Lp Formulation Problems And Solutions 2y1t20
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Linear Programming Formulation Exercises from Textbook
ISM 4400, Fall 2006: Page 1/14
SOLUTIONS TO SELECT PROBLEMS FROM CHAPTER 7 7-14
The Electrocomp Corporation manufactures two electrical products: air conditioners and large fans. The assembly process for each is similar in that both require a certain amount of wiring and drilling. Each air conditioner takes 3 hours of wiring and 2 hours of drilling. Each fan must go through 2 hours of wiring and 1 hour of drilling. During the next production period, 240 hours of wiring time are available and up to 140 hours of drilling time maybe used. Each air conditioner sold yields a profit of $25. Each fan assembled may be sold for a $15 profit. Formulate and solve this LP production mix situation to find the best combination of air conditioners and fans that yields the highest profit. Use the corner point graphical approach. Let X1 = the number of air conditioners scheduled to be produced X2 = the number of fans scheduled to be produced Maximize Subject to:
25X1 + 3X1 + 2X1 +
15X2 (maximize profit) 2X2 ≤ 240 (wiring capacity constraint) X2 ≤ 140 (drilling capacity constraint) X1, X2 ≥ 0 (non-negativity constraints)
Optimal Solution: X1 = 40 X2 = 60 Profit = $1,900 7-15
Electrocomp’s management realizes that it forgot to include two critical constraints (see Problem 7-14). In particular, management decides that to ensure an adequate supply of air conditioners for a contract, at least 20 air conditioners should be manufactured. Because Electrocomp incurred an oversupply of fans in the preceding period, management also insists that no more than 80 fans be produced during this production period. Resolve this product mix problem to find the new optimal solution. Let X1 = the number of air conditioners scheduled to be produced X2 = the number of fans scheduled to be produced Maximize Subject to:
25X1 + 3X1 + 2X1 + X1
15X2 (maximize profit) 2X2 ≤ 240 (wiring capacity constraint) X2 ≤ 140 (drilling capacity constraint) ≥ 20 (a/c contract constraint) X2 ≤ 80 (maximum # of fans constraint) X1, X2 ≥ 0 (non-negativity constraints)
Optimal Solution: X1 = 40 X2 = 60 Profit = $1,900
Linear Programming Formulation Exercises from Textbook
7-16
ISM 4400, Fall 2006: Page 2/14
A candidate for mayor in a small town has allocated $40,000 for last-minute advertising in the days preceding the election. Two types of ads will be used: radio and television. Each radio ad costs $200 and reaches an estimated 3,000 people. Each television ad costs $500 and reaches an estimated 7,000 people. In planning the advertising campaign, the campaign manager would like to reach as many people as possible, but she has stipulated that at least 10 ads of each type must be used. Also, the number of radio ads must be at least as great as the number of television ads. How many ads of each type should be used? How many people will this reach? Let X1 = the number of radio ads purchased X2 = the number of television ads purchased Maximize 3,000X1 + 7,000X2 Subject to: 200X1 + 500X2 X1 X2 X1 X1, X2
≤ ≥ ≥ ≥ ≥
(maximize exposure) 40,000 (budget constraint) 10 (at least 10 radio ads purchased) 10 (at least 10 television ads purchased) X2 (# of radio ads ≥ # of television ads) 0 (non-negativity constraints)
For solution purposes, the fourth constraint would be rewritten as: X1 − X2 ≥ 0 Optimal Solution: X1 = 175 7-17
X2 = 10
Exposure = 595,000 people
The Outdoor Furniture Corporation manufactures two products, benches and picnic tables, for use in yards and parks. The firm has two main resources: its carpenters (labor force) and a supply of redwood for use in the furniture. During the next production cycle, 1,200 hours of labor are available under a union agreement. The firm also has a stock of 3500 feet of good-quality redwood. Each bench that Outdoor Furniture produces requires 4 labor hours and 10 feet of redwood; each picnic table takes 6 labor hours and 35 feet of redwood. Completed benches will yield a profit of $9 each, and tables will result in a profit of $20 each. How many benches and tables should Outdoor Furniture produce to obtain the largest possible profit? Use the graphical LP approach. Let X1 = the number of benches produced X2 = the number of tables produced Maximize Subject to:
9X1 + 4X1 + 10X1 +
20X2 6X2 ≤ 1,200 35X2 ≤ 3,500 X1, X2 ≥ 0
Optimal Solution: X1 = 262.5
X2 = 25
(maximize profit) (labor hours constraint) (redwood capacity constraint) (non-negativity constraints) Profit = $2,862.50
Linear Programming Formulation Exercises from Textbook
7-18
ISM 4400, Fall 2006: Page 3/14
The dean of the Western College of Business must plan the school’s course offerings for the fall semester. Student demands make it necessary to offer at least 30 undergraduate and 20 graduate courses in the term. Faculty contracts also dictate that at least 60 courses be offered in total. Each undergraduate course taught costs the college an average of $2,500 in faculty wages, and each graduate course costs $3,000. How many undergraduate and graduate courses should be taught in the fall so that total faculty salaries are kept to a minimum? Let X1 = the number of undergraduate courses scheduled X2 = the number of graduate courses scheduled Minimize Subject to:
2,500X1 + 3,000X2 X1 X2 X1 + X2 X1, X2
Optimal Solution: X1 = 40 7-19
≥ ≥ ≥ ≥ 0
X2 = 20
(minimize faculty salaries) 30 (schedule at least 30 undergrad courses) 20 (schedule at least 20 grad courses) 60 (schedule at least 60 total courses) (non-negativity constraints) Cost = $160,000
MSA Computer Corporation manufactures two models of minicomputers, the Alpha 4 and the Beta 5. The firm employs five technicians, working 160 hours each per month, on its assembly line. Management insists that full employment (i.e., all 160 hours of time) be maintained for each worker during next month’s operations. It requires 20 labor hours to assemble each Alpha 4 computer and 25 labor hours to assemble each Beta 5 model. MSA wants to see at least 10 Alpha 4s and at least 15 Beta 5s produced during the production period. Alpha 4s generate $1,200 profit per unit, and Beta 5s yield $1,800 each. Determine the most profitable number of each model of minicomputer to produce during the coming month. Let X1 = the number of Alpha 4 computers scheduled for production next month X2 = the number of Beta 5 computers scheduled for production next month Maximize Subject to:
1,200X1 + 1,800X2 20X1 + 25X2 X1 X2 X1, X2
Optimal Solution: X1 = 10
(maximize profit) = 800 (full employment, 5 workers x 160 hours) ≥ 10 (make at least 10 Alpha 4 computers) ≥ 15 (make at least 15 Beta 5 computers) ≥ 0 (non-negativity constraints)
X2 = 24
Profit = $55,200
Linear Programming Formulation Exercises from Textbook
7-20
ISM 4400, Fall 2006: Page 4/14
A winner of the Texas Lotto has decided to invest $50,000 per year in the stock market. Under consideration are stocks for a petrochemical firm and a public utility. Although a long-range goal is to get the highest possible return, some consideration is given to the risk involved with the stocks. A risk index on a scale of 1–10 (with 10 being the most risky) is assigned to each of the two stocks. The total risk of the portfolio is found by multiplying the risk of each stock by the dollars invested in that stock. The following table provides a summary of the return and risk: Stock Petrochemical Utility
Estimated Return 12% 6%
Risk Index 9 4
The investor would like to maximize the return on the investment, but the average risk index of the investment should not be higher than 6. How much should be invested in each stock? What is the average risk for this investment? What is the estimated return for this investment? Let X1 = the number of dollars invested in petrochemical stocks X2 = the number of dollars invested in utility stocks Maximize Subject to:
.12X1 + X1 + 3X1 −
.06X2 X2 ≤ 50,000 2X2 ≤ 0 X1, X2 ≥ 0
(maximize return on investment) (limit on total investment) (average risk cannot exceed 6) (non-negativity constraints)
Optimal Solution: X1 = $20,000 X2 = $30,000 Return = $4,200 The total risk is 300,000 (9 x $20,000 + 4 x $30,000), which yields an average risk of 6 (300,000/50,000 = 6). 7-21
Referring to the Texas Lotto situation in Problem 7-20, suppose the investor has changed his attitude about the investment and wishes to give greater emphasis to the risk of the investment. Now the investor wishes to minimize the risk of the investment as long as a return of at least 8% is generated. Formulate this as an LP problem and find the optimal solution. How much should be invested in each stock? What is the average risk for this investment? What is the estimated return for this investment? Let X1 = the number of dollars invested in petrochemical stocks X2 = the number of dollars invested in utility stocks Minimize Subject to:
9X1 + X1 + .04X1 −
4X2 X2 ≤ 50,000 .02X2 ≥ 0 X1, X2 ≥ 0
(minimize total risk) (limit on total investment) (average return must be at least 8%) (non-negativity constraints)
Optimal Solution: X1 = $16,666.67 X2 = $33,333.33 Total risk = 283,333.33 (which equates to an average risk of 283,333.33/50,000 = 5.67). The total return would be $4000 (.12 x 16,666.67 + .09 x 33,333.33), which just happens to be a return of exactly 8% ($4000/$50,000).
