Heart Valve Suppliers
By:
Brandon Blaydes
Kevin Meono
Raymond Valencia
Table of Contents
Problem Statement………………………………………………………. p. 3
Summary Table and Formulation……………………………………p. 4
Sensitivity Analysis…………………………………………………………p. 6
Report to Manager………………………………………………………..p. 9
Problem Statement
US Labs manufactures mechanical heart valves from the heart valves of pigs. Different heart operations require valves of different sizes. US Labs purchases pig valves from three different suppliers. The cost and size mix of the valves purchased from each supplier are given in Table 3. Each month, US Labs places one order with each supplier. At least 500 large, 300 medium, and 300 small valves must be purchased each month. Because of limited availability of pig valves, at most 700 valves per month can be purchased from each supplier. Formulate an LP that can be used to minimize the cost of acquiring the needed valves.
Table 3
Supplier / Cost Per Valve ($) / Percent Large / Percent Medium / Percent Small1 / 5 / 40 / 40 / 20
2 / 4 / 30 / 35 / 35
3 / 3 / 20 / 20 / 60
(Figure 1)
Dr. Parisay’s comments are in red.
Summary Table and Formulation
Supplier / Cost Per Valve ($) / PercentLarge / Percent Medium / Percent
Small / Valves Purchased
1 / 5 / 40 / 40 / 20 / 700
2 / 4 / 30 / 35 / 35 / 700
3 / 3 / 20 / 20 / 60 / 700
Demand / 500 / 300 / 300
(Figure 2) Summary Table
Below is a list of our decision variables:
X1= number of valves from supplier 1
X2= number of valves from supplier 2
X3= number of valves from supplier 3
We chose these as our decision variables because our goal is to find out how many valves we should be buying from each supplier in order to minimize the total cost. Now we can formulate an objective function by multiplying each decision variable by the cost
Below is our objective function (OF):
Total Cost = Z = 5X1 + 4X2 + 3X3
It is also important to note the constraints given to us in the problem.
Constraint 1: order at least 500 large valves
0.40X1 + 0.30X2 + 0.20X3 >= 500
Constraint 2: order at least 300 medium valves
0.20X1 + 0.35X2 + 0.20X3 >= 300
Constraint 3: order at least 300 small valves
0.20X1 + 0.35X2 + 0.60X3 >= 300
Constraint 4-6: at most 700 valves from each supplier
0<=X1<=700
0<=X2<=700
0<=X3<=700
We entered all of our equations into WinQSB to find an optimal solution. Our objective function was placed in the first row, with the rest of the equations following. Notice how our bounds are defined from 0 to infinity (M). This means that our variables can only be positive solutions and that makes sense because it is infeasible to have a negative amount of supply.
(Figure 3) WinQSB Input
After inputting, the data and solving the problem, we were presented with the following figure:
(Figure 4) WinQSB Solution
From the table, the optimal solution would be to purchase 700 valves from Supplier 1, 700 valves from Supplier 2, and 50 valves from Supplier 3. The total cost would then be $6,450. The table below shows a summary of our optimal solution.
Amount / CostSupplier 1 / 700 / $3500
Supplier 2 / 700 / $2800
Supplier 3 / 50 / $150
Total / 1450 / $6450
(Figure 5) Summary of Optimal Solution
Sensitivity Analysis
We wanted to see how changing the constraints and costs would affect our optimal solution. We performed sensitivity analysis for the following:
1. Unit cost for supplier 1because its value in solution is one of the highest (700)
2. Demand for large valves
For our first scenario, we wanted to know how changing the unit cost for each supplier would affect our solution. For example, we know that supplier 1 is the most expensive. If the unit cost were to decrease to $4, how would that affect our optimal solution? How about a decrease to $3?
We entered each scenario into WinQSB and came up with the following results.
Supplier 1 Unit Cost $4
(Figure 6) Input for Unit Cost of $4
(Figure 7) Output for Unit Cost of $4
As you can see from the figure, our objective function or total cost actually decreased to $5,750. In our original solution, supplier 1 was listed as a basic variable. This means, anything within the allowable minimum and maximum range is reasonable. Notice how $5 fell within the range between 0 and 6. When we reduced the cost, $4 is also within that range. Therefore, reducing the unit cost for supplier 1 ended up reducing the total cost as well. The same can be said for reducing the unit cost to $3.
Supplier 1 Unit Cost $3
(Figure 8) Input for Unit Cost of $3
(Figure 9) Output for Unit Cost $3
In our second scenario, we wanted to see how changing one of the given constraints would affect our optimal solution. For example we decided to see what would happen if we changed the amount of large valves we needed to order. Our previous optimal solution showed that the order of large valves had the highest shadow price. That is if this demand changes it will have the highest impact on total cost. If demand increases the total cost will increase by $15 per unit increase in demand.
Once again, we inputted our scenario into WinQSB to come up with the following.
Change order of Large Valves to 495
(Figure 10) Input of Demand for large valves to 495
(Figure 11) Output of Demand for large valves to 495
As you can see, decreasing the demand for large valves decreased our total cost. That’s because for every 1 unit of demand we decrease, we save $15. So if we decrease by 5 units we save $75 and that’s how much our total cost decreased.
