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A Stochastic Optimization Case Study using ILOG CPLEX Concert Technology: Dual Sourcing Model for Managing Supply Chain Disruptions
Vikas Argod Department of Mechanical Engineering, The Pennsylvania State University, University Park, PA 16802 Email: [email protected] Anshuman Gupta, PhD Research Computing and Cyberinfrastructure (RCC), Information Technology Services (ITS), The Pennsylvania State University, University Park, Pennsylvania 16802 Phone: 814‐865‐6081, Email: [email protected] A two stage stochastic programming model for incorporating demand uncertainty and disruptions in supply chain is described and implemented using ILOG CPLEX Concert Technology. Exchange rate variation is also considered along with other uncertainties for foreign supplier selection. The model is focused at setting up order portfolio minimizing total costs and helps in predicting the emergency orders. Management of financial risks is also addressed by linking the optimization model with basic statistical and probabilistic methods.
1. Introduction
Continuous change, uncertain demand and intense competition are the defining characteristics of the today’s volatile business environment. In view of these conditions, meeting customers demand and making profit has become a challenge to supply chain managers. Most of the decisions have to be taken in highly uncertain environment. Operational and managerial flexibility are highly advocated in today’s supply chain. The uncertainty involved in most of these decisions can be linked to the basic structure underlying these activities, which is optimal usage of future resources on the basis of available present information and future scenarios1. In this age of globalization, supply chain is spread across countries. Especially developing countries are seen as main suppliers and exchange rate dynamics plays an important role in supplier selection. Corporations are using exchange rate variation for hedging supply chain risks2. But along with the higher profit due to selection of cheaper suppliers, risks of supply chain disruptions increase. As simple as uncertain lead time can affect the production plans which necessitates maintaining local supplier as a backup all times. Supply chain mangers have to optimize while selecting main supplier selection and also when placing orders for every planning horizon. In view of this observation, a versatile modeling and solution framework for aligning the internal planning decisions of a company under uncertainty is proposed in this study. Deterministic planning and scheduling models yield unrealistic results since it fails to recognize the effect of demand variability on the tradeoff between lost sales and inventory holding costs. Failure to consider stochasticity of future business conditions could lead to either unsatisfied
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customers causing loss of market share or high inventory holding costs2. Hence, following stochastic programming is justified.
There are lots of publications available addressing general issues of supply chain risk management focusing on demand uncertainty. There are some articles focusing precisely on disruptions and they suggest various ways of have safer supply chains and different methods of mitigating the consequences of disruptions.4 Lee and Wolfe5 suggest a list of possible methods to design supply chain with flexible sourcing strategies like developing multiple supply sources, maintaining local supplier with constant orders at all normal times etc. Single sourcing has allowed to reduce the sourcing cost and resulted in better quality, but this approach is risky. So, it is advisable for firms to consider multiple sourcing approaches to cover against disruptions in supply chain. In order to make choice between these two approaches, concept of real options is used by various literatures to evaluate and take into account flexibility1. Piaw6 laid down the comparison between single sourcing and multiple sourcing considering Kraljic’s portfolio approach of 4 types of commodities, which leads to differentiated purchasing strategies. Cohen and Huchzermeier showed how operational flexibility can be captured effectively through stochastic recourse (whereby the firm can respond to random outcomes by changing decisions). Newsvendor problem is one of simplest form of planning under demand uncertainty. McDonald and Karimi7 proposed midterm planning model with capacitated lot‐sizing problem. Deterministic formulation can be found in Gupta and Maranas8. MIT Research group on “Supply chain response to Global terrorism” (2003) have shown that firms usually focus on the type of disruption and not its source for securing the supply chain. They distinguish 6 types of disruptions: disruptions in supply, transportation, at facilities, freight breaches, in communication, and in demand.
The rest of paper is organized as follows. In the next section, general modeling considerations are explained followed by explanation of the model developed. Risk management strategies are given. Implementation methodology using CPLEX solver invoked using C++ API is briefly explained. Result for a simple case study is discussed for the completeness of the model. Methodology of approximating normal distributions by discrete ones and CPLEX code are given in the Appendix.
