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1 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 Inventory Optimization under Correlated Uncertainty Abhilasha Aswal G N S Prasanna, International.

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Presentation on theme: "1 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 Inventory Optimization under Correlated Uncertainty Abhilasha Aswal G N S Prasanna, International."— Presentation transcript:

1 1 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 Inventory Optimization under Correlated Uncertainty Abhilasha Aswal G N S Prasanna, International Institute of Information Technology – Bangalore

2 2 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 Outline  Motivation  Optimizing with correlated demands  Generalized EOQ  Related work  Some Extensions:  Generalized base stock  Geman Tank  Relational Algebra  Conclusions

3 3 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 The EOQ model  The EOQ model (Classical – Harris 1913)  C: fixed ordering cost per order  h: per unit holding cost  D: demand rate  Q * : optimal order quantity  f * : optimal order frequency Q*Q*

4 4 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 Inventory optimization for multiple products EOQ(K)?

5 5 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 Motivation  Inventory optimization example Automobile store Car type I Car type II Car type III Tyre type I Tyre type II Petrol Drivers Supplies

6 6 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 Motivation  Ordering and holding costs Product Ordering Cost in Rs. (per order) Holding Cost in Rs. (per unit) Car Type I100050 Car Type II100080 Car Type III100010 Tyre Type I2500.5 Tyre Type II500 (intl shipment)0.5 Petrol6001 Drivers750300

7 7 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 1 product versus 7 products  Exactly Known Demands, no uncertainty  EOQ solution and Constrained Optimization solution match exactly:  But… Product Demand per month EOQ SolutionConstrained Optimization Solution Order Frequency Order Quantity Cost Order Frequency Order Quantity Cost Car Type I401 20001402000 Car Type II251 20001252000 Car Type III500.510010000.51001000 Tyre Type I2500.55002500.5500250 Tyre Type II1250.255002500.25500250 Petrol3000.5600 0.5600 Drivers515150015 Total7600 UNREALISTIC!!! We cannot know the future demands exactly.

8 8 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 1 product versus 7 products  Bounded Uncorrelated Uncertainty  Assuming the range of variation of the demands is known, we can get bounds on the performance by optimizing for both the min value and the max value of the demands.  EOQ solution and Constrained Optimization solution are almost the same. Product EOQ solutionConstrained Optimization Order FrequencyOrder QuantityOrder FrequencyOrder Quantity MinMaxMinMaxMinMaxMinMax Car Type I0.5120400.512040 Car Type II01025010 Car Type III0.511002000.51100200 Tyre Type I0.250.5248.995000.250.5248500 Tyre Type II0.250.550010000.250.55001000 Petrol0.250.53006000.250.5300600 Drivers0.4512.2450.5125

9 9 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 1 product versus 7 products  Beyond EOQ: Correlated Uncertainty in Demand  Considering the substitutive effects between a class of products (cars, tyres etc.) 200 ≤ dem_tyre_1 + dem_tyre_2 ≤ 700 65 ≤ dem_car_1 + dem_car_2 + dem_car_3 ≤ 250  Considering the complementary effects between products that track each other 5 ≤ (dem_car_1 + dem_car_2 + dem_car_3) – dem_petrol ≤ 20 5 ≤ dem_car_2 – dem_drivers ≤ 20 EOQ cannot incorporate such forms of uncertainty.

10 10 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 1 product versus 7 products  Beyond EOQ: Correlated Uncertainty in Demand  Min-Max solution for different scenarios: Products With Substitutive Constraints With Complementary Constraints With both Substitutive and Complementary constraints Order Frequency Order Quantity Order Frequency Order Quantity Order Frequency Order Quantity Car Type I 0.7525 0.5380.540 Car Type II 0.513 0.522110 Car Type III 0.75125 0.751210.5180 Tyre Type I 0.25362 0.752500.75200 Tyre Type II 0.75500 0.753730.5400 Petrol 0.5400 0.52080.5222.5 Drivers 0.55 2 3 Cos t (Rs.) 4590.438 4593.6884654.188 EOQ Order Frequency Order Quantity 140 125 0.5100 0.5500 0.25500 0.5600 15 7600

11 11 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 1 product versus 7 products  Beyond EOQ: Correlated Uncertainty in Demand  Comparison of different uncertainty sets Scenario setsAbsolute Minimum CostAbsolute Maximum Cost Bounds only3349.59187.5 Bounds and Substitutive constraints 3412.59100 Bounds and Complementary constraints 4469.58972.5 Bounds, Substitutive and Complementary constraints 4482.58910

12 12 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 Optimizing with Correlated Demands Mathematical Programming Formalism

13 13 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 Optimal Inventory policy using “ILP”  Min-max optimization, not an LP.  Duality??  Fixed costs and breakpoints: non- convexities that preclude strong-duality from being achieved.  No breakpoints or fixed costs: min-max optimization  QP  Heuristics have to be used in general.

