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SOLUTIONS TO SELECT PROBLEMS FROM CHAPTER 7
Linear Programming Formulation Exercises from Textbook
ISM 4400, Fall 2006: Page /14
SOLUTIONS TO SELECT PROBLEMS FROM CHAPTER 7
7-14The 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.
LetX1 = the number of air conditioners scheduled to be produced
X2 = the number of fans scheduled to be produced
Maximize
25X1
+
15X2
(maximize profit)
Subject to:
3X1
+
2X2
240
(wiring capacity constraint)
2X1
+
X2
140
(drilling capacity constraint)
X1, X2
0
(non-negativity constraints)
Optimal Solution: X1 = 40 X2 = 60 Profit = $1,900
7-15Electrocomps 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
25X1
+
15X2
(maximize profit)
Subject to:
3X1
+
2X2
240
(wiring capacity constraint)
2X1
+
X2
140
(drilling capacity constraint)
X1
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
7-16A 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?
LetX1 = the number of radio ads purchased
X2 = the number of television ads purchased
Maximize
3,000X1
+
7,000X2
(maximize exposure)
Subject to:
200X1
+
500X2
40,000
(budget constraint)
X1
10
(at least 10 radio ads purchased)
X2
10
(at least 10 television ads purchased)
X1
X2
(# of radio ads # of television ads)
X1, X2
0
(non-negativity constraints)
For solution purposes, the fourth constraint would be rewritten as: X1X20
Optimal Solution: X1 = 175X2 = 10Exposure = 595,000 people
7-17The 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.
LetX1 = the number of benches produced
X2 = the number of tables produced
Maximize
9X1
+
20X2
(maximize profit)
Subject to:
4X1
+
6X2
1,200
(labor hours constraint)
10X1
+
35X2
3,500
(redwood capacity constraint)
X1, X2
0
(non-negativity constraints)
Optimal Solution: X1 = 262.5X2 = 25Profit = $2,862.50
7-18The dean of the Western College of Business must plan the schools 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?
LetX1 = the number of undergraduate courses scheduled
X2 = the number of graduate courses scheduled
Minimize
2,500X1
+
3,000X2
(minimize faculty salaries)
Subject to:
X1
30
(schedule at least 30 undergrad courses)
X2
20
(schedule at least 20 grad courses)
X1
+
X2
60
(schedule at least 60 total courses)
X1, X2
0
(non-negativity constraints)
Optimal Solution: X1 = 40X2 = 20Cost = $160,000
7-19MSA 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 months 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.
LetX1 = the number of Alpha 4 computers scheduled for production next month
X2 = the number of Beta 5 computers scheduled for production next month
Maximize
1,200X1
+
1,800X2
(maximize profit)
Subject to:
20X1
+
25X2
=
800
(full employment, 5 workers x 160 hours)
X1
10
(make at least 10 Alpha 4 computers)
X2
15
(make at least 15 Beta 5 computers)
X1, X2
0
(non-negativity constraints)
Optimal Solution: X1 = 10X2 = 24Profit = $55,200
7-20A 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 110 (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
Estimated Return
Risk Index
Petrochemical
12%
9
Utility 6% 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?
LetX1 = the number of dollars invested in petrochemical stocks
X2 = the number of dollars invested in utility stocks
Maximize
.12X1
+
.06X2
(maximize return on investment)
Subject to:
X1
+
X2
50,000
(limit on total investment)
3X1
2X2
0
(average risk cannot exceed 6)
X1, X2
0
(non-negativity constraints)
Optimal Solution: X1 = $20,000X2 = $30,000Return = $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).
7-21Referring 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?
LetX1 = the number of dollars invested in petrochemical stocks
X2 = the number of dollars invested in utility stocks
Minimize
9X1
+
4X2
(minimize total risk)
Subject to:
X1
+
X2
50,000
(limit on total investment)
.04X1
.02X2
0
(average return must be at least 8%)
X1, X2
0
(non-negativity constraints)
Optimal Solution: X1 = $16,666.67X2 = $33,333.33Total 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).
