LARGE SCALE SUPPLY CHAIN NETWORK DESIGN: AN EFFECTIVE HEURISTIC APPROACH

Galvez, Juan Leandro Andres Parra; Borenstein, Denis

doi:10.1590/0101-7438.2023.043.00269080

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ARTICLES • Pesqui. Oper. 43 • 2023 • https://doi.org/10.1590/0101-7438.2023.043.00269080 copy

LARGE SCALE SUPPLY CHAIN NETWORK DESIGN: AN EFFECTIVE HEURISTIC APPROACH

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

This work considers the strategic Supply Chain Network Design (SCND) problem, which is to define the number and location of facilities, and the flow of products among them to fulfilling a long-term deterministic demand. A two-phase heuristic approach was specially developed to solve large scale problems in reasonable time, extending a previous algorithm introduced in Farias et al. (2017)FARIAS E, LI JQ, GALVEZ JP & BORENSTEIN D. 2017. Simple heuristic for the strategic supply chain design of large-scale networks: A Brazilian case study. Computers & Industrial Engineering, 113: 746-756.. In the construction phase, a multi-start approach was developed to generate diversified initial solutions from each new iteration of a layered-based rounding heuristic. In the second phase, a local search heuristic improves the solution provided by the rounding method. The solution method is evaluated using randomly generated instances, and a evaluated strategic of marketing in a real case study applied to a company to redesigning the supply chain to two lines of products.The obtained results evidence the effectiveness and flexibility of the developed approach for handling very large instances.

Keywords:
strategic supply chain design; multi-start; layered-based

Symbol	Definition
Sets
C	the set of customer zones, indexed by c
W	the set of DCs, indexed by w
F	the set of factories, indexed by f
R	the set of raw materials, indexed by r
V	the set of suppliers, indexed by v
P	the set of products, indexed by p
Parameters
d _cp	the demand for product p ∈ P at CZ c ∈ C
Uw	the maximum number of DCs that are opened
Uf	the maximum number of factories that are opened
urp	the utilization rate of raw material r ∈ R per unit of finished product p ∈ P
up	capacity utilization rate per unit of product p ∈ P
CAP _w	the annual throughput at DC w ∈ W
CAP _vr	the supply capacity of raw material r ∈ R at vendor v ∈ V
CAP _f	the capacity of factory f ∈ F
$C T_{w}^{o}$	the annual fixed cost of operating DC w ∈ W
$C T_{f}^{o}$	the annual fixed cost of operating factory f ∈ F
$C T_{w p}^{g}$	the unit cost of throughput of product p ∈ P at DC w ∈ W
$C T_{f p}^{p}$	the unit production cost of product p ∈ P at factory f ∈ F
$C T_{f v r}^{t}$	the unit transportation cost of raw material r ∈ R from supplier v ∈ V to factory f ∈ F
$C T_{f w p}^{t}$	the unit transportation cost of product p ∈ P from factory f ∈ F to DC w ∈ W
$C T_{w c p}^{t}$	the unit transportation cost of product p ∈ P from DC w ∈ W to CZ c ∈ C
Variables
aw	a binary variable, and it is 1 if DC w ∈ W is selected, and 0 otherwise
bf	a binary variable, and it is 1 if factory f ∈ F is selected, and 0 otherwise
gwc	a binary variable, and it is 1 if customer c ∈ C is assigned to DC w ∈ W , and 0 otherwise
$z_{f w}^{p}$	the amount of product p ∈ P shipped from factory F to DC w ∈ W
$y_{v f}^{r}$	the amount of raw materials r ∈ R shipped from supplier v ∈ V to factory f ∈ F