14 14 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 Optimal Inventory policy by Sampling  A simple statistical sampling heuristic Begin for i = 1 to maxIteration { parameterSample = getParameterSample(constraint Set) bestPolicy = getBestPolicy(parameterSample) findCostBounds(bestPolicy) } chooseBestSolution() End

15 15 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 Optimizing with Correlated Demands: Analytical Formulation: Generalized EOQ(K)

16 16 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 Classical EOQ model  Per order fixed cost = f(Q)  holding cost per unit time = h(Q)

17 17 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 EOQ(K) with multiple products, uncertain demands  Additive SKU costs Case with 2 commodities, generalized to n commodities

18 18 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 EOQ(K) with multiple products, uncertain demands  Holding cost linear, ordering cost fixed

19 19 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 Analytical solution: Substitutive constraints  Holding cost linear, ordering cost fixed  Under a substitutive constraint D 1 + D 2 <= D

20 20 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 Analytical solution: Substitutive constraints - Example  2 products, demands D 1 & D 2  Costs: h 1 = 2/unit h 2 = 3/unit f 1 = 5/order f 2 = 5/order  D 1 + D 2 = D = 100  Maximum cost  Minimum cost

21 21 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 Analytical solution: Complementary constraints  Holding cost linear, ordering cost fixed  Under a complementary constraint D 1 – D 2 <= D, with D 1 and D 2 limited to D max

22 22 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 Analytical solution: Complementary constraints - Example  2 products, demands D 1 & D 2  Costs: h 1 = 2/unit h 2 = 3/unit f 1 = 5/order f 2 = 5/order  Demand constraints: D 1 - D 2 = K = 20 D 1 <= D max = 50 D 2 <= D max = 50  Maximum cost  Minimum cost

23 23 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 Both substitutive & complementary constraints  Holding cost linear, ordering cost fixed  Under both substitutive and complementary constraints  Convex optimization techniques are required for this optimization.

24 24 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 Both substitutive & complementary constraints - Optimization  Objective function: concave  Minimization: HARD!  Envelope based bounding schemes  Heuristics to find upper bound.  Simulated annealing based

25 25 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 Both substitutive & complementary constraints - Example  2 products, demands D 1 & D 2  Costs: h 1 = 2/unit h 2 = 3/unit f 1 = 5/order f 2 = 5/order  Demand constraints: 150 <= D 1 + D 2 <= 200 -20 <= D 1 – D 2 <= 20  Maximum cost: 99.88  Minimum cost  Enumerating all vertices (exact) 85.39  Simulated annealing heuristic 85.48499  Error: 0.111247 %

26 26 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 Both substitutive & complementary constraints – Example (contd)  5 products, demands D 1, D 2, D 3, D 4 & D 5  Costs: h 1 = 2/unit h 2 = 3/unit h 3 = 4/unit h 4 = 5/unit h 5 = 6/unit f 1, f 2, f 3, f 4, f 5 = 5/order  Demand constraints : D 1 + D 2 + D 3 + D 4 + D 5 <= 1000 D 1 + D 2 + D 3 + D 4 + D 5 >= 500 2 D 1 - D 2 <= 400 2 D 1 - D 2 >= 100 5 D 5 - 2 D 4 <= 900 5 D 5 - 2 D 4 >= 150 D 2 + D 4 <= 400 D 2 + D 4 >= 250 D 1 <= 350 D 1 >= 100 D 3 >= 150 D 3 <= 300 D 4 >= 75 D 4 <= 200

27 27 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 Both substitutive & complementary constraints – Example (contd)  Maximum cost: 436.6448  Minimum cost:  Enumerating all vertices (exact) 323.5942  Simulated annealing heuristic 324.4728  Error: 0.271505 %

28 28 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 Inventory constraints  Constrained Inventory Levels  If the inventory levels Qi and demands Di, are constrained as  The vector constraint above can incorporate constraints like  Limits on total inventory capacity (Q1+Q2 <= Q tot )  Balanced inventories across SKUs (Q1-Q2) <= ∆   Inventories tracking demand (Q1-D1<=D max )

29 29 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 Inventory constraints  Constrained Inventory Levels

30 30 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 Related Work  McGill (1995)  Inderfurth (1995)  Dong & Lee (2003)  Stefanescu et. al. (2004)  Bertsimas, Sim, Thiele et. al.

31 31 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 Related work  Bertsimas, Sim, Thiele - “Budget of uncertainty”  Uncertainty:  Normalized deviation for a parameter:  Sum of all normalized deviations limited:  N uncertain parameters  polytope with 2 N sides  In contrast, our polyhedral uncertainty sets:  More general  Much fewer sides

32 32 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 Extensions: Generalized basestock German Tank

33 33 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 Basestock with correlated inventory

34 34 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 The German Tank Problem Classical German Tank  Biased estimators  Maximum likelihood  Unbiased estimators  Minimum Variance unbiased estimator (UMVU)  Maximum Spacing estimator  Bias-corrected maximum likelihood estimator Generalization  Given correlated data samples, drawn from a uniform distribution- estimating the bounded region formed by correlated constraints enclosing the samples.  Estimating the constraints without bias and with minimum variance.

35 35 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 Information Theory and Relational Algebra  Uncertainty can be identified with Information.  Information  polyhedral volume  Relational algebra between alternative constraint polyhedra

36 36 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 Conclusions  Generalized EOQ to Correlated Demands  Analytical Solutions  Computational Solutions  Enumerative versus Simulated Annealing  Extensions of formulations  Generalized Basestock  German Tank  Information Theory and Relational Algebra

37 37 Abhilasha Aswal & G N S Prasanna IIIT-B INFORMS 2010 Thank you

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