STOCHASTIC DISCRETE LOT-SIZING WITH LEAD TIMES FOR FUEL SUPPLY OPTIMIZATION

Testuri, Carlos E.; Cancela, Héctor; Albornoz, Víctor M.

doi:10.1590/0101-7438.2019.039.01.0037

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Articles • Pesqui. Oper. 39 (1) • Jan-Apr 2019 • https://doi.org/10.1590/0101-7438.2019.039.01.0037 copy

STOCHASTIC DISCRETE LOT-SIZING WITH LEAD TIMES FOR FUEL SUPPLY OPTIMIZATION

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

We address the problem of expected cost minimization of meeting the uncertain fuel demand during a time planning horizon, where supply is provided by selecting discrete shipments with lead times. Due to uncertainty and the passage of time, corrective actions can be taken such as cancellation and postponement on supply of shipments with associated costs and delays. This problem is modeled as a stochastic multi-stage capacitated discrete lot-sizing problem with lead times. Computational experiments were performed on the resolution of various instances of the model for four information structures of uncertainty. The experimental optimal values and stochastic rating measures obtained show the validity and interest of the stochastic model, as well as the benefits that can be obtained with respect to a deterministic variant of the model that considers average demand.

Keywords:
oil and gas procurement; oil supply chain; stochastic lot-sizing; multi-stage stochastic integer programming; postponement

Arity	Periods	Scenarios	Nodes
2	5	16	31
2	6	32	63
3	5	81	121
3	6	243	364

Arity	Periods	Cargoes $[\| C \| : (\| A \| + \| P \|)]$	Equations	Variables	(binary)
2	5	10 (2+8)	225	249	(124)
2	6	12 (3+9)	480	549	(296)
3	5	10 (3+7)	827	809	(324)
3	6	12 (4+8)	2542	2485	(1028)

Instance	Arity	Periods	Cargoes	RP	Time(s)	WS	EVPI
1	2	5	10	10,254	0.07	8,993	1,261
2				11,699	0.32	10,428	1,271
3				12,755	0.66	10,601	2,154
4	2	6	12	8,091	6.65	6,961	1,130
5				11,760	14.76	10,415	1,345
6				17,014	31.59	15,561	1,453
7	3	5	10	8,186	1.14	6,177	2,009
8				13,284	20.76	10,945	2,339
9				16,719	17.05	14,294	1,632
10	3	6	12	12,371	(2.34%)	10,008	2,363
11				6,346	202.68	4,468	1,878
12				11,580	(1.60%)	9,003	2,577

Instance	Arity	Periods	Cargoes	t	EEV	VSS
1	2	5	10	1	10,254
				2	10,524	270
				3	13,743	3,489
				4	infeas.	∞
				5	infeas.	∞
2	2	5	10	1	11,699
				2	11,699	0
				3	12,125	426
				4	12,146	21
				5	infeas.	∞

Inst.	Arity	Periods	Cargoes	#AcqCrg	#Acq	#Can	#Pos
1	2	5	10	4	8	0	0
2				4	12	0	0
3				5	11	0	0
4	2	6	12	7	32	1	0
5				6	36	1	0
6				7	30	0	0
7	3	5	10	4	36	0	0
8				6	36	1	0
9				6	31	0	0
10^†	3	6	12	7	74	3	0
11				6	87	1	1
12^†				6	81	2	1

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[1] ^*Corresponding author.

T	periods, $t \in T : = \{1, . . ., H\}$
A	already acquired cargoes
P	possible cargoes to be acquired
C	cargoes, $c \in C : = A \cup P$

d _t	demand volume in period $t \in T$
s ₀	initial storage volume
s, $s$	minimum and maximum storage capacities by period
τ^c	period in which already acquired cargo $c \in A$ is received
q ^c	volume of cargo $c \in C$
γ^c	delivery time of cargo $c \in P$ , such that $0 \leq γ^{c} \leq H - 1$
δ^c	cancellation minimum time of already acquired cargo $c \in A$ , such that $0 \leq δ^{c} \leq τ^{c} - 1$
ε^c	postponement minimum time of already acquired cargo $c \in A$ , such that $0 \leq ε^{c} \leq H - τ^{c}$
ca ^c	acquisition unit cost of cargo $c \in C$
cc ^c	cancellation unit cost of cargo $c \in C$
cp ^c	postponement unit cost of cargo $c \in C$
h _t	storage unit cost in period $t \in T$
a _t	already acquired volume that is received in period $t \in T$

s _t	storage volume at the end of period $t \in T$
u _t	acquired volume into period $t \in T$
$v_{t}^{c}$	if cargo $c \in P$ is acquired in period $t \in \{1, . . ., H - γ^{c}\}$ , (binary)
w _t	cancelled volume out of period $t \in T$
$x_{t}^{c}$	if already acquired cargo $c \in A$ is cancelled in period $t \in \{1, . . ., τ^{c} - δ^{c}\}$ (binary)
y _t	postponed volume into period $t \in T$
$z_{t}^{c}$	if already acquired cargo $c \in A$ is postponed to period $t \in \{τ^{c} + ε^{c}, . . ., H\}$ (binary)

N	nodes of the scenario tree
L	leaf nodes of the scenario tree, $L : = {n \in N \| t (n) = H}$
$N_{γ}^{c}$	nodes where it is posible to acquire cargo $c \in P N_{γ}^{c} : = {n \in N \| t (n) \leq H - γ^{c}}$
$N_{δ}^{c}$	nodes where it is posible to cancel and postpone cargo $c \in A N_{δ}^{c} : = {n \in N \| t (n) \leq τ^{c} + δ^{c}}$
$T_{ε}^{c}$	periods to where it is possible to postpone cargo $c \in A T_{ε}^{c} : = {t \in T \| t \geq τ^{c} + ε^{c}}$

d _n	demanded volume in node $n \in N$
t(n)	period of node $n \in N$
p(n)	immediate predecessor node of node $n \in N$ in the tree
p(n,t)	t-th predecesor node of node $n \in N$ in the tree
π(n)	probability of node $n \in N$
P(n)	nodes in the path from root node to node $n \in N$
S(n)	successor nodes of node $n \in N$ in the tree
P(n₁, n₂)	nodes in the path from node n₁ to a successor node $n_{2} \in S (n_{1})$

s _n	stored volume at the end of period in node $n \in N$
u _n	acquired volume into node $n \in N$
$v_{n}^{c}$	if cargo $c \in P$ is acquired in node $n \in N_{γ}^{c}$ , (binary)
w _n	cancelled volume out of node $n \in N$
$x_{n}^{c}$	if already acquired cargo $c \in A$ is cancelled in node $n \in N_{δ}^{c}$ (binary)
y _n	postponed volume into node $n \in N$
$z_{n t}^{c}$	if already acquired cargo $c \in A$ is postponed in node $n \in N_{δ}^{c}$ to period $t \in T_{ε}^{c}$ (binary)