extra credit assignment group 6 · web viewreport to manager10 part b | problem statement12 part b...
TRANSCRIPT
Page 104 Problem 1
Table of ContentsProblem Statement.....................................................................................................................................3
Assumptions................................................................................................................................................4
Set-Up..........................................................................................................................................................4
WinQSB Solution.........................................................................................................................................6
Sensitivity Analysis......................................................................................................................................8
Report to Manager....................................................................................................................................10
Part B | Problem Statement......................................................................................................................12
Part B | Assumptions.................................................................................................................................13
Part B | Set - Up.........................................................................................................................................13
Part B | WinQSB Solution..........................................................................................................................14
Part B | Analysis........................................................................................................................................15
Part C | Problem Statement......................................................................................................................16
Part C | Assumptions.................................................................................................................................17
Part C | Set – Up........................................................................................................................................17
Part C | WinQSB Solution..........................................................................................................................17
Part C | Analysis........................................................................................................................................18
Part D | Problem Statement......................................................................................................................19
Part D| Comparison between Part A and Part B........................................................................................20
Part D| Comparison between Part B and Part C........................................................................................21
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Problem Statement_____________________________________________________________________________________
A customer requires during the next four months, respectively, 50, 65, 100, and 70 units of a commodity
(no backlogging is allowed). Production costs are $5, $8, $4, and $7 per unit during these months. The
storage cost from one month to the next is $2 per unit (assessed on ending inventory). It is estimated
that each unit on hand at the end of month 4 could be sold for $6. Formulate an LP that will minimize
the net cost incurred in meeting the demands of the next four months.
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Dr. Parisay’s comments and additions are in red.
b) Assume that backorder is allowed along with inventory. Each unit backordered for each month will have an opportunity cost of $2.5. Formulate this problem as a LP problem. Solve using WinQSB. Write a report to a manager and explain all your findings. No sensitivity analysis is required.
c) Solve part (b) as a transportation problem. Solve using WinQSB and transportation option. Compare the result with the one on part (b)
d) Compare the result for part (b) and the original problem. Comment on differences.
Problem Summary
Assumptions
For this problem, we are not given an initial inventory, so we assume that we have no inventory
at the beginning of month 1. We also assume, for simplicity reasons, that commodities manufactured
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during a month can be used to meet demand for that month. The problem did not give a production
capacity, so we will approach the problem assuming an unlimited capacity. We also ignore that month-
to-month variation in production costs (do not over-ride the given cost variations), such as hiring more
workers to produce more goods in one month than the next, may be incurred.
Set-Up
To simplify, we construct the following table to summarize the given data:
MONTHCOST OF
PRODUCTION / UNITDEMAND
1 $5 502 $8 653 $4 1004 $7 70
For each month, we must determine the number of commodities that should be produced. We define
the following decision variables:
Xt = number of commodities produced each month during month t
it = number of commodities on hand at the end of month t
where t=1,2,3,4 for each month in the problem.
Our total cost for this problem can be determined as follows:
Total cost = cost of producing commodities per month + inventory costs – profit from selling remaining
commodities
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Considering that the cost of production changes monthly and also that we can sell the remaining
commodities at the end of month 4 for $6 per unit, our formula for total cost is:
Total cost = 5x1+8x2+4x3+7x4+2(i1+i2+i3)-6i4
Our objective function is therefore:
Min z = 5x1+8x2+4x3+7x4+2i1+2i2+2i3-6i4
We define the relation for the inventory it to formulate a multiperiod model. The inventory at the end of
the month is the inventory left over from the previous month (it-1) plus the units produced for that
month (xt), minus that month’s demand (dt). Demand for each of the four months is 50, 65, 100, and 70
respectively. We express this relationship as:
it = it-1 + xt – dt t = 1 , 2 , 3 , 4
d1 = 50, d2 = 65, d3 = 100, d4 = 70
We can then define the problem’s constraints:
i1 = 0 + x1 – 50 i2= i1 + x2 – 65 i3= i2 + x3 – 100 i4= i3 + x4 – 70
x1 > 0 x2 > 0 x3 > 0 x4 > 0 i1 > 0 i2 > 0 i3 > 0 i4 > 0
With these twelve constraints, we are now able to use WinQSB to determine the optimal solution.
