Quantitative insights into the integrated push and pull production problem for lean supply chain planning 4.0

Figures & data

Table 1. Survey of the papers that address LSCP through mathematical programming models.

					SC paradigms
Authors	Decision level	Modelling approach	PPC system	Industrial application	Lean	Resilient	Sustainable	Flexible	Agile
Fahimnia, Sarkis, and Eshragh (2015)	Tactical Operational	MINLP	MRP	Metal- mechanical	x		x	x
Das (2018)	Strategic	MILP	JIT	Not provided	x		x
Das (2019)	Strategic	MILP	JIT	Food	x	x	x
Oh and Jeong (2019)	Tactical	MINLP	JIT	Electronics	x			x	x
Vafaeenezhad, Tavakkoli-Moghaddam, and Cheikhrouhou (2019)	Tactical	MILP	MRP	Wood and paper	x		x
Zamanian et al. (2020)	Tactical	MILP	JIT	Energy	x	x	x
Shafiee, Zare Mehrjerdi, and Keshavarz (2021)	Tactical Operational	MILP	JIT	Not provided	x	x	x
Gholami-Zanjani, Jabalameli, and Pishvaee (2021)	Tactical	MILP	MRP	Food	x	x	x
Our model	Tactical	MILP	JIT/ MRP	Footwear	x	x	x

Figure 1. LSCP 4.0 modelling and experimentation methodology.

A process flow diagram is presented with the sequence of steps involved for the modelling and experimentation of LSCP 4.0.

Figure 2. LSCP 4.0 design.

A lean SC structure is presented with three-dimensional icons considering information flows of: raw materials at levels one and two for aggregate demand, finished goods at levels three to five for two types of demand.

Table 2. Notation of the LSCP 4.0 model.

