A DECOMPOSITION APPROACH FOR THE TWO-STAGE STOCHASTIC SUPPLY NETWORK PLANNING IN LIGHT OF THE ROLLING HORIZON PRACTICE

Almeida, João Flávio de Freitas; Conceição, Samuel Vieira

doi:10.1590/0101-7438.2021.041s1.00234451

João Flávio de Freitas Almeida ^**Corresponding author

Departamento de Engenharia Industrial, Universidade Federal de Minas Gerais. Av. Antônio Carlos, 6627, Belo Horizonte, MG, Brazil. E-mail: joao.flavio@dep.ufmg.br

http://orcid.org/0000-0002-3884-217X

Samuel Vieira Conceição

Departamento de Engenharia Industrial, Universidade Federal de Minas Gerais. Av. Antônio Carlos, 6627, Belo Horizonte, MG, Brazil. E-mail: svieira@dep.ufmg.br

http://orcid.org/0000-0002-8090-264X

*Corresponding author

Deterministic parameters		Unit
$T_{l r p}^{R}$	Matrix of technical route of product p on resource r at location l	Binary
B _{p' p}	Bill of materials p’ required to produce a unit of product p	Scalar
$M_{l r p}^{C}$	Time unit required to produce product p on resource r at location l	h
$E_{l r t}^{F}$	Resource efficiency r at location l in period t	%
$A_{t}^{H}$	Available hours in each period t	h
$A_{l r t}^{X}$	Extra hours available on resource r at location l in period t	h
$L_{l p}^{M}$	Lot size of product p at location l	Scalar
$P_{l r t}^{M}$	Preventive maintenance of resource r at location l in period t	h
$S_{l p t}^{S}$	Safety stock of product p at location l in period t	Scalar
$S_{l p t}^{X}$	Stock capacity of product p at location l in period t	Scalar
$A_{l p t}^{R}$	Availability of raw materials for product p at location l in period t	Scalar
$S_{l p}^{0}$	Initial inventory of product p at location l	Scalar
Y _lr	Raw material yield on resource r at location l	%
$N_{l r}^{M}$	Number of resources of type r at location l	Scalar
$T_{m l l'}^{C X}$	Transportation capacity of raw-material on modal m from location l to l’	Scalar
$T_{m l l'}^{C Y}$	Transportation capacity of products on modal m from location l to l’	Scalar
$C_{l t}^{I}$	Inbound handling capacity at location l in period t	Scalar
$C_{l t}^{O}$	Outbound handling capacity on location l and time period t	Scalar
$A_{l r t}^{V}$	$(A_{t}^{H} N_{l r}^{M} - P_{l r t}^{M}) (E_{l r t}^{F} Y_{l r})$ Availability of resource r at location l in period t	h
Stochastic parameters
D _cpts	Demand of customer c for product p in period t and scenario s
R _ps	Sales revenue of finished product p in scenario s
N _ps	Fictitious cost penalty for not meeting demand p in scenario s
$C_{l r s}^{F}$	Fixed cost of resource r at location l in scenario s
$C_{l p s}^{V}$	Variable cost of production of p at location l in scenario s
$C_{l r s}^{X}$	Extra capacity cost on resource r at location l in scenario s
$C_{l p s}^{S}$	Unit inventory cost of product p at location l in scenario s
$C_{m l l' s}^{L}$	Unit transport cost on modal m from location l to l' in scenario s
$C_{l p s}^{P}$	Unit procurement cost of raw material x at location l in scenario s
$T_{l p s}^{X}$	Tax over finished product y sold to customer c in scenario s
ρ _s	Probability of each scenario s, $\sum_{s \in S} ρ_{s} = 1$

