A stochastic optimisation approach to maintain supply chain viability under the ripple effect

Figures & data

Table 1. Notation.

Indices
i	=	supply chain node (e.g. supplier, assembly plant, retailer), $i\in I$
k	=	geographic region, $k\in K$
s	=	disruption scenario, $s\in S$
t	=	planning period, $t\in T$
Input parameters
$a_i$	=	fixed cost of reserve capacity for primary supplier, $i\le M$, or OEM plant, i = m, per unit of reserve free capacity
$b_i$	=	variable cost of reserve capacity for primary supplier, $i\le M$, or OEM plant, i = m, per unit of deployed capacity
${\alpha }_i$	=	fixed cost per part of pre-positioning RMI of part type $i\le M$ at OEM plant
${\text{β}}_i$	=	variable cost per part of usage RMI of part type $i\le M$ pre-positioned at OEM plant
$c_i$	=	nominal per period capacity of supplier, $i\le 2M$, or OEM plant i = m
${\tilde{d}}_t$	=	original market demand in period t
$d_t^s$	=	market demand in period t, under disruption scenario s
$e_i$	=	per unit price of parts purchased from supplier $i\in I:i\le 2M$
$f_i$	=	fixed ordering cost for backup supplier, $i\in I:M+1\le i\le 2M$
γ	=	per unit penalty cost of unfulfilled demand for products
$p_k$	=	probability of regional disruption for region k
${\delta }_{it}^s$	=	zero-one availability coefficient for supplier, $i\le 2M$, or OEM plant i = m, in period, $t\in T$, under disruption scenario, $s\in S$
${\sigma }_i$	=	manufacturing and transportation time from supplier i to OEM plant
${\tau }_i^s$	=	delivery lead time from supplier i to OEM plant under disruption scenario s
${\theta }_k$	=	duration of regional disruption in region, $k\in K$
${\vartheta }_k$	=	delay time for the ripple effect in region, $k\in K:k>1$

Figure 1. Supply chain.

Table 2. Problem variables.

$q_i$	$\ge$	0, reserve capacity production rate at primary supplier, $i\le M$ or OEM plant, i = m, (capacity reservation)
$r_i^s$	$\ge$	0, reserve capacity production rate deployed at primary supplier, $i\le M$ or OEM plant, i = m, under disruption scenario s (reserve capacity deployment)
$u_i^s$	$\in$	$\{0,1\}$, where the value 1 means that backup supplier, $i\in I:M+1\le i\le 2M$, is selected as a recovery supplier under disruption scenario s; otherwise $u_i^s=0$ (recovery supplier selection)
$v_i^s$	$\in$	$[0,1]$, fraction of total demand for parts, ordered from recovery supplier, $M+1\le i\le 2M$, under disruption scenario s (recovery supply portfolio)
$x_{it}^s$	$\ge$	0, production of facility, $i\in I$, in period $t\in T$, under disruption scenario s (production of parts/products)
$y_i$	$\ge$	0, RMI of part type $i\in I:i\le M$, pre-positioned at OEM plant (pre-positioning of RMI)
$z_i^s$	$\ge$	0, RMI of part type $i\in I:i\le M$, pre-positioned at OEM plant and used under disruption scenario s (usage of RMI)
Auxiliary variable
$w_t^s$	$\in$	$\mathrm{\Re }$, inventory (if $w_t^s\ge 0$) or shortage (if $w_t^s<0$) of products at OEM plant at the beginning of period $t\in T\bigcup \{h+1\}$ under disruption scenario s (finished product inventory)

Figure 2. Maintaining of supply chain viability.

Figure 3. Boundary production trajectories.

Table 3. Input parameters.