2. Modeling the decision making process
An important component of decision making under uncertain conditions is the way of representing stochastic parameters. There are two distinct ways of incorporating uncertainty. In scenario based approach9, a random parameter is considered with all its possible future outcomes. The main disadvantage of this technique is that the size of the problem increases exponentially with the increase in the number of scenarios. To overcome this difficulty, continuous probability distributions for the random parameters are often used.10 It introduces nonlinearities in the problem but decrease the size of the problem. In this study, scenario based approach is used. Product demand and disruptions are modeled as normally distributed random variables. This approach has been widely invoked in the literature as it captures the essential
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features of demand uncertainty and it is convenient to use. In the two stage stochastic approach considered, there are two sets of decision variables. The first stage variables, sometimes known as design variables, correspond to decisions that need to be made prior to resolution of uncertainty (“here‐and‐now” decisions). Based on the realization of the random events, the second‐stage or control decisions are made according to the restrictions of the second‐stage recourse problem (“wait‐and‐see” decisions). The uncertainty is translated into the stochastic nature of the costs related to second‐stage variables. Hence, objective function comprises of first stage decision costs and expected recourse cost. The problem of increasing size of the problem with the increase in the number of scenarios is well answered by Dantzig‐Wolfe Decomposition11 and Benders decomposition12. For continuous probability distributions, this challenge has been taken care by explicit/implicit discretization of probability space for approximating multivariate probability integrals13. In the model which will be explained in the next section first stage cost is also an expected cost which consists of exchange rate variation. But the demand and disruption scenario are resolved in second stage when exchange rate calculations are already over. Supply chain managers need to select the supplier considering the effect of exchange rate variation and order has to be placed in advance such that sufficient time is available in order to rectify the effect of supply chain disruptions with the help of local suppliers.
3. Dual Sourcing model with supply chain disruptions
The problem addressed in this study can be stated as follows:
Given a set of candidate foreign suppliers (main suppliers) characterized by their respective cost with exchange rate dynamics, demand scenarios, disruption scenarios as percentage of order received from main supplier, local supplier with minimum and maximum orders and costs associated with it, inventory holding cost, inventory limit, normal and emergency production costs, determine the supplier to be chosen, optimum order to main supplier and order portfolio‐that minimizes the total expected cost. Sets
J = { j } = Set of candidate foreign suppliers K1= { k1} = Set of demand scenarios K2 = { k2} = Set of exchange rate scenarios K3 = { k3} = Set of disruption scenarios
Parameters
cjnor = Cost per product by supplier j
clocal = Cost per product by local supplier cback = Back order cost h = Inventory holding cost I0 = Inventory limit cp
nor= Normal production cost cp
eme= Emergency production cost Θk1= Demand in scenario k1
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ωΘk1= Probability of demand scenario k1
Ek2j= Exchange rate in scenario k2j ωE
k2j= Probability of demand scenario k2j Qmin
local= Minimum order to local supplier Qmax
local= Maximum order to local supplier dk3= percentage of order supplied by the main supplier in disruption scenario k3 ωd
k3= Probability of disruption scenario k3
Variables
Yj = 1 if Supplier j is chosen; 0 otherwise Qj
0= Order placed with supplier j QL
0= Order placed with local supplier Qk1k3
eme= Emergency order in demand scenario k1 and disruption scenario k3 Ik3= Inventory in disruption scenario k3
Bk1k3exe= Backorder observed after demand is realized and ordering emergency products in
demand scenario k1 and disruption scenario k3 Utilizing the above‐described notations, the Basic Dual Sourcing (BDS) model is formulated as follows. (BDS)
min ∑ ∑
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Subject to ∑ 1 (1) ∑ (2)
max 0, , (3)
, (4) max 0, , (5) 0 (6) 0,1 , 0, 0 (7)
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In the considered approach, a stage corresponds to the description of the system, in terms of the variables and constraints, between two points of uncertainty resolution. In the considered framework, “here‐and‐now” decisions are supplier selection (Yj) , initial order placed with the supplier j (Qj0)and initial order to the local supplier (QL
0). The costs incurred in this stage include the initial ordering cost with both the suppliers (first and second term). In the case of main supplier, expected cost due to exchange rate variation is calculated. Selecting only one main supplier from the pool of available suppliers is ensured by eq 1. After the initial order, first level of uncertainty is resolved as the disruption scenario comes into picture. Due to disruption, only dk3 times Qj0 is supplied by the main supplier while local supplier supplies whatever order placed. Hence inventory is given by eq 2. Inventory holding cost is indicated in the fifth term of objective function. (Average inventory cost is considered, for simplicity). Uncertainty in product demand is resolved at this stage and it calls for emergency orders from local supplier. Demand uncertainty also affects the bottom line of the firm through variable production costs, as accounted by third and fourth term in the objective function equation. Higher cost incurred due to emergency orders and production is reflected in sixth term of objective function. Once all the uncertainties are resolved, emergency order is calculated by eq 3, while bounds on order placed with the local supplier at all levels are enforced through eq 4. is the “wait‐and‐see” decisions , hence it establishes the order portfolio. Excess backorder if any, which was not met even at the end (may be due to the fact that order to local supplier was greater than maximum limit) is determined through eq 5 and respective cost is taken into account through the last term in the objective function. Firm has an inventory limit which is enforced through eq 6. A scenario –based description of both demand and disruption forms the basis of the above described model. The nonanticipative14 resolution of these uncertainties can be represented through scenario tree. Nonanticipativity implies that all the inner stage decisions are based on the actual values realized by all uncertain parameters that are preceding that decision. This means that inner decision variables depend on the entire history of the process of uncertainty resolution. For example, emergency order in last stage depends on the disruption observed in stage two and demand scenario observed in stage three. This nonanticipativity characteristic is captured by the diverging scenario tree. This is the main reason for exponential increase in the number of overall scenarios with the number of individual scenarios. A stage is formed wherever branching occurs15, while each branch of the tree corresponds to a particular realization of demand and disruption uncertainty. Cost associated with each branch is different which causes a distribution of total cost at the end of planning horizon. Minimization of the expected cost of this distribution is achieved by probabilities of demand and disruption scenario through the objective function of the BDS model.
