1 1 Practice Exam #1 John H. Vande Vate Fall, 2002

2 2 Question 1 Our plant makes two products and ships them to customers across the country. Demand for the products is relatively constant at 10 units of each product each day at each customer. We have 100 customers across the country. Product Selling PriceWeight Product 1:$300 5 lbs Product 2:$ lbs Our trucks hold 30,000 lbs. The average distance to customers is 1,000 miles. We pay our carrier $1.00/mile. Our inventory carrying cost is 15%/year. Treat a year as 240 days.

3 3 Question #1 What is the optimal frequency for direct shipments to our customers?

4 4 Answer Frequency = Shipments/year = Annual Demand/Quantity Convert to packages: one unit of each product Q = packages in each shipment Total Cost =  Inventory Cost +  Transportation Cost Inventory Cost as a function of Q = 15%*Cost of Package*Q (or Q/2 debatable) = 15%*$400*Q = $60 Q

5 5 Shipping Cost = Cost per Shipment*D/Q D is a customer’s annual demand for pkgs = 2,400 Shipping Cost = $1,000*2,400/Q = $2,400,000/Q

6 6 Total Cost = Inventory Cost + Transp. Cost. = $60 Q + $2,400,000/Q Optimal Solution balances these costs Q* =  2,400,000/60 =  40,000 = 200 Will it fit? 200 pkgs * 35 lbs/pks = 7,000 lbs < 30,000 lbs. It fits.

7 7 And the answer is… 200. Right? Wrong! The FREQUENCY is Every 20 days Question to think about: ship to each our customers every 20 day? You think about it. We won’t say any more about this.

8 8 Question #2 Our carrier’s trucks average about 50 miles per hour and, because of Federal Transportation Regulations, about 10 hours per 24 hour day. Calculate the pipeline inventory costs inherent in the distribution system in Question 1

9 9 Answer Average Days in transit: 1,000 miles/500 miles/day = 2 days Daily Production Same as daily demand: 1,000 pkgs = 10 pkgs/customer*100 customers Pipeline Inventory 2,000 pkgs Value of the inventory $400/pkg*2,000 pkgs = $800,000 Costs of the inventory 15% /yr *$800,000= $120,000/year

10 Question #3 1 Develop a Flows-on-Paths formulation of th following problem: m factories 2 products. Fact. f provides up to S[f, p] units of prod. p n customers Customer c requires R[c, p] units of prod. p. 2 distribution centers Capacity of DC d is Capacity[d] units.

11 Projects: Schedule Channels 1.From the factories to the DCs. FCost[f,d] 2.Between the two DCs. DCost[d, d’] 3.From the DCs to the customers. TCost[d,c] Formulate a Flows-on-Paths model in an algebraic modeling language. You may assume the parameters S, R, Capacity, FCost, DCost and TCost are already defined and populated with data.

12 AMPL Model /* Not the most elegant version, but it works. A good introductory model */ set PLANTS; set DCS; set CUSTS; set PRODS; param S{PLANTS, PRODS}; param R{CUSTS, PRODS}; param FCost{PLANTS, DCS}; param DCost{d1 in DCS, d2 in DCS: d1 <> d2}; param TCost{DCS, CUSTS}; param Capacity{DCS};

13 AMPL Model /* Flow that use 1 dc. Flows1[p, d, c, prd] is the volume of product prd on the path from factory p to customer c via the single dc d. */ Var Flows1{PLANTS, DCS, CUSTS, PRODS} >=0;

14 AMPL Model Continued /* The paths that use both DCs */ Set LongPaths := setof {p in PLANTS, d1 in DCS, d2 in DCS, c in CUSTS: d1 <> d2}(p, d1, d2, c); /* This makes life easier later */ /* Here’s how we refer to the flows on these paths – by product */ Var Flows2{LongPaths, PROD} >= 0;

15 Project Topics /* Now it’s easy */ Minimize TransportCost: sum{p in PLANTS, d in DCS, c in CUSTS, prd in PRODS} (FCost[p, d]+TCost[d, c])*Flows1[p, d, c, prd] + Sum{(p, d1, d2, c) in LongPaths, prd in PRODS}(FCost[p, d1]+DCost[d1, d2]+TCost[d2, c])* Flows2[p, d1, d2, c, prd];

16 AMPL Model s.t. ObserveSupply{p in PLANTS, prd in PRODS}: Sum{d in DCS, c in CUSTS} Flows1[p, d, c, prd] + Sum{(p, d1, d2, c) in LongPaths}Flows2[p, d1, d2, c, prd] <= S[p, prd]; s.t. MeetDemand{c in CUSTS, prd in PRODS}: Sum{p in PLANTS, d in DCS} Flows1[p, d, c, prd] + Sum{(p, d1, d2, c) in LongPaths}Flows2[p, d1, d2, c, prd] >= R[c, prd];

17 AMPL Model s.t. DCCapacity{dc in DCS}: Sum{p in PLANTS, c in CUSTS} Flows1[p, dc, c, prd] + Sum{(p, dc, d2, c) in LongPaths}Flows2[p, dc, d2, c, prd] + Sum{(p, d1, dc, c) in LongPaths}Flows2[p, d1, dc, c, prd] <= Capacity[dc];

18 Final Question In class we discussed two models for the inventory at a production plant serving many identical customers with staggered shipments via a single loading dock. The final question on the exam will explore this issue. We won’t discuss it further.

19 Administration Seyhmus will bring exam to class ( or fax to video students) 60 copies Open book – Open notes exam YOUR OWN WORK Leave an empty chair between you and your neighbor. 4 questions 25 points each Show your work. Write clearly. Return SIGNED exam to Seyhmus