									Developed Algorithm		Farias’ Algorithm
I	V	R	F	DC	P	CZ	TL(s)	B&B	Solution	Gap	Solution	Gap
1	5	5	3	10	5	150	50	425108	427181	0.48%	435332	2.40%
2	5	5	3	20	5	150	70	158261	159772	0.95%	160637	1.50%
3	5	5	3	30	5	150	200	129608	130849	0.96%	133333	2.87%
4	5	5	3	40	5	150	300	19012505	19190346	0.93%	19226313	1.12%
5	5	5	3	20	10	170	150	1487115	1495372	0.56%	1495372	0.56%
6	5	5	3	10	10	150	100	292283	293185	0.30%	296266	1.36%
7	5	5	3	10	40	150	200	301464	298570	0.96%	297243	1.40%
8	5	5	3	10	100	150	200	25358151	25433384	0.29%	25476402	0.47%
9	5	5	3	20	10	150	100	319322	321423	0.65%	321476	0.67%
10	5	5	3	30	10	150	300	227527	229360	0.80%	236082	3.75%
11	5	5	3	40	10	150	100	297083	297740	0.22%	298715	0.54%
12	5	5	3	50	10	150	200	141258	141668	0.29%	142030	0.55%
13	5	5	3	20	10	150	200	249768	252119	0.94%	255173	2.16%
14	5	5	3	20	50	150	200	1306830	1312622	0.44%	1327892	1.61%
15	10	10	5	10	40	150	200	132466656	133347168	0.66%	134070925	1.21%
16	10	10	5	10	150	250	500	10954463	11063021	0.99%	11110108	1.42%
17	10	10	5	20	150	250	500	64622125	64754133	0.20%	64796606	0.27%
18	10	10	5	10	5	150	200	132769	133759	0.74%	134277	1.13%
19	5	5	3	5	150	250	250	8382428	8448909	0.79%	8512811	1.56%
20	25	25	20	15	20	270	200	1585938	1597115	0.70%	1617470	1.98%
21	20	20	20	10	15	200	100	20129199	20214454	0.42%	20214454	0.42%
22	15	15	10	5	10	150	300	8685207	8761766	0.88%	8761766	0.88%
23	20	20	5	5	170	300	400	14036308	14174991	0.98%	14206526	1.21%
24	5	3	1	5	3	250	200	180509	181017	0.28%	181878	0.75%
25	5	5	3	30	5	100	60	262573	263224	0.24%	269622	2.63%

Author	Maximum Dimensions of the Instances
Author	V	R	F	DC	P	CZ	#Inst	Average Gap	Min Gap	Max Gap
Vidal and Goetschalckx (2001)	50	35	8	10	12	80	5	1.53	0.02	3.78
Jayaraman and Pirkul (2001JAYARAMAN V & PIRKUL H. 2001. Planning and coordination of production and distribution facilities for multiple commodities. European Journal of Operational Research , 133: 394-408.)	3	2	10	30	5	150	5	2.71	1.56	3.78
Lee and Kwon (2010LEE YH & KWON SG. 2010. The hybrid planning algorithm for the distribution center operation using tabu search and decomposed optimization. Expert Systems with Applications, 37: 3093-3103.)	-	-	10	20	80	30	12	3.95	0.00	15.06
Thanh et al. (2012THANH PN, BOSTEL N & PÉTON O. 2012. A DC programming heuristic applied to the logistics network design problem. International Journal of Production Economics , 135(1): 94-105.)	27	27	22	13	18	270	15	0.96	0.20	3.00
Farias et al. (2017FARIAS E, LI JQ, GALVEZ JP & BORENSTEIN D. 2017. Simple heuristic for the strategic supply chain design of large-scale networks: A Brazilian case study. Computers & Industrial Engineering, 113: 746-756.)	20	20	20	50	170	300	25	1.36	0.27	3.75
Developed Approach	20	20	20	50	170	300	25	0.63	0.20	0.99

Scenario	#SCs	Capacity	Solution	CZ (%)	Flow(%)	AU (%)
1	-	-	296597	-	-	-
2	5	3600	287563	8.15	8.14	100.00
3	5	6000	273536	15.93	14.3	95.34
4	5	9000	287681	10.37	16.02	74.78
5	5	12000	289169	22.22	27.73	95.67
6	10	3600	442257	18.52	15.81	91.17
7	10	6000	413694	27.41	27.82	97.42
8	10	9000	416767	21.48	23.71	54.07
9	10	12000	1319856	55.18	56.29	97.59
10	15	3600	438304	25.55	21.80	84.25
11	15	6000	413952	31.11	30.89	70.71
12	15	9000	470563	49.26	50.55	78.67
13	15	12000	1784654	84.44	84.82	97.48

Scenario	#SCs	Capacity	Solution	CZ (%)	Flow(%)	AU (%)
14	4	9000	290177	7.04	10.56	61.62
15	5	7000	287866	4.67	6.29	87.06
16	5	8000	287960	13.70	15.87	97.93
17	5	10000	297507	17.78	17.08	71.76
18	5	11000	299829	20.37	20.38	77.84

