Page 371 Problem 4 /
Group 1
Vanessa Condie
Lucia Horta
Geoffrey Cheung
October 4, 2011
Industrial and Manufacturing Engineering Department
California State Polytechnic University, Pomona
Table of Contents
Problem Statement:
Calculate Plant Capacity
Initial Graphical Representation
Dummy Demand Point
Updated Graphical Representation
Transportation Tableau
Mathematical Representation
Initial Basic Feasible Solution (BFS) using Northwest Corner Method (NWC)
Solving Using WinQSB
Sensitivity Analysis
Report to Manager
Part B
Problem Statement:
Steelco manufactures three types of steel at different plants. The time required to manufacture 1 ton of steel (regardless of type) and the costs at each plant are shown in Table 8. Each week, 100 tons of each type of steel (1, 2, and 3) must be produced. Each plant is open 40 hours per week.
a)Formulate a balanced transportation problem to minimize the cost of meeting Steelco’s weekly requirements.
b)Suppose the time required to produce 1 ton of steel depends on the type of steel as well as on the plant at which it is produced (see Table 9, page 372). Could a transportation problem still be formulated?
Table 8
Table 8 / Cost ($)Plant / Steel 1 / Steel 2 / Steel 3 / Time (minutes)
1 / 60 / 40 / 28 / 20
2 / 50 / 30 / 30 / 16
3 / 43 / 20 / 20 / 15
Table 9
Table 9 / Time (minutes)Plant / Steel 1 / Steel 2 / Steel 3
1 / 15 / 12 / 15
2 / 15 / 15 / 20
3 / 10 / 10 / 15
Calculate Plant Capacity
We are given demand for each type of steel in units of tons. The capability of each plant is given in minutes. We must convert capability of each plant to units of tons so that we can solve.
40 hour work week
100 tons of each steel type per week
Plant 1
Plant 2
Plant 3
Plant / Supply / Steel Type / SupplyPlant 1 / 120 tons / Steel 1 / 100 tons
Plant 2 / 150 tons / Steel 2 / 100 tons
Plant 3 / 160 tons / Steel 3 / 100 tons
Total / 430 tons / Total / 300 tons
At this point it does not matter which plant produces which type of steel.
Total Supply: 430 tons
Total Demand: 300 tons
Initial Graphical Representation
Dummy Demand Point
Add Dummy Demand point
Capacity:
130 tons excess supply
Updated Graphical Representation
Transportation Tableau
Steel1 / Steel 2 / Steel 3 / DummyPlant 1 / 60 / 40 / 28 / 0 / 120
Plant 2 / 50 / 30 / 30 / 0 / 150
Plant 3 / 20 / 20 / 20 / 0 / 160
100 / 100 / 100 / 130 / 430
Mathematical Representation
Supply constraints
Demand constraints
(We do not formulate the dummy.)
Xij = # sent from I to j
Initial Basic Feasible Solution (BFS) using Northwest Corner Method (NWC)
Steel1 / Steel 2 / Steel 3 / Dummy / CapacityPlant 1 / 60 / 40 / 28 / 0 / 120
100 / 20
Plant 2 / 50 / 30 / 30 / 0 / 150
80 / 70
Plant 3 / 20 / 20 / 20 / 0 / 160
30 / 130
Capacity / 100 / 100 / 100 / 130 / 430
Total Cost = 11900
Solving Using WinQSB
Input data into WinQSB table.
WinQSBSolution have complete solution with NBV
Our optimal solution is 9320.
Graphical Solution
Summary Table
From / To / Amount / Unit Cost / Total CostPlant 1 / Steel 3 / 40 / 28 / 1120
Plant 2 / Steel 1 / 100 / 50 / 5000
Plant 3 / Steel 2 / 100 / 20 / 2000
Plant 3 / Steel 3 / 60 / 20 / 1200
Total / 9320
Our optimal solution is 9320.
Sensitivity Analysis
We are interested in performing sensitivity analysis from plant 1 to steel 3 because it contains the second highest cost and has a reduction price of $8 per unit increase.Not clear and may be not a good one ?????
Range of Optimality
Range of Feasibility
Sensitivity Analysis from $26 to $31 unit cost for Plant 1 to Steel 3.Explain motivation for this range
Results of this analysis are below:
Based on the results, we should not manufacture send steel 3 at from plant 1 if the unit cost is higher than $30.
Sensitivity analysis from 150 to 170 for capacity of Plant 3. Change heading
Plant 3
We are interested in the production capacity of Plant 3. If we purchase new faster machines we can increase capacity. If we purchase new cheaper machines, they will be slower and our capacity will decrease. Rewrite motivation
We can see that as we increase capacity of Plant 3, the total cost will continue to decrease.
Report to Manager
Our recommended plan to meet demand has a minimum total weekly cost of $9320.
A summary plan is shown below:
From / produce / Amount / Unit Cost / Total CostPlant 1 / Steel 3 / 40 / 28 / 1120
Plant 2 / Steel 1 / 100 / 50 / 5000
Plant 3 / Steel 2 / 100 / 20 / 2000
Plant 3 / Steel 3 / 60 / 20 / 1200
Plant 1 – Excess / 80
Plant 2 – Excess / 50
Total / 9320
From Plant 1 we should sendmanufacture 40 tons of Steel 3. From Plant 2 to Steel 1 we should send 100 tons of steel. From Plant 3 to Steel 2 we should send 100 tons of steel and from Plant 3 to Steel 3 we should send 60 tons of steel. There will be extra capacity (or unused capacity) at Plant 1 of 80 tons of steel and at Plant 2 of 50 tons of steel. All the capacity at Plant 3 will be used. There is not alternate solution (not for manager).
The table below is not suitable for a manager report.
Based on the table above, we can increase the cost of productionshipment from Plant 1 to Steel 3 up to $30, and still have the same amount of steel processed at this plant. However, any change of shipping cost will affect the final total cost. It is important that while the minimum cost of shipping from Plant 3 to Steel 2 is listed as -$30, this is not feasible as it would require us being paid to ship steel instead of paying to ship steel (not for manager). We can manufacture the same amount of Steel 2 at plant 3 for any unit cost of manufacturing that is below 21.
If we want to send steel from Plant 1 to Steel 1 we need to reduce the unit cost of $60. The cost should be reduced below $50 in order to have shipment from Plant 1 to Steel 1. rewrite
The graph above shows the effect of changes in cost of shipping from Plant 1 to Steel 3 on total cost. The detailed shipping plan changes depending on the cost and if we want such detailed information we need to solve the problem again at the desired point. The graph indicates that if the cost of shipment goes above $30, we will not ship any steel from Plant 1 to Steel 3.
Based on the table above, no amount of capacity change for Plant 1 and 2 will affect total cost. (Refer to the ranges above, your sentence is not true of all possible values.) For every 1 increase in capacity in Plant 3, we can reduce total cost by $8. We can increase our capacity in Plant 3 up to 200 tons of steel. Needs rewrite
The graph above shows the effect of changes in supply of Plant 3 on total cost. The detailed shipping plan changes depending on the cost and if we want such detailed information we need to solve the problem again at the desired point. The graph indicates that as we continue to increase capacity, total cost will continue to decrease.
Part B
Note: This page will be updated with a more detailed solution once we have learned more linear programming techniques.
No you cannot create balanced transportation problem because each type of steel has different production times. This means that the capacity will constantly change.
In order to solve this problem, use Linear Programming (to be taught in class later).
Brief explanation:
Setup decision variables and then setup objective function and constraints. There are 15 constraints because each constraint needs to take into account supply, demand, and time available to produce.
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