} \right) \ge 318$	$z_4 \left( \mathbf{x} \right) \ge -M$
$z_1 \left( \mathbf{x} \right) \le 302$	$z_1 \left( \mathbf{x} \right) \le 302$	$z_1 \left( \mathbf{x} \right) \le 302$
$z_1 \left( \mathbf{x} \right) \ge 96\quad \;$	$z_1 \left( \mathbf{x} \right) \ge 96\quad \;$	$z_1 \left( \mathbf{x} \right) \ge 96\quad \;$
$\mathbf{x}\in \mathbf{X}$	$\mathbf{x}\in \mathbf{X}$	$\mathbf{x}\in \mathbf{X}$
Optimal Solution: (237,325,301,357)	Optimal Solution: (237,325,301,357)*	Infeasible

1. *  No need to solve. The solution is identical to that of a previously solved model (see [6] for details)

$$\begin{aligned} \mathbf{dz}^{2}=\left( {237,325,301,357} \right) . \end{aligned}$$

Step 3 Solve $D_{1(2)}^2$. The solution yields $\mathbf{z}^{2}=\left( {237,325,301,357} \right)$. Set $lz_3^3 =302,\,lz_4^3 =358$. $\mathbf{z}^{2}=\mathbf{dz}^{2}$ since $\mathbf{dz}^{2}$ is nondominated. Set $uz_1 =237$.

Step 1. Since$\frac{\left( {uz_n -lz_n } \right) }{\left( {z_n^{ IP \left( n \right) } -lz_n } \right) }=\frac{\left( {237-96} \right) }{\left( {351-96} \right) }>g^{*}=0$, set $uz_1^*=uz_1 -1=236$ and go to Step 2.

1.4 Iteration 3

Step 2. $t = 3$. Partition $P_{1(2)}^3$ into four submodels using the procedure of Lokman and Köksalan [6]:

$$\begin{aligned}&P_{1(2)}^3 \\&\hbox {Max}\;z_2 \left( \mathbf{x} \right) +\varepsilon _1 z_1 \left( \mathbf{x} \right) +\varepsilon z_3 \left( \mathbf{x} \right) +\varepsilon z_4 \left( \mathbf{x} \right) \quad \\&\hbox {subject}\;\hbox {to} \\&z_3 \left( \mathbf{x} \right) \ge 290-M+My_{31} \\&z_4 \left( \mathbf{x} \right) \ge 288-M+M\left( {1-y_{31} } \right) \\&z_3 \left( \mathbf{x} \right) \ge 326-M+My_{32} \\&z_4 \left( \mathbf{x} \right) \ge 318-M+M\left( {1-y_{32} } \right) \\&z_3 \left( \mathbf{x} \right) \ge 302-M+My_{33} \\&z_4 \left( \mathbf{x} \right) \ge 358-M+M\left( {1-y_{33} } \right) \\&z_1 \left( \mathbf{x} \right) \le 236\quad \\&z_1 \left( \mathbf{x} \right) \ge 96 \\&\mathbf{x}\in \mathbf{X} \end{aligned}$$

Submodel 1	Submodel 2	Submodel 3	Submodel 4
$\hbox {Max}\;z_2 \left( \mathbf{x} \right) +\varepsilon _1 z_1 \left( \mathbf{x} \right)$	$\hbox {Max}\;z_2 \left( \mathbf{x} \right) +\varepsilon _1 z_1 \left( \mathbf{x} \right)$	$\hbox {Max}\;z_2 \left( \mathbf{x} \right) +\varepsilon _1 z_1 \left( \mathbf{x} \right)$	$\hbox {Max}\;z_2 \left( \mathbf{x} \right) +\varepsilon _1 z_1 \left( \mathbf{x} \right)$
$+\varepsilon z_3 \left( \mathbf{x} \right) +\varepsilon z_4 \left( \mathbf{x} \right)$	$+\varepsilon z_3 \left( \mathbf{x} \right) +\varepsilon z_4 \left( \mathbf{x} \right)$	$+\varepsilon z_3 \left( \mathbf{x} \right) +\varepsilon z_4 \left( \mathbf{x} \right)$	$+\varepsilon z_3 \left( \mathbf{x} \right) +\varepsilon z_4 \left( \mathbf{x} \right)$
subject to	subject to	subject to	subject to
$z_3 \left( \mathbf{x} \right) \ge -M$	$z_3 \left( \mathbf{x} \right) \ge 290$	$z_3 \left( \mathbf{x} \right) \ge 302$	$z_3 \left( \mathbf{x} \right) \ge 326$
$z_4 \left( \mathbf{x} \right) \ge 358$	$z_4 \left( \mathbf{x} \right) \ge 358$	$z_4 \left( \mathbf{x} \right) \ge 318$	$z_4 \left( \mathbf{x} \right) \ge -M$
$z_1 \left( \mathbf{x} \right) \le 236$	$z_1 \left( \mathbf{x} \right) \le 236$	$z_1 \left( \mathbf{x} \right) \le 236$	$z_1 \left( \mathbf{x} \right) \le 236$
$z_1 \left( \mathbf{x} \right) \ge 96\quad \;$	$z_1 \left( \mathbf{x} \right) \ge 96\quad \;$	$z_1 \left( \mathbf{x} \right) \ge 96$	$z_1 \left( \mathbf{x} \right) \ge 96$
$\mathbf{x}\in \mathbf{X}$	$\mathbf{x}\in \mathbf{X}$	$\mathbf{x}\in \mathbf{X}$	$\mathbf{x}\in \mathbf{X}$
Optimal solution: (184,209,271,362)	Infeasible	Infeasible	Infeasible*

1. *  No need to solve. The solution is identical to that of a previously solved model (see [6] for details)

$$\begin{aligned} \mathbf{dz}^{3}=\left( {184,209,271,362} \right) . \end{aligned}$$

Step 3 Solve $D_{1(2)}^3$. The solution yields $\mathbf{z}^{3}=\left( {\hbox {337, 322, 3}0\hbox {1, 37}0} \right)$. Set $lz_3^4 =302$, and $lz_4^4 =371$.