7-24The 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:
Stock
Factor
Louisiana Gas and
Power
Trimex Insulation
Company
Short term growth potential, per dollar invested Intermediate
growth potential (over next three years), per dollar invested Dividend rate
.36.24
1.671.5
4%8%
potential
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
X1
+
X2
(minimize total investment)
Subject to:
.36X1
+
.24X2
720
(appreciation in the short term)
1.67X1
+
1.50X2
5,000
(appreciation in next three years)
.04X1
+
.08X2
200
(dividend income per year)
X1, X2
0
(non-negativity constraints)
Optimal Solution: X1 = $1,359X2 = $1,818.18 Total investment = $3,177.18
7-25Woofer 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?
LetX1 = 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
.90X1
+
.60X2
(minimize cost per pound of dog food)
Subject to:
X1
+
X2
=
1
(total weight should be one pound)
10X1
+
6X2
9
(at least 9 units of vitamin 1 in a pound)
12X1
+
9X2
10
(at least 10 units of vitamin 2 in a pound)
X1, X2
0
(non-negativity constraints)
Optimal Solution: X1 = .75 X2 = .25 Cost = $.825
SOLUTIONS TO SELECT PROBLEMS FROM CHAPTER 8
8-1(Production problem) Winkler Furniture manufactures two different types of china cabinets: a French Provincial model and a Danish Modern model. Each cabinet produced must go through three departments: carpentry, painting, and finishing. The table below contains all relevant information concerning production times per cabinet produced and production capacities for each operation per day, along with net revenue per unit produced. The firm has a contract with
an Indiana distributor to produce a minimum of 300 of each cabinet per week (or 60 cabinets per day). Owner Bob Winkler would like to determine a product mix to maximize his daily revenue. (a) Formulate as an LP problem.
(b) Solve using an LP software program or spreadsheet.
Cabinet Style
Carpentry
(Hours/Cabinet)
Painting
(Hours/Cabinet)
Finishing
(Hours/Cabinet)
Net Revenue per Cabinet ($)
French Provincial
3
1.5
.75
28
Danish Modern
2
1
.75
25
Dept. capacity (hrs)
360
200
125
LetX1 = the number of French Provincial cabinets produced each day
X2 = the number of Danish Modern cabinets produced each day
Maximize
28X1
+
25X2
(maximize revenue)
Subject to:
3X1
+
2X2
360
(carpentry hours available)
1.5X1
+
X2
200
(painting hours available)
.75X1
+
.75X2
125
(finishing hours available)
X1
60
(contract requirement on F.P. cabinets)
X2
60
(contract requirement on D.M. cabinets)
X1, X2
0
(non-negativity constraints)
Optimal Solution: X1 = 60X2 = 90Revenue = $3,930
8-2(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 high- risk, high-yield nursing home stock.
Subject to these restraints, the clients 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.
Investment
Projected Rate of Return (%)
Los Angeles municipal bonds5.3
Thompson Electronics, Inc.6.8
United Aerospace Corp.4.9
Palmer Drugs8.4
Happy Days Nursing Homes 11.8
(a) Formulate this portfolio selection problem using LP. (b) Solve this problem. LetX1 = 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
(maximize return on investment)
Subject to:
X1
+
X2
+
X3
+
X4
+
X5
250,000
(total funds available)
.8X1
-
.2X2
-
.2X3
-
.2X4
-
.2X5
0
(municipal bond restriction)
-.4X1
+
.6X2
+
.6X3
+
.6X4
-
.4X5
0
(electronics, aerospace, drugs combo)
-.5X1
+
X5
0
(nursing home as a percent of bonds)
X1, X2, X3, X4, X5 0(non-negativity constraints)
Optimal Solution: X1 = $50,000 X2 = $0 X3 = $0 X4 = $175,000 X5 = $25,000 ROI = $20,300
8-3(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. Changs 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 days 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.)
