ORSA MAC I
FOUO- Sensitive Examination Materials
Math Programming
XXXXXXXXXXExam 2
Math Programming
NAME: ________________
Grade: _______________
ADMINISTRATIVE NOTES:
.
1. SHOW ALL WORK! No credit can be given for inco
ect answers if you do not show your work.
XXXXXXXXXXpts.) Sinclair Plastics operates two chemical plants which produce polyethylene; the Ohio Valley plant which can produce up to 10,000 tons per month and the Lakeview plant which can produce up to 7,000 tons per month. Sinclair sells its polyethylene to three different auto manufacturing plants, Grand Rapids (demand = 3000 tons per month), Blue Ridge (demand = 5000 tons per month), and Sunset (demand = 4000 tons per month). The costs of shipping between the respective plants is shown in the table below:
Grand Rapids
Blue Ridge
Sunset
Ohio Valley
50
40
100
Lakeview
60
50
75
Sinclair Plastics wants to develop a LP model that will enable them to minimize their shipping costs while meeting the auto manufacturing plant demands.
The alge
aic model is shown below.
Let the decision variables (quantities shipped between plants) be:
Grand Rapids (1)
Blue Ridge (2)
Sunset (3)
Ohio Valley (1)
x11
x12
x13
Lakeview (2)
x21
x22
x23
Minimize
50x11 + 40x12 + 100x13 + 60x21 + 50x22 + 75x23
Subject to:
x11 + x12+ x13 ≤ 10000
(Ohio Valley supply)
x21 + x22+ x23 ≤ 7000
(Lakeview supply)
x11 + x21 ≥ 3000
(Grand Rapids demand)
x12 + x22 ≥ 5000
(Blue Ridge demand)
x13 + x23 ≥ 4000
xij ≥ 0
(Sunset demand)
(Non-Negativity)
The Excel solution and sensitivity analysis reports are shown on the next page.
Questions: The questions below are independent of each other. For example, the answer to question c is not affected by the conditions of question b. Each question is to be answered based solely upon the Excel solution output and Sensitivity Analysis reports shown above.
a. (3 pts) Identify the set of basic variables and their values.
. (3 pts) The demand at Grand Rapids increases by 100 tons. What impact does that have on the shipping costs?
c. (3 pts) Management of Sinclair Plastics wants to ship polyethylene from the Ohio Valley plant to Sunset. What improvement to the shipping cost must be made before this change would be worthwhile?
d. (3 pts) The shipping costs from Lakeview to Sunset increases by $5. What impact does that have on the recommended shipping plan and total shipping costs?
e. (3 pts) The demand at Blue Ridge increase from 5000 to 8000 tons. What impact does that have on the shipping plan and total shipping costs?
2. (25 pts.) The country of Orsamania has 15,000,000 acres of publically controlled agricultural land in active use. The government cu
ently is planning a way to divide this land among three basic crops (wheat, corn, and barley) next year. A certain percentage of each crop is exported to obtain badly needed foreign capital (dollars), and the rest of each of these crops is used to feed the populace. Raising these crops also provides employment for a significant proportion of the population. The main factors to be considered in allocating land to these crops are (1) the amount of foreign capital generated, (2) the number of citizens fed, and (3) the number of citizens employed in raising these crops. The table below shows how much each 1,000 acres of each crop contributes toward these factors.
Contribution per 1,000 Acres
Facto
Wheat
Corn
Barley
Foreign Capital
$3,000
$5,000
$4,000
Citizens Fed
150
75
100
Citizens Employed
10
15
12
The government established the following goals and priorities:
Priority 1: Earn at least $70,000,000 in foreign capital.
Priority 2: Feed at least 1,750,000 people.
Priority 3: Employ exactly 200,000 workers.
Formulate the problem as a goal programming problem and identify the three objective functions associated with the three priorities. Ensure you identify your decision variables.
3. (20 points). For today's deliveries, the Disputanta Army Depot has available eight trucks with a 34 ton capacity, fifteen trucks with an 18 ton capacity, and twelve trucks with an 11 ton capacity. The depot must dispatch the trucks to Fort Benning and Fort Rucker. Fort Benning requires at least 100 tons of supplies and Fort Rucker requires at least 150 tons. Each truck can make only one trip during the day.
The dollar cost of sending a truck from the depot to each destination is given below:
DESTINATION
34 ton truck
18 ton truck
11 ton truck
Fort Benning
$120 per trip
$80 per trip
$65 per trip
Fort Rucke
$200 per trip
$130 per trip
$110 per trip
A trade-off analysis has been performed to analyze the trade-off between total tonnage delivered to Ft. Benning and Ft. Rucker and the cost of delivering to the two installations. The math programming formulation of this problem and the trade-off curve and table developed are provided below:
Decision Variables: Let Xij = the number of trucks of type i going to location j where i = 1 for 34 ton trucks, 2 for 18 ton trucks, and 3 for 11 ton trucks and j = 1 for FT Benning and 2 for FT Rucker.
