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Project #1Optimization Model

John H. Vande Vate

Spring, 2001

22

Careful about Words...

• Plants can ship to – DCs (direct) or– The Warehouse (indirect)

• All combined shipments come from the warehouse (not the Indiana plant).

• All Monitors, TV’s and Keyboards are shipped via the warehouse.

33

Without Optimization

• For each DC there are 4 options:– All parts direct– All parts via Indianapolis warehouse– Green Bay & Indianapolis ship via warehouse– Denver & Indianapolis ship via warehouse

• Easy to calculate which of these is cheaper ….

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But...

• Ignoring the inventory implications at the plants and warehouse for indirect shipments

• Why?

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Not per DC

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Inventory at the Warehouse

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Approaches

• Inbound: Half a truckload for each plant that ships to the warehouse

• Outbound: ?

• The problem of different headways

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Time

• How much does indirect shipment add to the time to market?

Production DemandProduction

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Major weakness of Model

• It ignores the pipeline inventory

• This can have a significant impact on inventory!

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EOQ

• How to calculate the optimal shipment size?

• Is the cost per unit:– Distance/Q + Interest Rate*Value or– Distance/Q + Interest Rate*Value/2?

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It Depends!

• The EOQ formula– Inventory at the DC: Proportional to Q/2– Transportation Cost: Proportional to Demand/Q– Inventory at the Plant: ?

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Inventory at the Plant

• item-days inventory at the plant accumulated for each shipment to DC #1, say, if the shipment size is Q?

• Q2/(2*Production Rate)

Q

Q/Production Rate

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Total Item-Days

• How many such shipments will there be?

• Annual Demand at DC #1/Q

• So, the total item-days per year from shipments to DC #1 will be…

• Q2/(2*Production Rate)*Demand at DC/Q

• Q*Demand at DC/(2*Production Rate)

• So, making shipments of size Q to DC #1 adds what to the average inventory at the plant?

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Effect on Average Inventory

• Q*Demand at DC/(2*Production Rate)

• Example: Q*2500/(2*100*2500) = Q/200

• Correct EOQ for Direct Shipments:

• Total Cost: carrying cost*Q*Demand at DC/(2*Production Rate)

+ carrying cost*Q/2

+transport cost*Demand at DC/Q

Q* = 2*transport*Demand/carrying cost P/(D+P)

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In Our Case

• Since Demands at the n DCs are equal P/(D+P) = nD/(nD+D) = n/(n+1) • Q* = 2*transport*Demand/carrying cost P/(D+P)• Q* = 2*transport*Demand/carrying cost n/(n+1)

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Will this Work at the Warehouse?• “Outbound” Inventory of monitors

• Consider shipments to a DC the gets CPU’s and Consoles direct from the plants -- shipping only Monitors, etc. from Warehouse

• Analysis is identical

• Q/2 * Q/P * D/Q = Q*D/(2*P)

Half the shipment size

How long to accumulate the shipment

The number of shipments/year

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What about the other products?

• Consider a shipment of Monitors and CPU’s

• What is the production rate?

• How long does it take to accumulate the shipment?

Q

Q/Production Rate

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Assumption• Q is the number of “assemblies” shipped• Assembly is Monitor, Keyboard, TV and CPU• How long does it take to accumulate the shipment?• How long does it take to accumulate the

assemblies?• Assumption: Allocate Monitors etc. (which arrive

faster) to assemblies at the rate the CPUs arrive and the remainder to pure shipments.

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Example

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Can’t Model it Exactly• “Outbound” inventory for part at Warehouse

from shipments to DC • EOQ/2 * EOQ/P * D/EOQ = EOQ*D/(2P)• P = annual vol. of option thru warehouse

1 ship opt to dc via warehouse 0 otherwise

• P = sum{dc in DCs}Demand for option at DC* x[dc]

• In the denominator! That’s not linear

{Binary variable x[dc] =

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But, ….

• If all the EOQ*D’s are roughly equal we might approximate

sum{dc in DCs} h * EOQ[dc]/2 *D*x[dc]/P

sum{dc in DCs} h * EOQ[dc]/2 *(x[dc]/sum x[k])

h *(avg EOQ/2)*y

Where y is a binary variable that models

sum{dc in DCs} x[dc]/sum{k in DCs} x[k]

1 if any x[dc] =1 and

0 otherwise. { Binary variable y =

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Without Optimization

• Ignore inventory at the plants and at the warehouse

• Calculate best strategy at each dc

• All Direct: $4.5 million

• Only Denver via Indianapolis: $4.1 million

• Only GB via Indianapolis: $3.5 million

• All via Indianapolis: $3.0 million

• Best at each DC: $3.0 million

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Overview

• Sets– PLANTS– DCS– OPTIONS: All direct, none direct, ...

• Parameters– Distances– EOQs

• for direct shipments and • each option via the warehouse

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Variables

• Select an option for each dc

• Whether or not each plant ships to the warehouse

• Optionally, indicate whether or not each plant serves each dc directly (can be inferred from the option chosen.

• Whether or not the warehouse shipped each option to some dc

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Objective

• Transportation Costs

• Inventory Costs– Carrying cost for 1/2 the appropriate EOQ

value at the DCs– 1/2 the average EOQ value by option at the

warehouse (outbound)– Carrying cost of 1/2 a truckload at the

warehouse for each plant that ships to the warehouse (inbound)

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Inventory at the Plants

Shipments of size Q induce inventory of

Q*Demand at Destination/(2*Production Rate)

at the Plant• Q is the

– the EOQ for the DCs

– a truckload for the warehouse

• sum{dc in DCs}EOQ[dc]/(2*100)*Did we ship direct? +• sum{dc in DCs}TL/(2*weight*100)*Did we ship via warehouse?

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Constraints

• Select a single option at each dc

• Whether or not each plant ships to the warehouse

• Whether or not the warehouse ships each option.

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Reports

• To a manager, not a professor

• Executive Summary -- the business!

• Solution Details

• Appendix– Modeling Assumptions– Model Description– Model itself

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Executive Summary• Overview of Solution• Take solution back through “exact” calculations and

report– Total Cost– Breakdown of

• Transportation Cost• Inventory Cost

– Transportation Cost• Breakdown by

– Plant to Warehouse

– Plant to DCs

– Warehouse to DCs

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Exec Summary Cont’d

– Inventory (Volume, Value, Carrying Cost)• Breakdown by

– Plant

– Warehouse

» CPU

» Monitor

» Console

– DCs (overall average, min and max)

» CPU

» Monitor

» Console

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Exec Summary - Impacts

• Time to market (total days in inventory)– Console

– CPU

– Monitors

• Days in Inventory at DCs (avg, min, max)– Console

– CPU

– Monitor

• Any significant regional differences..

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What if...

• All via Indianapolis …

• Warehouses at other plants (above and beyond for class,…)

• ...

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Solution Details

• More detailed analysis of results summarized in Exec. Summary

• Charts, Tables, Graphs

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Appendix

• Assumptions, Approximations, Observations

• How we handled Indianapolis vs. Warehouse

• Approach to Inventory at the Warehouse and at Plants

• EOQ further discussion

• ...