Models and Simulation for Bulk Gas Production/Distribution

Transcription

1 Models and Simulation for Bulk Gas Production and Distribution Wasu Glankwamdee Jackie Griffin Jeff Linderoth Jierui Shen ISE Department Lehigh University Jim Hutton Peter Connard Air Products EWO Annual Meeting Pittsburgh, Pennsylvania November 9, 2006

2 Liquid Bulk Gas Production-Distribution LIN LOX LOX Sites S LOX Products P = {LOX, LIN} Customers C LIN LOX LIN LIN LOX LIN LOX LOX LIN LOX Planning Problem How should one set production levels at the sites s S and sourcing decisions (amount delivered from s S to c C) in order to meet customer demand at minimum cost? Research Scope We are interested in understanding the benefit of considering finer-grain and stochastic/robust planning models.

3 Bulk Gas Wrinkles Production Most sites operate in two modes: Regular Mode Extended Mode (Costs more than regular) Physics of Production: Maximum total production (LOX + LIN) Individual production limit (Fraction of total) Competitor Arrangements Enter contractual take-or-pay arrangements with competitors Allowed to remove (equal) fixed amount of product from each other s sites: Q S : Set of Pick up locations R S : Set of Taken out locations

4 A Simple Planning Model Objective Production Cost + Distribution Cost min ps x ps + β ps e ps ) + p P s S(α (f sc y psc ) p P s S c C Variables x ps : Regular production amount of p P at s S e ps : Extended production amount of p P at s S y psc : Amount of p P shipped from s S to c C Parameters α ps : Regular mode per unit production cost of p P at s S β ps : Extended mode per unit production cost of p P at s S f sc per-unit delivery cost from s S to c C

5 Constraints Maximum Production Level x ps M s, e ps N s s S p P p P x ps Λ p M s, e ps Λ p N s p P, s S Parameters M s : Regular mode maximum total production at s S N s : Extended mode maximum total production at s S Λ P : Maximum air-fraction of p P This is a fairly crude model of production.

6 Constraints Contract Amount Limit x pq Φ p q Q Customer Demand y psc B pc, s S p P p P, c C Parameters Φ p : Contract amount for p P B pc : Customer c C demand for p P Wrinkles Can add (penalized) shortage variable v pc : y psc + v pc B pc s S

7 Constraints Site Inventory Balance x ps + e ps c C y psc z ps = I ps, p P, s S Resource: Driver Hours and Truck Hours d sc y psc D s, s S p P c C d sc y psc K ps, p P, s S c C Parameters I ps : Change in inventory of p P at s S z ps : Amount of p P competitor removes from s S D s : Available driver hours at s S K ps : Available truck hours of p P at s S

8 Using the Production-Distribution Model Model is used to set monthly production levels and customer sourcing decisions. Sometimes, during the course of the month, things get out of skew. There can be a disconnect between planning and scheduling! A customer is about to be run out. Plants don t have enough product to meet short-term customer demand. What Happens in Practice Daily planners attempt to do a manual adjustment to the monthly schedule in order to meet customer demand. Sometimes, the planning model will be re-run given the current (changed) input conditions.

9 Why? One Possible Explanation Known variation in plant supply and customer demand during the course of the month I(t) Customer Usage Aggregate/Prorated Delivery Volume True Customer Inventory t What s the Cure? The rescheduling burden can be alleviated by solving the model at a finer time aggregation.

10 Why? Another Possible Explanation Unknown (random) variation in components of the model A plant just went down for an unscheduled maintenance. A customer used three times as much as forecasted. I(t) t What s the Cure? In those cases, a planning model that directly deals with the inherent uncertainty might be warranted: Stochastic Programming Robust Optimization

11 What to Do? Research Scope We are interested in understanding the benefit of considering finer-grain and stochastic/robust planning models. Step #1 Build and experiment with a multi-period version of the planning model Allow us to experiment with instances solved at variety of time grains Need a multi-period model in order to create a stochastic model anyway

12 A Multi-Period Planning Model T = {1, 2,..., T }: Set of time periods Essentially add a time index to all the variables and parameters Make (site) inventory a variable that can be carried from one period to the next: x pst + e pst c C y psct z pst + I ps,t 1 I pst = 0 p, s, t Also add inventory cost to objective: γ ps I pst p P s S t T

13 Creating the Model Model is built and created in Mosel modeling language. If we wish to build a stochastic programming model later on, Mosel has modules for building stochastic programs. Model reads instance data from (properly formatted) text files.

14 The Facts 1 Models need data. 2 Data is hard to come by. 3 We will create our own data. Instance Generation-Simulation code created in C++. Data is random, but reasonable. Sites Daily Production Rate With Random Outage Customers: Normally Distributed On-Off Specific Needs Call-In As an added bonus, an entire distribution of performance statistics can be produced.

15 Normally Distributed Daily demand is N (µ, σ 2 ). Daily demand is independent.

16 Zero Or Normally Distributed (On-Off) With probability p > 0, daily demand is 0. Else, daily demand is N (µ, σ 2 ). Daily demand is independent.

17 Specific Needs Demand is typically N (µ, σ 2 ). But at certain points in time, the customer try a new process, requiring significantly more consumption. Daily demand is not independent.