Linear Programming Formulation Exercises from Textbook
7-24
ISM 4400, Fall 2006: Page 5/14
The stock brokerage firm of Blank, Leibowitz, and Weinberger has analyzed and recommended two stocks to an investors’ club of college professors. The professors were interested in factors such as short term growth, intermediate growth, and dividend rates. These data on each stock are as follows:
Factor Short term growth potential, per dollar invested Intermediate growth potential (over next three years), per dollar invested Dividend rate potential
Stock Louisiana Gas and Trimex Insulation Power Company .36 .24 1.67
1.5
4%
8%
Each member of the club has an investment goal of (1) an appreciation of no less than $720 in the short term, (2) an appreciation of at least $5,000 in the next three years, and (3) a dividend income of at least $200 per year. What is the smallest investment that a professor can make to meet these three goals? Let X1 = the number of dollars invested in Louisiana Gas and Power X2 = the number of dollars invested in Trimex Insulation Co. Minimize Subject to:
X1 .36X1 1.67X1 .04X1
+ + + +
X2 .24X2 1.50X2 .08X2 X1, X2
≥ 720 ≥ 5,000 ≥ 200 ≥ 0
Optimal Solution: X1 = $1,359
(minimize total investment) (appreciation in the short term) (appreciation in next three years) (dividend income per year) (non-negativity constraints)
X2 = $1,818.18 Total investment = $3,177.18
Linear Programming Formulation Exercises from Textbook
7-25
ISM 4400, Fall 2006: Page 6/14
Woofer Pet Foods produces a low-calorie dog food for overweight dogs. This product is made from beef products and grain. Each pound of beef costs $0.90, and each pound of grain costs $0.60. A pound of the dog food must contain at least 9 units of Vitamin 1 and 10 units of Vitamin 2. A pound of beef contains 10 units of Vitamin 1 and 12 units of Vitamin 2. A pound of grain contains 6 units of Vitamin 1 and 9 units of Vitamin 2. Formulate this as an LP problem to minimize the cost of the dog food. How many pounds of beef and grain should be included in each pound of dog food? What is the cost and vitamin content of the final product? Let X1 = the number of pounds of beef in each pound of dog food X2 = the number of pounds of grain in each pound of dog food Minimize Subject to:
.90X1 X1 10X1 12X1
+ + + +
.60X2 X2 6X2 9X2 X1, X2
= ≥ ≥ ≥ 0
(minimize cost per pound of dog food) 1 (total weight should be one pound) 9 (at least 9 units of vitamin 1 in a pound) 10 (at least 10 units of vitamin 2 in a pound) (non-negativity constraints)
Optimal Solution: X1 = .75 X2 = .25 Cost = $.825
Linear Programming Formulation Exercises from Textbook
ISM 4400, Fall 2006: Page 7/14
SOLUTIONS TO SELECT PROBLEMS FROM CHAPTER 8 8-1
(Production problem) Winkler Furniture manufactures two different types of china cabinets: a French Provincial model and a Danish Modern model. Each cabinet produced must go through three departments: carpentry, painting, and finishing. The table below contains all relevant information concerning production times per cabinet produced and production capacities for each operation per day, along with net revenue per unit produced. The firm has a contract with an Indiana distributor to produce a minimum of 300 of each cabinet per week (or 60 cabinets per day). Owner Bob Winkler would like to determine a product mix to maximize his daily revenue. (a) Formulate as an LP problem. (b) Solve using an LP software program or spreadsheet. Cabinet Style French Provincial Danish Modern Dept. capacity (hrs)
Carpentry (Hours/Cabinet) 3 2 360
Painting (Hours/Cabinet) 1.5 1 200
Finishing (Hours/Cabinet) .75 .75 125
Net Revenue per Cabinet ($) 28 25
Let X1 = the number of French Provincial cabinets produced each day X2 = the number of Danish Modern cabinets produced each day Maximize Subject to:
28X1 3X1 1.5X1 .75X1 X1
+ + + +
25X2 (maximize revenue) 2X2 ≤ 360 (carpentry hours available) X2 ≤ 200 (painting hours available) .75X2 ≤ 125 (finishing hours available) ≥ 60 (contract requirement on F.P. cabinets) X2 ≥ 60 (contract requirement on D.M. cabinets) X1, X2 ≥ 0 (non-negativity constraints)
Optimal Solution: X1 = 60
X2 = 90
Revenue = $3,930
Linear Programming Formulation Exercises from Textbook
8-2
ISM 4400, Fall 2006: Page 8/14
(Investment decision problem) The Heinlein and Krarnpf Brokerage firm has just been instructed by one of its clients to invest $250,000 for her money obtained recently through the sale of land holdings in Ohio. The client has a good deal of trust in the investment house, but she also has her own ideas about the distribution of the funds being invested. In particular, she requests that the firm select whatever stocks and bonds they believe are well rated, but within the following guidelines: (a) Municipal bonds should constitute at least 20% of the investment. (b) At least 40% of the funds should be placed in a combination of electronic firms, aerospace firms, and drug manufacturers. (c) No more than 50% of the amount invested in municipal bonds should be placed in a highrisk, high-yield nursing home stock. Subject to these restraints, the client’s goal is to maximize projected return on investments. The analysts at Heinlein and Krampf, aware of these guidelines, prepare a list of high-quality stocks and bonds and their corresponding rates of return. Projected Rate of Return (%) 5.3 6.8 4.9 8.4 11.8
Investment Los Angeles municipal bonds Thompson Electronics, Inc. United Aerospace Corp. Palmer Drugs Happy Days Nursing Homes
(a) Formulate this portfolio selection problem using LP. (b) Solve this problem. Let X1 = dollars invested in Los Angeles municipal bonds X2 = dollars invested in Thompson Electronics X3 = dollars invested in United Aerospace X4 = dollars invested in Palmer Drugs X5 = dollars invested in Happy Days Nursing Homes Maximize
.053X1
+
.068X2
+
.049X3
+
.084X4
+
.118X5
Subject to:
X1
+
X2
+
X3
+
X4
+
X5
≤
250,000
.8X1
-
.2X2
-
.2X3
-
.2X4
-
.2X5
≥
0
(municipal bond restriction)
-.4X1
+
.6X2
+
.6X3
+
.6X4
-
.4X5
≥
0
(electronics, aerospace, drugs combo)
+
X5
≤
0
(nursing home as a percent of bonds)