WinQSB SolutionData input for WinQSB “Linear and Integer Programming”
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WinQSB solution:
Our minimized cost is $1,525. We put these results into a simple table to easily understand the
production plan the program suggests:
Month Units to Produce
Production Costper Unit
TotalProduction Cost
Demand
Units Remainingat End of Month
InventoryCost Total Cost
1 115 $ 5.00 $ 575.00 50 65 $ 130.00 $ 705.00
2 0 $ 8.00 $ - 65 0 $ - $ -
3 170 $ 4.00 $ 680.00 100 70 $ 140.00 $ 820.00
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4 0 $ 7.00 $ - 70 0 $ - $ -
GRAND TOTAL $ 1,525.00
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Sensitivity AnalysisFor our first sensitivity analysis, we chose to do a parametric analysis on the coefficient of
month 2 in the objective function (a NBV). More specifically, it is the cost of production during the
second month. We chose to do an analysis for this value because month 2 has the highest production
cost per unit, and we would like to see if a change in the coefficient would result in any production for
month 2. Highest production cost is not a suitable motivation in this case. It is better to consider
lowest RC, which it happens it is 1 in this case.
The horizontal line on the sensitivity analysis indicates the point where there should be no
production for month 2. In other words, the horizontal line represents the point where we keep the
suggested production plan. We see a horizontal line between the values of about 7.00 to infinity. This
means that if the production price of month 2 is at least $7.00 or higher per unit, then the suggested
optimal solution holds. Note that for coefficients below 7, the total production cost starts decreasing as
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well. This tells us that if the unit cost of production for month 2 goes below $7.00, it would be optimal to
produce units in month 2 to meet that month’s demand. The slope of the line determines the value of
this variable in the objective function. It is zero from 7 and upward (not clear), and we can calculate the
slope from around 2 to 6.99 (Slope = y2− y1x2−x1
=1525−1199.506.99−2
=65.23). (This is a good effort
showing your basic understanding of this graph. However, from practical point of view, you can use
WinQSB and solve the problem to come up with the exact value.) This value of 65.23 means that if the
production cost in month 2 is reduced to a value between $2 and $6.99, then we should produce 65
(65.23 rounded to the nearest whole number) units of commodity in order to achieve an optimal
solution.
Perform a SA for a BV.
Four our second sensitivity analysis, we chose to parametrically analyze the demand for month
3 (a binding constraint), which has the highest demand out of the 4 months (This is not a good reason.
You need to select highest SP). We would like to see how a change in this demand would affect our
optimal solution.
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In this sensitivity analysis graph, the dot marks the original value of the RHS value on the x-axis, 100, and
the corresponding total cost on the y-axis. It is unrealistic to have a negative demand for month 3, and
so we disregard values that cross into the negative axes. (This is a good observation, however, not in a
practical range. Basically, it is not practical to go much lower than 100, may be at least 80 is practical.)
Based on the analysis, we can see that the total cost is directly related to the demand for month 3 – as
the month 3 demand increases or decreases, so does the total cost.
Report to ManagerTo: Management
The minimum cost we calculated is $1,525. We summarize our findings in the following table:
Month Units to Produce
Production Costper Unit
TotalProduction Cost
Demand
Units Remainingat End of Month
InventoryCost Total Cost
1 115 $ 5.00 $ 575.00 50 65 $ 130.00 $ 705.00
2 0 $ 8.00 $ - 65 0 $ - $ -
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3 170 $ 4.00 $ 680.00 100 70 $ 140.00 $ 820.00
4 0 $ 7.00 $ - 70 0 $ - $ -
GRAND TOTAL $ 1,525.00
We recommend producing 115 units in month 1. It will cost us $575.00 to produce that many
commodities. We will then have enough products to supply our customers the 50 units they need for
month 1. We will also have 65 units left over, which will fulfill the customer’s demand for month 2. This
means we do not recommend producing any units in month 2. We recommend producing extra units in
month 1 because the cost of producing the 65 units in month 1 and keeping them in inventory is only
$455 ($5.00 for each unit, plus $2.00 per unit to keep). If we were to produce 65 units in month 2, the
cost would be $520 ($8.00 per unit). In this manner, we save the company $65.00. good observation
We also apply this same principle for the next two months. We recommend producing 170 units
of commodities in month 3. Producing the 170 units will cost us $680.00. However, we will have enough
units left over to fulfill the customer demands for months 3 and 4. Again, we do this because the cost to
produce the extra 70 units in month 3 to fulfill month 4 demands is only $4.00 per unit to produce and
$2.00 per unit to keep in inventory. This price is much cheaper than the $7.00 cost to produce each unit
in month 4. In summary, the cost of producing this extra amount of 70 units in month 3 is $420 while
the cost of producing 70 units in month 4 is $490. We are saving the company $70.00.
