18th European Symposium on Computer Aided Process Engineering – ESCAPE 18
Bertrand Braunschweig and Xavier Joulia (Editors)
© 2008 Elsevier B.V./Ltd. All rights reserved.
Towards Resilient Supply Chains: Uncertainty
Analysis Using Fuzzy Mathematical Programming
Kishalay Mitra,a Ravindra D. Gudi,b Sachin C. Patwardhan,b Gautam Sardara
a
Engineering and Industrial Services, Tata Consultancy Services Limited, 1, Mangaldas
Road, Pune 411001, Maharashtra, INDIA
b
Indian Institute of Technology, Bombay, Powai, Mumbai 400076,Maharashtra, INDIA
Abstract
A multi-site, multi-product supply chain planning problem under demand as well as
machine uptime uncertainty has been analyzed in multi-objective Pareto sense in this
paper using the fuzzy mathematical programming approach. Fuzzy programming not
only keeps the problem size in control which is the prime lacuna of the conventional
two stage scenario based stochastic programming approach but also free from any
assumption regarding the nature of the distribution of the uncertain parameters. It is
seen that the fuzzy approach is generic, relatively simple to use, and can be adapted for
bigger size planning problems of industrial scale. We demonstrate the proposed
approach on a relatively moderate size planning problem taken from the work of
McDonald and Karimi (1997) and discuss various aspects of uncertainty in context of
this problem.
Keywords: Supply Chain Optimization, Uncertainty,
Programming, Multi-objective Optimization, Pareto.
Fuzzy
Mathematical
1. Introduction
In the recent era of global economy, competitive pressures force big enterprises to
maintain their supply chain spread across the globe where the enterprise having more
responsive supply chain network, not the mere size of the network, performs better in
meeting customer satisfaction. Planning and thereby leveraging the best out of a supply
chain, therefore, is the primary focus of most of the enterprises. Effective coordination
and integration of the key business activities undertaken by an enterprise, starting from
the procurement of raw materials to the distribution of the final products to the
customers is the key in a supply chain network (Shapiro, 2001). Most of the supply
chain models assume parameters (cost components etc.) as well as various predictions
(demands etc.) are accurately known and hence the models available in commercial
enterprise software are deterministic in nature. But in real life situations, the enterprises
have to face the volatile market conditions where one has to meet customer satisfaction
under changing market conditions. Under these conditions, it is more realistic to
consider the effect of uncertainties on supply chain planning and thereby minimize their
impact on the final supply chain performance. In this work, we, therefore, present a
multi-objective supply chain planning framework under demand as well as machine
uptime uncertainty using fuzzy mathematical programming which is one of the three
popular methods (two stage stochastic and chance constraint programming being the
other two) of handling uncertainty in systems engineering literature (Birge and
2
K.Mitra et al.
Louveaux, 1994; Sahinidis, 2004). One of the most important merits of fuzzy
mathematical programming over other existing approaches is related to the fact that it
does not require the knowledge of distributions associated with the uncertain
parameters. The second important advantage that fuzzy programming has is related to
keeping the problem size small and tractable in the presence of an increase in the
number of uncertain parameters (unlike two stage stochastic programming) enabling the
fuzzy approach to handle large scale industrial problems without any further
modification in the solution techniques. In view of these merits, in this paper, the
uncertainty issues associated with a multi-site, multi-product supply chain mid term
planning problem, has been analyzed in detail using the fuzzy mathematical
programming approach taking the mid term planning model of McDonald and Karimi
(1997) as the basis of this work.
2. Adaptation of Deterministic Planning Model Under Fuzzy Framework
As opposed to other methods, fuzzy programming considers uncertain parameters as
fuzzy numbers and the constraints associated with those uncertain parameters as fuzzy
sets. A degree of satisfaction of a constraint is defined as the normalized membership
function (0λ1) of the constraint where the uncertain parameter is allowed to vary
within a given range (prefixed by  with the uncertain parameter) and the value of this
membership function signifies the extent of constraint violation (λ = 0 being the
complete violation and λ =1 being no violation, other values of  linearly varying
between 0 and 1). In this way, some amount of constraint violation is allowed. Finally
we write the  optimization problem of Bellman and Zadeh (1970) that attempts to
maximize the satisfactions of the constraints including the objective function.
The final midterm planning model of McDonald and Karimi (1997) when adapted under
fuzzy framework, can be described as follows:

Max
0  1, Pi , j , s , t , RL i , j , s , t ,Ci , s , t ,Si , s , c , t ,
I i , s , t , I i, c , t ,i , s , s  , t , I i, s , t , Yf , j , s , t
Pi, j,s,t  R i, j,s,t RL i, j,s,t
RL f , j,s ,t 
Ci ,s,t 
 RL
i i , f
(1)
i  I \ I RM
(3)
i , j,s , t
  P
i,i ,s
ii ,i ,s  0
(2)
i  I \ I FP
i, j,s , t
(4)
j
C i ,s,t    i ,s,s,t
i  I IP
(5)
s
Ii,s, t  Ii,s, t 1   Pi, j,s, t   i,s,s, t   Si,s,cti  I \ IRM
j
I i,s,t  I iL,s,t  I i,s,t
s
(6)
c
i  I FP
(7)
Towards Resilient Supply Chains: Uncertainty Analysis Using Fuzzy Mathematical
Programming
3
RL f , j,s , t  MRL f , j,s , t Yf , j,s , t  0
(8)
I i,s,t  I iL,s,t
(9)
 RL
i , j,s , t
 H j,s,t  0
(10a)
i



  RL i, j,s, t  H j,s, t  H j,s, t  H j,s, t     1  0

 i

(10b)
 RL f , j,s,t  H j,s,t  0
(11a)



  RL f , j,s, t  H j,s, t  H j,s, t  H j,s, t     1  0

 f

(11b)
f
 




 I i ,c, t  I i ,c, t 1   Si ,s ,c, t   d i ,c, t  d i ,c, t  d i ,c, t     1  0
s




(12b)


  S i ,s ,c , t    d i ,c, t  
t  t
 s , t  t

(13b)
 d
t  t
i ,c , t 

   1 0

Pi , j,s , t  R i , j,s , t H j,s , t
P
i , j, s , t
(14a)

 R i, j,s, t H j,s, t  H j,s, t  R i, j,s, t H j,s, t    1  0
(14b)
RL i , j,s , t  H j,s , t  0
RL
i , j,s , t
(15a)

 H j,s, t  H j,s, t  H j,s , t    1  0
(15b)
RL f , j,s , t  H j,s , t Yf , j,s , t  0
RL
f , j,s , t
(16a)

 H j,s , t  H j,s , t Yf , j,s , t  H j,s , t Yf , j,s , t    1  0


S i ,s,c, t   d i,c, t  
t  t


 

 I i,c, t   d i,c, t  

t

t



 d
t  t
 d
t  t
cos t  cos t  cos t
min
i ,c, t 
i ,c, t 
max

   1  0


   1  0


 cos t min    1  0
(16b)
(17b)
(18b)
(19)
4
K.Mitra et al.
Cost 
Where
 FC
f , j,s , t
Yf , j,s, t    i, j,s Pi, j,s, t   p i,s C i,s, t   h i,s, t I i,s, t 
f , j,s , t
t
i ,s ,c , t
S
s , c i ,s , c, t
i , j,s , t