X1, X2, X3, X4, X5
≥
-.5X1
(maximize return on investment)
0
(total funds available)
(non-negativity constraints)
Optimal Solution: X1 = $50,000 X2 = $0 X3 = $0 X4 = $175,000 X5 = $25,000 ROI = $20,300
Linear Programming Formulation Exercises from Textbook
8-3
ISM 4400, Fall 2006: Page 9/14
(Restaurant work scheduling problem). The famous Y. S. Chang Restaurant is open 24 hours a day. Waiters and busboys report for duty at 3AM., 7 AM., 11 AM., 3 P.M., 7 P.M., or 11 P.M., and each works an 8-hour shift. The following table shows the minimum number of workers needed during the six periods into which the day is divided. Chang’s scheduling problem is to determine how many waiters and busboys should report for work at the start of each time period to minimize the total staff required for one day’s operation. (Hint: Let Xi equal the number of waiters and busboys beginning work in time period i, where i = 1, 2,3,4,5,6.) Period 1 2 3 4 5 6
Time 3 A.M–7 A.M. 7 A.M–11 A.M. 11 A.M–3 P.M. 3 P.M–7 P.M. 7 P.M–11 P.M. 11 P.M–3 A.M.
Number of Waiters and Busboys Required 3 12 16 9 11 4
Let Xi = the number workers beginning work at the start of time period i (i=1,2,3,4,5,6) Minimize Subject to:
X1
+
X2
+
X3
+
X4
+
X5
(min. staff size) 3
(period 1)
X2
≥
12
(period 2)
X2
≥
16
(period 3)
X4
≥
9
(period 4)
X4
X5
≥
11
(period 5)
X5
X6
≥
4
(period 6)
X1, X2, X3, X4, X5, X6
≥
+ +
X6 ≥
X1 X1
+
+
X6
X3 X3
+
+
+
0
(non-negativity)
Linear Programming Formulation Exercises from Textbook
8-4
ISM 4400, Fall 2006: Page 10/14
(Animal feed mix problem) The Battery Park Stable feeds and houses the horses used to pull tourist-filled carriages through the streets of Charleston’s historic waterfront area. The stable owner, an ex-racehorse trainer, recognizes the need to set a nutritional diet for the horses in his care. At the same time, he would like to keep the overall daily cost of feed to a minimum. The feed mixes available for the horses’ diet are an oat product, a highly enriched grain, and a mineral product. Each of these mixes contains a certain amount of five ingredients needed daily to keep the average horse healthy. The table below shows these minimum requirements, units of each ingredient per pound of feed mix, and costs for the three mixes. In addition, the stable owner is aware that an overfed horse is a sluggish worker. Consequently, he determines that 6 pounds of feed per day are the most that any horse needs to function properly. Formulate this problem and solve for the optimal daily mix of the three feeds. Diet Requirement (Ingredients) A B C D E Cost/lb Let
Oat Product (units/lb) 2 .5 3 1 .5 $0.09
Feed Mix Enriched Grain (units/lb) 3 1 5 1.5 .5 $0.14
Mineral Product (units/lb) 1 .5 6 2 1.5 $0.17
Minimum Daily Requirement (units) 6 2 9 8 5
X1 = the number pounds of oat product per horse each day X2 = the number pounds of enriched grain per horse each day X3 = the number pounds of mineral product per horse each day
Minimize s.t.
.09X1 2X1 .5X1 3X1 X1 .5X1 X1
+ + + + + + +
.14X2 3X2 X2 5X2 1.5X2 .5X2 X2
+ + + + + + +
.17X3 X3 .5X3 6X3 2X3 1.5X3 X3 X1, X2, X3
≥ ≥ ≥ ≥ ≥ ≤
≥
6 2 9 8 5 6 0
(minimize cost) (ingredient A) (ingredient B) (ingredient C) (ingredient D) (ingredient E) (maximum feed per day) (non-negativity constraints)
Linear Programming Formulation Exercises from Textbook
8-6
ISM 4400, Fall 2006: Page 11/14
(Media selection problem) The advertising director for Diversey Paint and Supply, a chain of four retail stores on Chicago’s North Side, is considering two media possibilities. One plan is for a series of half- page ads in the Sunday Chicago Tribune newspaper, and the other is for advertising time on Chicago TV. The stores are expanding their lines of do-it-yourself tools, and the advertising director is interested in an exposure level of at least 40% within the city’s neighborhoods and 60% in northwest suburban areas. The TV viewing time under consideration has an exposure rating per spot of 5% in city homes and 3% in the northwest suburbs. The Sunday newspaper has corresponding exposure rates of 4% and 3% per ad. The cost of a half-page Tribune ment is $925; a television spot costs $2,000. Diversey Paint would like to select the least costly advertising strategy that would meet desired exposure levels. (a) Formulate using LP. (b) Solve the problem. Let
X1 = the number of newspaper ads placed X2 = the number of TV spots purchased
Minimize Subject to:
925X1 + 2,000X2 .04X1 + .05X2 ≥ .03X1 + .03X2 ≥ X1, X2 ≥ 0
(minimize cost) .4 (city exposure) .6 (suburb exposure) (non-negativity constraints)
Linear Programming Formulation Exercises from Textbook
8-11
ISM 4400, Fall 2006: Page 12/14
(College meal selection problem) Kathy Roniger, campus dietician for a small Idaho college, is responsible for formulating a nutritious meal plan for students. For an evening meal, she feels that the following five meal-content requirements should be met: (1) between 900 and 1,500 calories; (2) at least 4 milligrams of iron; (3) no more than 50 grams of fat; (4) at least 26 grams of protein; and (5) no more than 50 grams of carbohydrates. On a particular day, Roniger’s food stock includes seven items that can be prepared and served for supper to meet these requirements. The cost per pound for each food item and the contribution to each of the five nutritional requirements are given in the accompanying table: Calories/ Pound 295 1216 394 358 128 118 279
Food Item Milk Ground Meat Chicken Fish Beans Spinach Potatoes
Iron (mg/lb) 0.2 0.2 4.3 3.2 3.2 14.1 2.2
Table of Food Values and Costs Fat Protein Carbs. (gm/lb) (gm/lb) (gm/lb) 16 16 22 96 81 0 9 74 0 0.5 83 0 0.8 7 28 1.4 14 19 0.5 8 63
Cost/ Pound ($) 0.60 2.35 1.15 2.25 0.58 1.17 0.33
What combination and amounts of food items will provide the nutrition Roniger requires at the least total food cost? Let X1 = the number of pounds of milk per student in the evening meal X2 = the number of pounds of ground meat per student in the evening meal Etc., down to X7 = the number of pounds of potatoes per student in the evening meal Minimize S.T. (Cal.)