Set of indexes
G	Set of second-level raw materials (indexed by g: g = 1, … ,$|G|$)
I	Set of finished goods (indexed by i: i = 1, … ,$|I|$)
J	Set of retailers of finished goods (indexed by j: j = 1, … ,$|J|$)
K	Set of first-level raw materials (indexed by k: k = 1, … ,$|K|$)
L	Set of first-level local suppliers (indexed by l: l = 1, … , $|L|$)
M	Set of production plants for finished goods (indexed by m: m = 1, … ,$|M|$)
N	Set of alternative first-level suppliers (indexed by n: n = 1, … ,$|N|$)
O	Set of firm orders for finished goods (indexed by o: o = 1, … ,$|O|$)
T	Set of time periods (indexed by t: t = 1, … ,$|T|$)
X	Set of second-level suppliers (indexed by x: x = 1, … ,$|X|$)
W	Set of warehouses for finished goods (indexed by w: w = 1, … ,$|W|$)
Data
${\beta }_{\mathrm{iw}}$	Level of risk of shortage of supply defined for i in w
$\rho a_{\mathrm{kl}}$	Cost of carbon emissions in production processes and logistics of supplier l to produce k
$\rho b_{\mathrm{kn}}$	Cost of carbon emissions in production processes and logistics of supplier n to produce k
$A{a}_{\mathrm{gk}}$	Required quantity of g to produce a unit of product k
$A{b}_{\mathrm{ki}}$	Required quantity of k to produce a unit of product i
$b{i}_{\mathrm{im}}$	Unit profit for producing i of firm orders in m
$\mathrm{Cc}{g}_{\mathrm{gx}}$	Penalty cost of a soft JIT production coverage constraint for material g on x
$\mathrm{Cc}{l}_{\mathrm{kl}}$	Penalty cost of a soft JIT production coverage constraint for material k on l
$\mathrm{Cc}{n}_{\mathrm{kn}}$	Penalty cost of a soft JIT production coverage constraint for material k on n
$\mathrm{Cc}{m}_{\mathrm{im}}$	Penalty cost of a soft JIT production coverage constraint for the product i on m
$C{f}_{\mathrm{km}}$	Backorder cost per unit of demand of k in m for the demand forecasts of finished goods
$\mathrm{Cf}{h}_{\mathrm{km}}$	Backorder cost per unit of demand of k in m for firm orders of finished goods
$\mathrm{Cf}{i}_{\mathrm{ij}}$	Backorder cost per unit of firm order demand of i in j
$\mathrm{Cf}{j}_{\mathrm{ij}}$	Backorder cost per unit of forecasting demand of i in j
$\mathrm{Cf}{l}_{\mathrm{gl}}$	Backorder cost per unit of first-level raw material demand g in l
$\mathrm{Cfl}{a}_{\mathrm{kl}}$	Backorder cost per unit of demand of k in l for demand forecasts of finished goods
$\mathrm{Cfl}{b}_{\mathrm{kl}}$	Backorder cost per unit of demand of k in l for firm orders of finished goods
$\mathrm{Cf}{n}_{\mathrm{gn}}$	Backorder cost per unit of first-level raw material demand g in n
$\mathrm{Cfn}{a}_{\mathrm{kl}}$	Backorder cost per unit of demand of k in n for demand forecasts of finished goods
$\mathrm{Cfn}{b}_{\mathrm{kl}}$	Backorder cost per unit of demand of k in n for firm orders of finished goods
$\mathrm{Cf}{w}_{\mathrm{iw}}$	Backorder cost per unit of forecasting demand for i in w
$\mathrm{Cf}{x}_{\mathrm{iw}}$	Backorder cost per unit of firm order demand for i in w
$\mathrm{Cf}{y}_{\mathrm{gx}}$	Backorder cost per unit of second-level raw material demand g in x
$C{j}_{\mathrm{ij}}$	Inventory storage unit cost i in j
$C{k}_{\mathrm{km}}$	Inventory storage unit cost of k in m
$\mathrm{Ck}{l}_{\mathrm{kl}}$	Inventory storage unit cost of k in l
$\mathrm{Ck}{n}_{\mathrm{kn}}$	Inventory storage unit cost of k in n
$C{l}_{\mathrm{klm}}$	Transport and distribution unit cost of k from l to m
$\mathrm{Cl}{x}_{\mathrm{gxl}}$	Transport and distribution unit cost of g from x to l for first-level raw material orders
$\mathrm{Cl}{y}_{\mathrm{gxn}}$	Transport and distribution unit cost of g from x to n for first-level raw material orders
$C{m}_{\mathrm{imw}}$	Transport and distribution unit cost of i from m to w
$C{n}_{\mathrm{knm}}$	Transport and distribution unit cost of k from n to m
$C{p}_{\mathrm{im}}$	Labour cost per hour for the regular time production of i in m
$\mathrm{Cp}{e}_{\mathrm{im}}$	Labour cost per hour for the overtime production of i in m
$\mathrm{Cp}{l}_{\mathrm{kl}}$	Labour cost per hour for the production of k in l
$\mathrm{Cp}{n}_{\mathrm{kn}}$	Labour cost per hour for the production of k in n
$\mathrm{Cp}{x}_{\mathrm{gx}}$	Labour cost per hour for the production of g in x for second-level raw material orders
$C{s}_{\mathrm{iwj}}$	Transport and distribution unit cost of i from w to j
$C{w}_{\mathrm{iw}}$	Inventory storage unit cost of i in w
