Real-World Applications
3.10 Using Linear Programming to Solve Multiperiod Decision Problems: An Inventory Model
additional cost of $3 and the use of 3 hours of skilled labor, 100 liters of A can be transformed into 40 liters of C and 60 liters of B. Chemical C can either be sold or processed further. It costs $1 and takes 1 hour of skilled labor to process 100 liters of C into 60 liters of D and 40 liters of B. For each chemical the sales price per 100 liters and the maximum amount (in 100s of liters) that can be sold are given in Table 30.
A maximum of 200 labor hours are available. Formulate an LP whose solution will tell Chemco how to maximize their profit.
13 Carrington Oil produces two types of gasoline, gas 1 and gas 2, from two types of crude oil, crude 1 and crude 2. Gas 1 is allowed to contain up to 4% impurities, and gas 2 is allowed to contain up to 3% impurities. Gas 1 sells for
$8 per barrel, whereas gas 2 sells for $12 per barrel. Up to 4,200 barrels of gas 1 and up to 4,300 barrels of gas 2 can be sold. The cost per barrel of each crude, availability, and the level of impurities in each crude are as shown in Table 31. Before blending the crude oil into gas, any amount of each crude can be “purified” for a cost of $0.50 per barrel.
Purification eliminates half the impurities in the crude oil.
Determine how to maximize profit.
14 You have been put in charge of the Melrose oil refinery.
The refinery produces gas and heating oil from crude oil.
Gas sells for $8 per barrel and must have an average “grade level” of at least 9. Heating oil sells for $6 a barrel and must
T A B L E 30
B C D
Price ($) 12 16 26
Maximum demand 30 60 40
have an average grade level of at least 7. At most, 2,000 barrels of gas and 600 barrels of heating oil can be sold.
Incoming crude can be processed by one of three methods.
The per barrel yield and per barrel cost of each processing method are shown in Table 32. For example, if we refine 1 barrel of incoming crude by method 1, it costs us $3.40 and yields .2 barrels of grade 6, .2 barrels of grade 8, and .6 barrels of grade 10.
Before being processed into gas and heating oil, processed grades 6 and 8 may be sent through the catalytic cracker to improve their quality. For $1.30 per barrel, a bar-rel of grade 6 may be “cracked” into a barbar-rel of grade 8. For
$2 per barrel, a barrel of grade 8 may be cracked into a bar-rel of grade 10. Any leftover processed or cracked oil that cannot be used for heating oil or gas must be disposed of at a cost of $0.20 per barrel. Determine how to maximize the refinery’s profit.
T A B L E 31
Cost per Impurity Availability
Oil Barrel ($) Level (%) (Barrels)
Crude 1 6 10% 5,000
Crude 2 8 2% 4,500
T A B L E 32
Method Grade 6 Grade 8 Grade 10 Cost ($)
1 .2 .2 .6 3.40
2 .3 .3 .4 3.00
3 .4 .4 .2 2.60
3.10 Using Linear Programming to Solve Multiperiod
Sailco Corporation must determine how many sailboats should be produced during each of the next four quarters (one quarter three months). The demand during each of the next four quarters is as follows: first quarter, 40 sailboats; second quarter, 60 sailboats;
third quarter, 75 sailboats; fourth quarter, 25 sailboats. Sailco must meet demands on time. At the beginning of the first quarter, Sailco has an inventory of 10 sailboats. At the beginning of each quarter, Sailco must decide how many sailboats should be produced during that quarter. For simplicity, we assume that sailboats manufactured during a quar-ter can be used to meet demand for that quarquar-ter. During each quarquar-ter, Sailco can produce up to 40 sailboats with regular-time labor at a total cost of $400 per sailboat. By having employees work overtime during a quarter, Sailco can produce additional sailboats with overtime labor at a total cost of $450 per sailboat.
At the end of each quarter (after production has occurred and the current quarter’s de-mand has been satisfied), a carrying or holding cost of $20 per sailboat is incurred. Use linear programming to determine a production schedule to minimize the sum of produc-tion and inventory costs during the next four quarters.
Solution For each quarter, Sailco must determine the number of sailboats that should be produced by regular-time and by overtime labor. Thus, we define the following decision variables:
xt number of sailboats produced by regular-time labor (at $400/boat) during quarter t (t 1, 2, 3, 4)
yt number of sailboats produced by overtime labor (at $450/boat) during quarter t (t 1, 2, 3, 4)
It is convenient to define decision variables for the inventory (number of sailboats on hand) at the end of each quarter:
it number of sailboats on hand at end of quarter t (t 1, 2, 3, 4) Sailco’s total cost may be determined from
Total cost cost of producing regular-time boats
cost of producing overtime boats inventory costs
400(x1 x2 x3 x4) 450(y1 y2 y3 y4)
20(i1 i2 i3 i4) Thus, Sailco’s objective function is
min z 400x1 400x2 400x3 400x4 450y1 450y2
450y3 450y4 20i1 20i2 20i3 20i4 (60) Before determining Sailco’s constraints, we make two observations that will aid in for-mulating multiperiod production-scheduling models.