If you adhere to our recommendations, the total cost to the company will be $1,525.00. You
might, however, want to reduce production costs in a certain month to see if the total cost would go
even lower. For your convenience, we have studied the effects of lowering unit production cost for
month 2, which has the highest unit cost of production, on the total cost. We summarize our findings in
the following table:
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Unit Cost of Production
Total Cost (All 4 Months)
Units to Produce
$7.00 and above $1,525 0$2.00 to $6.99 $1200 to $1524.35 65
This table shows that if the unit cost of production in month 2 remains at $7.00 or increases to a
value above $7.00, then we should not produce any units this month and just produce it in the prior
month. However, if we do some cost-cutting and are able to reduce the unit production cost to a value
between $2.00 and $6.99, then our total cost for all four months would fall anywhere in between $1200
and $1524.35 based on how much we can reduce the unit production cost by. This enables us to
produce the require amount of units in month 2 (65) to fulfill demand rather than producing surplus
units in month 1 and taking on inventory cost.
Need to add from other SA.
Part B
Part B | Problem StatementApproach the same problem as the original, but now consider that you can backlog orders. We
can produce goods to meet the previous month’s demand for $2.50 per unit.
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Part B | AssumptionsFor this new addition to the problem, we keep the same assumptions as before. However, we
also make the assumption that we cannot backlog anything for month 4, since that is when the financial
period ends. We have no month 5 to produce anything for month 4, so this assumption must be made.
Good observation and assumption
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Part B | Set - UpWe add a new variable into our objective, y i. This variable represents the number of
commodities that will be produced through backlogging instead of regular production. Our updated
variables and constraints are therefore:
Xt = number of commodities produced each month during month t
it = number of commodities on hand at the end of month t
Yt = number of commodities produced for month t during the next month (ti+1) This definition implicitly
prevents a situation for two or three months of backorder. Is this what you want?
where t=1,2,3,4 for each month in the problem.
Constraints
X1 + y1 – i1 = 50 i1 + x2 + y2 – i2 = 65 Notice that with your definitions, for example, we should
not have a positive value for y2 and i2 at the same time in our solution i2 + x3 + y3 – i3 = 100
i3 + x4 + y4 - i4 = 70 y4=0
Objective Function
Min z = 5x1 + 8x2 + 4x3 + 7x4 + 2i1 + 2i2 + 2i3 – 6i4 + (8+2.5)y1 + (4+2.5)y2 + (7+2.5)y3
Pay attention to your definition of Yi. You need to modify OF coefficients. That is why your solution does not make sense. Your definition of Yi is different in nature with your definition of Ii. Everything below here is wrong.
Part B | WinQSB SolutionWinQSB Input: what are the constraints C5 to C11?
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WinQSB Solution: needs modification
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Part B | AnalysisOur updated solution for minimized cost for the four months is $957.50. From the solution, our
new production plan changes. Because a $2.50 cost to produce per unit (backlogged) is less expensive
than the cost to produce enough to meet the demand for any month, the program suggests to backlog
orders for all months except for month 4. This is because there’s no month 5 to produce month 4
supplies. Also, because producing goods to supply month 4’s demands during month 3 and keeping it in
inventory for one month is cheaper than producing month 4’s goods in month 4, the program suggests
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producing 70 units in month 3. All other orders for each month is backlogged, however. We can
represent this new production plan:
MONTH Demand
# of Units to Produce (Regular)
# of Units to Produce(For Backlogged
Orders)
# of Units to Keep in
InventoryProduction
CostInventory
Cost Total Cost
1 50 0 0 0 $ - $ - $ -
2 65 0 50 0 $ 125.00 $ - $ 125.00
3 100 70 65 70 $ 442.50 $ 140.00 $ 582.50
4 70 0 100 0 $ 250.00 $ - $ 250.00
GRAND TOTAL COST $ 957.50
The program’s suggestion is as follows: Do not produce anything in Month 1. In Month 2,
produce 50 units to meet Month 1’s demand. In Month 3, produce 135 units (65 for Month 2’s
backlogged orders and 70 to store in inventory). In Month 4, it suggests to produce 100 units to meet
Month 3’s demand.