.6X1
+
2.35X2
+
1.15X3
+
2.25X4
+
.58X5
+
1.17X6
+
295X1
+
1216X2
+
394X3
+
358X4
+
128X5
+
118X6
+
.33X7 279X7
≥
900 1500
(Cal.)
295X1
+
1216X2
+
394X3
+
358X4
+
128X5
+
118X6
+
279X7
≤
(Iron)
.2X1
+
.2X2
+
4.3X3
+
3.2X4
+
3.2X5
+
14.1X6
+
2.2X7
≥
4
(Fat)
16X1
+
96X2
+
9X3
+
.5X4
+
.8X5
+
1.4X6
+
.5X7
≤
50 26 50
(Protein)
16X1
(Carbs.)
22X1
+
81X2
+
74X3
+
83X4
+
7X5
+
14X6
+
8X7
≥
+
28X5
+
19X6
+
63X7
≤
X1, X2, X3, X4, X5, X6, X7
≥
0
Linear Programming Formulation Exercises from Textbook
8-12
ISM 4400, Fall 2006: Page 13/14
(High tech production problem) Quitmeyer Electronics Incorporated manufactures the following six microcomputer peripheral devices: internal modems, external modems, graphics circuit boards, CD drives, hard disk drives, and memory expansion boards. Each of these technical products requires time, in minutes, on three types of electronic testing equipment, as shown in the table the following table: Internal Modem Test device 1 7 Test device 2 2 Test device 3 5
External Modem 3 5 1
Circuit Board 12 3 3
CD Drive 6 2 2
Hard Drive 18 15 9
Memory Board 17 17 2
The first two test devices are available 120 hours per week. The third (device 3) requires more preventive maintenance and may be used only 100 hours each week. The market for all six computer components is vast, and Quitmeyer Electronics believes that it can sell as many units of each product as it can manufacture. The table that follows summarizes the revenues and material costs for each product: Device Internal modem External modem Graphics circuit board CD drive Hard disk drive Memory expansion board
Revenue Per Unit Sold ($) 200 120 180 130 430 260
Material Cost Per Unit ($) 35 25 40 45 170 60
In addition, variable labor costs are $15 per hour for test device 1, $12 per hour for test device 2. and $18 per hour for test device 3. Quitmeyer Electronics wants to maximize its profits. (a) Formulate this problem as an LP model. (b) Solve the problem by computer. What is the best product mix? (c) What is the value of an additional minute of time per week on test device 1? Test device 2? Test device 3? Should Quitmeyer Electronics add more test device time? If so, on which equipment? Let X1 = the number of internal modems scheduled for manufacture each week X2 = the number of external modems scheduled for manufacture each week Etc., down to X6 = the number of mem. expansion boards scheduled for mfg. each week Maximize
161.35X1
+
92.95X2
+
135.50X3
+
82.50X4
+
249.80X5
+
191.75X6
S.T.
7X1
+
3X2
+
12X3
+
6X4
+
18X5
+
17X6
≤
7200
2X1
+
5X2
+
3X3
+
2X4
+
15X5
+
17X6
≤
7200
5X1
+
1X2
+
3X3
+
2X4
+
9X5
+
2X6
≤
6000
X1, X2, X3, X4, X5, X6
≥
0
Linear Programming Formulation Exercises from Textbook
8-15
ISM 4400, Fall 2006: Page 14/14
(Material blending problem) Amalgamated Products has just received a contract to construct steel body frames for automobiles that are to be produced at the new Japanese factory in Tennessee. The Japanese auto manufacturer has strict quality control standards for all of its component subcontractors and has informed Amalgamated that each frame must have the following steel content: Material Manganese Silicon Carbon
Minimum Percent 2.1 4.3 5.05
Maximum Percent 2.3 4.6 5.35
Amalgamated mixes batches of eight different available materials to produce one ton of steel used in the body frames. The table below details these materials. Formulate and solve the LP model that will indicate how much of each of the eight materials should be blended into a 1-ton load of steel so that Amalgamated meets its requirements while minimizing cost. Material Available Alloy 1 Alloy 2 Alloy 3 Iron 1 Iron 2 Carbide 1 Carbide 2 Carbide 3
Manganese (%) 70.0 55.0 12.0 1.0 5.0 0 0 0
Silicon (%) 15.0 30.0 26.0 10.0 2.5 24.0 25.0 23.00
Carbon (%) 3.0 1.0 0 3.0 0 18.0 20.0 25.0
Pounds Available No limit 300 No limit No limit No limit 50 200 100
Cost Per Pound ($) 0.12 0.13 0.15 0.09 0.07 0.10 0.12 0.09
Let X1 = the number of pounds of alloy 1 in one ton of steel X2 = the number of pounds of alloy 2 in one ton of steel Etc., down to X8 = the number of pounds of carbide 3 in one ton of steel Minimize S.T. (Mnmin) (Mn-max)
.12X1
+
.13X2
+
.15X3
+
.09X4
+
.07X5
.7X1
+
.55X2
+
.12X3
+
.01X4
+
.05X5
≥
42
.7X1
+
.55X2
+
.12X3
+
.01X4
+
.05X5
≤
46
(Si-min)
.15X1
+
.30X2
+
.26X3
+
.10X4
+
.025X5
.23X8
≥
86
.23X8
≤
92 101
(Si-max)
.15X1
+
.30X2
+
.26X3
+
.10X4
+
.025X5
+
+ +
.10X6
.24X6 .24X6
+
+ +
.12X7
.25X7 .25X7
+
+ +
.09X8
(C-min)
.03X1
+
.01X2
+
.03X4
+
.18X6
+
.20X7
+
.25X8
≥
(C-max)
.03X1
+
.01X2
+
.03X4
+
.18X6
+
.20X7
+
.25X8
≤
107
≤
300
≤
50
≤
200
Alloy 2 lim.
X2
Carbide 1 lim.
X6
Carbide 2 lim.
