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Towards resilient supply chains: Uncertainty analysis using fuzzy mathematical programming
18th European Symposium on Computer Aided Process Engineering – ESCAPE 18 Bertrand Braunschweig and Xavier Joulia (Editors) © 2008 Elsevier B.V. All rights reserved.
701
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
K. Mitra et al.
702
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 Δ 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 λ linearly varying between 0 and 1). In this way, some amount of constraint violation is allowed. Finally we write the λ 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:
λ
Max
0 ≤ λ ≤1, Pi , j , s , t , RL i , j , s , t ,C i , s , t ,Si , s , c , t , I i , s , t , I i−, c , t ,σ i , s , s ′ , t , I iΔ, s , t , Yf , j , s , t
Pi , j,s,t = R i , j,s ,t RLi , j,s,t RL f , j,s , t =
C i ,s , t =
∑ RL
i∈Φ i , f
(1)
∀i ∈ I \ I RM
(3)
i , j,s , t
∑β ∑ P
i′ ,i ,s i′ ∋ β i ′ , i , s ≠ 0
(2)
∀i ∈ I \ I FP
i′, j,s , t
(4)
j
C i ,s , t = ∑ σ i ,s ′ ,s , t
∀i ∈ I IP
(5)
s′
Ii,s,t = Ii,s,t−1 + ∑Pi, j,s,t − ∑σi,s′,s,t − ∑Si,s,ct∀i ∈I \ IRM
(6)
I iΔ,s , t ≥ I iL,s ,t − I i ,s, t
(7)
j
s′
c
∀i ∈ I FP
Towards Resilient Supply Chains: Uncertainty Analysis Using Fuzzy Mathematical Programming
703
RL f , j,s , t − MRL f , j,s , t Yf , j,s , t ≥ 0
(8)
I iΔ,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 − ΔH j,s , t )⎟ ΔH j,s , t ⎥ + λ − 1 ≤ 0 ⎠ ⎣⎝ i ⎦
(10b)
∑ RL f , j,s , t − H j,s , t ≤ 0
(11a)
⎡⎛ ⎤ ⎞ ⎢⎜ ∑ RL f , j,s , t − (H j,s , t − ΔH j,s , t )⎟ ΔH j,s , t ⎥ + λ − 1 ≤ 0 ⎠ ⎣⎝ f ⎦
(11b)
f
⎡⎧⎛ − ⎤ ⎫ ⎞ − ⎢⎨⎜ I i ,c,t − I i ,c, t −1 + ∑ Si ,s,c,t ⎟ − (d i ,c,t + Δd i ,c,t )⎬ Δd i ,c, t ⎥ − λ + 1 ≥ 0 s ⎠ ⎭ ⎣⎢⎩⎝ ⎦⎥ ⎡⎛ ⎞ ⎢⎜⎜ ∑ S i ,s ,c , t ′ − ∑ d i ,c, t′ ⎟⎟ t ′≤ t ⎢⎣⎝ s , t′≤ t ⎠
∑ Δd t ′≤ t
i ,c, t ′
⎤ ⎥ + λ −1≤ 0 ⎥⎦
Pi , j,s , t ≤ R i , j,s , t H j,s , t
[{P
i, j,s,t
]
−Ri, j,s,t (Hj,s,t −ΔHj,s,t )} Ri, j,s,tΔHj,s,t +λ−1≤0
i , j,s , t
(14b) (15a)
]
− (H j,s, t − ΔH j,s, t )} ΔH j,s , t + λ − 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 − ΔH j,s , t )Yf , j,s , t } ΔH j,s , t Yf , j,s , t + λ − 1 ≤ 0
⎡⎧ ⎫ ⎢⎨S i ,s ,c, t − ∑ d i ,c , t ′ ⎬ t ′≤ t ⎭ ⎣⎩ ⎡⎛ − ⎞ ⎢⎜ I i , c, t − ∑ d i ,c , t ′ ⎟ t ′≤ t ⎠ ⎣⎝
∑ Δd t ′≤ t
∑ Δd t ′≤ t
[(cos t − cos t ) (cos t min
(13b)
(14a)
RL i , j,s , t − H j,s , t ≤ 0
[{RL
(12b)
i,c , t ′
i ,c, t ′
max
⎤ ⎥ + λ −1 ≤ 0 ⎦
⎤ ⎥ + λ −1 ≤ 0 ⎦
(16b) (17b)
(18b)
)]
− cos t min + λ − 1 ≤ 0
(19)