4. Risk Management
The central idea of any risk management is the incorporation of the classic tradeoff between risk and return within the framework of decision process15. As formulated, BDS model takes purely expected cost‐minimization perspective; hence it assumes that the firm is risk‐neutral. In general, firms prefer lower, as opposed to high variability of the cost for a given value of return. Consequently, controlling variability is well recognized in the financial community mainly when firm deals with portfolio management applications.16 Variance in returns is widely used in financial decision making in corporate policy formulation settings.17. But variance being a
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symmetric measure of the dispersion of the actual cost around the mean, a reduction in its value decreases downside risk but it decreases upside potential. In view of this limitation, in literature there are different approaches. A probabilistic approach and limit on the maximum cost is implemented in the current framework. In the former approach, probability that cost at any scenario (TC) is controlled by the addition of following constraint in the model description.
Pr (8) This ensures that the probability of the total cost (TC) in every scenario exceeding some
maximum cost ( ) is less than some specified level ( ). Using this probabilistic constraint within the current model framework requires an additional binary variable defined as follows.
1 0
(9)
∑ ∑
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(10) Addition of following set of constraints enforces the probabilistic constraint explained 1 1 (11) ∑ (12) where M is an upper bound on the total cost TC.
The probabilistic approach described above has an added complexity associated with additional binary variables. In light of this increasing complexity, a more computationally better approach which targets the worst‐case cost scenario can be implemented by adding following constraint. max (13) or (14) With this approach, the formulation becomes computationally less expensive and risk elimination criteria are also met.
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5. Implementation and solution using ILOG CPLEX Concert Technology
The model is implemented in C++ using CPLEX Concert Technology. Concert Technology is a set of modeling objects that allow a high‐level representation that mirrors the optimization model’s algebraic representation. Since CPLEX can solve linear problem and the presented
model involves nonlinearity (observe 0 ), linearization procedure is used. Literature
describes various methods of linearization for 0‐1 polynomial programs. This introduces one more set of binary variables and constraints as follows.
(15)
1 1 (16)
0,1 (17)
Thus, can be replaced with in the model described above thereby making it linear which can be extracted and solved using CPLEX.
One more interesting constraint is eq 3 which is a max constraint (eq 5 is also similar). This was implemented using “IloMax” operator available in CPLEX. As per the available documentation, syntax of IloMax is as follows18. public IloNumToNumStepFunction IloMax(const IloNumToNumStepFunction f1, const IloNumToNumStepFunction f2) This operator creates and returns a function equal to the maximal value of the functions f1 and f2. Usage of this operator was also cross checked with two greater than constraints and found consistency in the results. Scenarios were created by approximating continuous distributions of the random variables by discrete ones.19 Explanation of this method is given in the Appendix‐A. A standalone C++ code created this realizations and writes to an ASCII file which is taken as input by the CPLEX Concert Technology code. All C++ codes are given in the Appendix‐B with a sample input file.
6. Case Study
A manufacturing/assembly firm is in its starting phase of planning horizon. It has to select one main supplier from available two different suppliers. Since the price quoted by the suppliers will not take into account of exchange rate variation, decision should consider the best price case. The parameters characterizing each supplier, production charges, and demand and disruption variations are listed in Table 1. Note that all cost parameters are assumed to be relative monetary units (RMU) while demand levels are relative product units (RPU).