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[1] *Corresponding author

Step 1 (Construction phase): Find a good feasible solution
	Step 1.1 (Initialization) Define the parameters for the multi-start mechanism, namely number of factories, DCs, and assignment arcs to be disabled.
	Step 1.2 (Initial Solution) Obtain a lower bound for the problem by solving model (1)-(11), relaxing the binary condition on the decision variables in constraints (11).
	Step 1.3 (Rounding Initial Solution) Run the rounding heuristic (Algorithm 2) for the current solution for a feasible good solution. If no feasible solution is obtained, terminate. Otherwise, go to Step 2.
	Step 1.4 (Multi-start mechanism) Randomly disable factory, DCs and a percentage of assigned arcs of CZs that appear in the current solution, obtaining a new model.
	Step 1.5 (Rounding Solutions) Run the rounding heuristic (Algorithm 2) for the several instances generated in the previous step. If no feasible solution is obtained, go to Step 1.4. Otherwise, go to Step 2.
Step 2 (Improvement phase): Perform a Local Search, as follows:
	Step 2.1 Apply the DC Exchange routine (Algorithm 3) in the current solution from Step 1.
	Step 2.2 Apply the Arc Exchange routine (Algorithm 4) in the solution from Step 2.1.
	Step 2.3 Update the overall upper bound if a better solution is found. Store this as the best overall solution.
Step 3 (Termination checking): If the given time limit is reached or an acceptable gap is reached, output the overall best solution and stop. Otherwise, go to Step 1.4.

Step 1 (Initialization): Consider the linear relaxation solution of the model.
Step 2 (Locating the DCs): Let dc be the number of DC location variables. Set dc←0.
	Step 2.1: Sort binary variables a _w in a non-increasing order.
	Step 2.2: For all a _w do: If a _w is almost 1 ( more than 0.95), fix it to 1, and set $d c = d c + 1$ . If dc > U _w , go to step 3. If a _w < 0.95 then exit the loop.
	Step 2.3: If no a _w was fixed in Step 2.2, set a _w with the biggest value to 1.
	Step 2.4: With fixed a _w , solve the updated linear programming model again.
Step 3 (Assign customers to DCs): Let cz be the number of CZs that have been assigned. Set cz←0.
	Step 3.1: Sort binary variables g _wc in a non-increasing order.
	Step 3.2: For all g _wc do: If g _wc is almost 1 (more than 0.95), fix it to 1, and set $c z = c z + 1$ . If all CZs are fixed, go to step 4. If g _wc < 0.95, exit the loop.
	Step 3.3: If no g _wc was fixed in Step 3.2, choose the CZ c ^′ with the largest total demand. Choose then the DC w′with the largest remaining capacity. Fix gw′ c′ = 1.
	Step 3.4: With fixed g _wc in Steps 3.2 or 3.3, solve the updated linear programming model again.
Step 4 (Locate plants): If (b_f =0, 1, ∀f ∈ F), go to Step 5. Otherwise, solve model RTM, fixing the plants, DCs locations, and DCs-CZs assignments of the current solution. Define the total costs.
Step 5: Output the best solution.

Step 1 Considering the current solution, compute RDC for open DCs.
Step 2 Set O as the sorted set of open DCs in decreasing order of RDC _w .
Step 3 For all ow ∈ O do
	Step 3.1 Compute RDC _w for all closed DCs, using NF _ow , NC _ow , and Q _ow in Equation (15).
	Step 3.2 Set C as the sorted set of closed DCs in increasing order of RDC _w .
	Step 3.3 For all cw ∈ C do
		Step 3.3.1 If (CAP _cw > CAP _ow ) then exchange the DCs in the current model, by closing ow and opening cw, creating a temporary model. Adjust the flow of products to the new open DC in the temporary model. Otherwise, go to Step 3.3.
		Step 3.3.2 Run RTM for the temporary model. If the total cost of the new model is smaller than the current solution then update the current solution and the current model. Go to Step 3.
Step 4 Return the current solution and model.

Step 1 Considering the current solution and model, for all used arcs (w, c), ∀w, c in the current solution compute RAR.
Step 2 Set U as the sorted set of used arcs in decreasing order of RAR.
Step 3 For all u = (w, c) ∈ U do
	Step 3.1 Set n = (w ^′ , c ^′) ← the last element of set U
		Step 3.1.1 If (CAP _w ′ > CAP _w ) then exchange the arcs in the current model, creating a temporary model. Adjust the flow of products, transferring all product flows from arc (w, c) to arc (w ^′ , c ^′), and vice-versa in the temporary model. Otherwise, set u ← Next(u) and go to Step 4.
		Step 3.1.2 Run Model RTM with the new temporary model. If the total cost of the temporary model is smaller than the current solution then update the current solution and current model.
		Step 3.1.3 Set u ← Next(u) and delete n from set U .
Step 4 Return the current solution and current model.