and workforce planning functions. 1. No single AP or WP module is right for every situation. As the examples in this chapter show, aggregate and workforce planning can incorporate many different decision problems. A good AP or WP module is one that is tailored to address the speci?c issues faced by the ?rm. 2. Simplicity promotes understanding. Although it is desirable to address different issues in the AP/WP module, it is even more important to keep the model understandable. In general, these


s in the AP/WP module, it is even more important to keep the model understandable. In general, these modules are used to generate candidate production and workforce plans, which will be examined, combined, and altered manually before being published as “The Plan.” To generate a spectrum of plans (and explain them to others), the user must be able to trace changes in the model to changes in the plan. Because of this, it makes sense to start with as simple a formulation as possible. Additional de


lan. Because of this, it makes sense to start with as simple a formulation as possible. Additional detail (e.g., constraints) can be added later. 3. Linear programming is a useful AP/WP tool. The long planning horizon used for aggregate and workforce planning justi?es ignoring many production details; therefore, capacity checks, sales restrictions, and inventory balances can be expressed as linear constraints. As long as we are willing to approximate actual costs with linear functions, an LP so


ar constraints. As long as we are willing to approximate actual costs with linear functions, an LP solver is a very ef?cient method for solving many problems related to the AP and WP modules. Because we are working with speculative long-range data, it generally does not make sense to use anything more sophisticated than LP (e.g., nonlinear or integer programming) in most aggregate and workforce planning situations. 4. Robustness matters more than precision. No matter how accurate the data and h


planning situations. 4. Robustness matters more than precision. No matter how accurate the data and how sophisticated the model, the plan generated by the AP or WP module will never be followed exactly. The actual production sequence will be affected by unforeseen events that could not possibly have been factored into the module. This means that the mark of a good long-range production plan is that it enables us to do a reasonably good job even in the face of such contingencies. To ?nd such a p


hat it enables us to do a reasonably good job even in the face of such contingencies. To ?nd such a plan, the user of the AP module must be able to examine the consequences of various scenarios. This is another reason to keep the model reasonably simple.  Appendix 16A Linear Programming Linear programming is a powerful mathematical tool for solving constrained optimization problems. The name derives from the fact that LP was ?rst applied to ?nd optimal schedules or “programs” of resource alloc


rives from the fact that LP was ?rst applied to ?nd optimal schedules or “programs” of resource allocation. Hence, although LP generally does involve using a computer program, it does not entail programming on the part of the user in the sense of writing code. In this appendix, we provide enough background to give the user of an LP package a basic idea of what the software is doing. Readers interested in more details should consult one of the many good texts on the subject (e.g., Eppen, Gould,


rested in more details should consult one of the many good texts on the subject (e.g., Eppen, Gould, and Schmidt 1988 for an application-oriented overview, Murty 1983 for more technical coverage).  Formulation The ?rst step in using linear programming is to formulate a practical problem in mathematical terms. There are three basic choices we must make to do this: 1. Decision variables are quantities under our control. Typical examples for aggregate planning and workforce planning applications o


ties under our control. Typical examples for aggregate planning and workforce planning applications of LP are production quantities, number of workers to hire, and levels of inventory to hold. 2. Objective function is what we want to maximize or minimize. In most AP/WP applications, this is typically either to maximize pro?t or minimize cost. Beyond simply stating the objective, however, we must specify it in terms of the decision variables we have de?ned. 3. Constraints are restrictions on our


specify it in terms of the decision variables we have de?ned. 3. Constraints are restrictions on our choices of the decision variables. Typical examples for AP/WP applications include capacity constraints, raw materials limitations, restrictions on how fast we can add workers due to limitations on training capacity, and restrictions on physical ?ow (e.g., inventory levels as a direct result of how much we produce/procure and how much we sell). When one is formulating an LP, it is often useful


how much we produce/procure and how much we sell). When one is formulating an LP, it is often useful to try to specify the necessary inputs in the order in which they are listed. However, in realistic problems, one virtually never gets the “right” formulation in a single pass. The example in Section 16.4.2 illustrates some of the changes that may be required as a model evolves. To describe the process of formulating an LP, let us consider the problem presented in Table 16.2. We begin by selecti


rocess of formulating an LP, let us consider the problem presented in Table 16.2. We begin by selecting decision variables. Since there are only two products and because demand and capacity are assumed stationary over time, the only decisions to make concern how much of each product to produce per week. Thus, we let X 1 and X 2 represent the weekly production quantities of products 1 and 2, respectively. Next, we choose to maximize pro?t as our objective function. Since product 1 sells for $90


ectively. Next, we choose to maximize pro?t as our objective function. Since product 1 sells for $90 but costs $45 in raw material, its net pro?t is $45 per unit.9 Similarly, product 2 sells for $100 but costs $40 in raw material, so its net unit pro?t is $60. Thus, weekly pro?t will be 45X 1 + 60X 2 - weekly labor costs - weekly overhead costs But since we assume that labor and overhead costs are not affected by the choice of X 1 and X 2 , we can use the following as our objective function for


