Introduction to Optimization Introduction to Optimization

ancmat02.qxd 11/16/06 3:26 PM Page B24B24 • SUPPLEMENT BINTRODUCTION TO OPTIMIZATIONProblems1. DynaTune is trying to determine how to allocate its$145,000 advertising budget for a new product. Radio and televisionare the two media outlets selected for advertising. The followingtable shows the costs of advertising in these two mediaand the estimated number of new customers reached by increasingamounts of advertising.Media Type & Number NewNumber of Ads Customers Reached Cost per AdRadio 1–10 900 $1,000Radio 11–20 700 $900Radio 21–30 400 $800Television 1–5 10,000 $12,000Television 6–10 7,500 $10,000Television 11–15 5,000 $8,000For example, each of the first 10 ads the company places on theradio will cost $1000 and is expected to reach 900 new customers.Each of the next 10 will cost $900 and is expected to reach 700new customers. Assume the company will purchase no more than30 radio ads and no more than 15 television ads.(a) Formulate a LP to maximize the number of new customersreached.(b) Set up and solve the problem on a spreadsheet.(c) What is the optimal solution? Explain the rationale forthe solution.(d) Suppose the number of new customers reached by 11–15television ads is 7500 and the number reached by 6–10ads is 5000. Resolve the problem. What is wrong with thissolution? Is Solver incorrect, or is there some missinglogic in the model?2. Montana Pulp (MP) produces recycled paperboard by combiningfour grades of recycled paper stock. Each grade of stock hasa different strength, color, and texture. The strength, color, and textureof the finished paperboard are a weighted average of thosecharacteristics of the paper stock inputs. The table provides thecharacteristics of the paper inputs and their costs per ton. An orderfor 500 tons of paperboard with a strength of at least 7, a color ofat least 5, and a texture of at least 6 has been received. MP needs tofind the minimal-cost mix of inputs required to satisfy this order.Paper Stock Strength Color Texture Cost/Ton1 8 9 8 $1502 6 7 5 $1103 5 5 6 $904 3 4 5 $50(a) Formulate a LP to minimize the cost of the order.(b) Set up and solve the problem on a spreadsheet.(c) What is the optimal solution? Explain the rationale forthe solution.3. A company has three manufacturing plants (in Atlanta,Tulsa, and Springfield) that produce a product that is then shippedto one of four distribution centers. The three plants can producethirteen, eighteen, and twelve truckloads of product each week, respectively.Each distribution center needs ten truckloads of prod-uct each week. The shipping costs per truckload between the plantsand distribution centers are given in the table. The company needsto determine how much to ship from each plant to each distributioncenter and would like to minimize total shipping costs.Distribution CenterPlant A B C DAtlanta $800 $1300 $400 $700Tulsa $1100 $1400 $600 $1000Springfield $600 $1200 $800 $900(a) Formulate an LP to minimize the total shipping costs.(b) Set up and solve the problem on a spreadsheet.(c) What is the optimal solution? Explain the rationale forthe solution.4. FiberTech makes newsprint for newspapers at three mills, A,B, and C. The cost of producing newsprint is estimated to be $210at mill A, $225 at B, and $220 at C. Five primary geographical marketsare served from these mills. The monthly demand at each market,the shipping cost (per ton) between each mill and market, andthe monthly production capacity of each mill are given in the table.MarketCapacityMill 1 2 3 4 5 (tons/month)Shipping A $20 $25 $30 $15 $35 1200Cost/Ton B $30 $20 $32 $28 $19 1500C $25 $18 $28 $23 $31 900Monthly 600 100 500 800 500Demand(tons)FiberTech would like to assign production responsibilities tothe mills, and also specify how much should be shipped fromeach plant to each market, so as to minimize total productionand distribution costs.(a) Formulate a LP to minimize total production and distributioncosts.(b) Set up and solve the problem on a spreadsheet.(c) What is the optimal solution? Explain the rationale forthe solution.5. Northwest Pipe (NP) makes water pipe. NP is planning productionfor the next seven months, March through September. Theforecast demands (in thousands of feet) are, respectively, 40, 60, 70,80, 90, 100, and 80. NP can make 75,000 feet of pipe per monthusing regular-time production, at a cost of $1.25 per foot. They canmake up to an additional 15,000 feet using overtime production ata cost of $1.50 per foot. Any pipe made in one month and sold in alater month incurs an inventory holding cost of $0.15 per foot, permonth. NP expects to end February with 5000 feet of pipe, andwould like to plan to end September with 10,000 feet in inventory.NP would like to plan their production schedule to minimize totalcost during the next seven months.(a) Formulate a LP to minimize total costs.(b) Set up and solve the problem on a spreadsheet.(c) What is the optimal solution? Explain the rationale forthe solution.