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Review of Part IPreparation for Exam

John H. Vande VateFall 2009

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Review• Transportation costs are generally concave

– Economies of scale– Consolidation reduces transport costs

• But there are other financial/operational issues to balance– Operating Expense

• We focus on transport, but handling, labor, …

– Capital• We focus on working capital in inventory

– Time• We focus on OTD

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Review• Illustrated (some of) these trade-offs in our

“Case Study”– Estimating inventory costs

• “cycle” inventory driven by mode• Pipeline inventory driven by time and total demand

– The impact of crossdocking/consolidation on inventory

– Trading off inventory vs transportation• EOQ for 1-to-1• EPQ for 1-to-many

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Review

• After that we have to work harder …• Review of network models and useful extensions

– Modeling!• Pool points – Consolidating for speed

– Load driven systems• Zone skipping – Consolidating for cost (and

speed)– Service or schedule driven systems

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Review

• Multi-Stop routes & Milk runs– Digression into column generation – Application to multi-stop routes

• Location– Where to put consolidation/distribution

facilities• Landed cost models

– Incremental vs systems views

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Exam Overview

• Typically 4-5 questions• There will be modeling!• Open books, open notes• Calculator ok • No computers• No internet• No collaboration

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There will be modeling

• /* The objective: Minimize Transportation Cost in $/year */

• Minimize TransportCost: • Sum{(orig, dest) in LANES}TruckCosts[orig,

dest]*Trucks[orig, dest];

• Minimize

• You may provide answers using either notation.• Be sure your answers are clear and unambiguous.

– Define your variables – what are they?– Set out the units (e.g., $/mile, lbs/Truckload, …) – Be specific about indices of summation– Comment on what a constraint is designed to do

inLANESji

jiTrucksjiTruckCost),(

],[*],[

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Last Year’s Exam

• Question 1 (25 points) Basic understanding. Can you perform the kind of simple analysis we did for our case study company

• Question 2 (25 points) Elaboration of basic concepts. Apply the EPQ idea when costs include weight breaks and freight includes a mix of products

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Last Year’s Exam

• Question 3: Extending a basic model. Expand our basic consolidation model to address different products with different weights and dimensions

• Question 4: Theory through example. Did you understand the basic tenets of location?

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Last Year’s Exam• Consider the operations of a company, similar to the one we

discussed in the lecture of August 25th, that sells computers and TVs through 100 stores across the country.

• Assume the average distance to a store (from Indianapolis) is 1,000 miles and that a truck can travel 500 miles per day.

• As consumers adopted the new flat panel televisions, the business of the company has changed so that its stores sell 20 TVs and 10 computers (consisting of a CPU and a monitor) each day. (Assume 250 days in the year).

• The company closed down the operations in Denver and now produces – CPUs weighing 5 lbs and costing $300 each in Green Bay and – Flat Panel TVs and Monitors weighing 10 lbs and costing $400 each

in Indianapolis. • The distance from Green Bay to Indianapolis is 500 miles. • The company uses all full truck load shipments (a truck holds

35,000 lbs.) to ship everything to Indianapolis where it is consolidated and shipped in full truckload shipments to the stores.

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Question 1• Assuming an inventory holding cost of 15% and a

transportation cost of $1.50/mile compute:• The capital required to run the system including the capital:• at Green Bay: __$1,050,000 _____________ • at the cross dock in

Indianapolis:__$1,800,000_____________• at each store:__$750,000______________• in-transit between

– Green Bay and Indianapolis: __$300,000 (or $240,000 is more accurate)___________

– Indianapolis and each store: __$30,000____________

Units/Truck = 35000 lbs per truck 5 lbs per unit

= 7000 unitsValue of TL = $300*7000 = $2.1million

Inventory in Green Bay $1.05million

Units/day = 100 stores * 10 cpus/day = 1000 cpus

Worth $300,000Time units spend in transit is 1 days

So pipeline inventory $300,000

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What’s in truck to store?20 TVs weigh 200lbs and cost $ 8,000 10 Computers weigh 150lbs and cost $ 7,000 So this "basket" weighs 350lbs and costs $ 15,000

Since a truck holds 35,000 lbs, it can hold 100

• Value on a truck to a store– 100*$15,000 = $1.5 million

• Capital at a store: $750,000• Capital at Indianapolis cross dock

– $1.05 million from Green Bay– $0.75 million staged for store– $1.8 million total

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Question 1• Assuming an inventory holding cost of 15% and a

transportation cost of $1.50/mile compute:• The capital required to run the system including the capital:• at Green Bay: __$1,050,000 _____________ • at the cross dock in

