CHAPTER 3: INTRODUCTION TO LINEAR PROGRAMMING
3.1-1.
Swift & Company solved a series of LP problems to identify an optimal production schedule.
The first in this series is the scheduling model, which generates a shift-level schedule for a
28-day horizon. The objective is to minimize the difference of the total cost and the revenue. The
total cost includes the operating costs and the penalties for shortage and capacity violation. The
constraints include carcass availability, production, inventory and demand balance equations,
and limits on the production and inventory. The second LP problem solved is that of capable-to-
promise models. This is basically the same LP as the first one, but excludes coproduct and
inventory. The third type of LP problem arises from the available-to-promise models. The
objective is to maximize the total available production subject to production and inventory
balance equations.
As a result of this study, the key performance measure, namely the weekly percent-sold position
has increased by 22%. The company can now allocate resources to the production of required
products rather than wasting them. The inventory resulting from this approach is much lower
than what it used to be before. Since the resources are used effectively to satisfy the demand, the
production is sold out. The company does not need to offer discounts as often as before. The
customers order earlier to make sure that they can get what they want by the time they want. This
in turn allows Swift to operate even more efficiently. The temporary storage costs are reduced by
90%. The customers are now more satisfied with Swift. With this study, Swift gained a
considerable competitive advantage. The monetary benefits in the first years was $12.74 million,
including the increase in the profit from optimizing the product mix, the decrease in the cost of
lost sales, in the frequency of discount offers and in the number of lost customers. The main
nonfinancial benefits are the increased reliability and a good reputation in the business.
3.1-2.
(a) (b)
3-1
,(c) (d)
3.1-3.
(a)
(b)
Slope-Intercept Form Slope Intercept
2
Z= 6 x2 = - 2
3 x1 + 2 - 3 2
2
Z = 12 x2 = - 2
3 x1 + 4 - 3 4
2
Z = 18 x2 = - 2
3 x1 + 6 - 3 6
3-2
,3.1-4.
(a) x2 = - 1
2 x1 + 10
(b) The slope is - 1 2, the x2 intercept is 10.
(c)
3.1-5.
Optimal Solution: (x1* , x2* ) = (13,5) and Z * = 31
3-3
, 3.1-6.
Optimal Solution: (x1* , x2* )= (3,9) and Z * = 210
3.1-7.
Optimal Solution: (x1* , x2* )= (2, 4) and Z * = 110
3-4
3.1-1.
Swift & Company solved a series of LP problems to identify an optimal production schedule.
The first in this series is the scheduling model, which generates a shift-level schedule for a
28-day horizon. The objective is to minimize the difference of the total cost and the revenue. The
total cost includes the operating costs and the penalties for shortage and capacity violation. The
constraints include carcass availability, production, inventory and demand balance equations,
and limits on the production and inventory. The second LP problem solved is that of capable-to-
promise models. This is basically the same LP as the first one, but excludes coproduct and
inventory. The third type of LP problem arises from the available-to-promise models. The
objective is to maximize the total available production subject to production and inventory
balance equations.
As a result of this study, the key performance measure, namely the weekly percent-sold position
has increased by 22%. The company can now allocate resources to the production of required
products rather than wasting them. The inventory resulting from this approach is much lower
than what it used to be before. Since the resources are used effectively to satisfy the demand, the
production is sold out. The company does not need to offer discounts as often as before. The
customers order earlier to make sure that they can get what they want by the time they want. This
in turn allows Swift to operate even more efficiently. The temporary storage costs are reduced by
90%. The customers are now more satisfied with Swift. With this study, Swift gained a
considerable competitive advantage. The monetary benefits in the first years was $12.74 million,
including the increase in the profit from optimizing the product mix, the decrease in the cost of
lost sales, in the frequency of discount offers and in the number of lost customers. The main
nonfinancial benefits are the increased reliability and a good reputation in the business.
3.1-2.
(a) (b)
3-1
,(c) (d)
3.1-3.
(a)
(b)
Slope-Intercept Form Slope Intercept
2
Z= 6 x2 = - 2
3 x1 + 2 - 3 2
2
Z = 12 x2 = - 2
3 x1 + 4 - 3 4
2
Z = 18 x2 = - 2
3 x1 + 6 - 3 6
3-2
,3.1-4.
(a) x2 = - 1
2 x1 + 10
(b) The slope is - 1 2, the x2 intercept is 10.
(c)
3.1-5.
Optimal Solution: (x1* , x2* ) = (13,5) and Z * = 31
3-3
, 3.1-6.
Optimal Solution: (x1* , x2* )= (3,9) and Z * = 210
3.1-7.
Optimal Solution: (x1* , x2* )= (2, 4) and Z * = 110
3-4