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20 scenarios of each of the distributions are considered for the case study. This results in around 428 variables and 2455 constraints. It is to be noted that this size increased exponentially as the number of scenarios increase. This was solved in one of Penn state clusters, Hammer (64 bit, AMD Opteron processor). It took approximately 8 CPU s. Solution of this model is given in Table 2 and distribution of emergency order is given in Figure 1. The cheaper cost of main supplier is exploited and initial order is placed taking into account of disruptions. Probability of emergency order to the local supplier below 100RPUs is about 60%. Rest of the cases is higher emergency orders with very less probabilities. This is clear from the Figure 1. Next, the impact of proposed risk management techniques on the risk scenarios of the firm is studied. The results obtained for probabilistic methods and worst case cost control are presented in Figure 2‐3, respectively. The tradeoff curves are shown in these figures are obtained by first appending the corresponding risk management constraints to the BDS model followed by parametrically varying the appropriate risk metric.
Table 1. Parameter values used for the case study
Parameter Value
cjnor [10,9]
clocal 25 cback 200 H 15 I0 20,000 cp
no 50 cp
eme 125 Θ Normal(1500,100) Ek21( Exchange rate for supplier 1) Uniform(0.71,0.93) Ek22( Exchange rate for supplier 2) Uniform(0.7,1.1) Qmin
local, Qmaxlocal 100, 150000
dk3 Normal(0.8,0.1) Table 2. Result of the case study
Variable Value
Supplier Selected 2nd
Initial order placed with that supplier 1756.03 (RPU)
Net Expected cost 119341 (RMU)
Initial order to local supplier 100 (RPU)
Maximum cost in the distribution 175259 (RMU)
The conclusion is based on the observation that, as the risk metric decreases, expected cost increases which is obvious from the principles from Economics. In the current setting, expected cost increases or profit decreases for low risk cases.
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Figure 1. Emergency order distribution
Figure 2.Variation of the total expected cost with the probability level
Figure 3.Variation of the total expected cost with the worst‐case
total cost scenario
7. Summary
0%
10%
20%
30%
40%
50%
60%
70%
30.3 90.9 151.5 212.2 272.7 333.3 393.9 454.5 515.1 575.7
Prob
ability
Emergency orders
113000
114000
115000
116000
117000
118000
119000
120000
0% 1% 2% 3% 4% 5% 6%
Expe
cted
cost
Probability level
100000
105000
110000
115000
120000
125000
130000
135000
140000
140000 150000 160000 170000 180000 190000
Expe
cted
cost
Worst case total cost scenario
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In this work, the supplier selection and initial order calculation problem faced by a firm was addressed. In particular, this work focused on the incorporation of these three following issues in the planning process: (1) Inclusion of exchange rate variation while selecting suppliers, (2) inclusion of supply chain disruptions and demand uncertainty and (3) risk management for reducing cost variability. A multistage stochastic programming approach was considered for studying demand and disruption uncertainties in the decision making process. The partitioning of decision phases into “here‐and‐now” and “wait‐and‐see” provided the appropriate structure of stochastic stage formulation. More importantly, supplier selection and initial order to the main supplier was made prior to realizations of uncertainties involved in the framework. The issue of actively managing the risk exposure of the firm was also addressed. Two alternate approaches namely (1) probabilistic approach and (2) worst‐case total cost for limiting the downside risk maintaining the upside potential. To highlight the effectiveness and validity of the developed model, it was applied to a planning case study. Risk management techniques were also used in this case study. In conclusion, future work on the problem addressed in this work could proceed in several different directions. The model could be extended to a multiperiod setting where supplier selection and ordering depends on many periods. Usually, there will be supplier changing cost and local supplier quotes different price for different percentage of initial order. This model is from the side of firm while sudden variation in the demand for local supplier is not studied. The same model can be extended to studying impact of exchange rates in case of demand from many countries thereby using exchange rate dynamics to hedge the supply chain risks. Finally, the proposed model could also be integrated with contracts with suppliers for studying the cost saving compared to the present system of dual suppliers.
References:
1. Gupta A and Maranas, C.D. “Real‐Options‐based planning strategies Under Uncertainty”. Ind. Eng.Chem Res. 2004, 43, 3870‐3878
2. Cohen M.A and Huchzermeier. A, “Global supply chain network management under price/exchange rate risk and demand uncertainty".
3. Petkov, S. B.; Maranas, C. D. “Multiperiod Planning and Scheduling of Multiproduct Batch Plants under Demand Uncertainty”. Ind. Eng. Chem. Res. 1997, 36, 4864.