For quarter t,
Inventory at end of quarter t inventory at end of quarter (t 1)
quarter t production quarter t demand This relation plays a key role in formulating almost all multiperiod production-scheduling models. If we let dtbe the demand during period t (thus, d1 40, d2 60, d3 75, and d4 25), our observation may be expressed in the following compact form:
it it1 (xt yt) dt (t 1, 2, 3, 4) (61) Sailco Inventory
E X A M P L E 1 4
In (61), i0 inventory at end of quarter 0 inventory at beginning of quarter 1 10.
For example, if we had 20 sailboats on hand at the end of quarter 2 (i2 20) and pro-duced 65 sailboats during quarter 3 (this means x3 y3 65), what would be our end-ing third-quarter inventory? Simply the number of sailboats on hand at the end of quar-ter 2 plus the sailboats produced during quarquar-ter 3, less quarquar-ter 3’s demand of 75. In this case, i3 20 65 75 10, which agrees with (61). Equation (61) relates decision variables associated with different time periods. In formulating any multiperiod LP model, the hardest step is usually finding the relation (such as (61)) that relates decision variables from different periods.
We also note that quarter t’s demand will be met on time if and only if (sometimes written iff ) it 0. To see this, observe that it1 (xt yt) is available to meet period t’s demand, so that period t’s demand will be met if and only if
it1 (xt yt) dt or it it1 (xt yt) dt 0
This means that the sign restrictions it 0 (t 1, 2, 3, 4) will ensure that each quarter’s demand will be met on time.
We can now determine Sailco’s constraints. First, we use the following four constraints to ensure that each period’s regular-time production will not exceed 40: x1, x2, x3, x4 40. Then we add constraints of the form (61) for each time period (t 1, 2, 3, 4). This yields the following four constraints:
i1 10 x1 y1 40 i2 i1 x2 y2 60 i3 i2 x3 y3 75 i4 i3 x4 y4 25
Adding the sign restrictions xt 0 (to rule out negative production levels) and it 0 (to ensure that each period’s demand is met on time) yields the following formulation:
min z 400x1 400x2 400x3 400x4 450y1 450y2 450y3 450y4
20i1 20i2 20i3 20i4
s.t. x1 40, x2 40, x3 40, x4 40
i1 10 x1 y1 40, i2 i1 x2 y2 60 i3 i2 x3 y3 75, i4 i3 x4 y4 25 it 0, yt 0, and xt 0 (t 1, 2, 3, 4)
The optimal solution to this problem is z 78,450; x1 x2 x3 40; x4 25; y1 0; y2 10; y3 35; y4 0; i1 10; i2 i3 i4 0. Thus, the minimum total cost that Sailco can incur is $78,450. To incur this cost, Sailco should produce 40 sailboats with regular-time labor during quarters 1–3 and 25 sailboats with regular-time labor dur-ing quarter 4. Sailco should also produce 10 sailboats with overtime labor durdur-ing quarter 2 and 35 sailboats with overtime labor during quarter 3. Inventory costs will be incurred only during quarter 1.
Some readers might worry that our formulation allows Sailco to use overtime produc-tion during quarter t even if period t’s regular producproduc-tion is less than 40. True, our for-mulation does not make such a schedule infeasible, but any production plan that had yt 0 and xt 40 could not be optimal. For example, consider the following two production schedules:
Production schedule A x1 x2 x3 40; x4 25;
y2 10; y3 25; y4 0
Production schedule B x1 40; x2 30; x3 30; x4 25;
y2 20; y3 35; y4 0
Schedules A and B both have the same production level during each period. This means that both schedules will have identical inventory costs. Also, both schedules are feasible, but schedule B incurs more overtime costs than schedule A. Thus, in minimizing costs, schedule B (or any schedule having yt 0 and xt 40) would never be chosen.
In reality, an LP such as Example 14 would be implemented by using a rolling hori-zon, which works in the following fashion. After solving Example 14, Sailco would im-plement only the quarter 1 production strategy (produce 40 boats with regular-time labor).
Then the company would observe quarter 1’s actual demand. Suppose quarter 1’s actual demand is 35 boats. Then quarter 2 begins with an inventory of 10 40 35 15 boats.
