Suez Canal. Pipeline Suez

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1 Integer Programming Formulation Set II Richard S. Barr January 20, 1999 Contents 1 Network Optimization Megacorp Oil Transport Power Generation Stuart Furniture Integer Programming Graphical Solution Capital Budgeting Supplier Selection Pro Athlete Selection Truck Delivery Assignment HMO Site Selection Airline Schedule Site Selection Assembly Line Scheduling Superbowl Camera Location Facility Location Warehouse Location

2 1 Network Optimization 1.1 Megacorp Megacorp's monthly aggregate demand for the next three months is forecast to be: Month: Demand, units: The company has 50 workers, each capable of making 10 standard units per month. Therefore the company can product 500 units/month on regular time at $100 per unit, and 50 additional units/month on overtime at an added cost of $25/unit. Inventories are permitted (holding cost = $20/unit/month) as are backorders (at $30/unit/month). The beginning inventory for month 1 is 50 units, and the target ending inventory for month 3 is also 50 units. Assignment: draw a network model that describes the problem of meeting the demands at minimum cost. 1.2 Oil Transport Twenty million barrels of oil must be transported from Dhahran in Saudi Arabia to the ports of Rotterdam, Marseilles, and Naples in Europe. The demands of these ports are, respectively, 4, 12, and 4 million barrels. The following three alternative routes are possible (see Figure 1). ffl From Dhahran, around Africa to Rotterdam, Marseilles, and Naples. The average transportation and handling cost per barrell is $1.20, $1.40, and $1.40, respectively. ffl From Dhahran to the city of Suez, and then through the Suez Canal to Port Said. From Port Said the oil is shipped to Rotterdam, Marseilles and Naples. The average transportation and handling cost from Dhahran to the city of Suez is $0.30, and the additional unit cost of transporting through the canal is $0.20. Finally, the unit transportation costs from Port Said to Rotterdam, Marseilles, and Naples are $0.25, $0.20, and $0.15, respectively. ffl From Dhahran to the city of Suez, and then through the proposed pipeline system from Suez to Alexandria. The average transportation cost per barrel through the pipeline is $0.15, and the unit transportation costs from Alexandria to Rotterdam, Marseilles, and Naples are $0.22, $0.20, and $0.15, respectively. Thirty percent of the oil is transported by large tankers that cannot pass through the Suez Canal. Also the pipeline from Suez to Alexandria has a capacity of 10 million barrels of oil. 2

3 Rotterdam Marsailles Naples Alexandria Pipeline Suez Port Said Suez Canal Dhahran Figure 1: Oil Transportation Routes Formulate the problem as a network flow model and draw the associated diagram, indicating clearly the meaning of the nodes, and showing the supplies, demands, costs, upper bounds, and lower bounds. 1.3 Power Generation The network shown in Figure 2 represents an electrical power distribution network connecting power-generating points with power-consuming points. The arcs are undirected, since power may flow in either direction. Points 1, 4, 7, and 8 are generation points with generating capacities in kilowatt-hours (KWH) and unit costs as follows. Generating Point Capacity (000s of KWH) Unit cost ($/1000 KWH)

4 Figure 2: Electrical Power Grid Points 2, 5, 6, and 9 are consuming points with demands of 35, 50, 60, and 40 KWH, respectively. There is no upper bound on the transmission line capacities. The unit cost of transmission on each line segment is $11.0 per 1000 KWH. Formulate the power distribution problem as a network flow model. 1.4 Stuart Furniture The Stuart Corporation manufactures chairs at four plants located in various parts of the U.S. The cost of manufacture, excluding material, per chair and the minimum and maximum monthly regular and overtime production for each plant is shown in the table below. Regular Production Overtime Mfg. Maximum Minimum Additional Additional Plant Cost/chair Production Production Production Cost/chair 1 $5 500 ch 0ch 250 ch $ Twenty pounds of wood are required to produce each chair. The company obtains its wood from two sources. These sources can supply any amount of raw material, but contractual obligations require the company to buy at least four tons of wood from each supplier. The purchase costs for wood, FOB the source, are: 10 cents per pound from Source 1, and 7.5 cents per pound from Source 2. The shipping cost per pound of wood between each source and plant are shown below. 4