$C{x}_{\mathrm{gx}}$	Inventory storage unit cost of the second-level raw material inventory of g in x
$\mathrm{Cx}{l}_{\mathrm{gl}}$	Inventory storage unit cost of the second-level raw material inventory of g in l
$\mathrm{Cx}{n}_{\mathrm{gn}}$	Inventory storage unit cost of the second-level raw material inventory of g in n
$D_{\mathrm{ijt}}$	Forecasted demand for i in j during time period t
$D{o}_{\mathrm{ijot}}$	Demand of product i in j to satisfy firm orders o during time period t
$E{C}_m$	Capacity hours for over time production of plant m
$F{l}_{\mathrm{kl}}$	Estimated carbon emissions generation of $\text{C}{\text{O}}_2$ as kg of first-level supplier l to produce k
$F{l}_{\mathrm{kn}}$	Estimated carbon emissions generation of $\text{C}{\text{O}}_2$ as kg of first-level supplier n to produce k
$E_{\mathrm{oimt}}$	Big number
$H{x}_{\mathrm{gx}}$	Programmed receptions of raw material g in x
$H{l}_{\mathrm{gl}}$	Programmed receptions of raw material g in l
$\mathrm{Hl}{a}_{\mathrm{kl}}$	Programmed receptions of raw material k in l for finished goods demand forecasts
$\mathrm{Hl}{b}_{\mathrm{okl}}$	Programmed receptions for firm orders o of finished goods of material k in l
$H{n}_{\mathrm{gn}}$	Programmed receptions of raw material g in n
$\mathrm{Hn}{a}_{\mathrm{kn}}$	Programmed receptions of raw material k in n for finished goods demand forecasts
$\mathrm{Hn}{b}_{\mathrm{okn}}$	Programmed receptions for firm orders o of finished goods of material k in n
$\mathrm{Hm}{a}_{\mathrm{km}}$	Programmed receptions of material k in m for finished goods demand forecasts
$\mathrm{Hm}{b}_{\mathrm{okm}}$	Programmed receptions for firm orders o of finished goods of material k in m
$\mathrm{Hm}{c}_{\mathrm{im}}$	Programmed receptions of the product i in m for finished goods demand forecasts
$\mathrm{Hm}{d}_{\mathrm{oim}}$	Programmed receptions for firm orders o of finished goods i in m
$\mathrm{Hw}{a}_{\mathrm{iw}}$	Programmed receptions of the product i in w for finished goods demand forecasts
$\mathrm{Hw}{b}_{\mathrm{oiw}}$	Programmed receptions for firm orders o of finished goods i in w
$\mathrm{Hj}{a}_{\mathrm{ij}}$	Programmed receptions of the product i in j for finished goods demand forecasts
$\mathrm{Hj}{b}_{\mathrm{oij}}$	Programmed receptions for firm orders o of finished goods i in j
$I{L}_l$	Inventory storage capacity of g in l
$\mathrm{IL}{A}_l$	Inventory storage capacity of k in l
$I{M}_m$	Inventory storage capacity of k in m
$\mathrm{IM}{A}_m$	Inventory storage capacity of i in m
$I{N}_n$	Inventory storage capacity of g in n
$\mathrm{IN}{A}_n$	Inventory storage capacity of k in n
$I{W}_w$	Inventory storage capacity of i in w
$I{X}_x$	Inventory storage capacity of g in x
$J{C}_j$	Inventory storage capacity of i in j
$L{C}_l$	Capacity hours for the regular-time production of first-level supplier l
$L{D}_n$	Capacity hours for the regular-time production of first-level supplier n
$L{X}_x$	Capacity hours for the regular-time production of second-level supplier x
$\mathrm{Lo}{l}_{\mathrm{klm}}$	Maximum transport capacity of k from l to m
$\mathrm{Lo}{n}_{\mathrm{knm}}$	Maximum transport capacity of k from n to m
$\mathrm{Lot}{g}_g$	Production lot size of raw material g
$\mathrm{Lot}{i}_i$	Production lot size of product i
$\mathrm{Lot}{k}_k$	Production lot size of raw material k
$\mathrm{LT}{j}_{\mathrm{iwj}}$	Lead time to deliver one unit of i from w to j
$\mathrm{LT}{l}_{\mathrm{klm}}$	Lead time to deliver one unit of k from l to m
$\mathrm{LT}{n}_{\mathrm{knm}}$	Lead time to deliver one unit of k from n to m
$\mathrm{LT}{x}_{\mathrm{gxl}}$	Lead time to deliver one unit of g from x to l
$\mathrm{LT}{y}_{\mathrm{gxn}}$	Lead time to deliver one unit of g from x to n
$N{C}_n$	Capacity hours for the regular-time production of supplier n
$P{C}_m$	Capacity hours for the regular-time production of plant m
$S{L}_{\mathrm{klm}}$	Available suppliers l that can supply materials k to m
$S{N}_{\mathrm{knm}}$	Available suppliers n that can supply materials k to m
$\mathrm{TS}{G}_{\mathrm{gx}}$	Required processing time (hours) to produce one unit of g in x
$\mathrm{TS}{L}_{\mathrm{kl}}$	Required processing time (hours) to produce one unit of k in l
$\mathrm{TS}{N}_{\mathrm{kn}}$	Required processing time (hours) to produce one unit of k in n
$T{S}_{\mathrm{im}}$	Required processing time (hours) to produce one unit of in m
$V{p}_{\mathrm{ij}}$	Price of product i agreed with retailer j