We now make a forecast for quarter 5 demand (suppose the forecast is 36). Next deter-mine production for quarter 2 by solving an LP in which quarter 2 is the first quarter, quarter 5 is the final quarter, and beginning inventory is 15 boats. Then quarter 2’s pro-duction would be determined by solving the following LP:
min z 400(x2 x3 x4 x5) 450( y2 y3 y4 y5) 20(i2 i3 i4 i5) s.t. x2 40, x3 40, x4 40, x5 40
i2 15 x2 y2 60, i3 i2 x3 y3 75 i4 i3 x4 y4 25, i5 i4 x5 y5 36 it 0, yt 0, and xt 0 (t 2, 3, 4, 5)
Here, x5 quarter 5’s regular-time production, y5 quarter 5’s overtime production, and i5 quarter 5’s ending inventory. The optimal values of x2and y2for this LP are then used to determine quarter 2’s production. Thus, each quarter, an LP (with a planning hori-zon of four quarters) is solved to determine the current quarter’s production. Then current demand is observed, demand is forecasted for the next four quarters, and the process re-peats itself. This technique of “rolling planning horizon” is the method by which most dy-namic or multiperiod LP models are implemented in real-world applications.
Our formulation of the Sailco problem has several other limitations.
1 Production cost may not be a linear function of the quantity produced. This would vi-olate the Proportionality Assumption. We discuss how to deal with this problem in Chap-ters 9 and 13.
2 Future demands may not be known with certainty. In this situation, the Certainty As-sumption is violated.
3 We have required Sailco to meet all demands on time. Often companies can meet de-mands during later periods but are assessed a penalty cost for dede-mands that are not met on time. For example, if demand is not met on time, then customer displeasure may re-sult in a loss of future revenues. If demand can be met during later periods, then we say that demands can be backlogged. Our current LP formulation can be modified to incor-porate backlogging (see Problem 1 of Section 4.12).
4 We have ignored the fact that quarter-to-quarter variations in the quantity produced may result in extra costs (called production-smoothing costs.) For example, if we in-crease production a great deal from one quarter to the next, this will probably require the costly training of new workers. On the other hand, if production is greatly decreased from one quarter to the next, extra costs resulting from laying off workers may be incurred. In Section 4.12, we modify the present model to account for smoothing costs.
5 If any sailboats are left at the end of the last quarter, we have assigned them a value of zero. This is clearly unrealistic. In any inventory model with a finite horizon, the ventory left at the end of the last period should be assigned a salvage value that is in-dicative of the worth of the final period’s inventory. For example, if Sailco feels that each sailboat left at the end of quarter 4 is worth $400, then a term 400i4 (measuring the worth of quarter 4’s inventory) should be added to the objective function.
P R O B L E M S
Group A
1 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.
2 A company faces the following demands during the next three periods: period 1, 20 units; period 2, 10 units; period 3, 15 units. The unit production cost during each period is as follows: period 1—$13; period 2—$14; period 3—$15.
A holding cost of $2 per unit is assessed against each period’s ending inventory. At the beginning of period 1, the company has 5 units on hand.
In reality, not all goods produced during a month can be used to meet the current month’s demand. To model this fact, we assume that only one half of the goods produced during a period can be used to meet the current period’s de-mands. Formulate an LP to minimize the cost of meeting the demand for the next three periods. (Hint: Constraints such as i1 x1 5 20 are certainly needed. Unlike our example, however, the constraint i1 0 will not ensure that period 1’s demand is met. For example, if x1 20, then i1
0 will hold, but because only 12(20) 10 units of period 1 production can be used to meet period 1’s demand, x1 20 would not be feasible. Try to think of a type of constraint that will ensure that what is available to meet each period’s demand is at least as large as that period’s demand.) Group B
3 James Beerd bakes cheesecakes and Black Forest cakes.
During any month, he can bake at most 65 cakes. The costs per cake and the demands for cakes, which must be met on time, are listed in Table 33. It costs 50¢ to hold a cheesecake, and 40¢ to hold a Black Forest cake, in inventory for a month. Formulate an LP to minimize the total cost of meeting the next three months’ demands.
4 A manufacturing company produces two types of products: A and B. The company has agreed to deliver the products on the schedule shown in Table 34. The company has two assembly lines, 1 and 2, with the available production hours shown in Table 35. The production rates for each assembly line and product combination, in terms of
hours per product, are shown in Table 36. It takes 0.15 hour to manufacture 1 unit of product A on line 1, and so on. It costs $5 per hour of line time to produce any product. The inventory carrying cost per month for each product is 20¢
per unit (charged on each month’s ending inventory).
Currently, there are 500 units of A and 750 units of B in inventory. Management would like at least 1,000 units of each product in inventory at the end of April. Formulate an LP to determine the production schedule that minimizes the total cost incurred in meeting demands on time.
