Extra Credit Assignment Group 6

IE 416 - Parisay /
Extra Credit Assignment Group 6 /
Operations Research I /
Stephen Gonzales, Amandeep Tamber, Ross Nakata, Jonathan Gutierrez /
10/20/2011 /

Table of Contents

Problem Statement

Assumptions

Set-Up

WinQSB Solution

Sensitivity Analysis

Report to Manager

Problem Statement

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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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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 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, 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:

MONTH / COST OF
PRODUCTION / UNIT / DEMAND
1 / $5 / 50
2 / $8 / 65
3 / $4 / 100
4 / $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

wheret=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

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 y = 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–dtt = 1 , 2 , 3 , 4

d1 = 50, d2 = 65, d3 = 100, d4 = 70

We can then define the problem’s constraints:

i1 = 0 + x1 – 50i2= i1 + x2 – 65i3= i2 + x3 – 100i4= i3 + x3 – 70

x1 0 x2 0x3 0x4 0i1 0 i2 0i3 0i4 0

With these twelve constraints, we are now able to use WinQSB to determine the optimal solution.

WinQSB Solution

Data input for WinQSB “Linear and Integer Programming”

WinQSB solution:

Our minimized cost is $1,525.We put these results into a simple table to easily understand theproduction plan the program suggests:

Month / Units to Produce / Production Cost
per Unit / Total
Production Cost / Demand / Units Remaining
at End of Month / Inventory
Cost / 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
4 / 0 / $ 7.00 / $ - / 70 / 0 / $ - / $ -
GRAND TOTAL / $ 1,525.00

Sensitivity Analysis

For our first sensitivity analysis, we chose to do a parametric analysis on the coefficient of month 2 in the objective function. 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.

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 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, and we can calculate the slope from around 2 to 6.99 (Slope = ). 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.

Four our second sensitivity analysis, we chose to parametrically analyze the demand for month 3, which has the highest demand out of the 4 months. We would like to see how a change in this demand would affect our optimal solution.

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. 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 Manager

To: Management

The minimum cost we calculated is $1,525. We summarize our findings in the following table:

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.

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:

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.