Question 1171735
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.