Decision variables
$b{a}_{\mathrm{oklmt}}$	1 if supplier l qualifies to ship at time k in t, 0 otherwise
$b{b}_{\mathrm{oknmt}}$	1 if supplier n qualifies to ship at time k in t, 0 otherwise
$g_{\mathrm{oimt}}$	Number of units to satisfy firm orders o of i produced at a regular time in m and t
$g{e}_{\mathrm{oimt}}$	Number of units to satisfy firm orders o of i produced overtime in m and t
$i{j}_{\mathrm{ijt}}$	Inventory quantity available of i in j to satisfy forecast demand and inventory to be used for the firm orders during t
$\mathrm{ij}{a}_{\mathrm{ijt}}$	Inventory quantity used to satisfy the forecast demand of i in j during t
$\mathrm{ij}{b}_{\mathrm{oijt}}$	Inventory quantity available to satisfy firm orders o of i in j during t
$\mathrm{ij}{c}_{\mathrm{oijt}}$	Inventory quantity used to satisfy firm orders o of i in j during t
$i{l}_{\mathrm{glt}}$	Inventory quantity of available g in l to satisfy the demand for the first-level raw materials during t
$\mathrm{il}{a}_{\mathrm{oklt}}$	Inventory quantity used to satisfy firm orders o of k in l during t
$\mathrm{il}{b}_{\mathrm{klt}}$	Inventory quantity available of k in l to satisfy the forecast demand and inventory to be used for the firm orders during t
$\mathrm{il}{c}_{\mathrm{oklt}}$	Inventory quantity available to satisfy firm orders o of k in l during t
$\mathrm{il}{d}_{\mathrm{klt}}$	Inventory quantity used to satisfy the forecast demand of k in l during t
$i{n}_{\mathrm{gnt}}$	Inventory quantity of available g in n to satisfy the demand for the first-level raw materials during t
$\mathrm{in}{a}_{\mathrm{oknt}}$	Inventory quantity used to satisfy firm orders o of k in n during t
$\mathrm{in}{b}_{\mathrm{knt}}$	Inventory quantity available of k in n to satisfy the forecast demand and inventory to be used for the firm orders during t
$\mathrm{in}{c}_{\mathrm{oknt}}$	Inventory quantity available to satisfy firm orders o of k in n during t
$\mathrm{in}{d}_{\mathrm{knt}}$	Inventory quantity used to satisfy the forecast demand of k in n during t
$i{m}_{\mathrm{kmt}}$	Inventory quantity available of k in m to satisfy the forecast demand and inventory to be used for the firm orders during t
$\mathrm{im}{a}_{\mathrm{kmt}}$	Inventory quantity used to satisfy the forecast demand of k in m during t
$\mathrm{im}{b}_{\mathrm{okmt}}$	Inventory quantity available to satisfy firm orders o of k in m during t
$\mathrm{im}{c}_{\mathrm{okmt}}$	Inventory quantity used to satisfy firm orders o of k in m during t
$i{p}_{\mathrm{imt}}$	Inventory quantity available of i in m to satisfy the forecast demand and inventory to be used for the firm orders during t
$\mathrm{ip}{a}_{\mathrm{imt}}$	Inventory quantity used to satisfy the forecast demand of i in m during t
$\mathrm{ip}{b}_{\mathrm{oimt}}$	Inventory quantity available to satisfy firm orders o of i in m during t
$\mathrm{ip}{c}_{\mathrm{oimt}}$	Inventory quantity used to satisfy firm orders o of i in m during t
$i{w}_{\mathrm{iwt}}$	Inventory quantity available of i in w to satisfy the forecast demand and inventory to be used for the firm orders during t
$\mathrm{iw}{a}_{\mathrm{iwt}}$	Inventory quantity used to satisfy the forecast demand of i in w during t
$\mathrm{iw}{b}_{\mathrm{oiwt}}$	Inventory quantity available to satisfy firm orders o of i in w during t
$\mathrm{iw}{c}_{\mathrm{oiwt}}$	Inventory quantity used to satisfy firm orders o of i in w during t
$i{x}_{\mathrm{gxt}}$	Inventory quantity of g in x to satisfy the demand for the second-level raw materials in t
$j{a}_{\mathrm{gxt}}$	Auxiliary variable to generate a soft production constraint for raw material g in x during t
$j{b}_{\mathrm{oklmt}}$	Auxiliary variable to generate a soft production constraint to satisfy firm orders o of k in l in m during t
$j{c}_{\mathrm{oknmt}}$	Auxiliary variable to generate a soft production constraint to satisfy firm orders o of k in n in m during t
$j{d}_{\mathrm{oimt}}$	Auxiliary variable to generate a soft production constraint to satisfy firm orders o of i in l in m during t
$l{g}_{\mathrm{gxt}}$	Number of lots to be produced of raw material g in x during t
$l{i}_{\mathrm{imt}}$	Number of lots to be produced of product i in m to satisfy the forecasted demand for the finished goods during t
$\mathrm{li}{a}_{\mathrm{oimt}}$	Number of lots to be produced of product i in m to satisfy firm orders o of the finished goods during t
