Question 1165403
This is a classic linear programming (LP) problem. We need to define the decision variables, the objective function (minimizing cost), and the constraints based on the client and supplier requirements.

## 1. Decision Variables

Let $x_i$ be the number of reams of paper ordered from **Supplier $i$**, where $i \in \{A, B, C, D, E\}$.

* $x_A$: Reams ordered from Supplier A
* $x_B$: Reams ordered from Supplier B
* $x_C$: Reams ordered from Supplier C
* $x_D$: Reams ordered from Supplier D
* $x_E$: Reams ordered from Supplier E

Since the number of reams must be non-negative:
$$x_A, x_B, x_C, x_D, x_E \ge 0$$

## 2. Objective Function (Minimize Cost)

The objective is to minimize the total procurement cost. This is the sum of (Cost per Ream $\times$ Reams Ordered) for each supplier.

| Supplier | Cost per Ream |
| :---: | :---: |
| A | 3.50 |
| B | 2.00 |
| C | 6.50 |
| D | 5.00 |
| E | 4.00 |

$$\text{Minimize } Z = 3.50x_A + 2.00x_B + 6.50x_C + 5.00x_D + 4.00x_E$$

## 3. Constraints

### a) Total Demand Constraint

The client requires a total of 1500 reams of paper.
$$x_A + x_B + x_C + x_D + x_E = 1500$$

### b) Delivery Time Constraint (Within 7 Days)

The client requires at least 500 reams to be delivered within 7 days. We must identify suppliers with a delivery time $\le 7$ days: Suppliers A (5 days), D (4 days), and E (6 days).

$$x_A + x_D + x_E \ge 500$$

### c) Supplier C Relationship Constraint

Dunder Mifflin must order at least 100 reams from Supplier C.
$$x_C \ge 100$$

### d) Rivalry Constraint (Supplier D vs. B)

The order from Supplier D must be at least as many reams as the order from Supplier B.
$$x_D \ge x_B$$
This is typically written as:
$$x_D - x_B \ge 0$$

### e) Availability Constraints

The order from each supplier cannot exceed the reams available.

| Supplier | Reams Available |
| :---: | :---: |
| A | 200 |
| B | 600 |
| C | 600 |
| D | 200 |
| E | 200 |

$$x_A \le 200$$
$$x_B \le 600$$
$$x_C \le 600$$
$$x_D \le 200$$
$$x_E \le 200$$

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## Linear Optimization Model Summary

$$\text{Minimize } Z = 3.50x_A + 2.00x_B + 6.50x_C + 5.00x_D + 4.00x_E$$

$$\text{Subject to:}$$
1.  $$x_A + x_B + x_C + x_D + x_E = 1500$$ (Total Demand)
2.  $$x_A + x_D + x_E \ge 500$$ (7-Day Delivery)
3.  $$x_C \ge 100$$ (Supplier C Minimum)
4.  $$x_D - x_B \ge 0$$ (Rivalry)
5.  $$x_A \le 200$$
6.  $$x_B \le 600$$
7.  $$x_C \le 600$$
8.  $$x_D \le 200$$
9.  $$x_E \le 200$$
10. $$x_A, x_B, x_C, x_D, x_E \ge 0$$ (Non-negativity)