Problems for Presentation Problem 1: The Philbrick Company has two plants on opposite sides of the United States. Each of these plants produces the same two products and then sells them to wholesalers within its half of the country. The orders from wholesalers have already been received for the next 2 months (February and March), where the number of units requested are shown below. (The company is not obligated to completely fill these orders but will do so if it can without decreasing its profits.)
Each plant has 20 production days available in February and 23 production days available in March to produce and ship these products. Inventories are depleted at the end of January, but each plant has enough inventory capacity to hold 1,000 units total of the two products if an excess amount is produced in February for sale in March. In either plant, the cost of holding inventory in this way is $3 per unit of product 1 and $4 per unit of product 2. Each plant has the same two production processes, each of which can be used to produce either of the two products. The production cost per unit produced of each product is shown below for each process in each plant.
The production rate for each product (number of units produced per day devoted to that product) also is given for each process in each plant below.
The net sales revenue (selling price minus normal shipping costs) the company receives when a plant sells the products to its own customers (the wholesalers in its half of the country) is $83 per unit of product 1 and $112 per unit of product 2. However, it also is possible (and occasionally desirable) for a plant to make a shipment to the other half of the country to help fill the sales of the other plant. When this happens, an extra shipping cost of $9 per unit of product 1 and $7 per unit of product 2 is incurred. Management now needs to determine how much of each product should be produced by each production process in each plant during each month, as well as how much each plant should sell of each product in each month and how much each plant should ship of each product in each month to the other plant’s customers. The objective is to determine which feasible plan would maximize the total profit (total net sales revenue minus the sum of the production costs, inventory costs, and extra shipping costs). Formulate a complete linear programming model in algebraic form that shows the individual constraints and decision variables for this problem. Solve the problem using Excel Solver. Problem 2: Case 3.3 (Hillier and Lieberman)
Problem 3: Case 3.4 (Hillier and Lieberman)
Problem 4: Case 3.5 (Hillier and Lieberman)
Problem 5: Case 4.2 (Hillier and Lieberman)
Problem 6: The Hi-V Company manufactures and cans three orange extracts: juice concentrate, regular juice, and jam. The products, which are intended for commercial use, are manufactured in 5 – gallon cans. Jam uses Grade I oranges, and the remaining two products use Grade II. Table 1 lists the usages of oranges as well as next year’s demand. A market survey shows that the demand for regular juice is at least twice as high as that for the concentrate.
In the past, Hi-V bought Grade I and Grade II oranges separately at the respective prices of 25 cents and 20 cents per pound. This year, an unexpected frost forced growers to harvest and sell the crop early without sorting them into Grade I and Grade II. It is estimated that 30% of the 3,000,000-Ib crop falls into Grade I and only 60% into Grade II. For this reason, the crop is being offered at the uniform discount price of 19 cents per pound. Hi-V estimates that it will cost the company about 2.15 cents per pound to sort the
oranges into Grade I and Grade II. The below-standard oranges (10% of the crop) will be discarded.
For the purpose of cost allocation, the accounting department uses the following argument to estimate the cost per pound of Grade I and Grade II oranges. Because 10% of the purchased crop will fall below the Grade II standard, the effective average cost per (19 2.15) 23.5 cents. Given that the ratio of Grade I to pound can be computed as .9 Grade II in the purchased lot is 1 to 2, the corresponding average cost per pound based on (20 2 25 1) 21.67 cents. Thus the increase in the average price the old prices is 3 (=23.5 cents – 21.67 cents = 1.83 cents) should be reallocated to the two grades by a 1:2 1 ratio, yielding a Grade I cost per pound of 20 +1.83 ( ) =21.22 cents and a Grade II cost 3 2 of 25 + 1.83 ( ) =25.61 cents. Using this information, the accounting department 3 compiles the profitability sheet for the three in Table 2. Establish a production plan for the Hi-V Company. Table 1 Product
