Math 428 Operation Research
Linear Programming Project

This project is mainly to report on our analysis of the mathematical model established by the three-month production plan from the bicycle manufacturer. It includes what suggestions we’ve had for the client, details on how the model was set up and also how future change influences the results.

Firstly, we’ll explain how and why the model was set up in this way:
Our goal is to help the company minimize their cost on the production and the inventory.
Let x4m be the amount of mountain bikes produced in April
Let x5m be the amount of mountain bikes produced in May
Let x6m be the amount of mountain bikes produced in June
Let x4r be the amount of road bikes produced in April
Let x5r be the amount of road bikes produced in May
Let x6r be the amount of road bikes produced in June

Since each mountain bike frame costs $200 and each road bike costs $250, then the total production cost would be 200x4m+x5m+x6m+250(x4r+x5r+x6r). The carried-over inventory costs $10 for each bike frame. We got 150 in store before the three-month production plan and it costs 150∙10=$1500. Then for each month, we have 50+x4m+100+x4r-1300 for the carryover from April to May; 50+x4m+100+x4r-1300+x5m+x5r-1700 for the carryover from May to June. Hence these carryovers are going to cost $10 for each as well. Put everything together and simplies it we have the objective function:
Minimize Z=220x4m+210x5m+200x6m+270x4r+260x5r+250x6r-38500
For resources constraints, the machine time and labor cannot exceed the available amount.
We have the following constraint functions:
1.5x4m+1.6x4r≤3000;
1.5x5m+1.6x5r≤3000;
1.5x6m+1.6x6r≤3000;
x4m+1.2x4r≤2400; x5m+1.2x5r≤2400; x6m+1.2x6r≤1800;
For inventory constraints, it wants to end the month of June with inventories of 100 of each frame.
Because we already have 50 mountain bikes and 100 road bikes before this production plan, this should also be under the consideration. Excluding the expected sales, these inventories together with all bikes produce during this period should at least have 100 left for each frame (If 108 bikes left for example, the unnecessary production will cost more but the requirement is also satisfied. Hence we use great-or-equal-to sign here just in case the emergency pops up). So, 50+x4m+x5m+x6m-(700+900+1000)≥100 for mountain bike and 100+x4r+x5r+x6r-(600+800+900)≥100 for road bikes. Simplifying it, we got x4m+x5m+x6m≥2650; x4r+x5r+x6r≥2300;

Now, another limitation we have to concern is that for each month and each frame, the amount was carried over from last month plus the amount it produces in the current month should be sufficient. That is, the amount we have in hand is supposed to be larger than the expected sales so that we have enough bikes for the customers. Therefore,
Mountain bike in April: x4m+50≥700;
Road bike in April: x4r+100≥600;
Mountain bike in May: 50+x4m+x5m-700≥900;
Road bike in May: 100+x4r+x5r-600≥800;
Mountain bike in June: 50+x4m+x5m+x6m-700-900≥1000;
Road bike in June: 100+x4r+x5r+x6r-600-800≥900;

After simplification, we have x4m≥650; x4r≥500; x4m+x5m≥1550; x4r+x5r≥1300; x4m+x5m+x6m≥2550; x4r+x5r+x6r≥2200;

Combining all the information above, the complete model would be:
Minimize Z=220x4m+210x5m+200x6m+270x4r+260x5r+250x6r-38500 such that
1.5x4m+1.6x4r≤3000;
1.5x5m+1.6x5r≤3000;
1.5x6m+1.6x6r≤3000;
x4m+1.2x4r≤2400; x5m+1.2x5r≤2400; x6m+1.2x6r≤1800; x4m+x5m+x6m≥2650; x4r+x5r+x6r≥2300; x4m≥650; x4r≥500; x4m+x5m≥1550; x4r+x5r≥1300; x4m+x5m+x6m≥2550; x4r+x5r+x6r≥2200; xim≥0;i=4,5,6 xjr≥0;j=4,5,6

Next, we’ll analyze the model with the reinforcement of computation software LINGO.
(Electronic file screen captures are attached at the end.)
According to the LINGO, we may have the following recommendation to our client:

April: Produce 650 mountain bikes and 685 road bikes and plus 50 mountain bikes and 100 road bikes in inventory, we have 700 mountian bikes and 785 road bikes available. After a month, we expect to sell 700 mountain bikes and 600 road bikes which means we have no mountain bike in store and 185 road bikes left for May. May: Produce 900 mountain bikes and 1031 road bikes. Since we expect to sell 900 mountain bikes, the mountain bike is in the right amount. As for road bikes, we have 185 carryovers from April and 1031 new bikes which are 1216 in total. Subtracting the expected sales from it, we have 416 available for next month.

June: Produce 1100 mountain bikes and 584 road bikes. Because the expected sales in June are 1000 and we want 100 inventories, 1100 mountain bikes should work fine. 584 newly-produced and plus 416 carryovers come to 1000 road bikes in total. Since 900 out of 1000 are expected be sold, then 100 road bikes are left. All requirements are met and this three-month production plan will cost $1112510.
(Note that our recommendation is slightly different from the LINGO results because of the integer issue.)

The limiting resources are machine time in May and labor in June. For the increase of limiting resources, the company is willing to pay 6.25 per hour for the machine time and 16.67 for the labor work. If we change the cost of production or the carry-over fee, it certainly influences the face value of the cost but the machanism behind it is the same. Since models vary from person to person, in our model all parameters are relavent. But if we create new variables for carryovers instead of expressing it in terms of other parameters, the model would change but the results would remain the same. From the results we can see that the carryovers for mountain bikes did not have a lot to do with the optimization only the 50 inventories that we already have need to be concerned.

The model below is original one (which is not simplified but essentially the same).