K. Mitra et al.
704
Where
Cost= ∑FCf , j,s,t Yf , j,s,t + ∑νi, j,s Pi, j,s,t + ∑pi,s Ci,s,t + ∑hi,s,t Ii,s,t + f , j,s,t
i, j,s,t
i,s,t
i,s,t
∑t s,cSi,s,c,t + ∑t s,s′σi,s,s′,t + ∑ζi,s IiΔ,s,t + ∑μi,cIi−,c,t
i,s,c,t
i,s,s′,t
i,s,t
i,c,t
Pi , j,s ,t , RL i , j,s, t , RL f , j,s , t , C i , s , t , S i ,s , c , t , I i,s, t , I i−,c , t , σ i ,s ,s′, t , I iΔ,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 , S i ,s , c , t , I i,s, t , I i−,c , t , σ i ,s ,s′, t , I iΔ,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 (μic), raw material price (pis), penalty for dipping below safety level (ζis), fixed production cost (FCf,j,s,t) and variable production cost(νijs) 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
705
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
K. Mitra et al.
706
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
Certain
4
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 (534×12×12 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 J. F. Shapiro, 2001, Modeling the Supply Chain, Duxbury Thormson Learning, USA. J. R. Birge, F. Louveaux, 1994, Introduction to Stochastic Programming, Springer-Verlag, New York, USA. N. V. Sahinidis, 2004, Optimization Under Uncertainty: state-of-the-art and opportunities, Comp. Chem. Eng., 28, 971. C. M. McDonad, I. A. Karimi, 1997, Planning and Scheduling of parallel semicontinuous processes. 1. Production planning. Ind. & Engg. Chem. Res., 36, 2691. R.Bellman, L. A. Zadeh, 1970, Decision-making in a fuzzy environment. Man. Sci., 17, 141. A, Brooke, D. Kendrick and A. Meeraus, 1998, GAMS – A User’s Guide, GAMS Dev. Corp. Washington DC.
701
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
K. Mitra et al.
702
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 Δ 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 λ linearly varying between 0 and 1). In this way, some amount of constraint violation is allowed. Finally we write the λ 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:
λ
Max
0 ≤ λ ≤1, Pi , j , s , t , RL i , j , s , t ,C i , s , t ,Si , s , c , t , I i , s , t , I i−, c , t ,σ i , s , s ′ , t , I iΔ, s , t , Yf , j , s , t
Pi , j,s,t = R i , j,s ,t RLi , j,s,t RL f , j,s , t =
C i ,s , t =
∑ RL
i∈Φ i , f
(1)
∀i ∈ I \ I RM
(3)
i , j,s , t
∑β ∑ P
i′ ,i ,s i′ ∋ β i ′ , i , s ≠ 0
(2)
∀i ∈ I \ I FP
i′, j,s , t
(4)
j
C i ,s , t = ∑ σ i ,s ′ ,s , t
∀i ∈ I IP
(5)
s′
Ii,s,t = Ii,s,t−1 + ∑Pi, j,s,t − ∑σi,s′,s,t − ∑Si,s,ct∀i ∈I \ IRM
(6)
I iΔ,s , t ≥ I iL,s ,t − I i ,s, t
(7)
j
s′
c
∀i ∈ I FP
Towards Resilient Supply Chains: Uncertainty Analysis Using Fuzzy Mathematical Programming
703
RL f , j,s , t − MRL f , j,s , t Yf , j,s , t ≥ 0
(8)
I iΔ,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 − ΔH j,s , t )⎟ ΔH j,s , t ⎥ + λ − 1 ≤ 0 ⎠ ⎣⎝ i ⎦
(10b)
∑ RL f , j,s , t − H j,s , t ≤ 0
(11a)