Orange grade
Jam Concentrate Juice
I II II
Pounds of oranges per 5-gal can 5 30 15
Maximum demand (cans) 10,000 12,000 40,000
Table 2
Sales price Variable costs Allocated fixed overhead Total cost Net profit
Product (5-gal can) Jam Concentrate $15.50 $30.25 9.85 21.05 1.05 2.15 $10.90 $23.20 4.60 7.05
Juice $20.75 13.28 1.96 $15.24 5.51
Problem 7: The Elk Hills oil field has a majority ownership (80%) by the U.S. Federal Government. The Department of Energy (DOE) is authorized by law to sell the government’s share of the oil produced to the highest qualified bidders. At the same time, the law limits the quantity of oil delivered to any one bidder. The oil field has six delivery points with different production capacities (bbl/day). The amounts of daily production (in bbl/day) at each of the delivery points are presented daily as line items, and a bidder may submit bids on any number of line items. DOE collects the bids and evaluates them, starting with line item 1 and terminating with line item 6, awarding delivery to the highest bidder but taking into account the caps set by law on the quantity of oil any one bidder can receive. To be specific, Table 1 provides a summary of bonus prices bid on a certain day. A bonus is an increment over the highest price offered for similar grade oil produced in the delivery point area. No bidder can receive more than 20% of the total daily production of 180,000 bbl from all delivery points. TABLE 1
Line item
1
2
3
1 2 3 4 5 6
1.10 1.05 1.00 1.30 1.09 .89
.99 1.02 .95 1.25 1.12 .87
1.20 1.12 .97 1.31 1.15 .90
Bonus in $/bbl bid by bidder 4 5 6 7
1.10 1.08 .94 1.27 1.07 .86
.95 1.09 .93 1.28 1.08 .85
1.00 1.06 1.01 1.26 1.11 .91
1.05 1.11 1.02 1.32 1.05 .88
8
1.02 1.07 .98 1.32 1.10 .91
Production (1000 bbl/day) 20 30 25 40 35 30
DOE uses a ranking scheme for awarding the bids. Starting with line item 1, bidder 3 has the highest bid (bonus = $1.20) and hence is awarded the maximum amount allowed by both line item 1 production and the 20% limit imposed by law (=.2 x 180,000 = 36,000 bbl). From the data in the table, all line 1 item production (20,000 bbl) is allocated to bidder 3. Moving to line item 2, bidder 3 again offers the highest bonus but can only be awarded a maximum of 16,000 bbl because of the 20% limit. The remaining quantity is assigned to the bidder with the next-best bonus (=$1.11), thus allocating 14,000 bbl (=30,000-16,000) to bidder 7. The process is repeated until line item 6 is awarded. Does the proposed scheme guarantee maximum daily revenue for the government? Can the government do better by changing the 20% limit either up or down?
Problem 8: ABC Cola operates a plant in the northern section of the island nation of Tawanda. The plant produces soft drinks in three types of packages that include returnable glass bottles, aluminum cans, and nonreturnable plastic bottles. Returnable (empty) bottles are shipped to the distribution warehouses for reuse in the plant. Because of the continued growth in demand, ABC wants to build another plant. The demand for the soft drinks (expressed in cases) over the next 5 years is given in Table 1. The planned production capacities for the existing plant extrapolated over the same 5-year horizon are given in Table 2. The company owns six distribution warehouses: N1 and N2 are located in the north, C1 and C2 in the central section, and S1 and S2 in the south. The share of sales by each warehouse within its zone is given in Table 3. Approximately 60% of the sales occur in the north, 15% in the central section, and 25% in the south.
The company wants to construct the new plant either in the central section or in the south. The transportation cost per case of returnable bottles is given in Table 4. It is estimated that the transportation costs per case of cans and per case of non-returnables are, respectively, 60% and 70% of that of the returnable bottles. Should the new plant be located in the central or the southern section of the country?