$\mathrm{lk}{l}_{\mathrm{klmt}}$	Number of lots to be produced of raw material k in l to satisfy the forecasted demand for the finished goods in m during t
$\mathrm{lkl}{a}_{\mathrm{oklmt}}$	Number of lots to be produced of raw material k in l to satisfy firm orders o of the finished goods in m during t
$\mathrm{lk}{n}_{\mathrm{knmt}}$	Number of lots to be produced of raw material k in n to satisfy the forecasted demand for the finished goods in m during t
$\mathrm{lkn}{a}_{\mathrm{oknmt}}$	Number of lots to be produced of raw material k in n to satisfy firm orders o of the finished goods in m during t
$p{l}_{\mathrm{klmt}}$	Number of units k produced in l to satisfy the forecasted demand for the finished goods in m during t
$\mathrm{pl}{a}_{\mathrm{oklmt}}$	Number of k units produced in l to satisfy firm orders o of the finished goods in m during t
$p{n}_{\mathrm{knmt}}$	Number of k units produced in n to satisfy firm orders o of the finished goods in m during t
$\mathrm{pn}{a}_{\mathrm{oknmt}}$	Number of k units produced in n to satisfy firm orders o of the finished goods in m during t
$p_{\mathrm{imt}}$	Number of i units produced in m in a regular time to satisfy the forecast demand during t
$p{e}_{\mathrm{imt}}$	Number of i units produced in m in overtime to satisfy the forecast demand during t
$p{x}_{\mathrm{gxt}}$	Number of g units produced in x to satisfy the demand for the second-level raw materials during t
$q_{\mathrm{klmt}}$	Number of units of k supplied from l to m to satisfy the forecasted demand for the finished goods during t
$q{a}_{\mathrm{knmt}}$	Number of units of k supplied from n to m to satisfy the forecasted demand for the finished goods during t
$\mathrm{qb}{L}_{\mathrm{oklmt}}$	Number of units of k supplied to satisfy firm orders o from l to m during t
$\mathrm{qb}{N}_{\mathrm{oknmt}}$	Number of units of k supplied to satisfy firm orders o from n to m during t
$q{x}_{\mathrm{gxlt}}$	Number of units of g supplied from x to l to satisfy the demand for the first-level raw materials during t
$q{y}_{\mathrm{gxnt}}$	Number of units of g supplied from x to n to satisfy the demand for the first-level raw materials during t
$r{j}_{\mathrm{ijt}}$	Number of units of i in j required to satisfy the forecast demand during t
$\mathrm{rj}{a}_{\mathrm{oijt}}$	Number of units of i in j required to satisfy firm orders o during t
$r{l}_{\mathrm{glt}}$	Number of units of g in l required to satisfy the aggregate demand for the first-level raw materials during t
$\mathrm{rl}{a}_{\mathrm{klt}}$	Number of units of k in l required to satisfy the demand for the first-level raw materials during t
$\mathrm{rl}{b}_{\mathrm{oklt}}$	Number of units of k in l required to satisfy firm orders o during t
$r{m}_{\mathrm{kmt}}$	Number of units of k in m required to satisfy forecast demand t
$\mathrm{rm}{a}_{\mathrm{okmt}}$	Number of units of k in m required to satisfy firm orders o during t
$r{n}_{\mathrm{gnt}}$	Number of units of g in n required to satisfy the aggregate demand for the first-level raw materials during t
$\mathrm{rn}{a}_{\mathrm{knt}}$	Number of units of k in n required to satisfy forecast demand t
$\mathrm{rn}{b}_{\mathrm{oknt}}$	Number of units of k in n required to satisfy firm orders o during t
$r{p}_{\mathrm{imt}}$	Number of units of i in m required to satisfy forecast demand t
$\mathrm{rp}{a}_{\mathrm{oimt}}$	Number of units of i in m required to satisfy firm orders o during t
$r{w}_{\mathrm{iwt}}$	Number of units of i in w required to satisfy forecast demand t
$\mathrm{rw}{b}_{\mathrm{oiwt}}$	Number of units of i in w required to satisfy firm orders o during t
$r{y}_{\mathrm{gxt}}$	Number of units of g in x required to satisfy the demand for the second-level raw materials during t
$s{j}_{\mathrm{iwjt}}$	Number of units of i shipped from w to j to satisfy the forecast demand during t
$\mathrm{sj}{a}_{\mathrm{oiwjt}}$	Number of units of i shipped from w to j to satisfy firm orders o during t
$s{m}_{\mathrm{imwt}}$	Number of units of i shipped from m to w to satisfy the forecast demand during t
$\mathrm{sm}{a}_{\mathrm{oimwt}}$	Number of units of i shipped from m to w to satisfy firm orders o during t
$\mathrm{ss}{o}_{\mathrm{iw}}$	Quantity of the safety inventory of i in w to satisfy firm orders
$y_{\mathrm{oimt}}$	1 if the firm order o of i is produced in m during t, 0 otherwise