Objectives:
Min total cost = 120X11 + 80X21 + 65X31 + 200X12 + 130X32 + 110X32
Max total tonnage = 34X11 + 18X21 + 11X31 + 34X12 + 18X32 + 11X32
Subject to
X11 + X12 ≤ XXXXXXXXXXton trucks available)
X21 + X22 ≤ XXXXXXXXXXton trucks available)
X31 + X32 ≤ XXXXXXXXXXton trucks available)
34X11 + 18X21 + 11X31 ≥ 100 (FT Benning demand)
34X12 + 18X32 + 11X32 ≥ 150 (FT Rucker demand)
Xij ≥ 0 for i = 1, 2, 3 and j = 1,2
Solver table and Pareto optimal trade-off curve for cost vs. tonnage are shown below.
Change in
Tonnage
Cost
Tonnage
250
$1290
256
275
$1410
290
300
$1475
301
325
$1570
326
350
$1715
355
375
$1810
380
400
$1940
402
425
$2035
427
450
$2130
452
475
$2275
481
500
$2370
506
525
$2500
528
550
$2595
553
575
$2725
575
600
$2920
608
625
$3050
630
650
$3180
652
For the following questions (a through d), assume that the cu
ent planning situation is still valid and will not change. In other words, you only have eight 34 ton trucks, fifteen 18 ton trucks, and twelve 11 ton trucks available. Cost factors will not change and tonnage requirements still must be met. Any suggested options are simply based on a suggestion from outside sources and have not been checked yet to determine if the suggestion is good or even feasible. Use this table and chart as shown. Do not revise it.
a. (5 pts) What is the lowest total cost that will satisfy the solution? For that option, how many total tons will we send to the two forts?
. (5 pts) What is the largest total tonnage that we can send to the two forts? For that option, how much would it cost?
c. (5 pts) Can they ship 400 tons at a cost of $1500? Why or why not?
d. (5 pts) An option has been suggested to send 300 tons at a cost of $3000? Would you recommend that based on the table and the trade-off curve? Justify your answer.
4. (40 pts) The ORSA MAC Widget Company produces two types of widgets used in the Air Force’s new anti-gravity machine. Each widget 1 produced consumes three ounces of silver and two ounces of a base metal. Each widget 2 produced consumes two ounces of silver and four ounces of the base metal. The production line is allocated 28 ounces of silver and 32 ounces of the base metal per day, but these amounts can be exceeded if required. The Air Force requires at least four widget 1 per day and five widget 2 per day. ORSA MAC Widget Company charges the Air Force $40 for each widget 1 and $20 for each widget 2.
ORSA MAC Widgets established two goals:
Priority 1 Goal: To establish a daily production plan that will achieve at least $500 in daily revenue.
Priority 2 Goal: To stay within the silver and base metal allocations if at all possible. These are of equal importance.
Develop the Excel goal programming model and recommend a daily production plan. Identify how many widgets of each type are to be produced, the amounts of silver and base metal consumed, and the revenue.
1
MP Review
Solution
1. A manufacturing firm has discontinued the production of a certain unprofitable product line. This act created considerable excess production capacity. Management is considering devoting this excess capacity to one or more of three products; call them products 1, 2, and 3. The available capacity on the machines that might limit output is summarized in the following table:
Machine Type
Available Time
(in machine hours per week)
Milling Machine
500
Lathe
350
Grinde
150
The number of machine hours required for each unit of the respective products is shown in the following table:
Machine Type
Product 1
Product 2
Product 3
Milling Machine
9
3
5
Lathe
5
4
0
Grinde
3
0
2
The unit profit would be $30, $12, and $15, respectively, on products 1, 2, and 3. The objective is to determine how much of each product the firm should produce to maximize profit.
The alge
aic formulation is shown below.
Let Xi = the amount of product i to produce, where i = 1, 2, 3.
Max Z = 30X1 + 12X2 + 15X3
ST
9X1 + 3X2 + 5X3 ≤ 500 (milling machine availability)
5X1 + 4X2 ≤ 350 (lathe availability)
3X XXXXXXXXXX2X3 ≤ 150 (grinder availability)
Xi ≥ 0, for i = 1, 2, 3
Use the sensitivity report below to answer the following questions:
a. Identify the basic variables and their values.
Product 2 produced = 87.5
Product 3 produced = 47.5
Grinder slack = 55
. Why are the reduced costs for products 2 and 3 equal to 0?
They are basic variables and basic variables have a reduced cost = 0.
c. Why does the grinder constraint have a shadow price of 0?
There are excess grinder machine hours available and non-binding constraints have a shadow price = 0.
d. Describe what happens to the basis and objective function value if the profit