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1-to-1 Distribution

1-to-1 Distribution. John H. Vande Vate Spring, 2001. D(t) = cumulative demand to time t D’(t) = rate of demand at time t. Two cases: Only Rent Costs matter Only Inventory Costs matter. When Demand Varies Predictably. Make them all the same size

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1-to-1 Distribution

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  1. 1-to-1 Distribution John H. Vande Vate Spring, 2001 1

  2. D(t) = cumulative demand to time t D’(t) = rate of demand at time t. Two cases: Only Rent Costs matter Only Inventory Costs matter When Demand Varies Predictably 2

  3. Make them all the same size If we have n shipments in time t, make them all size D(t)/n Question reduces to n Trade off shipment cost (smaller n) vs Inventory cost (larger n) To Minimize the Maximum... 3

  4. Total cost with n shipments is Transportation cost (ignore variable portion) fixed*n Inventory Cost Rent Cost is $/unit/year Rent * (D(T)/n)*T Average Cost per unit Rent*T/n + fixed*n/D(T) So n is Rent*T*D(T)/fixed That’s just an EOQ problem 4

  5. More realistic - Ignore Rent • Wagner-Whitin Dynamic programming approach. • Computationally intensive • How accurate is the forecast of demand? • How sensitive is the cost to the answer? 5

  6. Discuss Later Wagner-Whitin 6

  7. The Continuous Approximation Approach There is some t’ where Area = (t3-t2)2D’(t’)/2 Fixed shipment cost + ci*area Area  (t3-t2)height/2 height  (t3-t2)D’(t’) t1 t3 t2 t0 7 t’

  8. H(t) = (ti - ti-1) if ti-1 t < ti So total cost = (Fixed + c*areai) = (Fixed + c*(ti - ti-1)2D’(ti’)/2) = (Fixed + c*H(ti’)2D’(ti’)/2) = (Fixed /H(t) + c*H(t)D’(t’)/2)dt Step Function An abuse of notation 8

  9. = (Fixed + c*H(ti’)2D’(ti’)/2) = (Fixed /H(t) + c*H(t)D’(t’)/2)dt Why? Fixed /H(t)dt = titi-1Fixed /H(t)dt = titi-1Fixed /(ti - ti-1)dt =  Fixed c*H(t)D’(t’)/2dt = titi-1c*H(t)D’(ti’)/2dt = titi-1c*(ti - ti-1)D’(ti’)/2dt =  c*(ti - ti-1)2D’(ti’)/2 =  c*H(t’i)2D’(ti’)/2 Equivalence 9

  10. Total cost = (Fixed + c*H(ti’)2D’(ti’)/2) = (Fixed /H(t) + c*H(t)D’(t’)/2)dt (Fixed /H(t) + c*H(t)D’(t)/2)dt Find a smooth function H(t) that minimizes the cost (an EOQ formula) H(t) = (2Fixed/cD’(t)) Approximation 10

  11. H(t) and Headways 11

  12. If Demand is constant with rate D’ We dispatch every t time units Cost per time = Fixed/t + ctD’/2 Best headway is t = (2Fixed/cD’) Compare with H(t) H(t) = (2Fixed/cD’(t)) What is H(t)? 12

  13. We have a continuous approximation H(t) to the discrete (step function) headways. How do we recover implementable headways from H(t)? Back to the Discrete World 13

  14. Finding Headways consistent with H(t) Headway = Avg of H(t) in [0, Headway] Avg = Integral of H(t) over the Headway/Headway Headway2 = Integral of H(t) over the Headway Find Headways so that the squares approximate the area under H(t) Consistent Headways 14

  15. Example 15

  16. H(t) and Headways 16

  17. Example Cont’d 17

  18. Inventory Cost 25.34 Shipment Cost 10 Total Cost 35.34 Is that any good? How’d we do? 18

  19. Optimum Answer set Periods; param Demand{Periods}; table DemandTable IN "ODBC" "DSN=Wagner" "Demand": Periods<-[Period], Demand; read table DemandTable; param FixedTransp := 1; param VarTransp := 1; param Holding := 1; /* $/unit/period */ var Inv{Periods} >= 0; /* Shipment quantity */ var Ship{Periods} binary; /* Whether or not we ship */ var Q{Periods} >= 0; /* Shipment size */ 19

  20. minimize TotalCost: sum{t in Periods} FixedTransp*Ship[t] + sum{t in Periods} VarTransp*Q[t] + sum{t in Periods} Holding*Inv[t]; s.t. InitialInventory: Q[1] - Inv[1] = Demand[1]; s.t. DefineInventory{t in Periods: t > 1}: Inv[t-1] + Q[t] - Inv[t] = Demand[t]; s.t. SetupOrNot{t in Periods}: Q[t] <= Ship[t]*sum{k in Periods: k >= t} Demand[k]; One Model 20

  21. Optimum Solution 25.5 Answer from CA Method 35.3 Comparison 21

  22. D’(t) changes pretty wildly Why so Bad? 22

  23. set Periods; param Demand{Periods}; table DemandTable IN "ODBC" "DSN=Wagner" "Demand": Periods<-[Period], Demand; read table DemandTable; param FixedTransp := 1; param VarTransp := 1; param Holding := 1; /* $/unit/period */ var Ship{Periods} binary; /* Amount we ship in period s that meets demand in period t */ var Q{s in Periods, t in Periods: t >= s} >= 0; More on the Optimization Model 23

  24. minimize TotalCost: sum{t in Periods} FixedTransp*Ship[t] + sum{s in Periods, t in Periods: t >= s} VarTransp*Q[s,t] + sum{s in Periods, t in Periods: t >= s} (t-s)*Q[s,t]; s.t. MeetDemand{t in Periods}: sum{s in Periods: s <= t} Q[s,t] = Demand[t]; s.t. ShipOrNot{s in Periods, t in Periods: t >=s}: Q[s,t] <= Ship[s]*Demand[t]; A Better Model 24

  25. Solves faster LP relaxation closer to MIP solutions Didn’t aggregate constraints Q[s,t] <= Ship[s]*Demand[t] Implies sum{t in Periods: t >= s} Q[s,t] <= Ship[s]*sum{t in Periods: t>= s} Demand[t]; Q[s] <= Ship[s]*sum{t in Periods: t >= s} Demand[t]; Why’s it Better 25

  26. A Computationally intensive Dynamic Programming Procedure for solving Why? Advantage/Disadvantage of CA over MIP Back To Wagner - Whitin 26

  27. ODBC = Open Data Base Connectivity ODBC Administrator : Control Panel DSN = Data Source Name Driver = Method for reading the DSN, e.g., Excel 97 Security and other features Aside on Importing Data 27

  28. Table <<tablename>> IN “ODBC” “DSN=<<dsnname>>” “tablename”: definedset <- [index], parametername~columnname, …; IN, OUT, INOUT SQL= sql statement With AMPL 28

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