Table 1 Year Package 1 Returnables 2400 Cans 1750 Nonreturnables 490
2
3
2450 2000 550
2600 2300 600
4 2800 2650 650
5 3100 3050 720
Table 2 Year Package 1 Returnables 1800 Cans 1250 Nonreturnables 350
2 1400 1350 380
3 1900 1400 400
4 2050 1500 400
5 2150 1800 450
Table 3 Warehouse N1 N2 C1 C2 S1 S2
Share percentage 85 15 60 40 80 20
Table 4
Warehouse N1 N2 C1 C2 S1 S2
Transportation cost per case ($) Existing plant Central plant 0.80 1.30 1.20 1.90 1.50 1.05 1.60 0.80 1.90 1.50 2.10 1.70
South plant 1.90 2.90 1.20 1.60 0.90 0.80
Problem 9: A steel company operates a foundry and two mills. The foundry casts three types of steel rolls that are machined in its machine stop before being shipped to the mills. Machined rolls are used by the mills to manufacture various products. At the beginning of each quarter, the mills prepare their monthly needs of rolls and submit them to the foundry. The foundry manager then draws a production plan that is essentially constrained by the machining capacity of the shop. Shortages are covered by direct purchase at a premium price from outside sources. A comparison between the cost per roll when acquired from the foundry and its outside purchase price is given in Table1. However, management points out that such shortage is not frequent and can be estimated to occur about 5% of the time. The processing times on the four different machines in the machine shop are given in Table 2. The demand for rolls by the two mills over the next 3 months is given in Table 3. Devise a production schedule for the machine shop
TABLE 1 Roll type price
Weight(lb)
1 2 3
800 1200 1650
Internal cost
External purchase
($ per roll)
($ per roll)
90 130 180
108 145 194
TABLE 2 Processing time per roll Roll 1 Roll 2 Roll 3
Number of machines
Available
hour per machine per month Machine type 1 2 3 4
1 0 6 3
5 4 3 6
7 6 0 9
10 8 9 5
320 310 300 310
TABLE 3 Demand in rolls Mill 1
Mill 2
Month Roll 1 1 2 3
500 0 100
Roll 2 Roll 3
Roll 1
200 300 0
200 300 0
400 500 300
Roll 2
Roll 3
100 200 400
0 200 200
Problem 10: The Construction of Brisbane International Airport requires the pipeline movement of about 1,355,000 m³ of sand dredged from five clusters at a nearby bay to nine sites at the airport location. The sand is used to help stabilize the swampy grounds at the proposed construction area. Some of the sites to which the sand is moved are dedicated to building roads both within and on the perimeter of the airport. Excess sand from a site will be moved by trucks to other outlying areas around the airport, where a perimeter road will be built. The distances (in 100 m) between the source clusters and the sites are summarized in Table 1. The table also shows the supply and demand quantities in 100 m³ at the different locations. (a) The project management has estimated a [volume (m³) × distance (100 m)] sand movement of 2,495,000 units at the cost of $ 0.65 per unit. Is the estimate given by the project management for sand movement on target?
TABLE 1 1
2
3
4
5
6
7
8
9
1 2 3 4 5
22 20 16 20 22
26 28 20 22 26
12 14 26 26 10
10 12 20 22 4
18 20 1.5 6 16
18 20 28 ∞ ∞
11 13 6 2 24
8.5 10 22 21 14
20 22 18 18 21
Demand
62
217
444
315
50
7
20
90
150
Supply 960 201 71 24 99
(b) The project management has realized that sand movement to certain sites cannot be carried out until some of the roads are built. In particular, the perimeter road (destination 9) must be built before movement to certain sites can be done. In Table 2, the blocked routes that require the completion of the perimeter road are marked with x. In view of these restrictions, how should the sand movement be made?
TABLE 2 1 2 3 4 5
1 x x
2 x x
3
4
5 x x
x x x
x
6
7
8
x x x
x
9
Problem 11: Ten years ago, a wholesale dealer started a business distributing pharmaceuticals from a central warehouse (CW). Orders were delivered to customers by vans. The warehouse has since been expanded in response to growing demand. Additionally, two new warehouses (W1 and W2) have been constructed. The central warehouse, traditionally well stocked, occasionally supplies the new warehouses with short items. The occasional supply of short items has grown into a large scale operation in which the two new warehouses receive for redistribution about one-third of their stock directly from the central warehouse. Table 1 gives the number of orders shipped out by each of the three warehouses to customer locations C1 to C6. A customer location is a town with several pharmacies. The dealer’s delivery schedule has evolved over the years to its present status. In essence, the schedule was devised in a rather decentralized fashion, with each warehouse determining its delivery zone based on “self-fulfilling” criteria. Indeed, in some instances, warehouses managers competed for new customers mainly to increase their “sphere of influence.” For instance, the manager of the central warehouse boast that their delivery zone includes not only regular customers but the other two warehouses as well. It is not unusual, then, that several warehouses deliver supplies to different pharmacies within the same town (customer location). The distances in miles traveled by vans between locations are given in Table 2. A vanload usually hauls 100 orders. Evaluate the present distribution policy of the dealer.