First-stage decision variables
α _lp	$\in ℤ^{+} :$ Multiple of lot-size for production of product p at location l in the first period
a _lrp	Production of product p on resource r at location l in the first period
b _lp	Consumption of raw material p at location l in the first period
s _lpt	Stock of product p at location l at the end of the first period
d _lp	Met demand of product p at location l in the first period
n _lp1s	Non-satisfied demand of product p at location l in the first period and scenario s
r _lp	$\in ℤ^{+} :$ Multiple of lot-size for procurement of raw material p at location l in the first period
t _mll'p	Transportation of product p on modal m from l to l’ in the first period
c _lr	Consumption of resource r at location l in the first period
$c_{l r}^{'}$	Overtime percentage for resource r at location l in the first period
y _lr	$\in \{0, 1\} :$ Activate (or not) resource r at location l in the first period
Second-stage decision variables
α _lpts	$\in ℤ^{+} :$ Production of product p at location l in period t and scenario s
a _lrpts	Production of product p on resource r at location l in period t and scenario s
b _lpts	Consumption of raw material p at location l in period t and scenario s
s _lpts	Stock of product p at location l at end of period t in scenario s
d _lpts	Met demand of product p at location l in period t and scenario s
n _lpts	Non-satisfied demand of product p at location l in period t and scenario s
r _lpts	$\in ℤ^{+} :$ Procurement of raw material p at location l in period t and scenario s
t _mll'pts	Transportation of product p on modal m from l to l’ in period t and scenario s
c _lrts	Consumption of resource r at location l in period t and scenario s
$c_{l r t s}^{'}$	Overtime percentage for resource r at location l in period t and scenario s
y _lrts	$\in \{0, 1\} :$ Activate (or not) resource r at location l in period t and scenario s

Elements^a	Description
$R_{1}^{P}$	After-tax revenue from sales in the first-stage
$C_{1}^{L}$	Logistics cost on different transport modes in the first-stage
$C_{1}^{F}$	Fixed cost for machine activation in each plant in the first-stage
$C_{1}^{V}$	Finished product cost in each plant in the first-stage
$C_{1}^{P}$	Procurement cost of raw material in the first-stage
$C_{1}^{S}$	Storage cost in each plant in the first-stage
$C_{1}^{X}$	Capacity expansion cost of resources in each plant in the first-stage
$C_{1}^{N}$	Non-delivery cost for each customer in the first-stage
$R_{2}^{P}$	Revenue after taxes from sales in the second-stage
$C_{2}^{L}$	Logistics cost on different transport modes in the second-stage
$C_{2}^{F}$	Fixed cost for machine activation in each plant in the second-stage
$C_{2}^{V}$	Finished product cost in each plant in the second-stage
$C_{2}^{P}$	Procurement cost of raw material in the second-stage
$C_{2}^{S}$	Storage cost in each plant in the second-stage
$C_{2}^{X}$	Capacity expansion cost of resources in each plant in the second-stage
$C_{2}^{N}$	Non-delivery cost for each customer in the second-stage

1	Let $GAP = \infty, θ_{s} = \infty \forall s \in S, t o l = 0.0001, n = 1$ , initialize parameters $s_{l p t}$ .
2	Relax integrality constraints of SP_[2] scenarios.
3	Relax integrality constraints of MP_[1].
4	If $GAP \leq t o l$ then
	Activate integrality constraints of SP_[2] scenarios.
	Solve the SP_[2] scenarios.
	Update variables of optimality cuts constraints.
	Activate integrality constraints of MP_[1].
	Solve the MP_[1]. The optimal solution to the original problem (1)-(48) was reached.
	Else
5	Solve SP_[2] scenarios.
6	Let $n = n + 1$ . Compute optimality cuts coefficients π _i , add (52) to MP_[1].
7	In n, solve the MP_[1] with previous iterations cuts and fix the values of s _{l pt} obtained from the first-stage solution.
8	Let $GAP = \min \{GAP, \sum_{s \in S} θ_{s} - Φ^{n}\}$ and return to step 4.

					LP (CPU(s))		MILP (CPU(s))
Scen	Const.	Var.	Int.	Bin.	M	MC Dec.	M^a	MC Dec.
20	71,964	151,082	3,319	221	3.50	7.09	124.62	56.49
40	143,704	301,582	6,619	441	8.95	14.95	1,131.98	117.86
60	215,444	452,082	9,919	661	14.31	22.82	3,373.13	341.52
80	287,184	602,582	13,219	881	21.72	30.72	5,609.30	524.31
100	358,924	753,082	16,519	1,101	28.39	39.75	10,000.00*	591.61
120	430,664	903,582	19,819	1,321	39.19	48.09	**	788.24
140	502,404	1,054,082	23,119	1,541	78.35	57.47	**	874.88
160	574,144	1,204,582	26,419	1,761	153.19	65.95	**	1,216.40
180	645,884	1,355,082	29,719	1,981	175.58	77.19	**	1,796.82
200	717,624	1,505,582	33,019	2,201	306.37	94.25	**	3,547.75