$m=11,I=\{1,\dots ,m,m+1\}$, 12 supply chain nodes (10 suppliers, one manufacturer and one retailer of finished products)
M = 5 primary suppliers, $i=1,\dots ,M$; M = 5 backup suppliers, $i=M+1,\dots ,2M$; assembly plant, i = m, retailer, i = m + 1
i = 1, 6 non-semiconductor suppliers, i = 2, 7 semiconductor foundry, i = 3, 8 semiconductor memory suppliers,
i = 4, 9 display suppliers, i = 5, 10 battery suppliers, i = 11 OEM assembly plant, i = 12 domestic retailer
$n=6,K=\{1,\dots ,n\}$, $2^n=64$	six geographic regions, 64 disruption scenarios
k = 1 China, k = 2 Taiwan, k = 3 Japan, k = 4 Korea, k = 5 EU, k = 6 USA
$I^1=\{1,11\},I^2=\{2,3\},I^3=\{4,5\},I^4=\{6,7\},I^5=\{8,9,10\},I^6=\left\{12\right\}$
$T=\{1,\dots ,12\}$, h = 12	12 planning periods (months)
${\tilde{d}}_t=1000$ $\mathrm{\forall }t\in T$	original demand per period for parts/products
$\tilde{D}=12000$	total original demand for parts/products
Regional disruption probabilities:
$p_1\in [0.65,0.55],p_2\in [0.55,0.45],p_3\in [0.55,0.45],p_4\in [0.45,0.35],p_5\in [0.30,0.20],p_6\in [0.30,0.20]$
$g_k=(h-{\theta }_k)/h;k\in K:k\le 2M$	yield rate of suppliers in region k
$c_i=\tilde{D},i=1,\dots ,M,c_i=0.75\tilde{D},i=M+1,\dots ,2M,c_m=1.2\tilde{D}$	nominal per period capacity of suppliers and OEM plant
${\phi }_i=0.25$	coefficient of reservation capacity
Cost parameters:
$a_i=0.5\gamma ,i\le M,i=m$, fixed cost of reserve capacity for facility i, per unit of reserving free capacity
$b_i=0.1\gamma ,i\le M,i=m$, variable cost of reserve capacity for facility i, per unit of deployed reserve capacity)
$e_i=\lceil \left(Uniform\right(6,10)\rceil ,i\le M$, $e_i=\lceil \left(Uniform\right(16,20)\rceil ,M+1\le i\le 2M$, price per part for primary and backup suppliers
$f_i=500\lceil \left(Uniform\right(16,20)\rceil$, fixed ordering cost for backup supplier, $M+1\le i\le 2M$
${\alpha }_i=1.5e_i,i\le M$, fixed cost per part of pre-positioning RMI of part type i at OEM plant
${\text{β}}_i=\lceil e_i/4\rceil ,i\le M$, variable cost per part of usage RMI of part type i at OEM plant
$\gamma =60$, per unit penalty cost of unfulfilled demand for products

Table 4. Computing boundary trajectories.