Table 3. Design of the experiments for each SC level.

Scenario code	Production strategy (MRP/JIT)	Backorders	Suppliers level 2 (x)	Local suppliers level 1 (l)	Alternative suppliers level 1 (n)	Plants (m)	Warehouses (w)	Retailers (j)
e1	MRP	+	X	X	X	X	SS	X
e2	MIXED	+	X	X	X	O	SS	X
e3	MIXED	+	O	X	X	X	SS	X
e4	MIXED	+	X	O	O	O	SS	X
e5	MIXED	+	X	O	O	X	SS	X
e6	JIT	+	O	O	O	O		X
e7	MIXED	+	X	X	X	O		X
e8	MIXED	+	O	X	X	X		X
e9	MIXED	+	X	O	O	X		X
e10	MIXED	+	O	O	O	X		X
e11	JIT	+	O	O	O	O	SS	X
e12	MRP ϟ	+	X	X	X	X	SS	X
e13	BEST MIXED ϟ	+	X	X	X	O		X
e14	MRP *	+	X	X	X	X	SS	X
e15	JIT *	+	O	O	O	O		X
e16	MRP	†	X	X	X	X	SS	X
e17	JIT	†	O	O	O	O		X
e18	BEST MIXED	†	X	X	X	O	SS	X
e19	MRP	++	X	X	X	X	SS	X
e20	JIT	++	O	O	O	O		X
e21	BEST MIXED	++	X	X	X	O	SS	X