Indianapolis:__$1,800,000_____________• at each store:__$750,000______________• in-transit between

– Green Bay and Indianapolis: __$300,000_________– Indianapolis and each store: __$30,000____________

1 Store sells $15,000/dayTime units spend in transit is 2 days

So pipeline inventory $30,000 per store

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Question 1 Cont’d

• The total cost of operating the system including:

– Annual transportation costs: __$402,000 (401,786)_ and

– Annual inventory holding costs: _$12,172,500 (or $12,163,500)_

From Green Bay to Indianapolis: Units per year = 100 stores * 10 units per day *250 days = 250,000

units/yearTrucks/year = 250,000/7000 = 35.7$/year = 35.7*$1.50/mile*500miles

= $26,800From Indianapolis to Store

A truck holds 100 days of salesWe visit each store 2.5 times per year

100 stores * $1500/visit *2.5 visits/year= $375,000

Capital At Green Bay: $1.05 millionAt Indianapolis: $1.8 millionAt each Store: $0.75 million

Between GB and Indianapolis: $0.3 millionBetween Indianapolis and Stores: $3 million

Total: $81.15 million

Holding cost: $12.2 million

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Question 2• Consider the company described in Question 1. The company is exploring

the option of replacing truck load shipments from Indianapolis to its stores with LTL shipments.

• As a first approximation to the magnitude of the opportunity, the company has assembled an estimate of average LTL rates for shipments to customers 1,000 miles away. For simplicity, they have averaged out geographic aspects (e.g., shipping to customers in Florida is more expensive) and come out with average costs based only on the weight of the shipment:

• Weight Cost per CWT ($/100lbs)• 500 – 1000 lbs $14• 1000 – 5000 lbs $12• 5000 – 20000 lbs $ 9• > 20000 lbs $ 7.25• The minimum charge for any shipment is $100. So for example, a 500 lb

shipment nominally costs $70 = $14/CWT*5 CWT, but the carrier won’t accept less than $100 for any shipment so the actual cost is $100.

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Thinking

• What are the options?• Claim: Ship either

– $100 shipping cost– 500 lbs– 1,000 lbs– 5,000 lbs or– 20,000 lbs

• Why?

If you ship 800 lbs, transport costs are the same as if you ship 500 lbs but

inventory costs are higher!

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Calculations

• What are we shipping? Basket of products– weighs 3.5 CWT– Worth $15,000– Holding cost/year: $2,250– Annual demand at a store: 250

• Minimum Charge: • $100/14 = 7.14 CWT

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With shipments of Unit Costs areTotal Transport to the

stores isShipment Size in

Baskets is Store Inventory is Total Cost

500 lbs 14 $ 1,225,000 1.43 1,071,429 1,385,714

1000 lbs 12 $ 1,050,000 2.86 2,142,857 1,371,429

5000 lbs 9 $ 787,500 14.29 10,714,286 2,394,643

20000 lbs 7.25 $ 634,375 57.14 42,857,143 7,062,946

The Numbers

Should we use LTL? Yes!

What constitutes a shipment?2.86 baskets

2.86*20 = 57 TVs2.86*10 = 29 computers

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Minimum Charge

• We exceeded it. No additional calculation needed.

• If we hadn’t? EPQ with fixed transport cost of $100

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Question 3• In class, we outlined a model to help identify which candidate consolidation points to use and to assign

customers to those consolidation points to minimize the costs of transportation while meeting a “service constraint” imposed in terms of a minimum number or frequency of trucks to each opened consolidation point.

• This question asks you to flesh out that formulation for a setting in which we sell several different products. Let PRODS denote the set of products we sell.

• Each customer has a projected demand for each product given in the parameter Demand. Let CUSTS denote the set of customers and, for each customer c and product p, let Demand[c, p] be the customers demand for that product in units per year.

• Each product has a cubic ft. per unit given in the parameter Cubes, i.e., Cubes[p] is the cubic feet occupied by one unit of product p.

• Each product has a unit weight given in the parameter Weight, i.e., Weight[p] is the weight in pounds of one unit of product p.

• Let CONSOLS denote the set of candidate consolidation points and suppose the • LTL (less-than-truckload) costs for shipping each customer’s annual demand for • each product from each candidate consolidation point are given in the parameter • LTL, i.e., LTL[c, p, k] is the LTL cost for shipping all of customer c’s annual • demand for product p to the customer from consolidation point k. • The LTL costs for direct shipments for the annual demand of each product from our plant to each customer are

given in the parameter Direct, i.e., Direct[c, p] is the LTL cost for shipping all of customer c’s annual demand for product p directly to the customer from the plant.