Scen.	20	40	60	80	100	120	140	160	180	200
Min.	8,272	7,329	7,932	7,487	7,660	7,438	7,569	7,132	7,461	7,255
Q1	8,798	8,725	8,537	8,335	8,470	8,650	8,344	8,303	8,456	8,327
Q2	9,130	8,964	8,860	8,759	8,756	9,027	8,805	8,720	8,873	8,717
$X$	9,198	8,952	8,858	8,768	8,804	8,994	8,756	8,710	8,829	8,702
σ	490	526	505	585	554	558	576	602	537	549
$σ_{X}$	109	83	65	65	55	51	49	48	40	39
Q3	9,436	9,312	9,137	9,148	9,173	9,443	9,130	9,119	9,133	9,046
Max.	10,262	9,883	10,097	10,084	9,998	10,100	10,143	10,342	10,340	10,406
C.I. _(95%)	215	133	128	128	109	100	95	93	79	76

Financial report	Value ($)	Operational report	Value (t.)
Sales revenue	1,910,739,797.99	Production on plant-[1]	2,979,420
Logistics cost	511,717,155.55	Production on plant-[2]	1,507,900
Production fixed-cost	5,987,185.18	Overtime on plant-[1](h)	3,593.52
Production variable-cost	7,780,054.66	Overtime on plant-[2](h)	2,241.92
Procurement cost	439,192,253.11	Transport on modal-[1]	7,557,859
Overtime cost	18,677.87	Transport on modal-[2]	5,245,359
Inventory cost	9,903,492.29	Transport on modal-[3]	5,242,080
Expected overall profit	936,140,979.30
Profit scenario-[1]	964,289,708.25	Total demand	4,929,647
Profit scenario-[2]	917,697,569.37	Satisfied demand	4,609,799
Profit scenario-[3]	926,435,660.28	Unsatisfied demand	319,848
Model statistics
Scenarios	3	Inventory on plant-[1]	2,527,276
Variables	1.062.079	Inventory on plant-[2]	2,436,179
Integers	6.936	Inventory on DCs	1,695,308
Binaries	1.632	Inventory on Ports	636,521
Constraints	1.085.067	Ore procurement-[S1]	4,320,000
Time CPU(s)	10.000,00	Ore procurement-[S2]	2,080,000
Gap (%)	0,97	Coal procurement-[S3]	3,150,000

S	Constraints	Variables	Integer	Binary	Monolithic^a	Decomposed^b
1	382.487	374.343	2.448	576	0,76%	1,60%
2	733.762	718.211	4.692	1.104	0,91%	1,14%
3	1.085.067	1.062.079	6.936	1.632	,*%	1,72%
4	1.436.372	1.405.947	9.180	2.160	,*%	1,57%
5	1.787.677	1.749.815	11.424	2.688	,*%	3,81%
6	2.138.982	2.093.683	13.668	3.216	,*%	1,53%

EVPI Analysis	Value ($)	VSS Analysis	Value ($)
Profit scenario-[1]	1,169,131,282.44	Profit²	749,142,546.11
Profit scenario-[2]	1,170,201,701.33
Profit scenario-[3]	1,169,201,815.94
(A ₁ ) Expected profit	1,169,511,599.93	(A ₂ ) Expected profit	749,142,546.11

2SSP model
Profit scenario-[1]	964,289,708.25
Profit scenario-[2]	917,697,569.37
Profit scenario-[3]	926,435,660.28
(B) Expected profit	936,140,979.30
EVPI (A ₁ -B):	233,370,620.63	VSS (B-A ₂ ):	186,998,433.19

Brasil

Brasil

A DECOMPOSITION APPROACH FOR THE TWO-STAGE STOCHASTIC SUPPLY NETWORK PLANNING IN LIGHT OF THE ROLLING HORIZON PRACTICE

ABSTRACT