4. Pochard S, “Managing risks of Supply chain disruptions: Dual sourcing as a real option”. Massachusetts Institute of Technology, August 2003
5. Lee, H.L., Wolfe, M. “Supply Chain Security without Tear”, Supply Chain Management Review, January‐February, 2003. pp 12‐20.
6. Piaw. T.C. “Purchasing and Supply chain management: Supplier management”. National University of Singapore.
7. McDonald, C ; Karimi, I.A. “Planning and Scheduling of parallel semi continuous processes” Ind. Eng. Chem. Res 1997,36,2691
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8. Gupta A, ;Maranas, C. D. “A hierarchical lagrangean relaxation procedure for solving midterm planning problems” Ind. Eng. Chem. Res 1999,38,1937
9. Liu, M. L.; Sahinidis, N. V.“Long Range Planning in the Process Industries: A Projection Approach”. Comp. Oper. Res. 1996, 23, 237.
10. Ierapetritou, M. G.; Pistikopoulos, E. N. “Batch plant design and operations under uncertainty.” Ind. Eng. Chem. Res. 1996, 35, 772.
11. Dantzig, G. B.; Wolfe, P. “The decomposition principle for linear programs”. Oper. Res. 1960, 8, 101.
12. Benders, J. F. “Partitioning procedures for solving mixed variables programming problems.” Numer. Math. 1962, 4, 238.
13. Straub, D. A.; Grossmann, I. E. “Integrated stochastic metric of flexibility for systems with discrete states and continuous parameter uncertainties”. Comp. Chem. Eng. 1990, 14, 967.
14. Birge J.R,; Louveaux, F. “Introduction to Stochastic Programming”, Springer: New York 1997 15. Gupta A,;Maranas C.D “Market Based Pollution abatement strategies : Risk management using
emission option contracts”. Ind. Eng. Chem Res 2003, 42, 802‐810 16. Luenberger, D.G. “Investment Science”, Oxford University Press: New York 1998 17. Damodaran, A. “Corporate Finance: Theory and practice”. John Wiley: New York 1998 18. http://terra.rice.edu/comp.res/apps/c/cplex/docs/refconcert/html/functions/IloMax.html 19. Peter Kall,; Stein W. Wallace. “Stochastic Programming” . First edition, John Wiley & Sons.
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Appendix –A
Discretizing the normal distribution19
If any random variable is following normal distribution, taking expected value is cumbersome if multivariate numerical integration is used. To avoid this difficulty, normal distribution can be approximated by discrete ones. Methodology for discretization is given below: Consider a random variables δi (i=1,2) which are normally distributed 1. Generate large samples δi
μ, μ=1,2….K, i=1,2, restricted to may be 99% intervals of the given distribution. Sample size K=100000 (say)
2. Choose equidistant partitions of the 99% intervals into ri, i=1,2 subintervals. (e., r1= r2=15) 3. Calculate for every subinterval Iiv, v = 1…ri, i=1,2, the arithmetic mean , Mi
v of the sample values δi
μ Є Iiv, yielding an estimate for the conditional expectation of δi and Iiv
4. Calculate for every subinterval Iiv the relative frequency piv for δiμ Є Iiv (i.e, piv = kiv/K, where k‐
iv is the number of sample values δiμ contained in Iiv). This yields an estimate for the
probability of {δiμ Є Iiv+}.
The discrete distributions, {( Mi
v,piv), v= 1,,…..ri}, i=1,2 are can be used as approximation for
the given normal distributions. Following figure shows these discrete distributions for N(0,12) and N(0.9) with 15 realizations each.
