# Math Help - Linear programming model

1. ## Linear programming model

AutoIgnite produces electronic ignition systems for automobiles at a plant in Cleveland, Ohio. Each ignition system is assembled from two components produced at Autoignites plants in Buffalo, NY and Dayton Ohio. The buffalo plant can produce 2000 units of component 1, 1000 units of component 2, or any combination of the two components each day. For instance, 60% of Buffalo's production time could be used to produce component 1 and 40% of Buffalos production time could be used to produce component 2; in this case, the buffalo plant would be able to produce 0.6(2000) =1200 units of component 1 each day and 0.4(1000) =400 units of component 2 each day. The dayton plant can produce 600 units of component 1, 1400 units of component 2, or any of the two components each day. At the end of each day, the component production at Buffalo and Dayton is sent to Clevland for assembly of the ignition systems on the following work days.

A. formulate a linear programming model that can be used to develop a daily production schedule fpr the buffalo and dayton plants that will maximize daily production of ignition systems at clevland.

B. Find the optimal solution

2. Originally Posted by stefmel2490
AutoIgnite produces electronic ignition systems for automobiles at a plant in Cleveland, Ohio. Each ignition system is assembled from two components produced at Autoignites plants in Buffalo, NY and Dayton Ohio. The buffalo plant can produce 2000 units of component 1, 1000 units of component 2, or any combination of the two components each day. For instance, 60% of Buffalo's production time could be used to produce component 1 and 40% of Buffalos production time could be used to produce component 2; in this case, the buffalo plant would be able to produce 0.6(2000) =1200 units of component 1 each day and 0.4(1000) =400 units of component 2 each day. The dayton plant can produce 600 units of component 1, 1400 units of component 2, or any of the two components each day. At the end of each day, the component production at Buffalo and Dayton is sent to Clevland for assembly of the ignition systems on the following work days.

A. formulate a linear programming model that can be used to develop a daily production schedule fpr the buffalo and dayton plants that will maximize daily production of ignition systems at clevland.

B. Find the optimal solution
Let the production of components 1 and 2 at Buffalo be $B_1$ and $B_2$, and at Dayton be $D_1$ and $D_2$.

We have non-negativity constraints:

$B_1 \ge 0,\ B_2\ge0,\ D_1\ge 0,\ D_2\ge 0$

We also have production capacity constraints:

$\frac{B_1}{2000}+\frac{B_2}{1000} \le 1$

and

$\frac{D_1}{600}+\frac{D_2}{1400} \le 1$

Now if we want no waste we have an equality constraint:

$(B_1+D_1)-(B_2+D_2)=0$

and our objective is to maximise either:

$B_1+D_1$

or

$B_2+D_2$

as these are the number of ignition systems

CB

3. The company currently has in stock: 1000 units which were produced in month 2; 2000 units which were produced in month 1; 500 units which were produced in month 0.
The company can only produce up to 6000 units per month and the managing director has stated that stocks must be built up to help meet demand in months 5, 6, 7 and 8. Each unit produced costs Mu 15 and the cost of holding stock is estimated to be Mu 0.75 per unit per month (based upon the stock held at the beginning of each month). The company has a major problem with deterioration of stock in that the stock inspection, which takes place at the end of each month regularly, identifies ruined stock (costing the company Mu 25 per unit). It is estimated that, on average, the stock inspection at the end of month t will show that 11% of the units in stock, which were produced in month t, are ruined; 47% of the units in stock, which were produced in month t-1, are ruined; 100% of the units in stock, which were produced in month t-2, are ruined. The stock inspection for month 2 is just about to take place.
The company wants a production plan for the next six months that avoids stockouts. Formulate their problem as a linear program.

on this site : Module 8:Aggregate Production Planning

there is partial solution , but how to run this on lpsolve?!!!

thanks