Part C
Part C | Problem StatementSolve the same problem in Part B using the Transportation Problem option in WinQSB.
_____________________________________________________________________________________
Part C | AssumptionsWe keep the same assumptions as we did in the previous parts.
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Part C | Set – UpFor this type of problem, we formulate having 3 sources. These sources are Regular Production,
Backlogged Production, and Produce + Keep (meaning that the goods are produced and then kept a
month for inventory). The WinQSB input is as follows:
Note: Because we have no month before Month 1 to produce goods and keep it in inventory to meet Month 1’s demand, we
input the same price as Normal Production, because having a 0 value in that field would cause difficulties. We also put a value
of 7 for the backlog option for Month 4, since a 0 value again would cause errors and because we have no Month 5 to fulfill
backlogged orders.
Part C | WinQSB SolutionThe WinQSB output:
Part C | AnalysisAccording to the program, our optimal solution is a cost of $957.50. Our production plan is
therefore: Backlog orders for Months 1, 2, and 3, meaning that their orders will be produced and their
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demands will be met on the next month. For Month 4, we have to use the option of “Produce + Keep”,
meaning we produce 70 units in Month 3 and then keep this to fill demand for Month 4.
Part D
Part D | Problem StatementCompare your solutions from part A with solutions from part B. Also compare your solutions
from part B to the solution from part C.
_____________________________________________________________________________________
Part D| Comparison between Part A and Part B
For part A where there was no backlogging allowed, we obtained an optimal solution with a
total minimum cost of $1525. Keeping in count that inventory hold cost was $2 in the first three months,
we opted to over produce on month 1 and month 3, where production cost was minimal. Since we were
not interested on maximizing our profits, we produced only enough to meet our demands, disregarding
the selling price of $6 per unit on month 4.
In part B, where backlogging was allowed at a cost of $2.5 per unit, we obtained a whole
different optimal solution with a total minimum cost of $957.50. The cost to backlog was minimal
compared to the cost to produce on several months, hence why we opted to produce at a backlogging
price on month 2 to fulfill the demand of month 1. During month 3 we produced at a backlogging cost to
fulfill the demand of month 2 but we also produced at regular production cost to fulfill the demand of
month 4. Lastly, we decided to produce on month 4 at a backlogging cost to fulfill the demand of month
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3. These decisions were based on the clear differences between production costs and inventory and
backlogging costs.
In comparison between part A and B, we have concluded that the option to backlog can have a
significant impact in our total minimum cost. If we were to choose part B over part A, where backlogging
is allowed, we would save $1525-$957.5= $567.5. Even though backlogging could save us a lot of
money, we could be jeopardizing customer satisfaction.
Part D| Comparison between Part B and Part C
For Part B, we got an optimal solution of $957.50 when we were allowed to backlog our
production. And for part C we entered our information into a transportation model. We assumed our
‘from’ fields to be our normal production costs, production costs for backlogging, and the cost of
producing the units with the $2 storage cost. The ‘to’ fields were the different months that we had
demand for. So, for the first line which was our normal production costs it was 5, 8, 4, 7 respectively for
each month. Then the next line was what it cost if we backlogged our production which the cost for all
months is $2.50. The third line is when we keep the units we produced and store them for the next
month. So, each cost is the previous months production cost plus a $2 cost for storing them in the
warehouse. Our demands are what were given to us in the problem statement for each month, which
was 50, 65, 100, and 70 respectively for each month. Our supply is any value above 0 because we
weren’t given a limit on how many units we could produce a month.
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In both models we got our total minimum cost to be $957.50. We assume them to be the same
because we took into consideration all the same costs. In the transportation model we had to add the
third line that included the storage cost because we wouldn’t have gotten the same answer if we didn’t
consider that cost in our total minimum cost. Even though they were two different forms of entering
the data into winQSB they got us the same answer. One got us the answer by having constraint
equations and told the program what the production values could be and couldn’t be. When we
entered our data into the transportation model we had to give it the costs per unit produced and our
demand and supply values. Then it gave us how many to produce and from which type of production to
which month we needed to produce. Either way we got the same total minimum cost but did it two
different ways.