PeriodTime
Number of Waiters and
Busboys Required
13 A.M7 A.M.3
27 A.M11 A.M.12
311 A.M3 P.M.16
43 P.M7 P.M.9
57 P.M11 P.M.11
6 11 P.M3 A.M. 4
LetXi = the number workers beginning work at the start of time period i (i=1,2,3,4,5,6)
(min. staff size)
3
(period 1)
12
(period 2)
16
(period 3)
9
(period 4)
11
(period 5)
0
4
(period 6)
(non-negativity)
MinimizeX1 +X2 +X3 +X4 +X5 +X6
Subject to:X1 +X6
X1 +X2
X2 +X3
X3 +X4
X4 +X5
X5 +X6
X1, X2, X3, X4, X5, X6
8-4(Animal feed mix problem) The Battery Park Stable feeds and houses the horses used to pull tourist-filled carriages through the streets of Charlestons 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.
Feed Mix
Diet Requirement (Ingredients)
Oat Product (units/lb)
Enriched Grain (units/lb)
Mineral Product (units/lb)
Minimum Daily Requirement (units)
A
2
3
1
6
B
.5
1
.5
2
C
3
5
6
9
D
1
1.5
2
8
E
.5
.5
1.5
5
Cost/lb
$0.09
$0.14
$0.17
LetX1 = 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
.09X1
+
.14X2
+
.17X3
(minimize cost)
s.t.
2X1
+
3X2
+
X3
6
(ingredient A)
.5X1
+
X2
+
.5X3
2
(ingredient B)
3X1
+
5X2
+
6X3
9
(ingredient C)
X1
+
1.5X2
+
2X3
8
(ingredient D)
.5X1
+
.5X2
+
1.5X3
5
(ingredient E)
X1
+
X2
+
X3
6
(maximum feed per day)
X1, X2, X3
0
(non-negativity constraints)
8-6(Media selection problem) The advertising director for Diversey Paint and Supply, a chain of four retail stores on Chicagos 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 citys 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 advertisement is $925; a television spot costs $2,000.
Diversey Paint would like to select the least costly advertising strategy that would meet desired exposure levels.
(a) Formulate using LP. (b) Solve the problem.
LetX1 = the number of newspaper ads placed
X2 = the number of TV spots purchased
Minimize
925X1
+
2,000X2
(minimize cost)
Subject to:
.04X1
+
.05X2
.4
(city exposure)
.03X1
+
.03X2
.6
(suburb exposure)
X1, X2
0
(non-negativity constraints)
8-11(College meal selection problem) Kathy Roniger, campus dietician for a small Idaho college, is responsible for formulating a nutritious meal plan for students. For an evening meal, she feels that the following five meal-content requirements should be met: (1) between 900 and 1,500 calories; (2) at least 4 milligrams of iron; (3) no more than 50 grams of fat; (4) at least 26 grams of protein; and (5) no more than 50 grams of carbohydrates. On a particular day, Ronigers food stock includes seven items that can be prepared and served for supper to meet these requirements. The cost per pound for each food item and the contribution to each of the five nutritional requirements are given in the accompanying table:
Table of Food Values and Costs
Food Item
Calories/ Pound
Iron
(mg/lb)
Fat
(gm/lb)
Protein
(gm/lb)
Carbs. (gm/lb)
Cost/ Pound ($)
Milk
295
0.2
16
16
22
0.60
Ground Meat
1216
0.2
96
81
0
2.35
Chicken
394
4.3
9
74
0
1.15
Fish
358
3.2
0.5
83
0
2.25
Beans
128
3.2
0.8
7
28
0.58
Spinach
118
14.1
1.4
14
19
1.17
Potatoes 279 2.2 0.5 8 63 0.33
What combination and amounts of food items will provide the nutrition Roniger requires at the least total food cost?
LetX1 = 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
900
1500
4
50
26
50
Minimize.6X1 +2.35X2 +1.15X3 +2.25X4 +.58X5 +1.17X6 +.33X7
S.T. (Cal.) 295X1 + 1216X2 + 394X3 + 358X4 + 128X5 + 118X6 + 279X7 (Cal.) 295X1 + 1216X2 + 394X3 + 358X4 + 128X5 + 118X6 + 279X7 (Iron).2X1 + .2X2 + 4.3X3 + 3.2X4 + 3.2X5 + 14.1X6 + 2.2X7 (Fat) 16X1 + 96X2 + 9X3 + .5X4 + .8X5 + 1.4X6 + .5X7
(Protein)16X1 +81X2 +74X3 +83X4 +7X5 +14X6 +8X7
(Carbs.)22X1 +28X5 +19X6 +63X7
X1, X2, X3, X4, X5, X6, X7 0
8-12(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
External
Circuit
CD
Hard
Memory
Modem
Modem
Board
Drive
Drive
Board
Test device 1
7
3
12
6
18
17
Test device 2
2
5
3
2
15
17
Test device 3 5 1 3 2 9 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
Revenue Per
Unit Sold ($)
Material Cost
Per Unit ($)
Internal modem20035
External modem12025
Graphics circuit board18040
CD drive13045
Hard disk drive430170
Memory expansion board 260 60
In addition, variable labor costs are $15 per hour for test device 1, $12 per hour for test device 2. and $18 per hour for test device 3. Quitmeyer Electronics wants to maximize its profits.