5 Shipping Cost/Pound from Plant Wood Source 1: $0.01/lb. $0.02/lb. $0.04/lb. $0.04/lb. Wood Source 2: $0.04/lb. $0.03/lb. $0.02/lb. $0.02/lb. The chairs are sold in four major cities: new York, Chicago, San Francisco, and Dallas. Transportation costs, in dollars per chair, between the plants and these cities, along with unit selling prices, maximum demands, and minimum chairs to be delivered are as follows. City NY Dal SF Chi Transportation 1: $1/ch $1/ch $2/ch $0/ch Cost from 2: $3/ch $6/ch $7/ch $3/ch This 3: $3/ch $1/ch $5/ch $3/ch Plant 4: $8/ch $2/ch $1/ch $4/ch Selling price: $20/ch $15/ch $20/ch $18/ch Max. demand: 200 ch 400 ch 1500 ch 1500 ch Min. delivery: 500 ch 100 ch 500 ch 500 ch 1. Design a network model to determine: (a) the source(s) from which each plant should buy its raw materials, (b) the production levels at each plant, (c) the volume to be sold at each city, and (d) the cities to which each plant should ship its product. 2. How would you handle: (a) the use of more expensive transportation, if profitable, and (b) multi-period planning with inventories? 2 Integer Programming 2.1 Graphical Solution Graphically solve the following integer program. 2.2 Capital Budgeting Maximize x 1 +5x 2 subject to: 4x 1 +3x 2» 6 3x 1 +2x 2» 18 x 1 ;x 2 0; integer The Board of Directors of a firm that produces industrial power tools has decided that the firm's original East Coast plant, which for some time has been cramped and outdated, must be either modernized or a new plant built. If the new plantis 5

6 Table 1: Investment Options Expenditure Estimated Net Project Year 1 Year 2 Year 3 Present Value Modernize Existing Facility $400,000 $150,000 $ 50,000 $200,000 Build New Facility 8,000, , ,000 Automate Line 150,000 50, ,000 Acquire Fleet 10, , , ,000 Build Shipping Dock 90,000 60, ,000 Buy Warehouse 200, , , ,000 built, a fully automated production line could be installed rather than continue with conventional, less capital intensive methods. The firm has also considered the possibility of investing in a private trucking fleet for delivery of its products. If the private fleet is acquired and the old facility is modernized rather than rebuilt, a new shipping dock would be required. A somewhat unrelated project, but one that would require a portion of the same capital resources, is the purchase of a local industrial distributor's warehouse to consolidate inventory held in several smaller leased warehouses. The alternative uses of capital described above are summarized in Table 1, along with each project's estimated net present value and the required capital expenditure for each over the next three years. The capital budget approved for these projects over the next three years is $1,200,000 in year 1, $600,000 in year 2, and $200,000 in year 3. Formulate a model to help the firm allocate capital to the available alternatives. 2.3 Supplier Selection A manufacturer of baby cereals is test marketing a new product next month and requires 20,000 boxes made to specification. The firm plans to purchase these from a number of small paper processors with whom it has dealt in the past and has requested each of these suppliers to submit a bid. Six suppliers have responded; each has specified the maximum order deliverable by the beginning of next month, and, to guarantee themselves some profit in the event that less than a break-even quantity is ordered, each has specified a setup charge. The bids are shown in Table 2. Formulate a model to determine the order that should be placed with each supplier. 6