Note: O: pull system with JIT production; X: push system with MRP tools; SS: safety inventory; +: null backorders during the last time period for demand forecasts; †: null backorders during the last time period for firm orders; ++: null backorders during the last time period for forecasts and firm orders; ϟ: all the demand consolidated for firm orders; *without lot-sizing production; BLANK: with no safety inventory.

Table 4. Objective function values and runtime per instance.

				Total costs (TC) ($)						Total incomes (TI) ($)
Size problems	Scenario code	Gap (%)	Runtime (seconds)	Production (AC)	Penalisation of JIT production coverage (CC)	Inventory (IC)	Transport (MC)	Backorder (BC)	CO2 emissions (LC)	Sales	Fixed charge for production of firm orders	Total profit ($)
Small	e1	0.02	696	93740	0	2061	16294	0	1980	160.495	92160	46420
	e2	0.02	1140	93744	0	2054	16294	0	1979	160.495	92160	46424
	e3	0.61	1265	93856	0	9495	16209	548	1979	160.495	92160	38408
	e4	3.57	32400	94285	620	6411	16361	10	2076	160.495	92160	40732
	e5	3.70	32400	94266	540	6239	16465	182	2067	160.447	92160	40688
	e6	4.46	32400	94208	685	13852	16154	184	2120	160.495	92160	33292
	e7	0.00	723	93617	0	1956	16279	0	1977	160.495	92160	46666
	e8	4.54	5160	94208	685	13852	16154	184	2120	160.495	92160	33292
	e9	5.02	32400	94468	380	6932	16445	119	2528	160.495	92160	39623
	e10	4.77	32400	94346	300	14029	16258	64	2182	160.495	92160	33316
	e11	4.10	32400	94307	765	13761	16221	0	2087	160.495	92160	33354
	e12	0.34	472	119211	0	3708	21051	936	4922	201.760	92160	51932
	e13	0.25	2275	118816	0	3444	21006	961	4804	201.760	92160	52729
	e14	0.00	725	93618	0	1952	16279	0	1977	160.495	92160	46669
	e15	3.7	32400	94227	690	12169	16506	279	2139	160.495	92160	34485
	e16	0.10	871	93739	0	2061	16294	101	1980	160.495	92160	46320
	e17	4.01	21780	94283	155	13922	16305	17	2070	160.495	92160	33743
	e18	0.03	935	93712	0	2062	16289	0	1979	160.452	92160	46410
	e19	0.06	234	93735	0	2062	16293	55	1979	160.495	92160	46371
	e20	4.37	21840	94311	375	14013	16243	36	2146	160.495	92160	33371
	e21	0.02	167	93738	0	2061	16293	0	1979	160.495	92160	46424
Medium	e1	0.19	671	302.676	0	7597	49818	581	8.679	481.430	92160	112079
	e2	0.21	244	302.489	0	7.644	49.788	793	8.706	481.430	92160	112010
	e3	0.01	1748	302538	0	15017	49688	340	8678	481.430	92160	105169
	e4	3.46	32400	303712	385	11990	50039	844	9041	481.430	92160	105419
	e5	3.20	32400	303853	305	11744	50000	782	8947	481.430	92160	105799
	e6	3.77	32400	303496	191	19583	49865	751	9091	481.430	92160	98453
	e7	0.00	151	302403	0	7165	49767	410	8636	481.430	92160	113049
	e8	0.00	90	302481	0	14474	49661	370	8667	481.430	92160	105777
	e9	4.94	32400	305733	310	12126	50404	442	9252	481.430	92160	103163
	e10	4.39	32400	304746	610	18609	49795	806	9552	481.430	92160	97312
	e11	3.57	32400	303899	390	19226	50044	822	8773	481.430	92160	98276
	e12	0.47	4740	311330	0	8750	51108	1522	8798	493.312	92160	111804
	e13	0.16	1250	310677	0	7449	50950	1582	8610	493.312	92160	114044
	e14	0.02	568	302556	0	7633	49798	360	8656	481.430	92160	112427
	e15	5.28	32400	304654	610	19872	50064	681	9881	481.430	92160	95668
	e16	0.10	195	302561	0	7618	49799	335	8681	481.430	92160	112436
	e17	4.17	32400	304647	80	18796	50086	494	9474	481.430	92160	97853
	e18	0.03	846	302484	0	7640	49788	373	8704	481.430	92160	112441
	e19	0.07	1088	302484	0	7640	49788	373	8704	481.430	92160	112321
	e20	4.51	19688	302563	0	7609	49799	460	8678	481.430	92160	96976
	e21	0.00	144	305261	305	19551	49983	130	9224	481.430	92160	112440
Large	e1	0.1	1292	980904	0	20916	150777	23270	30758	1.442.490	92160	235865
	e2	0.11	794	980963	0	20925	150784	23416	30766	1.442.498	92160	235644
	e3	0.01	317	981484	0	28313	150779	23133	30761	1.442.498	92160	228028
	e4	9.20	32400	987498	590	34500	151525	26172	35457	1.443.698	92160	207956
	e5	11.42	32400	991616	666	37917	151612	26042	34357	1.444.290	92160	202080
	e6	5.36	32400	984795	525	37750	151174	27715	32980	1.444.290	92160	209351
	e7	0.01	72	978634	0	20148	150548	23127	30814	1.442.498	92160	239227
	e8	0.00	65	979175	0	27568	150546	23086	30813	1.442.498	92160	231310
	e9	5.55	32400	984394	305	28623	150998	23213	33956	1.442.498	92160	221009
	e10	11.35	32400	996732	460	35572	151706	26340	35402	1.444.290	92160	198078
	e11	6.49	32400	988841	560	36522	151397	24384	34188	1.444.130	92160	208238
	e12	0.04	480	980904	0	20916	150777	23270	30758	1.442.490	92160	235865
	e13	0.01	62	978634	0	20148	150548	23127	30814	1.442.498	92160	239227
	e14	0.00	1826	980913	0	20932	150780	23580	30762	1.442.498	92160	235531
	e15	7.88	32400	984673	454	42738	151699	23744	33794	1.444.290	92160	207188
	e16	0.19	997	999563	0	22540	153067	15123	30358	1.460.616	92160	239965
	e17	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A
	e18	0.13	1560	999530	0	22550	153065	14939	30360	1.460.648	92160	240204
	e19	0.11	1183	999580	0	22602	153073	14855	30366	1.460.734	92160	240258
	e20	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A
	e21	0.08	1516	999493	0	22619	153059	14787	30347	1.460.648	92160	240343