X7
Carbide 3 lim. Weighs 1 ton
X1
+
X2
+
X3
+
X4
+
X5
+
X6
+
X1, X2, X3, X4, X5, X6, X7, X8
≥
X7 0
+
X8
≤
100
X8
=
2000
ISM 4400, Fall 2006: Page 1/14
SOLUTIONS TO SELECT PROBLEMS FROM CHAPTER 7 7-14
The Electrocomp Corporation manufactures two electrical products: air conditioners and large fans. The assembly process for each is similar in that both require a certain amount of wiring and drilling. Each air conditioner takes 3 hours of wiring and 2 hours of drilling. Each fan must go through 2 hours of wiring and 1 hour of drilling. During the next production period, 240 hours of wiring time are available and up to 140 hours of drilling time maybe used. Each air conditioner sold yields a profit of $25. Each fan assembled may be sold for a $15 profit. Formulate and solve this LP production mix situation to find the best combination of air conditioners and fans that yields the highest profit. Use the corner point graphical approach. Let X1 = the number of air conditioners scheduled to be produced X2 = the number of fans scheduled to be produced Maximize Subject to:
25X1 + 3X1 + 2X1 +
15X2 (maximize profit) 2X2 ≤ 240 (wiring capacity constraint) X2 ≤ 140 (drilling capacity constraint) X1, X2 ≥ 0 (non-negativity constraints)
Optimal Solution: X1 = 40 X2 = 60 Profit = $1,900 7-15
Electrocomp’s management realizes that it forgot to include two critical constraints (see Problem 7-14). In particular, management decides that to ensure an adequate supply of air conditioners for a contract, at least 20 air conditioners should be manufactured. Because Electrocomp incurred an oversupply of fans in the preceding period, management also insists that no more than 80 fans be produced during this production period. Resolve this product mix problem to find the new optimal solution. Let X1 = the number of air conditioners scheduled to be produced X2 = the number of fans scheduled to be produced Maximize Subject to:
25X1 + 3X1 + 2X1 + X1
15X2 (maximize profit) 2X2 ≤ 240 (wiring capacity constraint) X2 ≤ 140 (drilling capacity constraint) ≥ 20 (a/c contract constraint) X2 ≤ 80 (maximum # of fans constraint) X1, X2 ≥ 0 (non-negativity constraints)
Optimal Solution: X1 = 40 X2 = 60 Profit = $1,900
Linear Programming Formulation Exercises from Textbook
7-16
ISM 4400, Fall 2006: Page 2/14
A candidate for mayor in a small town has allocated $40,000 for last-minute advertising in the days preceding the election. Two types of ads will be used: radio and television. Each radio ad costs $200 and reaches an estimated 3,000 people. Each television ad costs $500 and reaches an estimated 7,000 people. In planning the advertising campaign, the campaign manager would like to reach as many people as possible, but she has stipulated that at least 10 ads of each type must be used. Also, the number of radio ads must be at least as great as the number of television ads. How many ads of each type should be used? How many people will this reach? Let X1 = the number of radio ads purchased X2 = the number of television ads purchased Maximize 3,000X1 + 7,000X2 Subject to: 200X1 + 500X2 X1 X2 X1 X1, X2
≤ ≥ ≥ ≥ ≥
(maximize exposure) 40,000 (budget constraint) 10 (at least 10 radio ads purchased) 10 (at least 10 television ads purchased) X2 (# of radio ads ≥ # of television ads) 0 (non-negativity constraints)
For solution purposes, the fourth constraint would be rewritten as: X1 − X2 ≥ 0 Optimal Solution: X1 = 175 7-17
X2 = 10
Exposure = 595,000 people
The Outdoor Furniture Corporation manufactures two products, benches and picnic tables, for use in yards and parks. The firm has two main resources: its carpenters (labor force) and a supply of redwood for use in the furniture. During the next production cycle, 1,200 hours of labor are available under a union agreement. The firm also has a stock of 3500 feet of good-quality redwood. Each bench that Outdoor Furniture produces requires 4 labor hours and 10 feet of redwood; each picnic table takes 6 labor hours and 35 feet of redwood. Completed benches will yield a profit of $9 each, and tables will result in a profit of $20 each. How many benches and tables should Outdoor Furniture produce to obtain the largest possible profit? Use the graphical LP approach. Let X1 = the number of benches produced X2 = the number of tables produced Maximize Subject to:
9X1 + 4X1 + 10X1 +
20X2 6X2 ≤ 1,200 35X2 ≤ 3,500 X1, X2 ≥ 0
Optimal Solution: X1 = 262.5
X2 = 25
(maximize profit) (labor hours constraint) (redwood capacity constraint) (non-negativity constraints) Profit = $2,862.50
Linear Programming Formulation Exercises from Textbook
7-18
ISM 4400, Fall 2006: Page 3/14
The dean of the Western College of Business must plan the school’s course offerings for the fall semester. Student demands make it necessary to offer at least 30 undergraduate and 20 graduate courses in the term. Faculty contracts also dictate that at least 60 courses be offered in total. Each undergraduate course taught costs the college an average of $2,500 in faculty wages, and each graduate course costs $3,000. How many undergraduate and graduate courses should be taught in the fall so that total faculty salaries are kept to a minimum? Let X1 = the number of undergraduate courses scheduled X2 = the number of graduate courses scheduled Minimize Subject to:
2,500X1 + 3,000X2 X1 X2 X1 + X2 X1, X2
Optimal Solution: X1 = 40 7-19
≥ ≥ ≥ ≥ 0
X2 = 20
(minimize faculty salaries) 30 (schedule at least 30 undergrad courses) 20 (schedule at least 20 grad courses) 60 (schedule at least 60 total courses) (non-negativity constraints) Cost = $160,000
MSA Computer Corporation manufactures two models of minicomputers, the Alpha 4 and the Beta 5. The firm employs five technicians, working 160 hours each per month, on its assembly line. Management insists that full employment (i.e., all 160 hours of time) be maintained for each worker during next month’s operations. It requires 20 labor hours to assemble each Alpha 4 computer and 25 labor hours to assemble each Beta 5 model. MSA wants to see at least 10 Alpha 4s and at least 15 Beta 5s produced during the production period. Alpha 4s generate $1,200 profit per unit, and Beta 5s yield $1,800 each. Determine the most profitable number of each model of minicomputer to produce during the coming month. Let X1 = the number of Alpha 4 computers scheduled for production next month X2 = the number of Beta 5 computers scheduled for production next month Maximize Subject to:
1,200X1 + 1,800X2 20X1 + 25X2 X1 X2 X1, X2
Optimal Solution: X1 = 10
(maximize profit) = 800 (full employment, 5 workers x 160 hours) ≥ 10 (make at least 10 Alpha 4 computers) ≥ 15 (make at least 15 Beta 5 computers) ≥ 0 (non-negativity constraints)
X2 = 24
Profit = $55,200
Linear Programming Formulation Exercises from Textbook
7-20
ISM 4400, Fall 2006: Page 4/14
A winner of the Texas Lotto has decided to invest $50,000 per year in the stock market. Under consideration are stocks for a petrochemical firm and a public utility. Although a long-range goal is to get the highest possible return, some consideration is given to the risk involved with the stocks. A risk index on a scale of 1–10 (with 10 being the most risky) is assigned to each of the two stocks. The total risk of the portfolio is found by multiplying the risk of each stock by the dollars invested in that stock. The following table provides a summary of the return and risk: Stock Petrochemical Utility
Estimated Return 12% 6%
Risk Index 9 4
The investor would like to maximize the return on the investment, but the average risk index of the investment should not be higher than 6. How much should be invested in each stock? What is the average risk for this investment? What is the estimated return for this investment? Let X1 = the number of dollars invested in petrochemical stocks X2 = the number of dollars invested in utility stocks Maximize Subject to:
.12X1 + X1 + 3X1 −
.06X2 X2 ≤ 50,000 2X2 ≤ 0 X1, X2 ≥ 0
(maximize return on investment) (limit on total investment) (average risk cannot exceed 6) (non-negativity constraints)
Optimal Solution: X1 = $20,000 X2 = $30,000 Return = $4,200 The total risk is 300,000 (9 x $20,000 + 4 x $30,000), which yields an average risk of 6 (300,000/50,000 = 6). 7-21
Referring to the Texas Lotto situation in Problem 7-20, suppose the investor has changed his attitude about the investment and wishes to give greater emphasis to the risk of the investment. Now the investor wishes to minimize the risk of the investment as long as a return of at least 8% is generated. Formulate this as an LP problem and find the optimal solution. How much should be invested in each stock? What is the average risk for this investment? What is the estimated return for this investment? Let X1 = the number of dollars invested in petrochemical stocks X2 = the number of dollars invested in utility stocks Minimize Subject to:
9X1 + X1 + .04X1 −
4X2 X2 ≤ 50,000 .02X2 ≥ 0 X1, X2 ≥ 0
(minimize total risk) (limit on total investment) (average return must be at least 8%) (non-negativity constraints)
Optimal Solution: X1 = $16,666.67 X2 = $33,333.33 Total risk = 283,333.33 (which equates to an average risk of 283,333.33/50,000 = 5.67). The total return would be $4000 (.12 x 16,666.67 + .09 x 33,333.33), which just happens to be a return of exactly 8% ($4000/$50,000).