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( Source: reference # 19)
Appendix –B CPLEX implementation and generation of input file
CPLEX code with probabilistic approach of risk management:dual.cpp #include <ilcplex/ilocplex.h> #include <math.h> #include <iostream> #include <fstream> ILOSTLBEGIN typedef IloArray<IloNumVarArray> IloNumVarArray2; int main(int argc, char **argv) { IloEnv env; try {
const char* filename = "dual_source.dat"; //input file if (argc >= 2) filename = argv[1]; ifstream file(filename); if( !file) { cerr << "No such file : " << filename << endl; throw(-1); } char dummy[100]; // dummy variable for reading comments in input file IloInt num_sup; //number of suppliers IloNum c_eme,c_back,h,I0,cp_nor,cp_eme,Q_max, TC_hat,alpha,QL_min,c_local;
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IloNumArray theta(env),c_nor(env),w_t(env),d(env),w_d(env);//parameters IloNumArray2 E(env),w_e(env); file >> dummy >> num_sup // reading the input fil >> dummy >> c_nor >> dummy >> c_local >> dummy >> c_back >> dummy >> h >> dummy >> I0 >> dummy >> cp_nor >> dummy >> cp_eme >> dummy >> theta >> dummy >> w_t >> dummy >> d >> dummy >> w_d >> dummy >> E >> dummy >> w_e >> dummy >> QL_min >> dummy >> Q_max >> dummy >> TC_hat >> dummy >> alpha; IloInt sup_num_check = c_nor.getSize(); // some error checks IloInt demand_size = theta.getSize(); IloInt theta_size = w_t.getSize(); IloInt dis_size = w_d.getSize(); if(sup_num_check != num_sup || demand_size !=theta_size) { cerr << " ERROR: Inconsistant Data " << endl; throw(-1); } IloInt j,k1,k2,k3; // different varibales for loops, printing etc IloNum Q_final,M,z,k; IloModel mod(env); M = 10000000.0; // BIG M : upper bound of Z, continuous variable of linearisation for(j=0; j<num_sup; j++) // error checking for E and its probability { if(E[j].getSize()!= w_e[j].getSize()) { cerr << " ERROR: Inconsistant Data for Exchanage rate scenario "<<endl; throw(-1); } } //Variables IloNumVarArray y(env,num_sup,0.0,1.0,ILOBOOL); //boolian variable for Supplier selection IloNumVarArray Q(env,num_sup,0.0,M); // Order placed with supplier j IloNumVarArray Z(env,num_sup,0.0,IloInfinity); // Continuous variable on account of //linearization IloNumVarArray ini_inv(env,dis_size,0.0,I0);
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IloNumVarArray2 Q_eme(env),B_exe(env),tc(env),Za(env); for (k1=0; k1<demand_size; k1++) {
Q_eme.add(IloNumVarArray(env,dis_size,0.0,IloInfinity)); //emergency shipments B_exe.add(IloNumVarArray(env,dis_size,0.0,IloInfinity)); //backorder
tc.add(IloNumVarArray(env,dis_size,0.0,TC_hat)); Za.add(IloNumVarArray(env,dis_size,0.0,1.0,ILOBOOL)); } IloNumVar QL(env,QL_min,Q_max); //Objective IloExpr sup_cost(env),local_cost(env),prod_cost(env),inv_cost(env),uncertain_cost(env); for (j=0; j<num_sup; j++) //supplier cost for(k2=0; k2<E[j].getSize(); k2++) sup_cost += (w_e[j][k2]*E[j][k2]*c_nor[j]*Z[j]); local_cost = QL*c_local; for (k3=0; k3<dis_size; k3++) for (j=0; j<num_sup; j++) //production cost prod_cost +=w_d[k3]*d[k3]*cp_nor*Z[j]; prod_cost += cp_nor*QL; for(k3=0;k3<dis_size; k3++) inv_cost += w_d[k3]*0.5*h*ini_inv[k3]; //inventory cost for (k3=0; k3<dis_size; k3++) for (k1=0; k1<demand_size; k1++) //uncertain cost due to demand variation
uncertain_cost += w_d[k3]*w_t[k1]*(Q_eme[k1][k3]*(c_local+cp_eme) + B_exe[k1][k3]*c_back);
mod.add(IloMinimize(env,sup_cost+prod_cost+local_cost+inv_cost+uncerta in_cost));
sup_cost.end(); prod_cost.end(); inv_cost.end(); uncertain_cost.end(); local_cost.end(); //Constraints IloExpr sup_cnst(env); //only one supplier is chosen for (j =0; j<num_sup; j++) sup_cnst += y[j]; mod.add(sup_cnst == 1); sup_cnst.end(); for (k3=0; k3<dis_size; k3++) { IloExpr inv_cnst(env); //inventory limit at any point for (j=0; j<num_sup; j++)