• The cost to send a truck from our plant to each candidate consolidation point are given in a parameter TruckCost, i.e., TruckCost[k] is the cost to send one truck from the plant to consolidation point k.

• A truck can hold (with the load factor calculated in) up to 30,000 lbs and up to 3,000 cubic feet (again this number incorporates the load factor).

• Our service requirement stipulates that we send at least 112 trucks a year to each open consolidation point. • Formulate a linear, mixed integer program to model the problem of minimizing the total cost of

transportation while meeting the service requirement. Do NOT consider multi-stop routes in your answer. Be sure to use the parameters described above. Be sure to clearly define your variables and their units (e.g., lbs, $, hours).

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AMPL Model

Set PRODS; /* The set of Products */Set CUSTS; /* The set of customers */Set CONSOLS; /* The set of candidate

consolidation points */Param Demand{CUSTS, PRODS}; /*

customer’s demand for each product in units per year. */

Param Cubes{PRODS}; /* the cubic feet occupied by one unit of each product */

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AMPL Model

Param Weight{PRODS}; /* the weight in pounds of one unit of each product */

Param LTL{CUSTS, PRODS, CONSOLS}; /* the LTL cost for shipping all of a customer annual demand for product p to the customer from consolidation point k. */

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AMPL ModelParam Direct{CUSTS, PRODS}; /* the LTL cost for shipping all of a

customer’s annual demand for each product directly to the customer from the plant. */

Param TruckCost{CONSOLS}; /* the cost to send one truck from the plant to consolidation point */.

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AMPL ModelVar Open{CONSOLS} binary; /* Whether each consol is open or not */Var Trucks{CONSOLS} integer >= 0; /* How many trucks we send to each

consol */Var Assign{CUSTS, PRODS, CONSOLS} >= 0; /* Fraction of demand for each

product to each customer we ship via each consol – note we allow fractions */

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AMPL Model

Var DirectShip{CUSTS, PRODS} >= 0; /* Fraction of demand for each product to each customer we ship direct from the plant */

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AMPL ModelMinimize Transport Cost: Sum{c in CUSTS, p in PRODS, k in CONSOLS} LTL[c, p, k]*Assign[c, p, k] + Sum{k in CONSOLS} TruckCost[c]*Trucks[c] + Sum{c in CUSTS, p in PRODS} Direct[c, p]*DirectShip[c, p];

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AMPL Models.t. MeetAllDemandForEachProductAtEachCustomer {c in CUSTS, p in PRODS}: sum{k in CONSOLS}Assign[c, p, k] + DirectShip[c, p] = 1;s.t. DontAssignCustomersToClosedConsols {c in CUSTS, p in PRODS, k in CONSOLS}: Assign[c, p, k] <= Open[k];

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AMPL Models.t. MeetAllDemandForEachProductAtEachCustomer {c in CUSTS, p in PRODS}: sum{k in CONSOLS}Assign[c, p, k] + DirectShip[c, p] = 1;s.t. DontAssignCustomersToClosedConsols {c in CUSTS, p in PRODS, k in CONSOLS}: Assign[c, p, k] <= Open[k];

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AMPL Models.t. SendEnoughTrucksToEachConsolToCarryCube {k in CONSOLS}: 3000*Trucks[k] >= sum{c in CUSTS, p in PRODS} Cubes[p]*Demand[c, p]*Assign[c, p, k];s.t. SendEnoughTrucksToEachConsolToCarryWeight {k in CONSOLS}: 30000*Trucks[k] >= sum{c in CUSTS, p in PRODS} Weight[p]*Demand[c, p]*Assign[c, p, k];

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AMPL Model

s.t. MeetServiceRequirementAtOpenConsols {k in CONSOLS}: Trucks[k] >= 112*Open[k];

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Question 4 • In our discussions on Location problems, we

observed that locating a facility at the “center of gravity” of a set of customers (average the x and y coordinates of the customers) does NOT minimize the sum of the Euclidean Distances to those customers.

• Does locating a facility at the “center of gravity” of a set of customers minimize the sum of the distances to those customers under the Manhattan Metric, where the distance between two points (x, y) and (x’, y’) is

|x – x’| + |y – y’|? No!

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Question 4

• If you answered “Yes” to part A, provide a brief argument supporting your conclusion. If you answered “No” to part A, provide an example showing the center of gravity is not the best location.

• Counterexample?

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Questions?

Good Luck!