Question 1171735: The corporate and strategic department has decided to add exactly 200 km of
highway and exactly 100 km of expressway to its road system this year. The
standard price for road construction is RM1 million per kilometer of highway and
RM5 million per kilometer of expressway. Only two contractors, company ABC and
company RST, are eligible to proceed with the construction, so the entire 300 km of
road must be built by these two companies.
However, company ABC can construct at most 200 km of roadway (highway and
expressway) and company RST can construct at most 150 km of roadway (highway
and expressway). Each company must be awarded a contract with a standard price
of at least RM250 million (before discounts).
Company ABC offers a discount of RM10,000 per km of highway and RM60,000 per
km of expressway. Meanwhile company RST offers a discount of RM20,000 per km
of highway and RM50,000 per km of expressway
a. Construct the formulas for the problem above by using 𝑥 for highway and 𝑦 for
expressway. Find the total discount to be named as D, received from both
companies in Ringgit Malaysia.
[15 marks]
b. The corporate and strategic department targets to maximize the total discount,
D. List out the SIX (6) constraints involved and evaluate the values of 𝑥 and 𝑦
that will maximize D.
[35 marks]
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Absolutely, let's break down this problem into its components and solve it.
**a. Constructing the Formulas**
**Variables:**
* Let x_ABC = km of highway built by ABC
* Let y_ABC = km of expressway built by ABC
* Let x_RST = km of highway built by RST
* Let y_RST = km of expressway built by RST
**Total Highway and Expressway Construction:**
* x_ABC + x_RST = 200 (Total highway)
* y_ABC + y_RST = 100 (Total expressway)
**Cost (Standard Price):**
* Total cost = 1,000,000(x_ABC + x_RST) + 5,000,000(y_ABC + y_RST)
* Total cost = 1,000,000(200) + 5,000,000(100) = 200,000,000 + 500,000,000 = RM 700,000,000
**Discount Formulas:**
* Discount from ABC = 10,000x_ABC + 60,000y_ABC
* Discount from RST = 20,000x_RST + 50,000y_RST
* Total Discount (D) = 10,000x_ABC + 60,000y_ABC + 20,000x_RST + 50,000y_RST
**Constraints:**
* ABC's Maximum Roadway: x_ABC + y_ABC ≤ 200
* RST's Maximum Roadway: x_RST + y_RST ≤ 150
* ABC's Minimum Contract Value: 1,000,000x_ABC + 5,000,000y_ABC ≥ 250,000,000 or x_ABC + 5y_ABC ≥ 250
* RST's Minimum Contract Value: 1,000,000x_RST + 5,000,000y_RST ≥ 250,000,000 or x_RST + 5y_RST ≥ 250
* x_ABC, y_ABC, x_RST, y_RST ≥ 0 (Non-negativity)
**b. Maximizing the Total Discount**
**Constraints:**
1. x_ABC + x_RST = 200
2. y_ABC + y_RST = 100
3. x_ABC + y_ABC ≤ 200
4. x_RST + y_RST ≤ 150
5. x_ABC + 5y_ABC ≥ 250
6. x_RST + 5y_RST ≥ 250
7. x_ABC, y_ABC, x_RST, y_RST >= 0
**Simplifying the Problem:**
* From (1), x_RST = 200 - x_ABC
* From (2), y_RST = 100 - y_ABC
**Substituting into the Discount Formula:**
* D = 10,000x_ABC + 60,000y_ABC + 20,000(200 - x_ABC) + 50,000(100 - y_ABC)
* D = 10,000x_ABC + 60,000y_ABC + 4,000,000 - 20,000x_ABC + 5,000,000 - 50,000y_ABC
* D = -10,000x_ABC + 10,000y_ABC + 9,000,000
**Substituting into the Remaining Constraints:**
* (3) x_ABC + y_ABC ≤ 200
* (4) (200 - x_ABC) + (100 - y_ABC) ≤ 150 => 300 - x_ABC - y_ABC ≤ 150 => x_ABC + y_ABC ≥ 150
* (5) x_ABC + 5y_ABC ≥ 250
* (6) (200 - x_ABC) + 5(100 - y_ABC) ≥ 250 => 200 - x_ABC + 500 - 5y_ABC ≥ 250 => x_ABC + 5y_ABC ≤ 450
**Revised Constraints:**
1. x_ABC + y_ABC ≤ 200
2. x_ABC + y_ABC ≥ 150
3. x_ABC + 5y_ABC ≥ 250
4. x_ABC + 5y_ABC ≤ 450
5. x_ABC, y_ABC >= 0
**Finding Corner Points:**
* x_ABC + y_ABC = 200
* x_ABC + y_ABC = 150
* x_ABC + 5y_ABC = 250
* x_ABC + 5y_ABC = 450
**Solving the System of Equations:**
1. **x_ABC + y_ABC = 200 and x_ABC + 5y_ABC = 450:**
* 4y_ABC = 250 => y_ABC = 62.5, x_ABC = 137.5
2. **x_ABC + y_ABC = 150 and x_ABC + 5y_ABC = 250:**
* 4y_ABC = 100 => y_ABC = 25, x_ABC = 125
3. **x_ABC + y_ABC = 200 and x_ABC + 5y_ABC = 250:**
* 4y_ABC = 50 => y_ABC = 12.5. x_ABC = 187.5
4. **x_ABC + y_ABC = 150 and x_ABC + 5y_ABC = 450:**
* 4y_ABC = 300 => y_ABC = 75. x_ABC = 75.
**Evaluating D:**
* D = -10,000x_ABC + 10,000y_ABC + 9,000,000
1. (137.5, 62.5): D = -1,375,000 + 625,000 + 9,000,000 = 8,250,000
2. (125, 25): D = -1,250,000 + 250,000 + 9,000,000 = 8,000,000
3. (187.5, 12.5): D=-1,875,000+125,000+9,000,000 = 7,250,000
4. (75, 75): D = -750,000 + 750,000 + 9,000,000 = 9,000,000
**Maximum Discount:**
The maximum discount is RM 9,000,000 when x_ABC = 75 and y_ABC = 75.
* x_RST = 200 - 75 = 125
* y_RST = 100 - 75 = 25
Therefore, ABC builds 75km of highway and 75km of expressway, and RST builds 125km of highway and 25km of expressway.
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