Perform SA on a nonbinding constraint, such as the demand for medium valve.
Report to the Manager
According to our research, the operations research department at US labs has found that the minimum purchasing cost from our suppliers if our recommendations are followed is $6450. To obtain that minimum cost, we have provided the following purchasing order for next month:
Purchasing Order:
Cost per Value ($) / # of Heart Valves to purchase / Total contributionSupplier 1 / 5 / 700 / $3500
Supplier 2 / 4 / 700 / $2800
Supplier 3 / 3 / 50 / $150
Total / 1450 / $6450
Minimum cost: $6450
Our recommendation suggests that you order 700 heart valves from supplier 1, 700 heart valves from supplier 2, 50 heart valves from supplier 3 in order to obtain the minimum cost of $6450. Even though Supplier 1 cost the most, it has the highest percentage of the large valves which constitutes 45% of the orders. (Good observation.)
Cost and Size Mix Table:
Supplier / Cost per Value ($) / Percent Large / Percent Medium / Percent Small1 / 5 / 40 / 40 / 20
2 / 4 / 30 / 35 / 35
3 / 3 / 20 / 20 / 60
Given the following purchase order and the cost and size mix table U.S. labs will obtain the following amount of heart valves from the suppliers:
Heart Valves received: This is a good summary for manager. You can add a last row indicating the demand for each one. Then you can eliminate the next table.
Total Valves / Large valves / Medium Valves / Small valvesSupplier 1 / 700 / 280 / 280 / 140
Supplier 2 / 700 / 210 / 245 / 245
Supplier 3 / 50 / 10 / 10 / 30
Total / 1450 / 500 / 535 / 415
Requirements met:
Size of Heart Valve / Minimum amount of heart valves required to fulfill demand. / Actual amount of heart valves received from suppliersLarge / 500 / 500 √ fulfilled
Medium / 300 / 535 √ fulfilled
Small / 300 / 415 √ fulfilled
As you can see U.S. labs have fulfilled the required amount of valves of each size that has to be purchased every month.
Due to the limited availability of the pig Heart Valves, at most 700 valves per month can be purchased from each supplier.
Maximum amount of heart valves available per supplier / Actual amount of heart valves received from suppliers.Supplier 1 / 700 / 700
Supplier 2 / 700 / 700
Supplier 3 / 700 / 50√ not used all
U.S. labs have not exceeded the supply that each supplier can provide.
This Purchasing order satisfies all the criteria and provides the lowest possible cost under the limitations and current conditions.
Sensitivity Analysis: This part should be rewritten and organized to fit for a manager’s report
If the limitations and conditions were to change, then the results would change accordingly. Although no change of suppliers or mix of suppliers would change since US labs are contractually obligated to make an order from each supplier the values of how much from each supplier would change in some situations.
If we changed the unit price of one of the suppliers, for example suppose that supplier 1 has decided to improve on its technologies and produce heart valves at a lower price in Suppliers 1’s warehouse resulting in a lower per unit price of 4, then the below result would appear in our computer program
Original, Supplier 1 Unit Cost $5 not this table in report to manager
Output if Supplier 1 Unit Cost $4 summarize this table for report to manager.
As you can see the only difference that can between the original and the new output is the total contribution Supplier 1 makes to the total minimum cost, the minimum cost and the shadow price. Shadow price is the price that it increases by for every 1 unit that is added. In this case the shadow price changes from -1 to -2. What this means is that for every one unit increase in units of heart valves that they sell from Supplier 1, the overall minimum price will decrease by $2. The shadow price effect can be seen in the Total change of the minimum price to $5,750. Rewrite this to fit a report for a manager.
This would be the case if we reduced the unit price to 3 instead.
Original, Supplier 1 Unit Cost $5 this is repeated
Supplier 1 Unit Cost $3
In this case the shadow price changes from -1 to -3. What this means is that for every one unit increase in units of heart valves that they sell from Supplier 1, the overall minimum price will decrease by $3. The shadow price effect can be seen in the Total change of the minimum price of 5,050.
As we decrease the price of the original supplier 1 by 1 we see that the minimum price of the total purchasing order decreases by $700. We did not increase the price to $6 because at that point it does not become an option and a new analysis must be formed. Unfortunately since we are legally bound by the contract that we have to purchase from each supplier, then this is not possible.
If we changed the limit on orders, for example suppose that supplier 1 has decided to improve on its technologies and has achieved better efficiency in Suppliers 1’s warehouse resulting in higher capacities of heart valves to 800 units per supplier, then the below result would appear in our computer program.
Availability of pig valves is 700 per supplier
Increase availability of pig valves to 800 per supplier
This scenario’s solution completely gets rid of one our suppliers meaning that we would have to change some limitations in order to abide with our contract to the suppliers. In this case we would have to find different limitations and make another analysis to find the minimal solution.
Some scenarios that may occur may be that our contract with the suppliers expires and we are able to utilize one supplier more than the others, thus yielding an even lower minimum cost. Another thing that may yield different results may be whether different sized Valves may actually cost more than the others. These are things that may be taken into consideration when making a decision.
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