re not affected by the choice of X 1 and X 2 , we can use the following as our objective function for the LP model: Maximize  45X 1 + 60X 2  Finally, we need to specify constraints. If we could produce as much of products 1 and 2 as we wanted, we could drive the above objective function, and hence weekly pro?t, to in?nity. This is not possible because of limitations on demand and capacity. 9 Note that we are neglecting labor and overhead costs in our estimates of unit pro?t. This is reasonable


e that we are neglecting labor and overhead costs in our estimates of unit pro?t. This is reasonable if  these costs are not affected by the choice of production quantities, that is, if we won’t change the size of the workforce or the number of machines in the shop.  590  Chapter 16  591  Aggregate and Workforce Planning  The demand constraints are easy. Since we can sell at most 100 units per week of product 1 and 50 units per week of product 2, our decision variables X 1 and X 2 must satisfy


week of product 1 and 50 units per week of product 2, our decision variables X 1 and X 2 must satisfy X 1 = 100 X 2 = 50 The capacity constraints are a little more work. Since there are four machines, which run at most 2,400 minutes per week, we must ensure that our production quantities do not violate this constraint on each machine. Consider workstation A. Each unit of product 1 we produce requires 15 minutes on this workstation, while each unit of product 2 we produce requires 10 minutes. He


uires 15 minutes on this workstation, while each unit of product 2 we produce requires 10 minutes. Hence, the total number of minutes of time required on workstation A to produce X 1 units of product 1 and X 2 units of product 2 is10 15X 1 + 10X 2 so the capacity constraint for workstation A is 15X 1 + 10X 2 = 2,400 Proceeding analogously for workstations B, C, and D, we can write the other capacity constraints as follows: 15X 1 + 35X 2 = 2,400  workstation B  15X 1 + 5X 2 = 2,400  workstation


city constraints as follows: 15X 1 + 35X 2 = 2,400  workstation B  15X 1 + 5X 2 = 2,400  workstation C  25X 1 + 14X 2 = 2,400  workstation D  We have now completely de?ned the following LP model of our optimization problem: Maximize  45X 1 + 60X 2  (16.107)  X 1 = 100  (16.108)  X 2 = 50  (16.109)  15X 1 + 10X 2 = 2,400  (16.110)  15X 1 + 35X 2 = 2,400  (16.111)  15X 1 + 5X 2 = 2,400  (16.112)  25X 1 + 14X 2 = 2,400  (16.113)  Subject to:  Some LP packages allow the user to enter the problem in


25X 1 + 14X 2 = 2,400  (16.113)  Subject to:  Some LP packages allow the user to enter the problem in a form almost identical to that shown in formulation (16.107)–(16.113). Spreadsheet programs generally require the decision variables to be entered into cells and the constraints speci?ed in terms of these cells. More sophisticated LP solvers allow the user to specify blocks of similar constraints in a concise form, which can substantially reduce modeling time for large problems. Finally, with


ts in a concise form, which can substantially reduce modeling time for large problems. Finally, with regard to formulation, we point out that we have not stated explicitly the constraints that X 1 and X 2 be non-negative. Of course, they must be, since negative production quantities make no sense. In many LP packages, decision variables are assumed to be non-negative unless 10 Note that this constraint does not address such detailed considerations as setup times that depend on the sequence of p


straint does not address such detailed considerations as setup times that depend on the sequence of products run on workstation A or whether full utilization of workstation A is possible given the WIP in the system. But as we discussed in Chapter 13, these issues are addressed at a lower level in the production planning and control hierarchy (e.g., in the sequencing and scheduling module).  592  Part III  Principles in Practice  the user speci?es otherwise. In other packages, the user must inc


Part III  Principles in Practice  the user speci?es otherwise. In other packages, the user must include the non-negativity constraints explicitly. This is something to beware of when using LP software.  Solution To get a general idea of how an LP package works, let us consider the above formulation from a mathematical perspective. First, note that any pair of X 1 and X 2 that satis?es 15X 1 + 35X 2 = 2,400  workstation B  15X 1 + 10X 2 = 2,400  workstation A  15X 1 + 5X 2 = 2,400  workstation


35X 2 = 2,400  workstation B  15X 1 + 10X 2 = 2,400  workstation A  15X 1 + 5X 2 = 2,400  workstation C  will also satisfy  because these differ only by having smaller coef?cients for X 2 . This means that the constraints for workstations A and C are redundant. Leaving them out will not affect the solution. In general, it does not hurt anything to have redundant constraints in an LP formulation. But to make our graphical illustration of how LP works as clear as possible, we will omit constraint


But to make our graphical illustration of how LP works as clear as possible, we will omit constraints (16.110) and (16.112) from here on. Figure 16.15 illustrates problem (16.107)–(16.113) in graphical form, where X 1 is plotted on the horizontal axis and X 2 is plotted on the vertical axis. The shaded area is the feasible region, consisting of all the pairs of X 1 and X 2 that satisfy the constraints. For instance, the demand constraints (16.108) and (16.109) simply state that X 1 cannot be l


traints. For instance, the demand constraints (16.108) and (16.109) simply state that X 1 cannot be larger than 100, and X 2 cannot be larger than 50. The capacity constraints are graphed by noting that, with a bit of algebra, we can write constraints (16.111) and (16.113) as X2 = -  X2 = -  X1 +  2,400 = -0.429X 1 + 68.57 35  (16.114)  X1 +  2,400 = -1.786X 1 + 171.43 14  (16.115)   15  35   25  14  If we replace the inequalities with equality signs in equations (16.114) and (16.115), then


