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Two raw materials are used to produce 2 products: a fuel additive, and a solvent base. The profit contributions are $40 per ton for the fuel additive, $30 per ton for the solvent base. Each ton of fuel additive is a blend of 0.4 tons of material 1, and 0.3 tons of material 2. Each ton of solvent base requires 0.5 tons of material 1,and 0.2 tons of material 2. RMC has 20 tons of material 1, 5 tons of material 2, and is interested in determining the optimal production quantities for the upcoming planning period. There is a fixed cost for production setup of the products, as well as a maximum production quantity for each of the two products. Product Setup Cost Maximum Production Fuel Additive $200 50 tons Solvent Base $50 25 tons a) Define the decision variables b) Define the objective function c) Define the constraints
a) F = tons of fuel additive produced S = tons of solvent base produced SF = 1 if the fuel additive is produced; SF = 0 if not SS = 1 if the solvent base is produced; SS = 0 if not b) Max 40F + 30S - 200SF - 50SS c) Material 1: 0.4F + 0.5S <= 20 Material 2: 0.3F + 0.2S <= 5 Maximize F: F - 50SF <= 0 Maximize S: S - 25SS <= 0 F, S >= 0 SF, SS = Binary
The Milton Lumber Company sells boards of various lengths (4-foot, 5-foot, 7-foot, and 9-foot wood boards). The customer's demand is 30 for 4-foot boards, 40 for 5-foot boards, 25 for 7-foot boards, and 60 for 9-foot boards per week. Milton Lumber cuts up boards of 22 feet in length to meet this demand and wants to determine how to satisfy its customers' demands with a minimal amount of waste. Assume that all boards share the same width and thickness. Formulate and solve an IP model. a) Define the decision variables b) Define the objective function c) Define the constraints
a) Xi = Number of pattern i to be produced ( i = 1,2,....,18) b) Minimize 2 x1 + x2 + ...... + 3 x17 + x18 c) 4 feet: 5x1 + .... + x12 >= 30 5 feet: x2 + ......+ x17 >= 40 7 feet: x4 + ..... + 3x18 >= 25 9 feet: x5 + .... + x16 >= 60 Xi = Integer
The following questions refer to a capital budgeting problem with six projects presented by 0-1 variables x1,x2,x3,x4,x5 and x6. Write constraints modeling a situation in which 1)Two of the projects 1,2,3, and 6 must be undertaken 2 )If projects 1 or 3 must be undertaken, they must be undertaken simultaneously. 3)Project 1 or 3 must be undertaken, but not both. 4)Project 1 cannot be undertaken unless project 2 and 3 also are undertaken. 5)Additional part (d), when project 2 and 3 are undertaken project 1 also must be undertaken.
1) x1 + x2 + x3 + x6 =2 2) x1 - x3 = 0 3) x1 + x3 = 1 4) x1 <= x2, x1 <= x3 5) x1 <= x2, x1 <= x3, x2+x3 -1 <= x1
Suppose a certain manufacturing company produces connecting rods for 4- and 6-cylinder automobile engines using the same production line. The cost required to set up the production line to produce the 4-cylinder connecting rods is $2,600, and the cost required to set up the production line for the 6-cylinder connecting rods is $3,800. Manufacturing costs are $12 for each 4-cylinder connecting rod and $17 for each 6-cylinder connecting rod. Hawkins makes a decision at the end of each week as to which product will be manufactured the following week. If a production changeover is necessary from one week to the next, the weekend is used to reconfigure the production line. Once the line has been set up, the weekly production capacities are 5,000 6-cylinder connecting rods and 7,000 4-cylinder connecting rods. a. Define the decision variables b. Define the objective function c. Define the constraints
a) X4 = # of 4 cylinder connecting rods produced next week X6 = # of 6 cylinder connecting rods produced next week S4 = 1 if the 4 cylinder rods are setup to be produced, S4 = 0 if not produced S6 = 1 if the 6 cylinder rods are setup to be produced, s6 = 0 if not produced b) Minimize 2600S4 + 3800S6 + 12X4 + 17X6 c) X4 <= 7000S4 X6 <= 5000S6 S4 + S6 = 1 X4, X6 are integers X4, X6 >=0 S4, S6 = Binary
The HP Company has a contract to produce 1000 notebook computers for a significant discount chain. ABC has two different machines that can make this kind of notebook computer. Because these machines are from other manufacturers and use differing technologies, their specifications are not the same. The company wants to minimize the total cost. Machine Fixed Cost Setup Variable Cost/notebook Capacity 1 $1,000 $300.00 1800 2 $1,500 $290.00 1900 a) Define the decision variables. b) Define the objective function. c) Define the constraints.