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inv_cnst += d[k3]*Z[j]; inv_cnst += QL; mod.add(ini_inv[k3] == inv_cnst); inv_cnst.end(); }
for (j=0; j<num_sup; j++) //linearising { mod.add(Z[j] <= M*y[j]); mod.add(Q[j]- M*(1-y[j]) <= Z[j] <= Q[j]); } for (k3=0; k3<dis_size; k3++) for (k1=0; k1<demand_size; k1++) //Emergency order calculation {
mod.add(Q_eme[k1][k3] == IloMax(0.0,(theta[k1]-ini_inv[k3]))); mod.add(Q_eme[k1][k3] <= Q_max);
mod.add(B_exe[k1][k3] == IloMax(0.0,(theta[k1]-ini_inv[k3]- Q_eme[k1][k3])));
} IloExpr first_stage(env); for (j=0; j<num_sup; j++) { for (k2=0; k2<E[j].getSize(); k2++) first_stage += (w_e[j][k2]*E[j][k2]*c_nor[j]*Z[j]); } first_stage += QL*(c_local+cp_nor); for (k3=0; k3<dis_size; k3++) { for (k1=0; k1<demand_size; k1++) { IloExpr TC(env),disruption_cost(env); for (j=0; j<num_sup; j++) disruption_cost += cp_nor*d[k3]*Z[j]; disruption_cost += 0.5*h*ini_inv[k3];
TC = first_stage + disruption_cost + Q_eme[k1][k3]*(c_local+cp_eme) + B_exe[k1][k3]*c_back;
mod.add(TC >= TC_hat - M*(1-Za[k1][k3])); mod.add(TC <= TC_hat*(1-Za[k1][k3]) + M*Za[k1][k3]); TC.end(); uncertain_cost.end(); } } first_stage.end(); IloExpr prob(env); for (k3=0; k3<dis_size; k3++) for (k1=0; k1<demand_size; k1++) prob += w_t[k1]*w_d[k3]*Za[k1][k3]; mod.add(prob <= alpha); prob.end(); IloCplex cplex(mod);
cplex.exportModel("dual_source1.lp"); //writing lp file
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std::ofstream out("dual_source.results"); //output file out << endl << "Minimum Cost = " << cplex.getObjValue() << endl; env.out() << endl << "Minimum Cost = " << cplex.getObjValue() << endl; for (j=0; j<num_sup; j++) { z = cplex.getValue(Z[j]); k = cplex.getValue(y[j]); if( z !=0.0) { out << endl << "Supplier choosen and the order placed\t"; env.out() << endl << "Supplier choosen and the order placed "; out << "j= " << j+1 << "\tQ = " << z << endl; env.out() << " j = " << j+1 << "\tQ = " << z << endl;
out << "Initial Order with the local supplier:\t" << cplex.getValue(QL) <<endl; env.out() << "Initial Order with the local supplier:\t" << cplex.getValue(QL) <<endl;
break; } else if(j== num_sup-1) {
out << "\n Emergency ordering is prefered..dual sourcing-> single sourcing "<<endl; env.out() << "\n Emergency ordering is prefered.. dual sourcing-> single sourcing " <<endl;
} } IloExpr add_cost(env); for (j=0; j<num_sup; j++) //supplier cost { for(k2=0; k2<E[j].getSize(); k2++) add_cost += (w_e[j][k2]*E[j][k2]*c_nor[j]*Z[j]); } add_cost += QL*(c_local);
out << endl << "k3\tk1\tprob\tOrder recieved\tDemand\t\tEmergency Order\tI- cost\t\tII-cost\t\tTotal Cost" <<endl;
for (k3=0; k3<dis_size; k3++) for (k1=0; k1<demand_size; k1++) { IloExpr tc,dis_cost(env),tc2,rece_order(env); for (j=0; j<num_sup; j++) { dis_cost += cp_nor*d[k3]*Z[j]; rece_order += d[k3]*Z[j]; } dis_cost += 0.5*h*ini_inv[k3] + QL*cp_nor;
tc2 = dis_cost +(Q_eme[k1][k3]*(c_local+cp_eme) + B_exe[k1][k3]*c_back);
tc = add_cost + tc2; out << k3 << "\t" << k1 <<"\t" << w_t[k1]*w_d[k3] << "\t"
<< cplex.getValue(rece_order) << "\t\t" << theta[k1] << "\t\t"
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<< cplex.getValue(Q_eme[k1][k3]) << "\t\t" << cplex.getValue(add_cost) << "\t\t" << cplex.getValue(tc2) << "\t\t" << cplex.getValue(tc) << endl; tc.end(); dis_cost.end(); rece_order.end(); tc2.end(); } add_cost.end(); } catch(IloException& ex) { //for any exceptions caught cerr << "Error : " << ex << endl; } catch(...) { cerr << "Error : Unknown exception caught!" << endl; } env.end(); return 0; }