Minimisation of Expected Cost (6(6) $\begin{array}{rl}{E}^c& =\sum _{s\in S}\sum _{i\in I:M+1\le i\le 2M}{P}_s{f}_i{u}_i^s/\tilde{D}\\ & +\sum _{s\in S}{P}_s(\sum _{i\in I:i\le M}{e}_i{g}_i^s+\sum _{i\in I:M+1\le i\le 2M}{e}_i{v}_i^s)\\ & +\sum _{i=1,\dots ,M,2M+1}(a_i{q}_i+\sum _{s\in S}{P}_s{b}_i{r}_i^s)/\tilde{D}\\ & +\sum _{i\in I:i\le M}({\alpha }_i{y}_i+\sum _{s\in S}{P}_s{\text{β}}_i{z}_i^s)/\tilde{D}\\ & +\gamma \sum _{s\in S}{P}_s(D_h^s-\sum _{t\in T}{x}_{mt}^s)/\tilde{D}.\end{array}$(6) ), model (33(33) $\begin{array}{rl}& min\left\{\left(6\right)\text{s.t. (12)--(21), (26)--(32)}\right\}\end{array}$(33) ): 6722 Var., 69 Bin., 8306 Cons. ${}^{\left(\mathrm{a}\right)}$
Minimum Expected Cost (${\underset{\_}{E}}^c$)	73.68
Minimum Expected Service Level (${\underset{\_}{E}}^{sl}$)	66.70%
Expected Primary Supply Portfolio (% of total demand for part type, i) ${}^{\left(\mathrm{b}\right)}$	1(70%), 2(90%), 3(90%), 4(90%), 5(90%)
Expected Recovery Supply Portfolio (% of total demand for part type, i−5) ${}^{\left(\mathrm{c}\right)}$	6(30%), 7(10%), 8(10%), 9(10%), 10(10%)
Expected Boundary Production Schedule (Period: Production Volume) ${}^{\left(\mathrm{d}\right)}$
t = 1:0, t = 2:0, t = 3:0, t = 4:500, t = 5:500, t = 6:500, t = 7:851, t = 8:965, t = 9:1202, t = 10:1088, t = 11:1219, t = 12:1179
Maximisation of Expected Service Level (7(7) $\begin{array}{rl}{E}^{sl}& =\frac{\sum _{t\in T}\sum _{s\in S}{P}_s{x}_{mt}^s}{\sum _{s\in S}{P}_s{D}_h^s}.\end{array}$(7) ), model (34(34) $\begin{array}{rl}& max\left\{\left(7\right)\text{s.t. (12)--(21), (26)--(32)}\right\}\end{array}$(34) ): 6722 Var., 69 Bin., 8306 Cons. ${}^{\left(\mathrm{a}\right)}$
Maximum Expected Service Level (${\overline{E}}^{sl}$)	93.47%
Maximum Expected Cost (${\overline{E}}^c$)	87.11
Expected Primary Supply Portfolio (% of total demand for part type, i) ${}^{\left(\mathrm{b}\right)}$	1(70%), 2(90%), 3(90%), 4(90%), 5(90%)
Expected Recovery Supply Portfolio (% of total demand for part type, i−5) ${}^{\left(\mathrm{c}\right)}$	6(30%), 7(10%), 8(10%), 9(10%), 10(10%)
Expected Boundary Production Schedule (Period: Production Volume) ${}^{\left(\mathrm{d}\right)}$
t = 1:240, t = 2:600, t = 3:600, t = 4:0, t = 5:600, t = 6:600, t = 7:1395, t = 8:1371, t = 9:1465, t = 10:1387, t = 11:1470, t = 12:1489

${}^{\left(\mathrm{a}\right)}$ Var.=total number of variables, Bin.=number of binary variables, Cons.=number of constraints.

${}^{\left(\mathrm{b}\right)}$ $\sum _{s\in S}{P}_s{g}_i^s;i\le M$.

${}^{\left(\mathrm{c}\right)}$ $\sum _{s\in S}{P}_s{v}_{M+i}^s;i\le M$.

${}^{\left(\mathrm{d}\right)}$ $\sum _{s\in S}{P}_s{x}_{mt}^s;t\in T$.

Table 5. Computational results for supply chain viability model Viabil_WD${}^{\left(\mathrm{a}\right)}$.

Weight factor, λ	0	0.25	0.50	0.75	1
Expected Cost ($E^c$)	80.38	80.38	80.38	80.38	80.38
Expected Service Level ($E^{sl}$)	93.47%	86.78%	80.09%	80.09%	80.09%
Expected Primary Supply Portfolio (% of total demand for part type, i) ${}^{\left(\mathrm{b}\right)}$
1(70%), 2(90%), 3(90%), 4(90%), 5(90%)
Expected Recovery Supply Portfolio (% of total demand for part type, i−5) ${}^{\left(\mathrm{c}\right)}$
6(30%), 7(10%), 8(10%), 9(10%), 10(10%)
Expected Viable Production Trajectory ${}^{\left(\mathrm{d}\right)}$
Period	Production	Production	Production	Production	Production
t = 1	240	180	120	98	76
t = 2	600	450	300	188	76
t = 3	600	450	300	188	76
t = 4	0	125	250	413	576
t = 5	600	575	550	563	576
t = 6	600	575	550	563	576
t = 7	1395	1259	1123	1083	1042
t = 8	1371	1270	1168	1163	1157
t = 9	1465	1399	1333	1363	1383
t = 10	1387	1312	1237	1258	1279
t = 11	1470	1407	1345	1378	1411
t = 12	1489	1411	1334	1352	1370

