A group of fresh industrial engineering graduates in the Sultanate of Oman are starting their own business, specializing in the production of two types of metal ingots, I and II. They are eager to apply what they learned in their Theory of Modeling and Optimization course to solve a real-world problem. The graduates operate a foundry that smelts steel, aluminum, and cast-iron scraps. The ingots have specific limits on the aluminum, graphite, and silicon contents. To meet these specifications, aluminum and silicon briquettes may be used in the smelting process. The problem is to determine the optimal blending of input materials that satisfies the demand requirements at minimum cost. This involves deciding how much of each input material to use in the production of each type of ingot. Input item Steel scrap Aluminum scrap Cast iron scrap Aluminum briquette Silicon briquette Contents (%) Aluminum Graphite 5 1 15 10 95 0 100 0 Ingredient Aluminum Graphite Silicon Demand (tons/day) 0 0 Ingot I (%) Minimum 8.1 1.5 2.5 Silicon 4 2 8 0 100 Maximum 10.8 3.0 130 Cost/ton ($) Available (tons/day) 100 1000 150 500 75 2500 Any amount Any amount 900 380 Ingot II (%) Minimum 6.2 4.1 2.8 250 Maximum 8.9 4.1

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question

1) Provide a schematic diagram for this alloy blending problem. (Indicate the problem data and

the controllable variables/decision variables on the diagram). 2) Formulate a Linear Programming

(LP) model that can be applied to determine the optimal input mix the foundry should smelt.

A group of fresh industrial engineering graduates in the Sultanate of Oman are starting their own business,
specializing in the production of two types of metal ingots, I and II. They are eager to apply what they
learned in their Theory of Modeling and Optimization course to solve a real-world problem. The graduates
operate a foundry that smelts steel, aluminum, and cast-iron scraps. The ingots have specific limits on the
aluminum, graphite, and silicon contents. To meet these specifications, aluminum and silicon briquettes
may be used in the smelting process. The problem is to determine the optimal blending of input materials
that satisfies the demand requirements at minimum cost. This involves deciding how much of each input
material to use in the production of each type of ingot.
Input item
Steel scrap
Aluminum scrap
Cast iron scrap
Aluminum briquette
Silicon briquette
Contents (%)
Aluminum Graphite
5
1
15
10
95
0
100
0
Ingredient
Aluminum
Graphite
Silicon
Demand (tons/day)
0
0
Ingot I (%)
Minimum
8.1
1.5
2.5
Silicon
4
2
8
0
100
Maximum
10.8
3.0
130
Cost/ton ($) Available (tons/day)
100
1000
150
500
75
2500
Any amount
Any amount
900
380
Ingot II (%)
Minimum
6.2
4.1
2.8
250
Maximum
8.9
4.1
Transcribed Image Text:A group of fresh industrial engineering graduates in the Sultanate of Oman are starting their own business, specializing in the production of two types of metal ingots, I and II. They are eager to apply what they learned in their Theory of Modeling and Optimization course to solve a real-world problem. The graduates operate a foundry that smelts steel, aluminum, and cast-iron scraps. The ingots have specific limits on the aluminum, graphite, and silicon contents. To meet these specifications, aluminum and silicon briquettes may be used in the smelting process. The problem is to determine the optimal blending of input materials that satisfies the demand requirements at minimum cost. This involves deciding how much of each input material to use in the production of each type of ingot. Input item Steel scrap Aluminum scrap Cast iron scrap Aluminum briquette Silicon briquette Contents (%) Aluminum Graphite 5 1 15 10 95 0 100 0 Ingredient Aluminum Graphite Silicon Demand (tons/day) 0 0 Ingot I (%) Minimum 8.1 1.5 2.5 Silicon 4 2 8 0 100 Maximum 10.8 3.0 130 Cost/ton ($) Available (tons/day) 100 1000 150 500 75 2500 Any amount Any amount 900 380 Ingot II (%) Minimum 6.2 4.1 2.8 250 Maximum 8.9 4.1
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