25  14  If we replace the inequalities with equality signs in equations (16.114) and (16.115), then these are simply equations of straight lines. Figure 16.15 plots these lines. The set of X 1 and X 2 points that satisfy these constraints includes all the points lying below both of these lines. The points marked by the shaded area are those satisfying all the demand, capacity, and non-negativity constraints. This type of feasible region de?ned by linear constraints is known as a polyhedron. N


ty constraints. This type of feasible region de?ned by linear constraints is known as a polyhedron. Now that we have characterized the feasible region, we turn to the objective. Let Z represent the value of the objective (i.e., net pro?t achieved by producing quantities X 1 and X 2 ). From objective  Figure 16.15  25X1 + 14X2 = 2,400  140.00  Feasible region for LP example.  X1 = 100  120.00  15X1 + 35X2 = 2,400  X2  100.00 80.00 60.00  X2 = 50  40.00 Feasible region  20.00 0.00  0  50  100 X1


X1 + 35X2 = 2,400  X2  100.00 80.00 60.00  X2 = 50  40.00 Feasible region  20.00 0.00  0  50  100 X1  150  Chapter 16  Figure 16.16  593  Aggregate and Workforce Planning  140.00  Solution to LP example.  120.00  Z = 5,557.94  X2  100.00 Z = 7,000  80.00  Optimal solution (75.79, 36.09)  60.00 40.00 Feasible region  20.00 0.00  0  Z = 3,000 50  100  150  X1  (16.107), X 1 and X 2 are related to Z by 45X 1 + 60X 2 = Z We can write this in the usual form for a straight line as   Z Z -45 X1 + =


Z by 45X 1 + 60X 2 = Z We can write this in the usual form for a straight line as   Z Z -45 X1 + = -0.75X 1 + X2 = 60 60 60  (16.116)  (16.117)  Figure 16.16 illustrates equation (16.117) for Z = 3,000, 5,557.94, and 7,000. Notice that for Z = 3,000, the line passes through the feasible region, leaving some points above it. Hence, we can feasibly increase pro?t (that is, Z ). For Z = 7,000 the line lies entirely above the feasible region. Hence, Z = 7,000 is not feasible. For Z = 5,557.94, t


e line lies entirely above the feasible region. Hence, Z = 7,000 is not feasible. For Z = 5,557.94, the objective function just touches the feasible region at a single point, the point (X 1 = 75.79, X 2 = 36.09). This is the optimal solution. Values of Z above 5,557.74 are infeasible, values below it are suboptimal. The optimal product mix, therefore, is to produce 75.79 (or 75, rounded to an integer value) units of product 1 and 36.09 (rounded to 36) units of product 2. We can think of ?nding


teger value) units of product 1 and 36.09 (rounded to 36) units of product 2. We can think of ?nding the solution to an LP by steadily increasing the objective value (Z ), moving the objective function up and to the right, until it is just about to leave the feasible region. Because the feasible region is a polyhedron whose sides are made up of linear constraints, the last point of contact between the objective function and the feasible region will be a corner, or extreme point, of the feasible


en the objective function and the feasible region will be a corner, or extreme point, of the feasible region.11 This observation allows the optimization algorithm to ignore the in?nitely many points inside the feasible region and search for a solution among the ?nite set of extreme points. The simplex algorithm, developed in the 1940s and still widely used, works in just this way, proceeding around the outside of the polyhedron, trying extreme points until an optimal one is found. Other, more m


und the outside of the polyhedron, trying extreme points until an optimal one is found. Other, more modern algorithms use different schemes to ?nd the optimal point, but will still converge to an extreme-point solution.  Sensitivity Analysis The fact that the optimal solution to an LP lies at an extreme point enables us to perform useful sensitivity analysis on the optimal solution. The principal sensitivity information available to us falls into the following three categories. 1. Coef?cients i


l sensitivity information available to us falls into the following three categories. 1. Coef?cients in the objective function. For instance, if we were to change the unit pro?t for product 1 from $45 to $60, then the equation for the objective function would change from 11 Actually, it is possible that the optimal objective function lies right along a ?at spot connecting two extreme points of the polyhedron. When this occurs, there are many pairs of X 1 and X 2 that attain the optimal value of


e polyhedron. When this occurs, there are many pairs of X 1 and X 2 that attain the optimal value of Z , and the solution is called degenerate. Even in this case, however, an extreme point (actually, at least two extreme points) will be among the optimal solutions.  594  Part III  Figure 16.17  Principles in Practice  140.00  Effect of changing objective coef?cients in LP example.  120.00 45X1 + 60X2 = 5,557.94  X2  100.00  60X1 + 60X2 = 6,712.80  80.00 60.00 40.00 Feasible region  20.00 0.00