Linear Programming Formulation Exercises from Textbook
7-24
ISM 4400, Fall 2006: Page 5/14
The stock brokerage firm of Blank, Leibowitz, and Weinberger has analyzed and recommended two stocks to an investors’ club of college professors. The professors were interested in factors such as short term growth, intermediate growth, and dividend rates. These data on each stock are as follows:
Factor Short term growth potential, per dollar invested Intermediate growth potential (over next three years), per dollar invested Dividend rate potential
Stock Louisiana Gas and Trimex Insulation Power Company .36 .24 1.67
1.5
4%
8%
Each member of the club has an investment goal of (1) an appreciation of no less than $720 in the short term, (2) an appreciation of at least $5,000 in the next three years, and (3) a dividend income of at least $200 per year. What is the smallest investment that a professor can make to meet these three goals? Let X1 = the number of dollars invested in Louisiana Gas and Power X2 = the number of dollars invested in Trimex Insulation Co. Minimize Subject to:
X1 .36X1 1.67X1 .04X1
+ + + +
X2 .24X2 1.50X2 .08X2 X1, X2
≥ 720 ≥ 5,000 ≥ 200 ≥ 0
Optimal Solution: X1 = $1,359
(minimize total investment) (appreciation in the short term) (appreciation in next three years) (dividend income per year) (non-negativity constraints)
X2 = $1,818.18 Total investment = $3,177.18
Linear Programming Formulation Exercises from Textbook
7-25
ISM 4400, Fall 2006: Page 6/14
Woofer Pet Foods produces a low-calorie dog food for overweight dogs. This product is made from beef products and grain. Each pound of beef costs $0.90, and each pound of grain costs $0.60. A pound of the dog food must contain at least 9 units of Vitamin 1 and 10 units of Vitamin 2. A pound of beef contains 10 units of Vitamin 1 and 12 units of Vitamin 2. A pound of grain contains 6 units of Vitamin 1 and 9 units of Vitamin 2. Formulate this as an LP problem to minimize the cost of the dog food. How many pounds of beef and grain should be included in each pound of dog food? What is the cost and vitamin content of the final product? Let X1 = the number of pounds of beef in each pound of dog food X2 = the number of pounds of grain in each pound of dog food Minimize Subject to:
.90X1 X1 10X1 12X1
+ + + +
.60X2 X2 6X2 9X2 X1, X2
= ≥ ≥ ≥ 0
(minimize cost per pound of dog food) 1 (total weight should be one pound) 9 (at least 9 units of vitamin 1 in a pound) 10 (at least 10 units of vitamin 2 in a pound) (non-negativity constraints)
Optimal Solution: X1 = .75 X2 = .25 Cost = $.825
Linear Programming Formulation Exercises from Textbook
ISM 4400, Fall 2006: Page 7/14
SOLUTIONS TO SELECT PROBLEMS FROM CHAPTER 8 8-1
(Production problem) Winkler Furniture manufactures two different types of china cabinets: a French Provincial model and a Danish Modern model. Each cabinet produced must go through three departments: carpentry, painting, and finishing. The table below contains all relevant information concerning production times per cabinet produced and production capacities for each operation per day, along with net revenue per unit produced. The firm has a contract with an Indiana distributor to produce a minimum of 300 of each cabinet per week (or 60 cabinets per day). Owner Bob Winkler would like to determine a product mix to maximize his daily revenue. (a) Formulate as an LP problem. (b) Solve using an LP software program or spreadsheet. Cabinet Style French Provincial Danish Modern Dept. capacity (hrs)
Carpentry (Hours/Cabinet) 3 2 360
Painting (Hours/Cabinet) 1.5 1 200
Finishing (Hours/Cabinet) .75 .75 125
Net Revenue per Cabinet ($) 28 25
Let X1 = the number of French Provincial cabinets produced each day X2 = the number of Danish Modern cabinets produced each day Maximize Subject to:
28X1 3X1 1.5X1 .75X1 X1
+ + + +
25X2 (maximize revenue) 2X2 ≤ 360 (carpentry hours available) X2 ≤ 200 (painting hours available) .75X2 ≤ 125 (finishing hours available) ≥ 60 (contract requirement on F.P. cabinets) X2 ≥ 60 (contract requirement on D.M. cabinets) X1, X2 ≥ 0 (non-negativity constraints)
Optimal Solution: X1 = 60
X2 = 90
Revenue = $3,930
Linear Programming Formulation Exercises from Textbook
8-2
ISM 4400, Fall 2006: Page 8/14
(Investment decision problem) The Heinlein and Krarnpf Brokerage firm has just been instructed by one of its clients to invest $250,000 for her money obtained recently through the sale of land holdings in Ohio. The client has a good deal of trust in the investment house, but she also has her own ideas about the distribution of the funds being invested. In particular, she requests that the firm select whatever stocks and bonds they believe are well rated, but within the following guidelines: (a) Municipal bonds should constitute at least 20% of the investment. (b) At least 40% of the funds should be placed in a combination of electronic firms, aerospace firms, and drug manufacturers. (c) No more than 50% of the amount invested in municipal bonds should be placed in a highrisk, high-yield nursing home stock. Subject to these restraints, the client’s goal is to maximize projected return on investments. The analysts at Heinlein and Krampf, aware of these guidelines, prepare a list of high-quality stocks and bonds and their corresponding rates of return. Projected Rate of Return (%) 5.3 6.8 4.9 8.4 11.8
Investment Los Angeles municipal bonds Thompson Electronics, Inc. United Aerospace Corp. Palmer Drugs Happy Days Nursing Homes
(a) Formulate this portfolio selection problem using LP. (b) Solve this problem. Let X1 = dollars invested in Los Angeles municipal bonds X2 = dollars invested in Thompson Electronics X3 = dollars invested in United Aerospace X4 = dollars invested in Palmer Drugs X5 = dollars invested in Happy Days Nursing Homes Maximize
.053X1
+
.068X2
+
.049X3
+
.084X4
+
.118X5
Subject to:
X1
+
X2
+
X3
+
X4
+
X5
≤
250,000
.8X1
-
.2X2
-
.2X3
-
.2X4
-
.2X5
≥
0
(municipal bond restriction)
-.4X1
+
.6X2
+
.6X3
+
.6X4
-
.4X5
≥
0
(electronics, aerospace, drugs combo)
+
X5
≤
0
(nursing home as a percent of bonds)