C++ code for input file generation:dual-input.cpp #include "randomlib1.h" #include <iostream> #include <fstream> #define N 100000 #define M 25 /* Read readme.txt before getting confused with anything in the program*/ using std::cout; using std::cin; using std::endl; void set_zero(double[][N],int, int); int main() { //declaration of variables double lower[M],upper[M], //lower and upper limits mean[M],stddev[M], //mean and standard deviation sum[M][N],binn[M][N], //sum for each bin and # of #'s in bin bin_size[M],binl[N],binu[N], //size of the bin, limits of the bin ran,rann,f[N],cost[M]; //random number generated, mean of the bin, frequency int s,size, //# of distributions, sample size i,j,k,si,sj; //some variables, seed for RNG int n[M],choice[M]; char dummy[128];
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//open the input and then output file std::ifstream in("dual_source.in"); std::ofstream out("dual_source.dat"); in >> dummy >> s; for (i=0; i<s; i++) in >> dummy >> choice[i] >> mean[i] >> stddev[i] >> lower[i] >> upper[i] >> n[i]; in >> dummy; for (i=0; i<s-2; i++) in >> cost[i]; in >> dummy >> size >> dummy >> si >> sj; in.close(); //initialization set_zero(sum,s,N); set_zero(binn,s,N); /* Make sure the initialisation routine has been called */ if (!test) RandomInitialise(si,sj); // start the real work for (i=0; i<s; i++) { bin_size[i] = (upper[i]-lower[i])/(float)n[i]; k = 0; binl[k] = lower[i]; for (k=0; k<n[i]; k++) { binu[k] = binl[k] + bin_size[i]; if(k !=(n[i]-1)) binl[k+1] = binu[k]; } j = 0; while (j!=size) { if(choice[i] == 1) ran = RandomGaussian(mean[i],stddev[i]); else ran = RandomDouble(lower[i], upper[i]); for (k=0; k<n[i]; k++) { if(ran > binl[k] && ran < binu[k]) { j++; sum[i][k] = sum[i][k]+ran; binn[i][k] = binn[i][k]+1; } } } }
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out << "*Number_of_suppliers\n" << (s-2) << "\n" << "*C_normal\n" << "["; for (i=0;i<s-2; i++) { if (i !=s-3) out << cost[i] << ","; else out << cost[i] << "]\n"; } out << "*C_local\n" << "25\n" << "*C_back\n" << "200\n" << "*inventory_holding_h\n" << "15\n" << "*inventory_limit\n" << "20000\n" << "*C_prod_normal\n" << "50\n" << "*C_prod_emergency\n" << "125\n"; i = 0; // Writing Demand realization and disruptions for (i=0; i<2; i++) { if (i==0) out << "*theta\n"; else out << "*disruptions\n"; for (k=0; k<n[i]; k++) { rann = sum[i][k]/binn[i][k]; f[k] = binn[i][k]/size; if(k==0) out << "[" << rann << ","; else if(k<n[i]-1) out << rann << ","; else out << rann << "]" << endl; } if (i==0) out << "*w_theta-k1\n"; else out << "*w_d-k3\n"; for (k=0; k<n[i]; k++) { if(k==0) out << "[" << f[k] << ","; else if(k<n[i]-1) out << f[k] << ","; else out << f[k] << "]" << endl; } } out << "*E-k2\n" << "["; for (i=2; i<s; i++) { for (k=0; k<n[i]; k++) { rann = sum[i][k]/binn[i][k]; f[k] = binn[i][k]/size; if(k==0) out << "[" << rann << ","; else if(k<n[i]-1) out << rann << ","; else if(i==s-1) out << rann << "]"; else out << rann << "],"; } } out << "]\n*w_theta-k1\n["; for (i=2; i<s; i++) { for (k=0; k<n[i]; k++) { rann = sum[i][k]/binn[i][k]; f[k] = binn[i][k]/size;
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if(k==0) out << "[" << f[k] << ","; else if(k<n[i]-1) out << f[k] << ","; else if(i==s-1) out << f[k] << "]"; else out << f[k] << "]," ; } } out << "]\n*QL_max\n" << "100" <<"\n*Q_max\n15000" << "\n*TC_hat\n" << "250000" << "\n*alpha\n" << "0.1" << endl; out.close(); } void set_zero(double setting[][N],int m,int n) { int i,j; for (j=0; j<m; j++) { for (i=0; i<n; i++) setting[j][i] = 0; } } Input file for the CPLEX code: *Number_of_suppliers 2 *C_normal [10,9] *C_local 25 *C_back 200 *inventory_holding_h 15 *inventory_limit 20000 *C_prod_normal 50 *C_prod_emergency 125 *theta [1217.2,1247.12,1276.66] *w_theta-k1 [0.1,0.8,0.1] *disruptions [0.8,0.7,0.9] *w_d-k3 [0.15,0.75,0.1] *E-k2 [[0.878488,0.901713,0.924972],[0.999965,1.03996,1.07997]] *w_theta-k1 [[0.33,0.33,0.33],[ 0.33,0.33,0.33]] *QL_max 100
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*Q_max 15000 *TC_hat 145000 *alpha 0.01