+ 60X2 = 5,557.94  X2  100.00  60X1 + 60X2 = 6,712.80  80.00 60.00 40.00 Feasible region  20.00 0.00  0  50  100  150  X1  equation (16.117) to   60 Z Z X2 = - X1 + = -X 1 + 60 60 60  (16.118)  so the slope changes from -0.75 to -1; that is, it gets steeper. Figure 16.17 illustrates the effect. Under this change, the optimal solution remains (X 1 = 75.79, X 2 = 36.09). Note, however that while the decision variables remain the same, the objective function does not. When the unit pro?t for pro


the decision variables remain the same, the objective function does not. When the unit pro?t for product 1 increases to $60, the pro?t becomes 60(75.79) + 60(36.09) = $6,712.80 The optimal decision variables remain unchanged until the coef?cient of X 1 in the objective function reaches 107.14. When this happens, the slope becomes so steep that the point where the objective function just touches the feasible region moves to the extreme point (X 1 = 96, X 2 = 0). Geometrically, the objective fun


the feasible region moves to the extreme point (X 1 = 96, X 2 = 0). Geometrically, the objective function “rocked around” to a new extreme point. Economically, the pro?t from product 1 reached a point where it became optimal to produce all product 1 and no product 2. In general, LP packages will report a range for each coef?cient in the objective function for which the optimal solution (in terms of the decision variables) remains unchanged. Note that these ranges are valid only for one-at-a-ti


s of the decision variables) remains unchanged. Note that these ranges are valid only for one-at-a-time changes. If two or more coef?cients are changed, the effect is more dif?cult to characterize. One has to rerun the model with multiple coef?cient changes to get a feel for their effect. 2. Coef?cients in the constraints. If the number of minutes required on workstation B by product 1 is changed from 15 to 20, then the equation de?ned by the capacity constraint for workstation B changes from e


d from 15 to 20, then the equation de?ned by the capacity constraint for workstation B changes from equation (16.114) to X2 = -    20 35    X1 +  2,400 = -0.571X 1 + 68.57 35  (16.119)  so the slope changes from -0.429 to -0.571; again, it becomes steeper. In a manner analogous to that described above for coef?cients in the objective function, LP packages can determine how much a given coef?cient can change before it ceases to de?ne the optimal extreme point. However, because changing the coe


ent can change before it ceases to de?ne the optimal extreme point. However, because changing the coef?cients in the constraints moves the extreme points themselves, the optimal decision variables will also change. For this reason, most LP packages do not report this sensitivity data, but rather make use of this product as part of a parametric programming option to quickly generate new solutions for speci?ed changes in the constraint coef?cients. 3. Right-hand side coef?cients. Probably the mos


for speci?ed changes in the constraint coef?cients. 3. Right-hand side coef?cients. Probably the most useful sensitivity information provided by LP models is for the right-hand side variables in the constraints. For instance, in formulation (16.107)–(16.113), if we run 100 minutes of overtime per week on machine B, then its right-hand  Chapter 16  595  Aggregate and Workforce Planning  side will increase from 2,400 to 2,500. Since this is something we might want to consider, we would like to


ll increase from 2,400 to 2,500. Since this is something we might want to consider, we would like to be able to determine its effect. We do this differently for two types of constraints: a. Slack constraints are constraints that do not de?ne the optimal extreme point. The capacity constraints for workstations A and C are slack, since we determined right at the outset that they could not affect the solution. The constraint X 2 = 50 is also slack, as can be seen in Figures 16.15 and 16.16, althou


he solution. The constraint X 2 = 50 is also slack, as can be seen in Figures 16.15 and 16.16, although we did not know this until we solved the problem. Small changes in slack constraints do not change the optimal decision variables or objective value at all. If we change the demand constraint on product 2 to X 2 = 49, it still won’t affect the optimal solution. Indeed, not until we reduce the constraint to X 2 = 36.09 will it have any effect. Likewise, increasing the right-hand side of this c


constraint to X 2 = 36.09 will it have any effect. Likewise, increasing the right-hand side of this constraint (above 50) will not affect the solution. Thus, for a slack constraint, the LP package tells us how far we can vary the right-hand side without changing the solution. These are referred to as the allowable increase and allowable decrease of the right-hand side coef?cients. b. Tight constraints are constraints that de?ne the optimal extreme point. Changing them changes the extreme point,


raints are constraints that de?ne the optimal extreme point. Changing them changes the extreme point, and hence the optimal solution. For instance, the constraint that the number of hours per week on workstation B not exceed 2,400, that is, 15X 1 + 35X 2 = 2,400 is a tight constraint in Figures 16.15 and 16.16. If we increase or decrease the right-hand side, the optimal solution will change. However, if the changes are small enough, then the optimal extreme point will still be de?ned by the sam


ever, if the changes are small enough, then the optimal extreme point will still be de?ned by the same constraints (i.e., the time on workstations B and D). Because of this, we are able to compute the following: Shadow prices are the amount by which the objective increases per unit increase in the right-hand side of a constraint. Since slack constraints do not affect the optimal solution, changing their right-hand sides has no effect, and hence their shadow prices are always zero. Tight constra