X1, X2, X3, X4, X5
≥
-.5X1
(maximize return on investment)
0
(total funds available)
(non-negativity constraints)
Optimal Solution: X1 = $50,000 X2 = $0 X3 = $0 X4 = $175,000 X5 = $25,000 ROI = $20,300
Linear Programming Formulation Exercises from Textbook
8-3
ISM 4400, Fall 2006: Page 9/14
(Restaurant work scheduling problem). The famous Y. S. Chang Restaurant is open 24 hours a day. Waiters and busboys report for duty at 3AM., 7 AM., 11 AM., 3 P.M., 7 P.M., or 11 P.M., and each works an 8-hour shift. The following table shows the minimum number of workers needed during the six periods into which the day is divided. Chang’s scheduling problem is to determine how many waiters and busboys should report for work at the start of each time period to minimize the total staff required for one day’s operation. (Hint: Let Xi equal the number of waiters and busboys beginning work in time period i, where i = 1, 2,3,4,5,6.) Period 1 2 3 4 5 6
Time 3 A.M–7 A.M. 7 A.M–11 A.M. 11 A.M–3 P.M. 3 P.M–7 P.M. 7 P.M–11 P.M. 11 P.M–3 A.M.
Number of Waiters and Busboys Required 3 12 16 9 11 4
Let Xi = the number workers beginning work at the start of time period i (i=1,2,3,4,5,6) Minimize Subject to:
X1
+
X2
+
X3
+
X4
+
X5
(min. staff size) 3
(period 1)
X2
≥
12
(period 2)
X2
≥
16
(period 3)
X4
≥
9
(period 4)
X4
X5
≥
11
(period 5)
X5
X6
≥
4
(period 6)
X1, X2, X3, X4, X5, X6
≥
+ +
X6 ≥
X1 X1
+
+
X6
X3 X3
+
+
+
0
(non-negativity)
Linear Programming Formulation Exercises from Textbook
8-4
ISM 4400, Fall 2006: Page 10/14
(Animal feed mix problem) The Battery Park Stable feeds and houses the horses used to pull tourist-filled carriages through the streets of Charleston’s historic waterfront area. The stable owner, an ex-racehorse trainer, recognizes the need to set a nutritional diet for the horses in his care. At the same time, he would like to keep the overall daily cost of feed to a minimum. The feed mixes available for the horses’ diet are an oat product, a highly enriched grain, and a mineral product. Each of these mixes contains a certain amount of five ingredients needed daily to keep the average horse healthy. The table below shows these minimum requirements, units of each ingredient per pound of feed mix, and costs for the three mixes. In addition, the stable owner is aware that an overfed horse is a sluggish worker. Consequently, he determines that 6 pounds of feed per day are the most that any horse needs to function properly. Formulate this problem and solve for the optimal daily mix of the three feeds. Diet Requirement (Ingredients) A B C D E Cost/lb Let
Oat Product (units/lb) 2 .5 3 1 .5 $0.09
Feed Mix Enriched Grain (units/lb) 3 1 5 1.5 .5 $0.14
Mineral Product (units/lb) 1 .5 6 2 1.5 $0.17
Minimum Daily Requirement (units) 6 2 9 8 5
X1 = the number pounds of oat product per horse each day X2 = the number pounds of enriched grain per horse each day X3 = the number pounds of mineral product per horse each day
Minimize s.t.
.09X1 2X1 .5X1 3X1 X1 .5X1 X1
+ + + + + + +
.14X2 3X2 X2 5X2 1.5X2 .5X2 X2
+ + + + + + +
.17X3 X3 .5X3 6X3 2X3 1.5X3 X3 X1, X2, X3
≥ ≥ ≥ ≥ ≥ ≤
≥
6 2 9 8 5 6 0
(minimize cost) (ingredient A) (ingredient B) (ingredient C) (ingredient D) (ingredient E) (maximum feed per day) (non-negativity constraints)
Linear Programming Formulation Exercises from Textbook
8-6
ISM 4400, Fall 2006: Page 11/14
(Media selection problem) The advertising director for Diversey Paint and Supply, a chain of four retail stores on Chicago’s North Side, is considering two media possibilities. One plan is for a series of half- page ads in the Sunday Chicago Tribune newspaper, and the other is for advertising time on Chicago TV. The stores are expanding their lines of do-it-yourself tools, and the advertising director is interested in an exposure level of at least 40% within the city’s neighborhoods and 60% in northwest suburban areas. The TV viewing time under consideration has an exposure rating per spot of 5% in city homes and 3% in the northwest suburbs. The Sunday newspaper has corresponding exposure rates of 4% and 3% per ad. The cost of a half-page Tribune ment is $925; a television spot costs $2,000. Diversey Paint would like to select the least costly advertising strategy that would meet desired exposure levels. (a) Formulate using LP. (b) Solve the problem. Let
X1 = the number of newspaper ads placed X2 = the number of TV spots purchased
Minimize Subject to:
925X1 + 2,000X2 .04X1 + .05X2 ≥ .03X1 + .03X2 ≥ X1, X2 ≥ 0
(minimize cost) .4 (city exposure) .6 (suburb exposure) (non-negativity constraints)
Linear Programming Formulation Exercises from Textbook
8-11
ISM 4400, Fall 2006: Page 12/14
(College meal selection problem) Kathy Roniger, campus dietician for a small Idaho college, is responsible for formulating a nutritious meal plan for students. For an evening meal, she feels that the following five meal-content requirements should be met: (1) between 900 and 1,500 calories; (2) at least 4 milligrams of iron; (3) no more than 50 grams of fat; (4) at least 26 grams of protein; and (5) no more than 50 grams of carbohydrates. On a particular day, Roniger’s food stock includes seven items that can be prepared and served for supper to meet these requirements. The cost per pound for each food item and the contribution to each of the five nutritional requirements are given in the accompanying table: Calories/ Pound 295 1216 394 358 128 118 279
Food Item Milk Ground Meat Chicken Fish Beans Spinach Potatoes
Iron (mg/lb) 0.2 0.2 4.3 3.2 3.2 14.1 2.2
Table of Food Values and Costs Fat Protein Carbs. (gm/lb) (gm/lb) (gm/lb) 16 16 22 96 81 0 9 74 0 0.5 83 0 0.8 7 28 1.4 14 19 0.5 8 63
Cost/ Pound ($) 0.60 2.35 1.15 2.25 0.58 1.17 0.33
What combination and amounts of food items will provide the nutrition Roniger requires at the least total food cost? Let X1 = the number of pounds of milk per student in the evening meal X2 = the number of pounds of ground meat per student in the evening meal Etc., down to X7 = the number of pounds of potatoes per student in the evening meal Minimize S.T. (Cal.)
