Chapter 4.16, Problem 5P

Explanation of Solution

Formulation of pre-emptive goal programming model for Deancorp:

Consider the below table,

	Head	Chuck	Mutton	Moisture
Fat(per lb)	0.05	0.24	0.11	0
Protein(per lb)	0.20	0.26	0.08	0
Cost(in $¢$)	0.12	9	8	0

Consider the below variables for solving the problem,

${\text{x}}_{\text{1}}$= Percentage of head used in the mixture

${\text{x}}_{\text{1}}$= Percentage of chuck used in the mixture

${\text{x}}_{\text{1}}$= Percentage of mutton used in the mixture

${\text{x}}_{\text{1}}$= Percentage of moisture used in the mixture

$s_{\text{i}}{}^{\text{-}}$= Amount by which the user under the i^th goal

$s_{\text{i}}{}^{+}$= Amount by which the user exceeds the i^th goal (i=1,2,3)

$\text{z}$= penalty associated with meeting the requirement.

As the sausage should consist of at least 15% protein, so

  $\text{0}{\text{.2x}}_{\text{1}}\text{+0}{\text{.26x}}_{\text{2}}\text{+0}{\text{.08x}}_{\text{3}}{\text{+s}}_{\text{1}}{}^{\text{-}}{\text{-s}}_{\text{1}}{}^{\text{+}}\text{=0}\text{.15}$

As the sausage should consist of at most 8% fat, so

  $\text{0}{\text{.5x}}_{\text{1}}\text{+0}{\text{.24x}}_{\text{2}}\text{+0}{\text{.11x}}_{\text{3}}{\text{+s}}_{\text{2}}{}^{\text{-}}{\text{-s}}_{\text{2}}{}^{\text{+}}\text{=0}$