1.5 Iteration 4

Step 2. $t = 4$. Partition $P_{1(2)}^4$ into five submodels using the procedure of Lokman and Köksalan [6]:

$$\begin{aligned}&P_{1(2)}^4 \\&\hbox {Max}\;z_2 \left( \mathbf{x} \right) +\varepsilon _1 z_1 \left( \mathbf{x} \right) +\varepsilon z_3 \left( \mathbf{x} \right) +\varepsilon z_4 \left( \mathbf{x} \right) \quad \\&\hbox {subject}\;\hbox {to} \\&z_3 \left( \mathbf{x} \right) \ge 290-M+My_{31} \\&z_4 \left( \mathbf{x} \right) \ge 288-M+M\left( {1-y_{31} } \right) \\&z_3 \left( \mathbf{x} \right) \ge 326-M+My_{32} \\&z_4 \left( \mathbf{x} \right) \ge 318-M+M\left( {1-y_{32} } \right) \quad \\&z_3 \left( \mathbf{x} \right) \ge 302-M+My_{33} \\&z_4 \left( \mathbf{x} \right) \ge 358-M+M\left( {1-y_{33} } \right) \\&z_3 \left( \mathbf{x} \right) \ge 302-M+My_{34} \\&z_4 \left( \mathbf{x} \right) \ge 371-M+M\left( {1-y_{34} } \right) \\&z_1 \left( \mathbf{x} \right) \le 236\\&z_1 \left( \mathbf{x} \right) \ge 96 \\&\mathbf{x}\in \mathbf{X} \end{aligned}$$

Submodel 1	Submodel 2	Submodel 3	Submodel 4	Submodel 5
$\hbox {Max}\;z_2 \left( \mathbf{x} \right) +\varepsilon _1 z_1 \left( \mathbf{x} \right)$	$\hbox {Max}\;z_2 \left( \mathbf{x} \right) +\varepsilon _1 z_1 \left( \mathbf{x} \right)$	$\hbox {Max}\;z_2 \left( \mathbf{x} \right) +\varepsilon _1 z_1 \left( \mathbf{x} \right)$	$\hbox {Max}\;z_2 \left( \mathbf{x} \right) +\varepsilon _1 z_1 \left( \mathbf{x} \right)$	$\hbox {Max}\;z_2 \left( \mathbf{x} \right) +\varepsilon _1 z_1 \left( \mathbf{x} \right)$
$+\varepsilon z_3 \left( \mathbf{x} \right) +\varepsilon z_4 \left( \mathbf{x} \right)$	$+\varepsilon z_3 \left( \mathbf{x} \right) +\varepsilon z_4 \left( \mathbf{x} \right)$	$+\varepsilon z_3 \left( \mathbf{x} \right) +\varepsilon z_4 \left( \mathbf{x} \right)$	$+\varepsilon z_3 \left( \mathbf{x} \right) +\varepsilon z_4 \left( \mathbf{x} \right)$	$+\varepsilon z_3 \left( \mathbf{x} \right) +\varepsilon z_4 \left( \mathbf{x} \right)$
subject to	subject to	subject to	subject to	subject to
$z_3 \left( \mathbf{x} \right) \ge -M$	$z_3 \left( \mathbf{x} \right) \ge 290$	$z_3 \left( \mathbf{x} \right) \ge 302$	$z_3 \left( \mathbf{x} \right) \ge 302$	$z_3 \left( \mathbf{x} \right) \ge 326$
$z_4 \left( \mathbf{x} \right) \ge 371$	$z_4 \left( \mathbf{x} \right) \ge 371$	$z_4 \left( \mathbf{x} \right) \ge 371$	$z_4 \left( \mathbf{x} \right) \ge 318$	$z_4 \left( \mathbf{x} \right) \ge -M$
$z_1 \left( \mathbf{x} \right) \le 236$	$z_1 \left( \mathbf{x} \right) \le 236$	$z_1 \left( \mathbf{x} \right) \le 236$	$z_1 \left( \mathbf{x} \right) \le 236$	$z_1 \left( \mathbf{x} \right) \le 236$
$z_1 \left( \mathbf{x} \right) \ge 96\quad \;$	$z_1 \left( \mathbf{x} \right) \ge 96\quad \;$	$z_1 \left( \mathbf{x} \right) \ge 96$	$z_1 \left( \mathbf{x} \right) \ge 96$	$z_1 \left( \mathbf{x} \right) \ge 96$
$\mathbf{x}\in \mathbf{X}$	$\mathbf{x}\in \mathbf{X}$	$\mathbf{x}\in \mathbf{X}$	$\mathbf{x}\in \mathbf{X}$	$\mathbf{x}\in \mathbf{X}$
Infeasible	Infeasible*	Infeasible*	Infeasible*	Infeasible*

1. *  No need to solve. The solution is identical to that of a previously solved model (see [6] for details)

$P_{1(2)}^4$ is infeasible. Set $lz_1 =uz_1^*+1=237$ and go to Step 4.

Step 4 Stop, $lz_1 =237\le z_1^{ NP \left( 1 \right) } \le 237=uz_1$ and therefore $z_1^{ NP \left( 1 \right) } =237$.