Note: Total profit = Sales – Total costs (TC); N/A: non-feasible solution.

Figure 3. Relation between inventory cost and profits for the medium-sized problem.

A relation between inventory costs and total model profits is presented for each scenario of the performed experiments for the medium-sized problem.

Table 5. Computational efficiency.

Size problems	Scenario code	Iterations	Constraints	Variables	Integers	Non-zeros	Density (%)
Small	e1	88317	238670	354192	71760	950706	11.00
	e6	5408987	276494	354192	71760	1108722	11.00
	e7	35299	237185	354195	71760	951525	11.00
Medium	e1	131338	237185	354195	71760	951525	11.00
	e6	6120129	276494	354192	71760	1108722	11.00
	e7	36270	238670	354192	71760	950706	11.00
Large	e1	98724	237185	354195	71760	951525	11.00
	e6	8981257	276494	354192	71760	1108722	11.00
	e7	38050	238670	354192	71760	950706	11.00

Table 6. Bullwhip effect measures.

Scenario code	Suppliers level 2 (X)	Suppliers level 1 (L + N)	Plants (M)	Warehouses (W)	Retailers (J)
e1 (MRP)	1.00	1.64	1.65	1.54	1.42
e6 (JIT)	1.00	1.58	1.64	1.63	1.33
e7 (BEST MIXED)	1.00	1.66	1.66	1.64	1.35
e16 (MRP^†)	1.00	1.63	1.65	1.54	1.42

Note: ^†values of scenario e1 (MRP) when considering null backorders during the last time period for firm orders.

Figure 4. Supply chain net inventory amplification in scenario e7.

Representation of the amplification of net inventory propagating through three SC levels from retailer to manufacturer in scenario e7.

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Data availability statement

The authors confirm that the data supporting the findings of this study are available in the article and in Appendix.