.6X1
+
2.35X2
+
1.15X3
+
2.25X4
+
.58X5
+
1.17X6
+
295X1
+
1216X2
+
394X3
+
358X4
+
128X5
+
118X6
+
.33X7 279X7
≥
900 1500
(Cal.)
295X1
+
1216X2
+
394X3
+
358X4
+
128X5
+
118X6
+
279X7
≤
(Iron)
.2X1
+
.2X2
+
4.3X3
+
3.2X4
+
3.2X5
+
14.1X6
+
2.2X7
≥
4
(Fat)
16X1
+
96X2
+
9X3
+
.5X4
+
.8X5
+
1.4X6
+
.5X7
≤
50 26 50
(Protein)
16X1
(Carbs.)
22X1
+
81X2
+
74X3
+
83X4
+
7X5
+
14X6
+
8X7
≥
+
28X5
+
19X6
+
63X7
≤
X1, X2, X3, X4, X5, X6, X7
≥
0
Linear Programming Formulation Exercises from Textbook
8-12
ISM 4400, Fall 2006: Page 13/14
(High tech production problem) Quitmeyer Electronics Incorporated manufactures the following six microcomputer peripheral devices: internal modems, external modems, graphics circuit boards, CD drives, hard disk drives, and memory expansion boards. Each of these technical products requires time, in minutes, on three types of electronic testing equipment, as shown in the table the following table: Internal Modem Test device 1 7 Test device 2 2 Test device 3 5
External Modem 3 5 1
Circuit Board 12 3 3
CD Drive 6 2 2
Hard Drive 18 15 9
Memory Board 17 17 2
The first two test devices are available 120 hours per week. The third (device 3) requires more preventive maintenance and may be used only 100 hours each week. The market for all six computer components is vast, and Quitmeyer Electronics believes that it can sell as many units of each product as it can manufacture. The table that follows summarizes the revenues and material costs for each product: Device Internal modem External modem Graphics circuit board CD drive Hard disk drive Memory expansion board
Revenue Per Unit Sold ($) 200 120 180 130 430 260
Material Cost Per Unit ($) 35 25 40 45 170 60
In addition, variable labor costs are $15 per hour for test device 1, $12 per hour for test device 2. and $18 per hour for test device 3. Quitmeyer Electronics wants to maximize its profits. (a) Formulate this problem as an LP model. (b) Solve the problem by computer. What is the best product mix? (c) What is the value of an additional minute of time per week on test device 1? Test device 2? Test device 3? Should Quitmeyer Electronics add more test device time? If so, on which equipment? Let X1 = the number of internal modems scheduled for manufacture each week X2 = the number of external modems scheduled for manufacture each week Etc., down to X6 = the number of mem. expansion boards scheduled for mfg. each week Maximize
161.35X1
+
92.95X2
+
135.50X3
+
82.50X4
+
249.80X5
+
191.75X6
S.T.
7X1
+
3X2
+
12X3
+
6X4
+
18X5
+
17X6
≤
7200
2X1
+
5X2
+
3X3
+
2X4
+
15X5
+
17X6
≤
7200
5X1
+
1X2
+
3X3
+
2X4
+
9X5
+
2X6
≤
6000
X1, X2, X3, X4, X5, X6
≥
0
Linear Programming Formulation Exercises from Textbook
8-15
ISM 4400, Fall 2006: Page 14/14
(Material blending problem) Amalgamated Products has just received a contract to construct steel body frames for automobiles that are to be produced at the new Japanese factory in Tennessee. The Japanese auto manufacturer has strict quality control standards for all of its component subcontractors and has informed Amalgamated that each frame must have the following steel content: Material Manganese Silicon Carbon
Minimum Percent 2.1 4.3 5.05
Maximum Percent 2.3 4.6 5.35
Amalgamated mixes batches of eight different available materials to produce one ton of steel used in the body frames. The table below details these materials. Formulate and solve the LP model that will indicate how much of each of the eight materials should be blended into a 1-ton load of steel so that Amalgamated meets its requirements while minimizing cost. Material Available Alloy 1 Alloy 2 Alloy 3 Iron 1 Iron 2 Carbide 1 Carbide 2 Carbide 3
Manganese (%) 70.0 55.0 12.0 1.0 5.0 0 0 0
Silicon (%) 15.0 30.0 26.0 10.0 2.5 24.0 25.0 23.00
Carbon (%) 3.0 1.0 0 3.0 0 18.0 20.0 25.0
Pounds Available No limit 300 No limit No limit No limit 50 200 100
Cost Per Pound ($) 0.12 0.13 0.15 0.09 0.07 0.10 0.12 0.09
Let X1 = the number of pounds of alloy 1 in one ton of steel X2 = the number of pounds of alloy 2 in one ton of steel Etc., down to X8 = the number of pounds of carbide 3 in one ton of steel Minimize S.T. (Mnmin) (Mn-max)
.12X1
+
.13X2
+
.15X3
+
.09X4
+
.07X5
.7X1
+
.55X2
+
.12X3
+
.01X4
+
.05X5
≥
42
.7X1
+
.55X2
+
.12X3
+
.01X4
+
.05X5
≤
46
(Si-min)
.15X1
+
.30X2
+
.26X3
+
.10X4
+
.025X5
.23X8
≥
86
.23X8
≤
92 101
(Si-max)
.15X1
+
.30X2
+
.26X3
+
.10X4
+
.025X5
+
+ +
.10X6
.24X6 .24X6
+
+ +
.12X7
.25X7 .25X7
+
+ +
.09X8
(C-min)
.03X1
+
.01X2
+
.03X4
+
.18X6
+
.20X7
+
.25X8
≥
(C-max)
.03X1
+
.01X2
+
.03X4
+
.18X6
+
.20X7
+
.25X8
≤
107
≤
300
≤
50
≤
200
Alloy 2 lim.
X2
Carbide 1 lim.
X6
Carbide 2 lim.
X7
Carbide 3 lim. Weighs 1 ton
X1
+
X2
+
X3
+
X4
+
X5
+
X6
+
X1, X2, X3, X4, X5, X6, X7, X8
≥
X7 0
+
X8
≤
100
X8
=
2000
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