5 During the next two months, General Cars must meet (on time) the following demands for trucks and cars: month 1—400 trucks, 800 cars; month 2—300 trucks, 300 cars.
During each month, at most 1,000 vehicles can be produced.
Each truck uses 2 tons of steel, and each car uses 1 ton of steel. During month 1, steel costs $400 per ton; during month 2, steel costs $600 per ton. At most, 1,500 tons of steel may be purchased each month (steel may only be used
T A B L E 33
Month 1 Month 2 Month 3
Item Demand Cost/Cake ($) Demand Cost/Cake ($) Demand Cost/Cake ($)
Cheesecake 40 3.00 30 3.40 20 3.80
Black Forest 20 2.50 30 2.80 10 3.40
T A B L E 34
Date A B
March 31 5,000 2,000
April 30 8,000 4,000
T A B L E 35
Production Hours Available
Month Line 1 Line 2
March 800 2,000
April 400 1,200
T A B L E 36
Production Rate
Product Line 1 Line 2
A 0.15 0.16
B 0.12 0.14
during the month in which it is purchased). At the beginning of month 1, 100 trucks and 200 cars are in inventory. At the end of each month, a holding cost of $150 per vehicle is assessed. Each car gets 20 mpg, and each truck gets 10 mpg. During each month, the vehicles produced by the company must average at least 16 mpg. Formulate an LP to meet the demand and mileage requirements at minimum cost (include steel costs and holding costs).
6 Gandhi Clothing Company produces shirts and pants.
Each shirt requires 2 sq yd of cloth, each pair of pants, 3.
During the next two months, the following demands for shirts and pants must be met (on time): month 1—10 shirts, 15 pairs of pants; month 2—12 shirts, 14 pairs of pants.
During each month, the following resources are available:
month 1—90 sq yd of cloth; month 2—60 sq yd. (Cloth that is available during month 1 may, if unused during month 1, be used during month 2.)
During each month, it costs $4 to make an article of clothing with regular-time labor and $8 with overtime labor.
During each month, a total of at most 25 articles of cloth-ing may be produced with regular-time labor, and an un-limited number of articles of clothing may be produced with overtime labor. At the end of each month, a holding cost of
$3 per article of clothing is assessed. Formulate an LP that can be used to meet demands for the next two months (on time) at minimum cost. Assume that at the beginning of month 1, 1 shirt and 2 pairs of pants are available.
7 Each year, Paynothing Shoes faces demands (which must be met on time) for pairs of shoes as shown in Table 37. Workers work three consecutive quarters and then receive one quarter off. For example, a worker may work during quarters 3 and 4 of one year and quarter 1 of the next year.
During a quarter in which a worker works, he or she can produce up to 50 pairs of shoes. Each worker is paid $500 per quarter. At the end of each quarter, a holding cost of $50 per pair of shoes is assessed. Formulate an LP that can be used to minimize the cost per year (labor holding) of meeting the demands for shoes. To simplify matters, assume
that at the end of each year, the ending inventory is zero.
(Hint: It is allowable to assume that a given worker will get the same quarter off during each year.)
8 A company must meet (on time) the following demands:
quarter 1—30 units; quarter 2—20 units; quarter 3—40 units. Each quarter, up to 27 units can be produced with regular-time labor, at a cost of $40 per unit. During each quarter, an unlimited number of units can be produced with overtime labor, at a cost of $60 per unit. Of all units produced, 20% are unsuitable and cannot be used to meet demand. Also, at the end of each quarter, 10% of all units on hand spoil and cannot be used to meet any future demands. After each quarter’s demand is satisfied and spoilage is accounted for, a cost of $15 per unit is assessed against the quarter’s ending inventory. Formulate an LP that can be used to minimize the total cost of meeting the next three quarters’ demands. Assume that 20 usable units are available at the beginning of quarter 1.
9 Donovan Enterprises produces electric mixers. During the next four quarters, the following demands for mixers must be met on time: quarter 1—4,000; quarter 2—2,000;
quarter 3—3,000; quarter 4—10,000. Each of Donovan’s workers works three quarters of the year and gets one quarter off. Thus, a worker may work during quarters 1, 2, and 4 and get quarter 3 off. Each worker is paid $30,000 per year and (if working) can produce up to 500 mixers during a quarter. At the end of each quarter, Donovan incurs a holding cost of $30 per mixer on each mixer in inventory. Formulate an LP to help Donovan minimize the cost (labor and inventory) of meeting the next year’s demand (on time). At the beginning of quarter 1, 600 mixers are available.
T A B L E 37
Quarter 1 Quarter 2 Quarter 3 Quarter 4
600 300 800 100