Solving a Sequencing Problem: Allocating Cost Savings

[Cyn]

<Moderator's note: LaTeX code edited.>

1. Homework Statement
Hi, I have a question.
I have a sequencing problem $(N, \sigma_{0}$, $\{p_{j}\}_{j\in J}, \{c_{i}\}_{i\in N})$ with $N = \{1,2,3\}$ the set of players, and $J = \{A,B,C,D,E,F\}$ the set of jobs. The processing times of the jobs are:
$$p_{A} =8\\
p_{B} =10 \\
p_{C} = 15\\
p_{D} = 5 \\
p_{E}=6\\
p_{F}= 12$$
De set jobs van de spelers, $J(i)$, zijn:
$$J(1) = \{B,E\}, J(2) = \{A,D\}, J(3) =\{C,F\},$$
$c_{1}(\sigma) = 7min\{C_{B}(\sigma),C_{E}(\sigma)\}$,
$c_{2}(\sigma) = 4min\{C_{A}(\sigma),C_{D}(\sigma)\}$,
$c_{3}(\sigma) = 12min\{C_{C}(\sigma),C_{F} (\sigma)\}$,
$\sigma_{0} = (A B C D E F).$

Homework Equations

What is the allocation of the cost savings?

The Attempt at a Solution

I have calculated the $\alpha$'s and the optimal order begins with (EFD). Now I want to calculate an allocation of the cost savings.
I have said that my new order becomes $(EFDCBA)$. So $MP(\sigma_{0})=\{(A,B), (A,C), (A,D), (A,E), (A,F), (B,C), (B,D), (B,E), (B,F), (C,D), (C,E), (C,F), (D, E), (D,F)\}$
Then, I have calcultated $g_{ij}. (g_{ij} = \alpha_{j}p_{i}-\alpha_{i}p_{j}).$

$g_{AB} = 16; g_{AE} = 32; g_{BC} = 15; g_{BD} = 5; g_{BE} = 28; g_{BF} = 36; g_{CE} = 33; g_{DE} = 11$
If we use EGS, is it correct that $EGS_{1}= 102$?​

 

[tnich]

Homework Statement

Hi, I have a question.
I have a sequencing problem (N, $\sigma_{0}$, $\{p_{j}\}_{j\in J}, \{c_{i}\}_{i\in N}$) with N = \{1,2,3\} the set of players, and J = \{A,B,C,D,E,F\} the set of jobs. The processing times of the jobs are:
$$p_{A} =8\\
p_{B} =10 \\
p_{C} = 15\\
p_{D} = 5 \\
p_{E}=6\\
p_{F}= 12$$
De set jobs van de spelers, $J(i)$, zijn:
$$J(1) = \{B,E\}, J(2) = \{A,D\}, J(3) =\{C,F\},$$
$c_{1}(\sigma) = 7min\{C_{B}(\sigma),C_{E}(\sigma)\}$,
$c_{2}(\sigma) = 4min\{C_{A}(\sigma),C_{D}(\sigma)\}$,
$c_{3}(\sigma) = 12min\{C_{C}(\sigma),C_{F} (\sigma)\}$,
$\sigma_{0}$ = (A B C D E F).

Homework Equations

What is the allocation of the cost savings?

The Attempt at a Solution

I have calculated the $\alpha$'s and the optimal order begins with (EFD). Now I want to calculate an allocation of the cost savings.
I have said that my new order becomes (EFDCBA). So MP($\sigma_{0}$)={(A,B), (A,C), (A,D), (A,E), (A,F), (B,C), (B,D), (B,E), (B,F), (C,D), (C,E), (C,F), (D, E), (D,F)}
Then, I have calcultated $g_{ij}$. ($g_{ij} = \alpha_{j}p_{i}-\alpha_{i}p_{j}$).

$g_{AB} = 16;
g_{AE} = 32;
g_{BC} = 15;
g_{BD} = 5;
g_{BE} = 28;
g_{BF} = 36;
g_{CE} = 33;
g_{DE} = 11$
If we use EGS, is it correct that $EGS_{1}= 102?​

Your problem statement is very difficult to understand.
1) One thing that would help a lot would be to use "##" instead of "$" to delimit your LaTeX notation. Then it would display correctly.
2) Please give definitions of the variables. What do p, c, g, σ and α represent?
3) You give as your goal that you want to allocate cost savings, but you have not clearly defined your objective in mathematical terms. Cost savings relative to what? What are you optimizing?