7 Table 2: Supplier Costs and Capacities Set-up Variable Cost/100 boxes Max. Quantity Supplier 1 $200 $2.83 5,000 Supplier 2 $250 $2.66 7,000 Supplier 3 $300 $1.87 4,000 Supplier 4 $350 $1.88 7,000 Supplier 5 $400 $ ,000 Supplier 6 $400 $1.81 9,000 Table 3: Player Data Batting First-year Annual Deferred Player Average Bonus Salary Compensation Dave Wonfield.321 $600,000 $600,000 $1,500,000 Reggie Traction , , ,000 Pete Nose , ,000 1,700,000 Carlton Kissed , , , Pro Athlete Selection The owner of the California Angles, concerned with his team's inability to win the pennant despite his continued efforts to improve the team, has decided to try to sign two more free agents. In preparation, he has started to negotiate with four players. Their career batting averages and his estimate for the bottom-line financial remuneration necessary to sign each is given in Table 3 The Angles owner has determined that, given the state of his current and estimated future financial resources, he can spend no more than $800,000 in first-year bonuses, no more than $1.3 million in salaries, and no more than $3- million in deferred payments. Formulate a model to determine the players to sign so that he gets the best hitting players subject to his financial constraints. 2.5 Truck Delivery Assignment Knighted Parcel Service has deliveries to be made today to five large industrial customers. These deliveries consist of several parcels of various sizes; their total weight is given below: 7

8 Customer Weight of Shipment (lb) 1 1, , , , ,000 The firm has four delivery vans free to make these deliveries; their capacities and operating costs are given below. Truck Capacity Operating Cost/Day 1 2,000 $ ,000 $ ,000 $ ,000 $67.00 In addition, due to distance considerations, a single van cannot deliver to customers 1 and 3 nor to customers 2 and 4. Also, company policy dictates that the shipments to a single customer not be divided among trucks; that is, a customer receives its deliveries from one truck. Formulate an integer programming model to determine the least cost allocation of delivery vans that guarantees that all deliveries will be made. 2.6 HMO Site Selection Heath Maintenance Organizations (HMOs) are being planning for a large metropolitan area. There are three potential sites and six census tracts to be served by the HMOs. City staff has estimated the number of subscribers from each of the six tracts who would go to a given site if the tract was assigned to the HMO at that site, as indicated by the data below: Census Tract Site A 50,000 20,000 30,000 12,000 25,000 35,000 B 14,000 17,000 37,000 45,000 46,000 13,000 C 32,000 45,000 41,000 22,000 31,000 10,000 The cost of building an HMO at site A is estimated to be $4.3 million, at site B, $4.1 million, and at site C, $4.7 million. The budget is $10 million. Also, each new HMO requires a minimum enrollment of 79,000 subscribers. Formulate a model to help city planners determine where HMOs should be built. 2.7 Airline Schedule An airline is preparing its schedule for counter personnel for the busy Christmas season. The number of people needed during each hour of the day is given in 8

9 Table 4: Personnel Requirements, by Time Period Time Personnel Needed 6 a.m. 7 a.m. 5 7 a.m. 8 a.m. 8 8 a.m. 9 a.m. 8 9 a.m. 10 a.m a.m. 11 a.m a.m. 12 noon noon 1 p.m p.m. 2 p.m p.m. 3 p.m p.m. 4 p.m. 1l 4 p.m. 5 p.m p.m. 6 p.m p.m. 7 p.m p.m. 8 p.m p.m. 9 p.m p.m. 10 p.m p.m. 11 p.m. 6 Table 4. Over the years, the company has established the 11 shifts shown in Table 5. Temporaries can be used for this seasonal schedule, so that split shifts (3, 4, and 11) can be used. Formulate a model to minimize the number of people needed to meet anticipated demand. Are integer restrictions on the decision variables necessary? 2.8 Site Selection Gentle Edison, Inc. plans to build six new nuclear power plants on sites which have been selected. The firm plans to start construction on all plants within a five-year period, and the director of planning plans to formulate an optimization model to determine how the expansion should proceed subject to a number of company policies and governmental regulations. The model is to be comprised of 0-1 variables y ij, defined as 1, if plant i is started in year j, 0 otherwise. To guarantee that the model requires each plant to be started in one and only one year, the formulation will contain the following constraints: 5X y ij = 1 for i =1;2;:::;6 j=1 9