t
i ,s ,s, t
s ,s
i ,s , t
 i ,s ,s, t    I

i ,s i ,s , t
i ,s , t
i ,s , t
  I

i ,c i,c, t
i ,c, t
Pi , j,s , t , RL i , j,s , t , RL f , j,s , t , C i ,s , t , Si ,s ,c , t , I i ,s , t , I i,c ,t ,  i ,s ,s, t , I i,s ,t ≥ 0, Yf , j,s,t  0,1
(20)
where Pi , j,s , t , RL i , j,s , t , RL f , j,s , t , C i ,s , t , Si ,s ,c , t , I i ,s , t , I i,c ,t ,  i ,s ,s, t , I i,s ,t represents the
production, run length for each product, run length for each product family,
consumption, supply to the market, inventory at production site, missing demand at
market, intermediate product, inventory below safety level for product i or family f of
several products to be produced at facility j at site s or customer c or market m at time
period t respectively. Here Yf , j,s , t represents the binary variable to decide a product
family f to be produced at machine j, site s and time period t or not. Few other important
parameters are demand (di,c,t), machine uptime (Hj,s,t), minimum run length for the
product family f (MRLf,j,,s,t), Safety stock target for product ( I iL,s ,t ), effective rate of
production (Rijst) whereas various unit cost parameters are inventory holding cost (hist),
revenue (ic), raw material price (pis), penalty for dipping below safety level (is), fixed
production cost (FCf,j,s,t) and variable production cost(ijs) and transportation costs (tss,
tsc). Based on whether the product or product family is chosen with or without fixed
associated cost, the final planning model formulation would be a LP / MILP / MINLP.
Observing an inherent trade off, we can generate the multi-objective Pareto optimal
points for the above formulation using overall planning cost, margin / variation allowed
for uncertain parameters and demand satisfaction as different trade-off criteria.
3. Results and Discussion
The motivating example considered here is taken from the first case study of McDonald
and Karimi (1997). There are two production locations (S1 and S2) having one unit in
each and each production unit has a single raw material supplier. Production units S1
and S2 are connected to market M1 and M2 respectively putting demands for 34
products. Unit S1 and S2 manufacture products P1 - P23 and P24 - P34 respectively.
Products at S2 are produced from a set of intermediate products produced at S1 e.g.
product P24 is produced from product P1, product P25 is produced from product P4 and
so on (see McDonald and Karimi, 1997). There are eleven product families F1 – F11
that are composed by clubbing the products at site S1 e.g. products P1, P2, P3 form
product family F1 and so on (see McDonald and Karimi, 1997). Market M1 has
customers who have demands for products P1-P23 and market M2 has customers
having demands for products P24-P34. The demand values for all 34 products are taken
from the original work for a 1 year planning horizon (with a time period of duration 1
month). To see the effect of sudden rise in demands, the demand values for time periods
6 and 12 are changed to 300% of the demand values given in the original work while all
other demand values for the rest of the 10 time periods are considered to be 20% of the
demand values reported in the original problem. These modified demands are
henceforth called the baseline values for fuzzy programming or nominal values for
deterministic problem. First we analyze the effect of demand uncertainty for two fuzzy
formulations: (i) multi-objective midterm planning product formulation without any
minimum run length restriction (henceforth called as Model 1 consisting of equations
Towards Resilient Supply Chains: Uncertainty Analysis Using Fuzzy Mathematical
Programming
5
1,2,4-7,9,10a,11a,12b,13b,14a,15a,17b,18b,19) and (ii) multi-objective midterm
planning product family formulation with minimum run length restriction (hence forth
called as Model 2 consisting of equations 1 - 9,10a,11a,12b,13b,14a,15a,16a,17b,18b,
19). The first problem results in an LP formulation (4789 single equations, 3989 single
variables) whereas the second problem is an MILP problem (5198 single equations,
4214 single variables, 132 binaries). The complete formulation was coded in the
modeling environment of GAMS© and solved using BDMLP (Brooke et al, 1998) as
well as COIN-OR SYMPHONY solver for LP and MILP respectively. Model 2 was
found to converge to 0.01% of the best possible value spanning over 25 different runs.