(a) Formulate this problem as an LP model.
(b) Solve the problem by computer. What is the best product mix?
(c) What is the value of an additional minute of time per week on test device 1? Test device 2? Test device 3? Should Quitmeyer Electronics add more test device time? If so, on which equipment?
LetX1 = the number of internal modems scheduled for manufacture each week
X2 = the number of external modems scheduled for manufacture each week
Etc., down to X6 = the number of mem. expansion boards scheduled for mfg. each week
Maximize
161.35X1
+
92.95X2
+
135.50X3
+
82.50X4
+
249.80X5
+
191.75X6
S.T.
7X1
+
3X2
+
12X3
+
6X4
+
18X5
+
17X6
7200
2X1
+
5X2
+
3X3
+
2X4
+
15X5
+
17X6
7200
5X1
+
1X2
+
3X3
+
2X4
+
9X5
+
2X6
6000
X1, X2, X3, X4, X5, X60
8-15(Material blending problem) Amalgamated Products has just received a contract to construct steel body frames for automobiles that are to be produced at the new Japanese factory in Tennessee. The Japanese auto manufacturer has strict quality control standards for all of its component subcontractors and has informed Amalgamated that each frame must have the following steel content:
Material
Minimum Percent
Maximum Percent
Manganese
2.1
2.3
Silicon
4.3
4.6
Carbon 5.05 5.35
Amalgamated mixes batches of eight different available materials to produce one ton of steel used in the body frames. The table below details these materials. Formulate and solve the LP model that will indicate how much of each of the eight materials should be blended into a 1-ton load of steel so that Amalgamated meets its requirements while minimizing cost.
Material
Available
Manganese
(%)
Silicon
(%)
Carbon
(%)
Pounds
Available
Cost Per
Pound ($)
Alloy 1
70.0
15.0
3.0
No limit
0.12
Alloy 2
55.0
30.0
1.0
300
0.13
Alloy 3
12.0
26.0
0
No limit
0.15
Iron 1
1.0
10.0
3.0
No limit
0.09
Iron 2
5.0
2.5
0
No limit
0.07
Carbide 1
0
24.0
18.0
50
0.10
Carbide 2
0
25.0
20.0
200
0.12
Carbide 3 0 23.00 25.0 100 0.09
LetX1 = 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
42
46
86
92
101
107
300
50
200
100
=
2000
Minimize.12X1 +.13X2 +.15X3 +.09X4 +.07X5 +.10X6 +.12X7 +.09X8
S.T. (Mn-.7X1 +.55X2 +.12X3 +.01X4 +.05X5
(Mn-max).7X1 +.55X2 +.12X3 +.01X4 +.05X5
(Si-min) .15X1 +.30X2 +.26X3 + .10X4 +.025X5 + .24X6 + .25X7 + .23X8 (Si-max) .15X1 +.30X2 +.26X3 + .10X4 +.025X5 + .24X6 + .25X7 + .23X8 (C-min).03X1 +.01X2 +.03X4 +.18X6 +.20X7 +.25X8 (C-max) .03X1 + .01X2 + .03X4 + .18X6 + .20X7 + .25X8
Alloy 2 lim.X2
Carbide 1 lim.X6
Carbide 2 lim.X7
Carbide 3 lim.X8
Weighs 1 tonX1 +X2 +X3 +X4 +X5 +X6 +X7 +X8
X1, X2, X3, X4, X5, X6, X7, X8 0