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

11

1-to-1 Distribution

John H. Vande Vate

Spring, 2001

22

When Demand Varies Predictably

• 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

33

To Minimize the Maximum...

• 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)

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That’s just an EOQ problem

• 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

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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?

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Wagner-Whitin

• Discuss Later

77

The Continuous Approximation Approach

t0t1 t2 t3

Fixed shipment cost + ci*area

t’

Area (t3-t2)height/2

There is some t’ where Area = (t3-t2)2D’(t’)/2

height (t3-t2)D’(t’)

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Step Function

• 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

An abuse of notation

99

Equivalence= (Fixed + c*H(ti’)2D’(ti’)/2)

= (Fixed /H(t) + c*H(t)D’(t’)/2)dt

Why?

Fixed /H(t)dt = ti ti-1

Fixed /H(t)dt

• = ti ti-1

Fixed /(ti - ti-1)dt

• = Fixed

c*H(t)D’(t’)/2dt = ti ti-1

c*H(t)D’(ti’)/2dt

• = ti ti-1

c*(ti - ti-1) D’(ti’)/2dt

• = c*(ti - ti-1)2 D’(ti’)/2

• = c*H(t’i)2 D’(ti’)/2

1010

Approximation

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))

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H(t) and Headways

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H(t)

h

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What is H(t)?

• 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))

1313

Back to the Discrete World

• We have a continuous approximation H(t) to the discrete (step function) headways.

• How do we recover implementable headways from H(t)?

1414

Consistent Headways

• 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)

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Example

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0.8

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

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D'(t)

D(t)

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H(t) and Headways

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H(t)

h

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Example Cont’dt D'(t) D(t) H(t) Integral of H(t) t*t h Integral of H(t) t*t h

1 0.12 0.60 4.09 4.09 1 52 0.12 1.20 4.09 8.18 4 53 0.12 1.79 4.09 12.27 9 54 0.05 2.07 6.04 18.31 16 55 0.05 2.34 6.04 24.35 25 56 0.05 2.62 6.04 30.40 36 6.04 1 37 0.38 4.53 2.29 32.68 49 8.33 4 38 0.38 6.44 2.29 34.97 64 10.62 9 39 0.38 8.35 2.29 37.26 81 12.91 16

10 0.95 13.08 1.45 38.71 100 14.36 2511 0.95 17.81 1.45 40.17 121 15.81 3612 0.95 22.55 1.45 41.62 144 17.27 4913 0.98 27.42 1.43 43.05 169 18.70 6414 0.98 32.30 1.43 44.48 196 20.13 8115 0.98 37.18 1.43 45.92 225 21.56 10016 0.75 40.94 1.63 47.55 256 23.19 12117 0.75 44.71 1.63 49.18 289 24.82 14418 0.75 48.47 1.63 50.80 324 26.45 16919 0.73 52.11 1.66 52.46 361 28.11 19620 0.73 55.75 1.66 54.12 400 29.77 225

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How’d we do?

• Inventory Cost 25.34

• Shipment Cost 10

• Total Cost 35.34

• Is that any good?

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Optimum Answerset 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 */

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One Model

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];

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Comparison

• Optimum Solution 25.5

• Answer from CA Method 35.3

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Why so Bad?

• D’(t) changes pretty wildly

t D'(t)1.00 0.12 2.00 0.12 3.00 0.12 4.00 0.05 5.00 0.05 6.00 0.05 7.00 0.38 8.00 0.38 9.00 0.38

10.00 0.95 11.00 0.95 12.00 0.95 13.00 0.98 14.00 0.98 15.00 0.98 16.00 0.75 17.00 0.75 18.00 0.75 19.00 0.73 20.00 0.73

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D'(t)

D(t)

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More on the Optimization Modelset 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;

2424

A Better Model

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];

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Why’s it Better• 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];

2626

Back To Wagner - Whitin

• A Computationally intensive Dynamic Programming Procedure for solving

• Why?

• Advantage/Disadvantage of CA over MIP

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Aside on Importing Data

• 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

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With AMPL

• Table <<tablename>> IN “ODBC”

• “DSN=<<dsnname>>”

• “tablename”:

• definedset <- [index], parametername~columnname, …;

• IN, OUT, INOUT

• SQL= sql statement