It is observed that there lies a trade-off between the overall planning cost and margins
provided on uncertain demands. These Pareto Optimal (PO) points for model 1 and
model 2 with different values of demand margins of the basis values are presented in
Figure 1. The PO front of model 2 (MILP) lies marginally above the PO front of model
1 as model 2 (MILP) is a more restrictive case of model 1 (LP) and hence leads to
higher cost.
Total Cost & Constraint Margin Pareto
10000
9000
8000
Cost
7000
6000
5000
4000
3000
2000
0
20
40
60
80
Demand Constraint Margin
Dmd 30% margin model 1
100
Dmd 30% margin model 2
Fig. 1. Total cost vs. demand margin Pareto fronts for model 1 and model 2 respectively
Next we focus on a few points of the model 1 and model 2 PO front (two points
correspond to 30% and 50% margin points). These two cases are compared with the
results of the deterministic planning model run for the nominal demands (henceforth
called as nominal case). As compared to the nominal case, the plan for the uncertainty
cases shows a trend of higher production to handle uncertainty in future (Fig 2). The
start of production only around the sixth time period can be explained as the demand till
that time period being met by the already existing initial inventory at the production
site. More accumulation of inventory for future uncertainty is not visible as there is a
cost associated with it. On a relative basis, the unit cost component of the McDonald
and Karimi (1997) model is defined in such a way that the model gives higher
preference for maintaining inventory at the safety level as long as there is no demand
miss and that happens during the time periods of 6 - 11. At the time period 12, the
model allows its safety level to get depleted to meet market demand because the cost
associated with a missing demand at market is higher than that of the inventory costs.
The case is more aggravated when the margin on the product demand uncertainty is
higher (50% margin). The relative results of uncertainty analysis for model 2 and model
1 shows products are made at all possible time nodes, if required, because of the
6
K.Mitra et al.
absence of the minimum run length as well as product family constraint in the case of
model 1.
Dem and Patterns
Amount
Amount
40
20
0
60
40
20
0
1
2
3
4
Certain
5
6
7
Month
Margin 30%
8
9
10
11
12
1
Margin 50%
2
3
4
Certain
Inventory Patterns
100
5
6
7
Month
Margin 30%
8
9
10
11
12
Margin 50%
Shortage Patterns
10
80
Amount
Amount
Production Patterns
80
60
60
40
20
5
0
0
1
1
2
3
4
Certain
5
6
7
Month
Margin 30%
8
9
10
11
Margin 50%
2
12
Certain
3
4
5
6
7
Month
Margin 30%
8
9
10
11
12
Margin 50%
Fig. 2. Demand, Production, Inventory and Shortage patterns for product P1 under various
demand margins over the entire planning horizon for model 2 ((a) when demand values are
certainly known, (b) demand margins are 30% of the nominal values, (c) demand margins are
50% of the nominal values)
The effect of machine uptime uncertainty when considered in addition to demand
uncertainty leads to LP (1,2,4-7,9,10b-15b,17b-19b) and MINLP (1-9,10b-19b)
formulations for product and product family formulations respectively. MINLP problem
is solved using BARON solver. These results are not presented here for the sake of
brevity and will be communicated in future publications.
4. Conclusion
In this paper, the multi-objective mid term supply chain planning problem is solved
using fuzzy mathematical programming. The slot based planning model of McDonald
and Karimi (1997) is adopted under fuzzy programming paradigm and solved for
various uncertain scenarios. If the same problem would have been solved using the
scenario based two-stage stochastic programming approach considering just 5 scenarios
for each of the 34 products for 12 time periods interlinked by inventory balance
equations, we would have to solve a very large problem (5 341212 scenarios) which was
not the case for fuzzy approach. In fuzzy approach, the problem formulation is quite
generic and easy to model, and the time involved in solving the problem is much
smaller in comparison. In addition, it does not require the knowledge of distributions
associated with the uncertain parameters. Due to these strong advantages, the fuzzy
programming promises a great potential in handling problems under uncertainty.
References
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