a) Xi = # of notebook computers produced on machine i (i= machines 1,2) Si = 1, if machine used, Si = 0 if not used (where i= machines 1,2) b) Minimize 1000S1 + 1500S2 + 300X1 + 290X2 c) X1 <= 1800S1 X2 <= 1900S2 X1 + X2 = 1000 Xi >= 0 (i= machines 1,2) Si = Binary (i= machines 1,2) X1, X2 = integer
The Milton Lumber Company sells boards of various lengths (3-foot and 7-foot wood boards). This company's customers demand 10 3-foot boards and 15 7-foot boards per week. Milton Lumber cuts up boards of 13 feet in length to meet this demand and wants to determine how to satisfy its customers' demands with a minimal amount of waste. Assume that all boards share the same width and thickness. Formulate and solve an IP model.
a) Xi = number of pattern i to be produced (i=1,2) b) Minimize 1x1 + 0x2 c) 3 feet: 4x1 + 2x2 >=10 7 feet: 0x1 + 1x2 >=15 Xi = integer
Premier Consulting's two consultants, Avery and Baker, can be scheduled to work for clients up to a maximum of 160 hours each over the next four weeks. A third consultant, Campbell, has some administrative assignments already planned and is available for clients up to a maximum of 140 hours over the next four weeks. The company has four clients with projects in process. The estimated hourly requirements for each of the clients over the four-week period are as follows. Client Hours A 180 B 75 C 100 D 85 Hourly rates vary for the consultant-client combination and are based on several factors, including project type and the consultant's experience. The rates (dollars per hour) for each consultant-client combination are as follows. A B C D Avery 100 125 115 100 Baker 120 135 115 120 Campbell 155 150 140 130 a) Define the decision variables b) Define the objective function c) Define the constraints
a) Xij = # of hours from consultant i assigned to client j i = 1,2,3 and j=1,2,3,4 b) Maximize 100x11 + 125x12 + 115x13 + 100x14 + 120x21 + 135x22 + 115x23 + 120x24 + 155x31 + 150x32 + 140x33 + 130x34 c) x11+x12+x13+x14 <= 160 x21+x22+x23+x24 <= 160 x31+x32+x33+x34 <= 140 x11+x21+x32 = 180 x12+x22+x32=75 x13+x23+x33 =100 x14+x24+x34 =85 xij ≥ 0 for all i, j
A B C Tom 100 120 80 John 150 70 120 (#s in table are costs) a) Define the decision variables b) Define the objective function c) Define the constraints
a) Xij = 1 if person i is assigned to project j, = 0 if otherwise. Where i = 1,2 j = 1,2,3 b) Minimize Cost: 100x11 + 120x12 + 80x13 + 150x21 + 70x22 + 120x23 c) Project A: x11 + x21 <=1 Project B: x12 + x22 <=1 Project C: x13 + x23 <=1 Tom: x11 + x12 + x13 = 1 John: x21 + x22 + x23 = 1 Xij = Binary
The distribution system for the Herman Company consists of three plants, two warehouses, and four customers. Plant capacities and shipping costs per unit (in $) from each plant to each warehouse are as follows. Plant Warehouse Capacity 1 2 1 4 7 450 2 8 5 600 3 5 6 380 Customer demand and shipping costs per unit (in $) from each warehouse to each customer are as follows. Warehouse Customer 1 2 3 4 1 6 4 8 4 2 3 6 7 7 Demand 300 300 300 400 a) Define the decision variables b) Define the objective function c) Define the constraints
a) Xij = the number of units shipped from node i to node j i=1,2,3,4,5 j= 4,5,6,7,8,9 b) Min 4x14 + 7x15 + 8x24 + 5x25 + 5x34 + 6x35 + 6x46 + 4x47 + 8x48 + 4x49 + 3x56 + 6x57 + 7x58 + 7x59 c) x14+x15 <=450 x24+x25 <=600 x34+x35<= 380 (x14+x24+x34)-(x46+x47+x48+x49)=0 (x15+x25+x35)-(x56+x57+x58+x59)=0 x46+x56=300 x47+x57 = 300 x48+x58 =300 x49+x59 =400 Xij = Integer.