⎡⎛ ⎤ ⎞ ⎢⎜ ∑ RL f , j,s , t − (H j,s , t − ΔH j,s , t )⎟ ΔH j,s , t ⎥ + λ − 1 ≤ 0 ⎠ ⎣⎝ f ⎦
(11b)
f
⎡⎧⎛ − ⎤ ⎫ ⎞ − ⎢⎨⎜ I i ,c,t − I i ,c, t −1 + ∑ Si ,s,c,t ⎟ − (d i ,c,t + Δd i ,c,t )⎬ Δd i ,c, t ⎥ − λ + 1 ≥ 0 s ⎠ ⎭ ⎣⎢⎩⎝ ⎦⎥ ⎡⎛ ⎞ ⎢⎜⎜ ∑ S i ,s ,c , t ′ − ∑ d i ,c, t′ ⎟⎟ t ′≤ t ⎢⎣⎝ s , t′≤ t ⎠
∑ Δd t ′≤ t
i ,c, t ′
⎤ ⎥ + λ −1≤ 0 ⎥⎦
Pi , j,s , t ≤ R i , j,s , t H j,s , t
[{P
i, j,s,t
]
−Ri, j,s,t (Hj,s,t −ΔHj,s,t )} Ri, j,s,tΔHj,s,t +λ−1≤0
i , j,s , t
(14b) (15a)
]
− (H j,s, t − ΔH j,s, t )} ΔH j,s , t + λ − 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 − ΔH j,s , t )Yf , j,s , t } ΔH j,s , t Yf , j,s , t + λ − 1 ≤ 0
⎡⎧ ⎫ ⎢⎨S i ,s ,c, t − ∑ d i ,c , t ′ ⎬ t ′≤ t ⎭ ⎣⎩ ⎡⎛ − ⎞ ⎢⎜ I i , c, t − ∑ d i ,c , t ′ ⎟ t ′≤ t ⎠ ⎣⎝
∑ Δd t ′≤ t
∑ Δd t ′≤ t
[(cos t − cos t ) (cos t min
(13b)
(14a)
RL i , j,s , t − H j,s , t ≤ 0
[{RL
(12b)
i,c , t ′
i ,c, t ′
max
⎤ ⎥ + λ −1 ≤ 0 ⎦
⎤ ⎥ + λ −1 ≤ 0 ⎦
(16b) (17b)
(18b)
)]
− cos t min + λ − 1 ≤ 0
(19)
K. Mitra et al.
704
Where
Cost= ∑FCf , j,s,t Yf , j,s,t + ∑νi, j,s Pi, j,s,t + ∑pi,s Ci,s,t + ∑hi,s,t Ii,s,t + f , j,s,t
i, j,s,t
i,s,t
i,s,t
∑t s,cSi,s,c,t + ∑t s,s′σi,s,s′,t + ∑ζi,s IiΔ,s,t + ∑μi,cIi−,c,t
i,s,c,t
i,s,s′,t
i,s,t
i,c,t
Pi , j,s ,t , RL i , j,s, t , RL f , j,s , t , C i , s , t , S i ,s , c , t , I i,s, t , I i−,c , t , σ i ,s ,s′, t , I iΔ,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 , S i ,s , c , t , I i,s, t , I i−,c , t , σ i ,s ,s′, t , I iΔ,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 (μic), raw material price (pis), penalty for dipping below safety level (ζis), fixed production cost (FCf,j,s,t) and variable production cost(νijs) 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
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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
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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
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absence of the minimum run length as well as product family constraint in the case of model 1. Dem and Patterns
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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 (534×12×12 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 J. F. Shapiro, 2001, Modeling the Supply Chain, Duxbury Thormson Learning, USA. J. R. Birge, F. Louveaux, 1994, Introduction to Stochastic Programming, Springer-Verlag, New York, USA. N. V. Sahinidis, 2004, Optimization Under Uncertainty: state-of-the-art and opportunities, Comp. Chem. Eng., 28, 971. C. M. McDonad, I. A. Karimi, 1997, Planning and Scheduling of parallel semicontinuous processes. 1. Production planning. Ind. & Engg. Chem. Res., 36, 2691. R.Bellman, L. A. Zadeh, 1970, Decision-making in a fuzzy environment. Man. Sci., 17, 141. A, Brooke, D. Kendrick and A. Meeraus, 1998, GAMS – A User’s Guide, GAMS Dev. Corp. Washington DC.