TABLE 1 Route From
To
CW CW CW CW CW W1 W1 W1 W1 W2 W2 W2
W1 W2 C1 C2 C3 C1 C3 C4 C5 C2 C5 C6
Number of orders 2000 1500 4800 3000 1200 1000 1100 1500 1800 1900 600 2200
TABLE 2 CW W1 W2 C1 C2 C3 C4 C5 C6
CW 0 5 45 50 30 30 60 75 80
W1 5 0 80 38 70 30 8 10 60
W2 45 80 0 85 35 60 55 7 90
C1 50 38 85 0 20 40 25 30 70
C2 30 70 35 20 0 40 90 15 10
C3 30 30 60 40 40 0 10 6 90
C4 60 8 55 25 90 10 0 80 40
C5 75 10 7 30 15 6 80 0 15
C6 80 60 90 70 10 90 40 15 0
Problem 12: Walsh’s Juice Company produces three products from unprocessed grape juice― bottled juice, frozen juice concentrate, and jelly. It purchases grape juice from three vineyards near Great Lakes. The grapes are harvested at the vineyards and immediately converted in to a juice at plants at the vineyards and immediately converted into juice at plants at the plants in Virginia, Michigan, Tennessee, and Indiana, where it is processed into bottled grape juice, frozen juice concentrate, and jelly. Vineyard output typically differs each month in the harvesting season, and the plants have different processing capacities. In a particular month the vineyard in New York has 1,400 tons of unprocessed grape juice available, whereas the vineyard into Ohio has 1,700 tons and the vineyard in Pennsylvania has 1,100 tons. The processing capacity per month is 1,200 tons of unprocessed juice at the plant in Virginia, 1,100 tons of juice at the plant in Indiana. The cost per ton of transporting unprocessed juice from the vineyards to the plant is as follows:
Plant Vineyard
Virginia
Michigan
Tennessee
New York Pennsylvania Ohio
$850 970 900
$720 790 830
$910 1,050 780
Indiana $750 880 820
The plants are different ages, have different equipment, and have different wage rates; thus, the cost of processing each product each product at each plant ($/ton) differs, as follows:
Plant Product Juice Concentrate Jelly
Virginia $2,100 4,100 2,600
Michigan $2,350 4,300 2,300
Tennessee $2,200 3,950 2,500
Indiana $1,900 3,900 2,800
This month the company needs to process a total of 1,200 tons of bottled juice, 900 tons of frozen concentrate, and 700 tons of jelly at the four plants combined. However, the production process for frozen concentrate results in some juice dehydration, and the process for jelly includes a cooking stage that evaporates water content. To process 1 ton of frozen concentrate requires 2 tons of unprocessed juice; 1 ton of jelly requires 1.5 tons of unprocessed juice; and 1 ton of bottled juice requires 1 ton of unprocessed juice. Walsh’s management wants to determine how many tons of grape juice to ship from each of the vineyards to each of the plants and the number of tons of each product to process at each plant. Thus, management needs a model that includes both the logistical aspects of this problem and the production processing aspects. It wants a solution that will minimize total costs, including the cost of transporting grape juice from the vineyards to plants and the product processing costs. Help Walsh’ Solve this problem by formulating a linear programming model and solve it by using the computer.
Problem 13: The Spring family has owned and operated a garden tool and implements manufacturing company since 1952. The company sells garden tools to distributors and also directly to hardware stores and home improvement discount chains. The Spring Company’s four most popular small garden tools are a trowel, a hoe, a rake, and a shovel. Each of these tools is made from durable steel and has a wooden handle. The Spring family prides itself on its high-quality tools The manufacturing process encompasses two stages. The first stage includes two operations―stamping out the metal tool heads and drilling screw holes in them. The completed tool heads then flow to the second stage, which includes an assembly operation where the handles
are attached to the tool heads, a finishing step, and packaging. The processing times per tool for each operation are provided in the following table:
Tool(hr./unit) Operation Stamping Drilling Assembly Finishing Packaging
Trowel 0.04 0.05 0.06 0.05 0.03
Hoe 0.17 0.14 0.13 0.21 0.15
Rake 0.06 ― 0.05 0.02 0.04
Shovel 0.12 0.14 0.10 0.10 0.15
Total Hours Available per Month 500 400 600 550 500
The Steel the company uses is ordered from an iron and steel works in Japan. The company has 10,000 square feet of sheet steel available each month. The metal required for tool and the monthly contracted production volume per tool are provided in the following table:
Trowel Hoe Rake Shovel
Sheet Metal (ft.²) 1.2 1.6 2.1 2.4
Monthly Contracted Sales 1,800 1,400 1,600 1,800
The primary reasons the company has survived and prospered are its ability to meet customer demand on time and its high quality. As a result, the Spring Company will produce on an overtime basis in order to meet its sales requirements, and it also has a long-standing arrangement with a local tool and die company to manufacture its tool heads. The Spring Company feels comfortable subcontracting the first-stage operations because it is easier to detect defects prior to assembly and finishing. For the same reason, the company will not subcontract for the entire tool because defects would be particularly hard to detect after the tool because defects would be particularly hard to detect after the tool is finished and packaged. However, the company does have 100 hours of overtime available each month for each operation in both stages. The regular production and overtime costs per tool for both stages are provided in the following table:
Stage 1
Trowel Hoe Rake Shovel
Regular Cost $6.20 10.00 8.00 10.00
Stage 2 Overtime Cost $6.20 10.70 8.50 10.70
Regular Cost $3.00 5.00 4.00 5.00
Overtime Cost $3.10 5.40 4.30 5.40
The cost of subcontracting in stage 1 adds 20% to the regular production cost. The Spring Company wants to establish a production schedule for regular and overtime production in each stage and for the number of tool heads subcontracted, at the minimum cost. Formulate a linear programming model for this problem and solve the model using the computer. Which resources appear to be most critical in the production process?