ng their right-hand sides has no effect, and hence their shadow prices are always zero. Tight constraints, however, generally have nonzero shadow prices. For instance, the shadow price for the constraint on workstation B is 1.31. (Any LP solver will automatically compute this value.) This means that the objective will increase by $1.31 for every extra minute per week on the workstation. So if we can work 2,500 minutes per week on workstation B, instead of 2,400, the objective will increase by 1


can work 2,500 minutes per week on workstation B, instead of 2,400, the objective will increase by 100 × 1.31 = $131. Maximum allowable increase/decrease gives the range over which the shadow prices are valid. If we change a right-hand side by more than the maximum allowable increase or decrease, then the set of constraints that de?ne the optimal extreme point may change, and hence the shadow price may also change. For example, as Figure 16.18 shows, if we increase the right-hand side of the c


rice may also change. For example, as Figure 16.18 shows, if we increase the right-hand side of the constraint on workstation B from 2,400 to 2,770, the constraint moves to the very edge of the feasible region de?ned by 25X 1 + 14X 2 = 2,400 (machine D) and X 2 = 50. Any further increases in the right-hand side will cause this constraint to become slack. Hence, the shadow price is $1.31 up to a maximum allowable  Figure 16.18  25X1 + 14X2 = 2,400  140.00  X1 = 100  15X1 + 35X2 = 2,770  120.00


maximum allowable  Figure 16.18  25X1 + 14X2 = 2,400  140.00  X1 = 100  15X1 + 35X2 = 2,770  120.00  15X1 + 35X2 = 2,400  100.00 X2  Feasible region when RHS of constraint of workstation B is increased to 2,770.  80.00 60.00  X2 = 50  40.00 Feasible region  20.00 0.00  0  50  100 X1  150  596  Part III  Principles in Practice  increase of 370 (that is, 2,770 - 2,400). In this example, the shadow price is zero for changes above the maximum allowable increase. This is not always the case, howeve


w price is zero for changes above the maximum allowable increase. This is not always the case, however, so in general we must resolve the LP to determine the shadow prices beyond the maximum allowable increase or decrease.  Study Questions 1. Although the technology for solving aggregate planning models (linear programming) is well established and AP modules are widely available in commercial systems (e.g., MRP II systems), aggregate planning does not occupy a central place in the planning func


stems (e.g., MRP II systems), aggregate planning does not occupy a central place in the planning function of many firms. Why do you think this is true? What difficulties in modeling, interpreting, and implementing AP models might be contributing to this?  2. Why does it make sense to consider workforce planning and aggregate planning simultaneously in many situations?  3. What is the difference between a chase production plan and a level production plan, with respect to the amount of inventory


between a chase production plan and a level production plan, with respect to the amount of inventory carried and the fluctuation in output quantity over time? How do the production plans generated by an LP model relate to these two types of plan?  4. In a basic LP formulation of the product mix aggregate planning problem, what information is provided by the following? (a) The optimal decision variables. (b) The optimal objective function. (c) Identification of which constraints are tight and wh


riables. (b) The optimal objective function. (c) Identification of which constraints are tight and which are slack. (d) Shadow prices for the right-hand sides of the constraints.  Problems 1. Suppose a plant can supplement its capacity by subcontracting part of or all the production of certain parts. (a) Show how to modify LP (16.28)-(16.32) to include this option, where we define  Vit  =  kit  =  units of product i received from a subcontractor in period t premium paid for subcontracting produ


=  units of product i received from a subcontractor in period t premium paid for subcontracting product i in period t (i.e., cost above variable cost of making it in-house)  ?it  =  minimum amount of product i that must be purchased in period t  =  maximum amount of product i that can be purchased in period t  (e.g., specified as part of long-term contract with supplier)  Vit  (e.g., due to capacity constraints on supplier, as specified in long-term contract) (b) How would you modify the formul


city constraints on supplier, as specified in long-term contract) (b) How would you modify the formulation in part (a) if the contract with a supplier stipulated only that total purchases of product i over the time horizon must be at least ? ? (c) How would you modify the formulation in part (a) if the supplier contract, instead of specifying ? and ii, stipulated that the firm specify a base amount of product i, to be purchased every month, and that the maximum purchase in a given month can exc


unt of product i, to be purchased every month, and that the maximum purchase in a given month can exceed the base amount by no more than 20 percent? (d) What role might models like those in parts (a) to (c) play in the process of negotiating contracts with suppliers?  2. Show how to modify LP (16.49)-(16.54) to represent the case where overtime on all the workstations must be scheduled simultaneously (i.e., if one resource runs overtime, all resources run overtime). Describe how you would handl


ously (i.e., if one resource runs overtime, all resources run overtime). Describe how you would handle the case where, in general, different workstations can have different amounts of overtime, but two workstations, say A and B, must always be scheduled for overtime together.  Chapter 16  597  Aggregate and Workforce Planning  3. Show how to modify LP (16.61)–(16.67) of the workforce planning problem to accommodate multiple products. 4. You have just been made corporate vice president in charg