10 Table 5: Available Time Shifts Shift Time Period 1 10 a.m. 6 p.m. 2 6 a.m. 2:30 p.m a.m. 1 p.m. and 4 p.m. 8 p.m a.m. 1 p.m. and 6 p.m. 10 p.m. 5 9:30 a.m. 5:30 p.m. 6 1l a.m. 7 p.m noon 8 p.m. 8 1 p.m. 9 p.m. 9 4 p.m. 10 p.m p.m. 11 p.m a.m. 9 a.m. and 5 p.m. 9 p.m. For each of the limitations below, formulate a constraint (or set of constraints) to incorporate that restriction into the integer programming model. Consider each stipulation separately. in other words, each restriction is independent of the previous ones. 1. Plants 1, 2, and 3 must be started no later than year No more than two plants can be started in any year. 3. No more than three plants can be started in the first 2 years, and no more than five plants in the first 4 years. 4. Plant 2 cannot be started before Plant 1, but they can be started in the same year. 5. If Plants 1 and 2 are started within the same year, then no other plants can be started in that year. 6. In the first 2 years, the company starts construction either on Plants 1 and 2 or on Plants 3 and Plant 1 can be started in year 1 if and only if either Plant 2 or Plant 3is started, but not if both 2 and 3 are started. 2.9 Assembly Line Scheduling An assembly line is comprised of several work stations where one or more jobs may be performed in the process of completing the manufacture of an item. The item typically travels from station to station (on a conveyor belt, for example) 10

11 Table 6: Camera Locations and Stadium Areas Covered Camera Location Stadium Areas 1 1,3,4,6,7 2 8,4,7,12 3 2,5,9,11,13 4 1,2,18,19,21 5 3,6,10,12,14 6 8,14,15,16, ,21,24,25 8 2,10,16,23 9 1,6, ,22,24, ,4,6,8 12 1,6,12,17 and the time it spends in each station is not permitted to exceed the cycle time of the assembly line. Consider, as an example, a product whose assembly requires six operations. Certain operations are required before others may be performed, and the table below gives the precedence relations and job durations. For this assembly line, the cycle time is 80 seconds. Formulate an optimization model to schedule the assembly operations such that a minimal number of work stations must be opened. Job Immediate Predecessor Duration (sec) , , Superbowl Camera Location Super Sunday, the once-a-year TV extravaganza focused upon the activities of the NFL Super Bowl, has almost become a tradition. CBS has received the TV contract for this year's family festivities. Producers have identified 12 potential camera locations within the stadium. They have also identified 25 stadium areas which may require camera coverage during the pregame, game, and postgame activities. The Table 6 indicates each camera location and the stadium areas the camera can cover. CBS executives are concerned about costs for the production. Consequently, they have set an objective of minimizing the number of camera locations. In 11

12 Table 7: Warehousing and Transportation Costs with Monthly Capacities Transportation Cost/Ton Monthly Monthly Ware- Demand City Supply Fixed house Capacity Cost A $1625 $400 $685 $1630 $1160 $ T $7650 B T $3500 C T $5000 D T $4100 E T $ T 8T 12 T 6T 7T 11 T ψ Monthly demand, tons seeking this objective they want at least one camera to be available to cover each stadium area. Camera location 9 is the blimp," and executives have decided that the blimp will be used because of viewer expectation and fascination with the shots from this location. Stadium areas 1 and 2 are locker room locations. The viewer interest in football personalities has led the executives to request that at least two camera locations be available to cover each of these areas. Formulate a model to determine the minimum number of cameras needed for the desired coverage. 3 Facility Location 3.1 Warehouse Location The Jacob Company of Jacob, CA, currently has a warehouse in each of cities A E, which supply customer regions throughout the US. It convenient to aggregate customer areas and consider the customers to be located in cities 1 5. There is some feeling that Jacob is overwarehoused;" that is, it may be able to save substantial fixed costs by closing some warehouses without duly increasing transportation and service costs. Relevant data have been collected and assembled on a per-month basis, as given in Table 7. Demand is given in tons of product. For example, closing warehouse A would result in a fixed-cost savings of $7,650. If City 5 receives all of its monthly demand from E, then the associated transportaiton cost for supplying City 5 would be = $2177 per month. A demand city need not receive all of its supply from a single source. Multiple sourcing may result from the limited capacity of each warehouse (e.g., B can process at most 24 tons per month). 12