# Linear Programming Inventory Problem

I'm still trying to get used to the nature of these problems and I'd appreciate some further explanation.

A manufacturing company produces two types of
products: A and B. The company has agreed to deliver the
products on the schedule shown in Table 34. The company
has two assembly lines, 1 and 2, with the available
production hours shown in Table 35. The production rates
for each assembly line and product combination, in terms of
hours per product, are shown in Table 36. It takes 0.15 hour
to manufacture 1 unit of product A on line 1, and so on. It
costs $5 per hour of line time to produce any product. The inventory carrying cost per month for each product is 20¢ per unit (charged on each month’s ending inventory). Currently, there are 500 units of A and 750 units of B in inventory. Management would like at least 1,000 units of each product in inventory at the end of April. Formulate an LP to determine the production schedule that minimizes the total cost incurred in meeting demands on time. So far, I've made my decision variables as follows:$A_i = $Amount of Product A produced in month$iB_i = $Amount of Product A produced in month$ii = 1,2$And I have some idea of what the constraints will look like. My issue is formulating the objective function. Any help would be appreciated. ## 1 Answer Let$A_{ij}$denote the amount of product A produced in month$i$on line$j$. Similarly for$B_{ij}\$. The constraints are as follows: \begin{align*} 500+A_{11}+A_{12}\geq&\,5\mathord{,}000\\ 750+B_{11}+B_{12}\geq&\,2\mathord{,}000\\ 500+A_{11}+A_{12}-5\mathord{,}000+A_{21}+A_{22}\geq&\,8\mathord{,}000+1\mathord,000\quad\text{(note: inventory demand included)}\\ 750+B_{11}+B_{12}-2\mathord{,}000+B_{21}+B_{22}\geq&\,4\mathord{,}000+1\mathord,000\quad\text{(note: inventory demand included)}\\ 0\mathord{.}15A_{11}+0\mathord{.}12 B_{11}\leq&\,800\\ 0\mathord{.}15A_{21}+0\mathord{.}12 B_{21}\leq&\,400\\ 0\mathord{.}16A_{12}+0\mathord{.}14 B_{12}\leq&\,2\mathord{,}000\\ 0\mathord{.}16A_{22}+0\mathord{.}14 B_{22}\leq&\,1\mathord{,}200,\\ \end{align*} and every variable must be non-negative. The objective to minimize is \begin{align*} &\,5\times\left[0\mathord{.}15(A_{11}+A_{21})+0\mathord{.}12(B_{11}+B_{21})+0\mathord.16(A_{12}+A_{22})+0\mathord.14(B_{12}+B_{22})\right]\\ +&\,0\mathord{.}2\times\left[(500+A_{11}+A_{12}-5\mathord,000)+(750+B_{11}+B_{12}-2\mathord,000)\right.\\ +&\,\left.(500+A_{11}+A_{12}-5\mathord,000+A_{21}+A_{22}-8\mathord,000)\right.\\ +&\,\left.(750+B_{11}+B_{12}-2\mathord,000+B_{21}+B_{22}-4\mathord,000)\right]. \end{align*} Can you follow the logic behind this formulation?

• Yes, it's crystal clear. Thanks for the explanation. I had a feeling I wasn't defining my decision variables correctly – audiFanatic Sep 17 '13 at 3:01