Q.Interpret the model in Example 2, in case

[!FORMULA] $A = \begin{pmatrix} 10 & 15 & 6 \\ 10 & 20 & 0 \end{pmatrix}, \qquad B = \begin{pmatrix} 3 & 4 & 0 \\ 7 & 9 & 3 \\ 5 & 12 & 7 \end{pmatrix}$

and the available raw materials are $330$ units of $R_1$, $455$ units of $R_2$ and $140$ units of $R_3$.

Column sums (total demand): $R_1 = 335$, $R_2 = 467$, $R_3 = 147$ — each above the stock $330, 455, 140$. Trimming to $A_1 = \begin{pmatrix} 9 & 12 & 6 \\ 10 & 20 & 0 \end{pmatrix}$ gives sums $311, 436, 138$, within stock.

Step 1 — Identify. Substitute the given numbers into the Example 2 model to see whether the firm's stock can fill both orders.

Step 2 — Set up. Here row $F_i$ of $A$ is client $i$'s order in products $P_1,P_2,P_3$; $B$ gives the raw material $R_1,R_2,R_3$ per unit of each product; stock is $R_1=330,\ R_2=455,\ R_3=140$.

Step 3 — Formulate. The raw material required is $AB$ (clients $\times$ raw materials).

Step 4 — Solve. Compute the product row by row.

Row $F_1$: $R_1 = 10\cdot3+15\cdot7+6\cdot5 = 30+105+30 = 165$; $R_2 = 10\cdot4+15\cdot9+6\cdot12 = 40+135+72 = 247$; $R_3 = 10\cdot0+15\cdot3+6\cdot7 = 0+45+42 = 87$.

Row $F_2$: $R_1 = 10\cdot3+20\cdot7+0 = 170$; $R_2 = 10\cdot4+20\cdot9+0 = 220$; $R_3 = 10\cdot0+20\cdot3+0 = 60$. Hence

Total raw material for both orders (column sums):

Step 5 — Interpret and validate. The demand $335,\,467,\,147$ exceeds the stock $330,\,455,\,140$ in every raw material, so both orders cannot be filled as they stand. Since the recipe $B$ is fixed, the firm must either buy more raw material or have the clients cut their orders. For instance, replacing $A$ by the reduced orders

gives $A_1 B = \begin{pmatrix} 141 & 216 & 78 \\ 170 & 220 & 60 \end{pmatrix}$, whose column sums $311,\,436,\,138$ lie comfortably below stock — so the trimmed orders can be supplied.

Method: Computing and Interpreting a Matrix Product Against a Resource Constraint

This method applies once the general orders-times-recipe model has been set up and specific numerical matrices are given.

Steps

Step 1: Confirm the matrices multiply in the intended order

Check the given $A$ (clients $\times$ products) and $B$ (products $\times$ raw materials) share the "products" dimension, so $AB$ is defined and gives a clients $\times$ raw-materials matrix.

Step 2: Compute each entry of $AB$ by row-times-column

For each client row $i$ and raw-material column $k$, compute $\sum_j A_{ij}B_{jk}$ — multiply matching entries and add. Do this systematically, one row of $A$ at a time.

Step 3: Sum the columns of $AB$ to get total demand

Add down each column of the resulting matrix to get the total amount of each raw material needed across all clients combined — this is what must be compared against stock.

Step 4: Compare total demand with the available stock

Check each raw material's total demand against the number given as available. If any demand figure exceeds its stock, the orders as placed cannot be fulfilled.

Step 5: If needed, test a reduced order against stock

If the original orders exceed stock, repeat Steps 2-3 with a trimmed orders matrix to see whether the reduced demand now fits within the available stock.

Applying to this problem: compute $AB=\begin{pmatrix}165&247&87\\170&220&60\end{pmatrix}$, sum columns to get demand $335,467,147$, compare against stock $330,455,140$ — every raw material is short, so the orders cannot be met as placed; the reduced order $A_1$ brings demand down to $311,436,138$, within stock.

Common Mistakes

Mistake 1: Arithmetic slip in one entry of the matrix product

Why it's wrong: Each entry of $AB$ requires three separate multiplications added together (since the shared dimension is 3); a single multiplication or addition error in one entry throws off both that client's requirement and the column total used for the stock comparison. Correct approach: compute each entry of $AB$ as a clearly written sum (e.g. $10\cdot3+15\cdot7+6\cdot5$) rather than mental arithmetic, and double check each term.

Mistake 2: Comparing a row total instead of a column total against stock

Why it's wrong: A row of $AB$ gives one client's requirement for all three raw materials, but the stock is a single pool shared by both clients — so it's the column sums (total demand per raw material, across all clients) that must be compared to stock, not any individual row. Correct approach: sum each column of $AB$ first, then compare that sum to the stock figure for that raw material.

Mistake 3: Concluding "orders can be met" from only some raw materials being sufficient

Why it's wrong: All required raw materials must be available simultaneously — if even one raw material's demand exceeds its stock, the orders as a whole cannot be fulfilled, regardless of whether the other two are within stock. Correct approach: check every raw material's demand against its stock individually, and the orders fail if any one of them is insufficient.