roblem to accommodate multiple products. 4. You have just been made corporate vice president in charge of manufacturing for an automotive components company and are directly in charge of assigning products to plants. Among many other products, the ?rm makes automotive batteries in three grades: heavyduty, standard, and economy. The unit net pro?ts and maximum daily demand for these products are given in the ?rst table below. The ?rm has three locations where the batteries can be produced. The m


given in the ?rst table below. The ?rm has three locations where the batteries can be produced. The maximum assembly capacities, for any mix of battery grades, are given in the second table below. The number of batteries that can be produced at a location is limited by the amount of suitably formulated lead the location can produce. The lead requirements for each grade of battery and the maximum lead production for each location are also given in the following tables.  Product  Unit Pro?t ($/ba


lead production for each location are also given in the following tables.  Product  Unit Pro?t ($/battery)  Maximum Demand (batteries/day)  Lead Requirements (lbs/battery)  Heavy-duty Standard Economy  12 10 7  700 900 450  21 17 14  Plant Location  Assembly Capacity (batteries/day)  Maximum Lead Production (lbs/day)  1 2 3  550 750 225  10,000 7,000 4,200  (a) Formulate a linear program that allocates production of the three grades among the three locations in a manner that maximizes pro?t. (


llocates production of the three grades among the three locations in a manner that maximizes pro?t. (b) Suppose company policy requires that the fraction of capacity (units scheduled/ assembly capacity) be the same at all locations. Show how to modify your LP to incorporate this constraint. (c) Suppose company policy dictates that at least 50 percent of the batteries produced must be heavy-duty. Show how to modify your LP to incorporate this constraint. 5. Youohimga, Inc., makes a variety of co


Show how to modify your LP to incorporate this constraint. 5. Youohimga, Inc., makes a variety of computer storage devices, which can be divided into two main families that we call A and B. All devices in family A have the same routing and similar processing requirements at each workstation; similarly for family B. There are a total of 10 machines used to produce the two families, where the routings for A and B have some workstations in common (i.e., shared) but also contain unique (unshared)


tings for A and B have some workstations in common (i.e., shared) but also contain unique (unshared) workstations. Because Youohimga does not always have suf?cient capacity to meet demand, especially during the peak demand period (i.e., the months near the start of the school year in September), in the past it has contracted out production of some of its products to vendors (i.e., the vendors manufacture devices that are shipped out under Youohimga’s label). This year, Youohimga has decided to


nufacture devices that are shipped out under Youohimga’s label). This year, Youohimga has decided to use a systematic aggregate planning process to determine vendoring needs and a long-term production plan. (a) Using the following notation X it = units of family i (i = A, B) produced in month t (t = 1, . . . , 24) and available to meet demand in month t  598  Part III  Principles in Practice  Vit = units of family i purchased from vendor in month t and available to meet demand in month t Iit =


it = units of family i purchased from vendor in month t and available to meet demand in month t Iit = ?nished goods inventory of family i at end of month t dit = units of family i demanded (and shipped) during month t c jt = hours available on work center j( j = 1, . . . , 10) in month t ai j = hours required at work center j per unit of family i v i = premium (i.e., extra cost) per unit of family i that is vendored instead of being produced in-house h i = holding cost to carry one unit of fami


ly i that is vendored instead of being produced in-house h i = holding cost to carry one unit of family i in inventory from one month to the next formulate a linear program that minimizes the cost (holding plus vendoring premium) over a two-year (24-month) planning horizon of meeting monthly demand (i.e., no backorders are permitted). You may assume that vendor capacity for both families is unlimited and that there is no inventory of either family on hand at the beginning of the planning horizo


mited and that there is no inventory of either family on hand at the beginning of the planning horizon. (b) Which of the following factors might make sense to examine in the aggregate planning model to help formulate a sensible vendoring strategy? r Altering machine capacities r Sequencing and scheduling r Varying size of workforce r Alternate shop ?oor control mechanisms r Vendoring individual operations rather than complete products r All the above (c) Suppose you run the model in part (a) an


operations rather than complete products r All the above (c) Suppose you run the model in part (a) and it suggests vendoring 50 percent of the total demand for family A and 50 percent of the demand for B. Vendoring 100 percent of A and 0 percent of B is capacity-feasible, but results in a higher cost in the model. Could the 100–0 plan be preferable to the 50–50 plan in practice? If so, explain why. 6. Mr. B. O’Problem of Rancid Industries must decide on a production strategy for two top-secret


hy. 6. Mr. B. O’Problem of Rancid Industries must decide on a production strategy for two top-secret products, which for security reasons we will call A and B. The questions concern (1) whether to produce these products at all and (2) how much of each to produce. Both products can be produced on a single machine, and there are three brands of machine that can be leased for this purpose. However, because of availability problems, Rancid can lease at most one of each brand of machine. Thus, O’Pro


because of availability problems, Rancid can lease at most one of each brand of machine. Thus, O’Problem must also decide which, if any, of the machines to lease. The relevant machine and product data are given below:  Machine  Hours to Produce One Unit of A  Hours to Produce One Unit of B  Weekly Capacity (hours)  Weekly Lease + Operating Cost ($)  Brand 1 Brand 2 Brand 3  0.5 0.4 0.6  1.2 1.2 0.8  80 80 80  20,000 22,000 18,000  Product  Maximum Demand (units/week)  Net Unit Pro?t ($/unit)