Problem 14: International textile Company, Ltd., is a Hong Kong- based firm that distributes textiles worldwide. The company is owned by the Lao family. Should the People’s Republic of China continue its economic renaissance, the company hopes to use its current base to expand operations to the mainland. International Textile has mills in the Bahamas, Hong Kong, Korea, Nigeria, and Venezuela, each weaving fabrics out of two or more raw fibers: cotton, polyester, and/or silk. The mills service eight company distribution centers located near the customer’s geographical centers of activity. Because transportation costs historically have been less than 10% of total expenses, management has paid little attention to extracting savings through judicious routing of shipments. Ching Lao is returning from the United States, where he has just completed his bachelor’s degree in marketing. He believes that each year he can save International Textile hundreds of thousands of dollars―perhaps millions―just by better routing of fabrics from mills to distribution centers. One glaring example of poor routing is the current assignment of fabric output to the Mexico City distribution center from Nigeria instead of from Venezuela, less than a third the distance. Similarly, the Manila center now gets most of its textiles from Nigeria and Venezuela, although the mills in Hong Kong itself are much closer. Of course, the cost of shipping a bolt of cloth does not depend on distance alone. Table 1 provides the actual costs supplied to Lao from company headquarters. Distribution center demands are seasonal, so a new shipment plan must be made each month. Table 2 provides the fabric
requirements for the month of March. International Textile’s mills have varying capacities for producing the various types of cloth. Table 3 provides the quantities that apply during March. Lao wants to schedule production and shipments in such a way that the most costly customers are shorted when there is insufficient capacity, and the least efficient plants operate at less than full capacity when demand falls below maximum production capacity.
Table 1
Shipping cost Data(Dollars per Bolt) Distribution Center
Mill Los Angles Chicago Bahamas 2 2 Hong Kong 6 7 Korea 5 6 Nigeria 14 12 Venezuela 4 3
London 3 8 8 6 5
Mexico City Manila 3 7 10 2 11 4 9 11 1 9
Rome 4 9 9 7 6
Tokyo 7 4 1 5 11
New York 1 8 7 10 4
Rome 100 700 200
Tokyo 200 900 700
New York 700 2,500 200
Table 2 Fabric Demands for March (Bolts) Distribution Center Fabric Los Angles Chicago Cotton 500 800 Polyester 1,000 2,000 Silk 100 100
Table 3
London 900 3,000 200
Mexico City Manila 900 800 1,500 400 50 400
March Production Capacities (Bolts) Production Capacity
Mill Bahamas Hong Kong Korea Nigeria Venezuela
Cotton 1,000 2,000 1,000 2,000 1,000
Polyester 3,000 2,500 3,500 0 2,000
Silk 0 1,000 500 0 0
1) Find the optional March shipment schedule and its total transportation cost for each of the following: a. Cotton b. Polyester c. silk 2) The company will be opening a silk-making department in the Nigeria mill. Although it will not be completed for several months, a current capacity of 1,000 bolts for that fabric might be used during March for an added one- time cost of $2,000. Find the new optional shipment and the total cost for that fabric. Should the Nigeria mill process silk in March?