${}^{\left(\mathrm{a}\right)}$ Var.=6722, Bin.=69, Cons.=8306, model has 204 quadratic objective terms

${}^{\left(\mathrm{b}\right)}$ $\sum _{s\in S}{P}_s{g}_i^s;i\le M$.

${}^{\left(\mathrm{c}\right)}$ $\sum _{s\in S}{P}_s{v}_{M+i}^s;i\le M$.

${}^{\left(\mathrm{d}\right)}$ $\sum _{s\in S}{P}_s{x}_{mt}^s;t\in T$.

Figure 4. Boundary trajectories and viable expected production for $\lambda =0.5$.

Figure 5. Boundary and viable RMI of part types for $\lambda =0.5$.

Figure 6. Boundary and viable reserve capacity for $\lambda =0.5$.

Figure 7. Boundary trajectories and viable expected production schedules for $\lambda =0,0.25,0.75,1$.

Figure 8. Expected shortage of products for viable solutions of Viabil_WD, for $\lambda =0,0.5,1$.

Figure 9. Pre-positioned RMI and reserve capacity for resilient solutions of Resil_WS, for $\lambda =0,0.5,1$.

Figure 10. Expected production and shortage for resilient solutions of Resil_WS, for $\lambda =0,0.5,1$.

Table 6. Solution results for supply chain resilience model Resil_WS

Weight factor, λ	0	0.25	0.50	0.75	1
Expected Cost ($E^c$)	80.33	79.23	77.50	75.74	75.04
Expected Service Level ($E^{sl}$)	93.47%	93.47%	90.93%	85.79%	80.09%
Expected Primary Supply Portfolio (% of total demand for part type, i) ${}^{\left(\mathrm{b}\right)}$
1(70%), 2(90%), 3(90%), 4(90%), 5(90%)
Expected Recovery Supply Portfolio (% of total demand for part type, i−5) ${}^{\left(\mathrm{c}\right)}$
6(30%), 7(10%), 8(10%), 9(10%), 10(10%)
Expected Resilient Production Trajectory ${}^{\left(\mathrm{d}\right)}$
Period	Production	Production	Production	Production	Production
t = 1	0	520	440	0	0
t = 2	240	520	500	500	209
t = 3	600	520	500	500	500
t = 4	600	520	500	500	500
t = 5	600	520	500	500	500
t = 6	600	520	500	500	500
t = 7	1500	1349	1319	1291	1185
t = 8	1430	1331	1310	1261	1159
t = 9	1320	1302	1278	1262	1265
t = 10	1410	1371	1354	1325	1261
t = 11	1500	1377	1356	1328	1252
t = 12	1417	1377	1356	1328	1281

${}^{\left(\mathrm{a}\right)}$ Var.=6722, Bin.=69, Cons.=8306.

${}^{\left(\mathrm{b}\right)}$ $\sum _{s\in S}{P}_s{g}_i^s;i\le M$.

${}^{\left(\mathrm{c}\right)}$ $\sum _{s\in S}{P}_s{v}_{M+i}^s;i\le M$.

${}^{\left(\mathrm{d}\right)}$ $\sum _{s\in S}{P}_s{x}_{mt}^s;t\in T$.

Data availability statement

The data that support the findings of this study are available from the corresponding author, upon reasonable request.