0.8  80 80 80  20,000 22,000 18,000  Product  Maximum Demand (units/week)  Net Unit Pro?t ($/unit)  A B  200 100  150 225  Chapter 16  599  Aggregate and Workforce Planning  (a) Letting X i j represent the number of units of product i produced per week on machine j (for example, X A1 is the number of units of A produced on the brand 1 machine), formulate an LP to maximize weekly pro?t (including leasing cost) subject to the capacity and demand constraints. (Hint: Observe that the leasing/oper


ng leasing cost) subject to the capacity and demand constraints. (Hint: Observe that the leasing/operating cost for a particular machine is only incurred if that machine is used and that this cost is ?xed for any nonzero production level. Carefully de?ne 0–1 integer variables to represent the all-or-nothing aspects of this decision.) (b) Suppose that the suppliers of brand 1 machines and brand 2 machines are feuding and will not service the same company. Show how to modify your formulation to e


machines are feuding and will not service the same company. Show how to modify your formulation to ensure that Rancid leases either brand 1 or brand 2 or neither, but not both. 7. All-Balsa, Inc., produces two models of bookcases, for which the relevant data are summarized as follows:  Selling price Labor required Bottleneck machine time required Raw material required  Bookcase 1  Bookcase 2  $15 0.75 hour/unit 1.5 hours/unit 2 bf/unit  $8 0.5 hour/unit 0.8 hour/unit 1 bf/unit  P1 = units of b


15 0.75 hour/unit 1.5 hours/unit 2 bf/unit  $8 0.5 hour/unit 0.8 hour/unit 1 bf/unit  P1 = units of bookcase 1 produced per week P2 = units of bookcase 2 produced per week OT = hours of overtime used per week RM = board-feet of raw material purchased per week A1 = dollars per week spent on advertising bookcase 1 A2 = dollars per week spent on advertising bookcase 2 Each week, up to 400 board feet (bf ) of raw material is available at a cost of $1.50/bf. The company employs four workers, who wor


t (bf ) of raw material is available at a cost of $1.50/bf. The company employs four workers, who work 40 hours per week for a total regular-time labor supply of 160 hours per week. They work regardless of production volumes, so their salaries are treated as a ?xed cost. Workers can be asked to work overtime and are paid $6 per hour for overtime work. There are 320 hours per week available on the bottleneck machine. In the absence of advertising, 50 units per week of bookcase 1 and 60 units per


e bottleneck machine. In the absence of advertising, 50 units per week of bookcase 1 and 60 units per week of bookcase 2 will be demanded. Advertising can be used to stimulate demand for each product. Experience shows that each dollar spent on advertising bookcase 1 increases demand for bookcase 1 by 10 units, while each dollar spent on advertising bookcase 2 increases demand for bookcase 2 by 15 units. At most, $100 per week can be spent on advertising. An LP formulation and solution of the pr


5 units. At most, $100 per week can be spent on advertising. An LP formulation and solution of the problem to determine how much of each product to produce each week, how much raw material to buy, how much overtime to use, and how much advertising to buy are given below. Answer the following on the basis of this output. MAX 15 P1 + 8 P2 - 6 OT - 1.5 RM - A1 - A2 SUBJECT TO 2) P1 - 10 A1 <= 50 3) P2 - 15 A2 <= 60 4) 0.75 P1 + 0.5 P2 - OT <= 160 5) 2 P1 + P2 - RM <= 0 6) RM <= 400 7) A1 + A2 <= 1


time costs only $4 per hour (and all other parameters remain unchanged), how much overtime should All-Balsa use? (b) If each unit of bookcase 1 sold for $15.50 (and all other parameters are unchanged), what will the optimal pro?t per week be—or can you not tell without resolving the LP? (c) What is the most All-Balsa should be willing to pay for another unit of raw material? (d) If each worker were required (as part of the regular workweek) to work 45 hours per week (and all other parameters re


ere required (as part of the regular workweek) to work 45 hours per week (and all other parameters remained unchanged), what would the company’s pro?t be? (e) If each unit of bookcase 2 sold for $10 (and all other parameters remained unchanged), what would be the optimal quantity of bookcase 2 to produce—or can you not tell without resolving the LP? (f) Reconsider the All-Balsa problem formulation and suppose that instead of having 400 bf of raw material available at $1.50/bf, All-Balsa faces a


on and suppose that instead of having 400 bf of raw material available at $1.50/bf, All-Balsa faces a two-tier pricing scheme such that the ?rst 200 bf/week costs $2.00/bf, but any amount above 200 bf/week up to a  Chapter 16  601  Aggregate and Workforce Planning  limit of an additional 300 bf/week costs $ p/bf. (Note: p is a constant, not a variable, and we cannot purchase the $ p/bf raw material unless we ?rst purchase 200 bf of the $2.00 raw material.) To modify the LP to compute an “optim