18 Call-In Demand is typically 0. But at certain points in time, the customer require significant (nearly constant) daily consumption. Examples: Cranberry season. Flash-freezing fish harvest. Daily demand is not independent.

19 Class Structure InstanceFamily NumDays vector<site> vector<customer> Instance Something that can be solved! create(instancefamily &if, vector<int> &daysperperiod) Each class has a makerandom() method to instantiate itself with random, reasonable, data. When real data is available, we can extended the classes to instantiate themselves by reading from a file.

20 So What? The $64 Question What should we do with this framework? This is a fundamental question for an enterprise-wide optimization (EWO) initiative. Just because XPRESS says a solution is better doesn t make it so. How do you convince a user that an enterprise-engineered solution really will have a big impact on the operation?

21 Experiment Horse Race 0. t = Generate a solution (policy) x p : Single Period Model Multi-Period Model 2. Implement x p for some simulation increment. This costs c p t. 3. Re-sample data (demand). 4. The re-sampled data generates a new instance with different input state. Let t = t + 1. Goto 1.

22 More Code Details Find solution policy x p XPRMrunmod(ewoMod, &result, NULL)); Retrieve solution from solver into Instance class // Set the solution from the Mosel model instance.setsolution(ewomod);

23 Follow The Solution 1 Produce at rate suggested by solution for simulation increment amount of time. 2 Make deliveries for (vector<delivery>::const_iterator it = instance.getdeliveriesbegin(); it!= instance.getdeliveriesend(); ++it) { amt = lpsolutionvalue (from XPRESS); if (period == 1) { int n1 = sim.getsimulincrementsize(); int n2 = daysperperiod[period-1]; assert(n1 <= n2); double tamt = (amt * n1)/n2; double dc = instancefamily.makedelivery(p, s, c, tamt, n1); sim.incrementdeliverycost(dc); } }

24 Make Delivery Like Simulation // Deliver only if customer inventory will be less than safety // stock in the next period if (ci - predictdemand[0] < CUSTOMER_DAILY_SAFETY_STOCK) { // Deliver in an increment of TRUCK_SIZE while(ci - predictdemand[0] + damt < CUSTOMER_DAILY_SAFETY_STOCK) { damt += TRUCK_SIZE; } } // Here increment the date in instancefamily -- updates true // customer inventories based on a new sample instancefamily.movetimeforward(sim.getsimulincrementsize());

25 Preliminary Results C = 200, S = 10. Horizon: 28 days. Simulation Increment: 7 days. Total Cost (M) Customer Shortages Base Policy Avg Median Stdev Avg Median Stdev Single Period Two Periods Four Periods Base Policy RPC (M) EPC (M) DC (M) Single Period Two Periods Four Periods

26 Cost Distributions for Different Base Policies Single Period Two Periods Four Periods

27 Conclusions Many planning methodologies result in a disconnect between strategic and tactical solutions. An important question for EWO: What real impact can you expect if you do things differently? To help answer this question, simulation can be a useful tool. The finer time aggregation model shows promising results. As an added bonus, you can produce an entire distribution of performance statistics. Optimization Under Uncertainty Helping a decision maker to find a good distribution.

28 What s Next? A Stochastic Model The World Is Random! In the planning model, make parameters functions of some random variable ω M st (ω), N st (ω): Production Capacity Uncertain B pct (ω): Demand for product p from customer c in period t is uncertain z prt (ω): Amount of product p contract partner takes out at site r in period t is uncertain. Typically r R t T z prt(ω) = Φ p

29 We Love Σ, Ye We Do! min X X X η X n(α psnx psn+β psne psn+γ psi psn)+ X X X η X n(f scy pscn)+ X X η n p P s S n N p P s S c C n N p P c C n N (1a) X s.t x psn M sn, s S, n N (1b) p P X e psn N sn, s S, n N (1c) p P x psn Λ pm sn, p P, s S, n N (1d) e psn Λ pn sn, p P, s S, n N (1e) X y pscn + u pc,ρ(n) v pc,ρ(n) B pcn = u pcn v pcn, p P, c C, n N(1f) s S X y prcn I pr,ρ(n) x prn e prn + I prn = z prn, p P, r R, n N (1g) c C X y pscn I ps,ρ(n) x psn e psn + I psn = 0, p P, s S\R, n N (1h) c C X x pqn + φ pn = φ p,ρ(n), p P, n N (1i) q Q X X d scy pscn D sn, s S, n N (1j) p P c C X d scy pscn K psn, p P, s S, n N (1k) c C I psn U ps, p P, s S, n N (1l) I psn Γ psi ps0, p P, s S, n N (T (1m) + 1) φ pn Models 0, p and Simulation P, n for N Bulk Gas Production/Distribution (1n)

30 What s Next? Transfer Technology to Air Products Run on Real Data Need to write code to instantiate objects from data files Keep Considering Uncertainty! Minimax Model Robust Optimization Model Stochastic model Disruption Planning: consider more extreme scenarios. Help build contingency lists for sourcing decisions.

31 Next Steps Tactical Decision Making Adapt to run simulation on various tactical models. (Smaller time grain) Cost of Regionalization Use simulation to estimate the cost of first doing sourcing assignment, then setting production level and devliery amounts