Question 1165403: Dunder Mifflin supplies paper to corporate clients. An important client needs 1500 reams of paper.
Dunder Mifflin can procure the paper from several suppliers. The characteristics of the suppliers
are as follows:
Cost per ream Delivery time Reams available
Supplier A 3.50 5 days 200
Supplier B 2.00 10 days 600
Supplier C 6.50 15 days 600
Supplier D 5.00 4 days 200
Supplier E 4.00 6 days 200
When procuring the paper, Dunder Mifflin needs to ensure the following conditions are satisfied:
(a) The client requires that at least 500 of the 1500 reams be delivered within the next 7 days.
(b) In order to maintain good relations with Supplier C, Dunder Mifflin must order at least 100
reams from them.
(c) Because of a rivalry between supplier B and D, Dunder Mifflin must order at least as many
reams from supplier D as they do from supplier B.
(d) Each supplier has a limited number of reams of paper available, as shown in the table.
Formulate a linear optimization model to help Dunder Mifflin procure the paper at the lowest cost.
(You don’t need to find the optimal solution to the model you formulate.)
Answer by CPhill(2138)   (Show Source):
You can put this solution on YOUR website! 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)
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