rial unless we ?rst purchase 200 bf of the $2.00 raw material.) To modify the LP to compute an “optimal” production/advertising policy, we de?ne RM1 = bf of raw material purchased at $2.00/bf RM2 = bf of raw material purchased at $ p/bf To formulate an appropriate LP to represent this new pricing scheme, we ?rst replace 1.5RM in the objective function by 2RM1 + pRM2. i. If p > 2, what other changes in the previous LP make it properly re?ect the new pricing scheme? ii. If p < 2, what other chang


ges in the previous LP make it properly re?ect the new pricing scheme? ii. If p < 2, what other changes in the previous LP make it properly re?ect the new pricing scheme? 8. Consider a production line with four workstations, labeled j = 1, 2, 3, and 4, in tandem (all products ?ow through all four machines in order). Three different products, labeled i = A, B, and C, are produced on the line. The hours required on each workstation for each product and the net pro?t per unit sold (ri ) are given


hours required on each workstation for each product and the net pro?t per unit sold (ri ) are given as follows:  j i  1  2  3  4  ri  A B C  2.4 2.0 0.9  1.1 2.2 0.9  0.8 1.2 1.0  3.0 2.1 2.5  $50 $65 $70  The number of hours available (c jt ) and the upper and lower limits on demand (d¯it and d it ) for each product over the next four quarters are as follows:  t  1  2  3  4  c1t c2t c3t c4t  640 640 1,920 1,280  640 640 1,920 1,280  1,280 640 1,920 1,280  1,280 640 1,920 2,560  d¯ At d At d¯


40 640 1,920 1,280  640 640 1,920 1,280  1,280 640 1,920 1,280  1,280 640 1,920 2,560  d¯ At d At d¯ Bt d Bt d¯Ct d Ct  100 0 100 20 300 0  50 0 100 20 250 0  50 0 100 20 250 0  75 0 100 25 400 50  (a) Suppose we use a quarterly holding cost of $5 and a quarterly backorder cost of $10 per item on all products and allow backordering. Formulate an LP to maximize pro?t minus holding and backorder costs subject to the constraints on workstation capacity and minimum/maximum sales. (b) Using the LP s


osts subject to the constraints on workstation capacity and minimum/maximum sales. (b) Using the LP solver of your choice, solve your formulation in part (a). Which constraints are binding in your solution? (c) Suppose that there is an inspect operation immediately after station 2 (which has plenty of capacity and therefore does not need to be modeled as an extra resource) and 20  602  Part III  Principles in Practice  percent of the parts (regardless of product type) are recycled back through


Principles in Practice  percent of the parts (regardless of product type) are recycled back through stations 1 and 2. Show how to modify your formulation in part a to model this. 9. A manufacturer of high-voltage switches projects demand (in units) for the upcoming year to be as follows.  Jan Feb Mar Apr May Jun  1,000 1,000 1,000 2,000 2,400 2,500  Jul Aug Sep Oct Nov Dec  3,200 2,000 1,000 900 800 800  The plant runs 160 hours per month and produces at an average rate of 10 switches per hou


00 800 800  The plant runs 160 hours per month and produces at an average rate of 10 switches per hour. Unit pro?t per switch sold is $50, and the estimated cost to hold a switch in inventory for 1 month is $5. There is no inventory at the start of the year. Overtime can be used at a cost of $300 per hour. (a) Compute the inventory-holding and overtime cost of a chase production strategy (i.e., producing the amount demanded in each month). (b) Compute the inventory holding and overtime cost of


producing the amount demanded in each month). (b) Compute the inventory holding and overtime cost of a level production strategy (i.e., producing the same amount each month). If the monthly production quantity is set equal to average monthly demand, how much inventory will be left at the end of the year? (c) Compute a production strategy by solving a linear program to maximize pro?t (i.e., net sales revenue minus inventory carrying cost minus overtime cost). Is the amount of overtime in the pla


ales revenue minus inventory carrying cost minus overtime cost). Is the amount of overtime in the plan reasonable? If not, what changes to the LP model could be made to generate a more reasonable solution? (d) How does the solution change if the inventory carrying cost is reduced to $3 per unit per month? If overtime costs are reduced to $200 per hour? Given that these costs are approximate, what do these results imply about the production plan? 10. Reconsider Problem 2 of Chapter 6 in which a


t do these results imply about the production plan? 10. Reconsider Problem 2 of Chapter 6 in which a manufacturer produced three models of vacuum cleaner on a three-station production line. (a) Use linear programming to compute a monthly production plan that maximizes monthly pro?t, and compare it to the pro?t resulting from the current plan given in Chapter 6 and those suggested by the labor hours and ABA cost accounting calculations. (b) Could this LP solution have been arrived at by rank-ord


urs and ABA cost accounting calculations. (b) Could this LP solution have been arrived at by rank-ordering the products according to pro?tability by a cost accounting scheme? What does this say about the effectiveness of using accounting methods to plan production schedules?