The national bank is working to develop an efficient work schedule for full-time tellers. The schedule must provide for efficient operation of the bank. On Fridays the bank is open from 9:00 A.M. to 7:00 P.M. The number of tellers necessary to provide adequate customer service during each hour of operation is summarized in Table. Time # of Tellers Time # of Tellers 9-10 5 2-3 6 10-11 3 3-4 3 11-12 7 4-5 6 12-1 9 5-6 5 1-2 8 6-7 6 Each full-time employee starts on the hour and works a 4-hour shift, followed by 1 hour for lunch and then a 3-hour shift. The full-time employees cost the bank $20 per hour ($140/day). Formulate an integer linear programming model that can be used to develop a schedule that will satisfy customer service needs at a minimum employee costs. a) Define the decision variables. b) Define the objective function. c) Define the constraints.
a) x1 = # of employees starting at 9am x2 = # of employees starting at10:00am x3 = # of employees starting at 11:00am b) Minimize 140x1 + 140x2 + 140x3 c) x1 >= 5 x1 + x2 >= 3 x1 + x2 + x3 >= 7 x1 + x2 + x3 >= 9 x2 + x3 >= 8 x1 + x3 >= 6 x1 + x2 >= 3 x1 + x2 + x3 >=6 x2 + x3 >= 5 x3 >= 6 x1, x2, x3 = integer Non-negative constraint?
The national bank is working to develop an efficient work schedule for full-time tellers. The schedule must provide for efficient operation of the bank. On Fridays, the bank is open from 9:00 A.M. to 8:00 P.M. The number of tellers necessary to provide adequate customer service during each hour of operation is summarized in Table. Each full-time employee starts on the hour and works a 4-hour shift, followed by 1 hour for lunch and then a 3-hour shift. The full-time employees cost the bank $30 per hour ($210 per day). Formulate an integer linear programming model that can be used to develop a schedule that will satisfy customer service needs at a minimum employee cost. Time # of Tellers Time # of Tellers 9-10 7 2-3 5 10-11 2 3-4 3 11-12 9 4-5 8 12-1 8 5-6 10 1-2 6 6-7 5 7-8 4 a) Define the decision variables b) Define the objective function c) Define the constraints
a) x1 = # of full-time employees starting at 9:00am x2 = # of full-time employees starting at 10:00am x3 = # of full-time employees starting at 11:00am x4 = # of full-time employees starting at Noon b) Minimize 210x1 +210x2 + 210x3 + 210x4 c) x1 >= 7 x1 + x2 >= 2 x1 + x2 + x3 >= 9 x1 +x2 + x3 + x4 >= 8 x2 + x3 + x4 >= 6 x1 + x3 + x4 >= 5 x1 + x2 + x4 >= 3 x1 + x2 + x3 >= 8 x2 + x3 + x4 >= 10 x3 + x4 >=5 x4 >= 4. x1,x2,x3, x4 = Integer
Metropolitan Microwaves, Inc. is planning to expand its sales operation by offering other electronic appliances. The company has identified seven new product lines it can carry. Initial Invest. Sq. Ft Return 1) TV/VCRs 6000 125 8.1% 2) TVs 12000 150 9.0 3) Projection TVs 20000 200 11.0 4) VCRs 14000 40 10.2 5) DVD Players 15000 40 10.5 6) Video Games 2000 20 14.1 7) Home Computers 32000 100 13.2 Metropolitan has decided that they should not stock projection TVs unless they stock either TV/VCRs or TVs. Also, they will not stock both VCRs and DVD players, and they may stock video games if they stock TVs. Finally, the company wishes to introduce at least three new product lines. If the company has $45,000 to invest and 420 sq. ft. of floor space available, formulate an integer linear program for Metropolitan to maximize its overall expected return. a) Define the decision variables b) Define the objective function c) Define the constraints
a) xi = 1 if product line i is introduced; = 0 otherwise. Where i =1,2,3,4,5,6,7 b) Max 0.081(6000)x1 + 0.09(12000)x2 + 0.11(20000)x3 + 0.102(14000)x4 + 0.105(15000)x5 + 0.141(2000)x6 + 0.132(32000)x7 c) Money: 6000x1 + 12000x2 + 20000x3 + 14000x4 + 15000x5 + 2000x6 + 32000x7 <= 45000 Space: 125x1 +150x2 +200x3 +40x4 +40x5 +20x6 +100x7 <= 420 Stock projection TVs only if stock TV/VCRs or TVs: x1 + x2 >= x3 Do not stock both VCRs and DVD players: x4 + x5 <= 1 May stock video games if they stock TV's: x2 - x6 >= 0 Introduce at least 3 new lines: x1 + x2 + x3 + x4 + x5 + x6 + x7 >= 3 Xi = Binary
