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I believe this is a linear algebra problem, but if not please let me know:

Say you have 4 suppliers. You want to order 4 different items. The 4 suppliers each have a different price for each item and charge shipping but only for orders under a given amount. How do you decide what to order from each supplier to get the lowest total cost including shipping?

Here is a Table of Sample Data. I need to buy all 4 items. I don't have to buy all the supplies from a single supplier.

          Supplier A    Supplier B    Supplier C     Supplier D
Item 1     $30             $40          $50             $60
Item 2     $40             $50          $60             $35
Item 3     $45             $60          $60             $65
Item 4     $90             $55          $35             $45
Shipping   $25             $20          $15             $10

Each Supplier offers free shipping for orders over $100

I tried each option and found a solution, but I'd like a way for a computer to answer the question so I can scale this to many items and many suppliers.

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  • $\begingroup$ This is not a linear programming problem, since the cost is not a linear function of quantity ordered. Looks like a hard algorithmic problem similar to knapsack. $\endgroup$ – user147263 Dec 14 '14 at 0:45
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The first action, which you have to do is to define the indices and the variables.

indices

i: Index for the ith supplier.

$i \in \{1,2,3,4\}$

i=1 stands for supplier A, i=2 stands for supplier B, ...

j: Index for the jth item.

$j \in \{1,2,3,4\}$

variables

$x_{ij}$: Amount of items j, which are bought at supplier i.

$x_{ij} \in \mathbb N_0$

$y_i$: $\begin{cases} 1, \text{if} \ c_i \leq 100 \\ 0, \text{if} \ c_i > 100 \end{cases}$

$c_i$: Costs of all items, which are bought at supplier i.

parameter

$t_i$: Shipping costs of supplier i.

$p_{ij}$: price of item j from supplier j.

constraints

An amount of $b_j$ items of category j have to be bought at least.

$\sum_{i=1}^4 x_{ij}\geq b_j \ \ \forall \ j$

Costs of the items, which are bought at supplier i

$\sum_{j=1}^4 p_{ij}\cdot x_{ij}=c_i \ \ \forall \ i$

objective function

The shipping costs are

$\sum_{i=1}^4 y_i\cdot t_ i$

The costs for the items are

$\sum_{i=1}^4 \sum_{i=1}^4 p_{ij}\cdot x_{ij}$

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