1 1 Review of Part I Preparation for Exam John H. Vande Vate Fall 2009

2 2 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

3 3 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

4 4 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

5 5 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

6 6 Exam Overview Typically 4-5 questions There will be modeling! Open books, open notes Calculator ok No computers No internet No collaboration

7 7 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

8 8 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

9 9 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?

10 Last Year’s Exam Consider the operations of a company, similar to the one we discussed in the lecture of August 25 th, 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.

11 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 = lbs per truck 5 lbs per unit = 7000 units Value of TL = $300*7000 = $2.1million Inventory in Green Bay $1.05million Units/day = 100 stores * 10 cpus/day = 1000 cpus Worth $300,000 Time units spend in transit is 1 days So pipeline inventory $300,000

12 What’s in truck to store? 20 TVs weigh200lbs and cost $ 8, Computers weigh150lbs and cost $ 7,000 So this "basket" weighs350lbs and costs $ 15,000 Since a truck holds 35,000lbs, 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

13 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/day Time units spend in transit is 2 days So pipeline inventory $30,000 per store

14 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/year Trucks/year = 250,000/7000 = 35.7 $/year = 35.7*$1.50/mile*500miles = $26,800 From Indianapolis to Store A truck holds 100 days of sales We visit each store 2.5 times per year 100 stores * $1500/visit *2.5 visits/year = $375,000 Capital At Green Bay: $1.05 million At Indianapolis: $1.8 million At each Store: $0.75 million Between GB and Indianapolis: $0.3 million Between Indianapolis and Stores: $3 million Total: $81.15 million Holding cost: $12.2 million

15 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: WeightCost per CWT ($/100lbs) 500 – 1000 lbs $ – 5000 lbs $ – lbs $ 9 > 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.

16 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!

17 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

18 With shipments ofUnit Costs are Total Transport to the stores is Shipment Size in Baskets isStore Inventory isTotal Cost 500 lbs14 $ 1,225, ,071,429 1,385, lbs12 $ 1,050, ,142,857 1,371, lbs9 $ 787, ,714,286 2,394, lbs7.25 $ 634, ,857,143 7,062,946 The Numbers Should we use LTL? Yes! What constitutes a shipment? 2.86 baskets 2.86*20 = 57 TVs 2.86*10 = 29 computers

19 Minimum Charge We exceeded it. No additional calculation needed. If we hadn’t? EPQ with fixed transport cost of $100

20 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).

21 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 */

22 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. */

23 AMPL Model Param 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 */.

24 AMPL Model Var 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 */

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

26 AMPL Model Minimize 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];

27 AMPL Model s.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];

28 AMPL Model s.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];

29 AMPL Model s.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];

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

31 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!

32 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?

33 Questions? Good Luck!