Chapter 4.16, Problem 9P

Explanation of Solution

Formulation of pre-emptive goal programming model:

Let’s take the following variables,

${\text{x}}_{\text{t}}$= The number of globots produced by regular time labour during quarter $\text{t}$ $\text{(t=1,2,3,4)}$

${\text{y}}_{\text{t}}$= The number of globots produced by overtime labour during quarter $\text{t}$ $\text{(t=1,2,3,4)}$

${\text{i}}_{\text{t}}$= Number of globots on hand at end of quarter $\text{t}$.

Then,

$\text{Total cost }\left(\text{TC}\right)\text{ =cost of producing regular time globots+cost of producing overtime globots}$

$\text{Inventory at end of quarter t=inventory at end of quarter(t-1)+quarter t production-demand of t quarter}$

Now, consider the below

${\text{i}}_{\text{t}}{\text{=i}}_{\text{t-1}}{\text{+(x}}_{\text{t}}{\text{+y}}_{\text{t}}{\text{)-d}}_{\text{t}}$

Here,

${\text{d}}_{\text{t}}$=demand during time $\text{t}$

For the first quarter,

${\text{i}}_1{\text{=i}}_0{\text{+(x}}_1{\text{+y}}_1\text{)-13}$

For the second quarter,

${\text{i}}_2{\text{=i}}_1{\text{+(x}}_2{\text{+y}}_2\text{)-14}$

For the third quarter,

${\text{i}}_3{\text{=i}}_2{\text{+(x}}_3{\text{+y}}_3\text{)-12}$

For the fourth quarter,

${\text{i}}_4{\text{=i}}_3{\text{+(x}}_4{\text{+y}}_4\text{)-15}$

The pre-emptive goal programming can be formulated as given below,

Minimize, ${\text{z=P}}_{\text{1}}{\text{d}}_{\text{1}}{}^{\text{-}}{\text{+P}}_{\text{2}}{\text{d}}_{\text{2}}{}^{\text{+}}{\text{+P}}_{\text{3}}{\text{d}}_{\text{3}}{}^{\text{+}}$

Subject to,