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 proï¬t, to inï¬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 proï¬t. This is reasonable if  these costs are not affected by the choice of production quantities, that is, if we wonât 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 X 1 â¤ 100 X 2 â¤ 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. 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 â¤ 2,400 Proceeding analogously for workstations B, C, and D, we can write the other capacity constraints as follows: 15X 1 + 35X 2 â¤ 2,400  workstation B  15X 1 + 5X 2 â¤ 2,400  workstation C  25X 1 + 14X 2 â¤ 2,400  workstation D  We have now completely deï¬ned the following LP model of our optimization problem: Maximize  45X 1 + 60X 2  (16.107)  X 1 â¤ 100  (16.108)  X 2 â¤ 50  (16.109)  15X 1 + 10X 2 â¤ 2,400  (16.110)  15X 1 + 35X 2 â¤ 2,400  (16.111)  15X 1 + 5X 2 â¤ 2,400  (16.112)  25X 1 + 14X 2 â¤ 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)â(16.113). Spreadsheet programs generally require the decision variables to be entered into cells and the constraints speciï¬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 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 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 speciï¬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 satisï¬es 15X 1 + 35X 2 â¤ 2,400  workstation B  15X 1 + 10X 2 â¤ 2,400  workstation A  15X 1 + 5X 2 â¤ 2,400  workstation C  will also satisfy


592  Part III  Principles in Practice  the user speciï¬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 satisï¬es 15X 1 + 35X 2 â¤ 2,400  workstation B  15X 1 + 10X 2 â¤ 2,400  workstation A  15X 1 + 5X 2 â¤ 2,400  workstation C  will also satisfy  because these differ only by having smaller coefï¬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 constraints (16.110) and (16.112) from here on. Figure 16.15 illustrates problem (16.107)â(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 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 â¤ â  X2 â¤ â  X1 +  2,400 = â0.429X 1 + 68.57 35  (16.114)  X1 +  2,400 = â1.786X 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 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 deï¬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 proï¬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  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 â45 X1 + = â0.75X 1 + X2 = 60 60 60  (16.116)  (16.117)


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 â45 X1 + = â0.75X 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 proï¬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, 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 ï¬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 region.11 This observation allows the optimization algorithm to ignore the inï¬nitely many points inside the feasible region and search for a solution among the ï¬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 modern algorithms use different schemes to ï¬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. Coefï¬cients in the objective function. For instance, if we were to change the unit proï¬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 ï¬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 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 coefï¬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  0  50  100  150  X1


594  Part III  Figure 16.17  Principles in Practice  140.00  Effect of changing objective coefï¬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  0  50  100  150  X1  equation (16.117) to   60 Z Z X2 = â X1 + = âX 1 + 60 60 60  (16.118)  so the slope changes from â0.75 to â1; 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 proï¬t for product 1 increases to $60, the proï¬t becomes 60(75.79) + 60(36.09) = $6,712.80 The optimal decision variables remain unchanged until the coefï¬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 function ârocked aroundâ to a new extreme point. Economically, the proï¬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 coefï¬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-time changes. If two or more coefï¬cients are changed, the effect is more difï¬cult to characterize. One has to rerun the model with multiple coefï¬cient changes to get a feel for their effect. 2. Coefï¬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 deï¬ned by the capacity constraint for workstation B changes from equation (16.114) to X2 â¤ â    20 35    X1 +  2,400 = â0.571X 1 + 68.57 35  (16.119)  so the slope changes from â0.429 to â0.571; again, it becomes steeper. In a manner analogous to that described above for coefï¬cients in the objective function, LP packages can determine how much a given coefï¬cient can change before it ceases to deï¬ne the optimal extreme point. However, because changing the coefï¬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 speciï¬ed changes in the constraint coefï¬cients. 3. Right-hand side coefï¬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)â(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


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 be able to determine its effect. We do this differently for two types of constraints: a. Slack constraints are constraints that do not deï¬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 â¤ 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 â¤ 49, it still wonât affect the optimal solution. Indeed, not until we reduce the constraint to X 2 â¤ 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 coefï¬cients. b. Tight constraints are constraints that deï¬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 â¤ 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 deï¬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 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 100 Ã 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 deï¬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 constraint on workstation B from 2,400 to 2,770, the constraint moves to the very edge of the feasible region deï¬ned by 25X 1 + 14X 2 â¤ 2,400 (machine D) and X 2 â¤ 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  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, however, so in general we must resolve the LP to determine the shadow prices beyond the maximum allowable increase or decrease.


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, 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 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 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 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 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 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 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 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


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 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)â(16.67) of the workforce planning problem 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 ï¬rm makes automotive batteries in three grades: heavyduty, standard, and economy. The unit net proï¬ts and maximum daily demand for these products are given in the ï¬rst table below. The ï¬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 Proï¬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 proï¬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 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) workstations. Because Youohimga does not always have sufï¬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 Youohimgaâs 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


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 = ï¬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 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 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 ï¬oor control mechanisms r Vendoring individual 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 100â0 plan be preferable to the 50â50 plan in practice? If so, explain why. 6. Mr. B. OâProblem 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, OâProblem 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 ($)  Brand 1 Brand 2 Brand 3  0.5 0.4 0.6  1.2 1.2 0.8  80 80 80  20,000 22,000 18,000  Product  Maximum Demand (units/week)  Net Unit Proï¬t ($/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 proï¬t (including 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 ï¬xed for any nonzero production level. Carefully deï¬ne 0â1 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 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


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 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 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 ï¬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 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 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 <= 100 8) 1.5 P1 + 0.8 P2 <= 320 END  600  Part III  Principles in Practice  OBJECTIVE FUNCTION VALUE 1)  2427.66700  VARIABLE P1 P2 OT RM A1 A2  VALUE 160.000000 80.000000 .000000 400.000000 11.000000 1.333333  REDUCED COST .000000 .000000 2.133334 .000000 .000000 .000000  ROW 2) 3) 4) 5) 6) 7) 8)  SLACK OR SURPLUS .000000 .000000 .000000 .000000 .000000 87.666660 16.000000  DUAL PRICES .100000 .066667 3.866666 6.000000 4.500000 .000000 .000000  NO. ITERATIONS =  5  RANGES IN WHICH THE BASIS IS UNCHANGED:  VARIABLE P1 P2 OT RM A1 A2  ROW 2 3 4 5 6 7 8  CURRENT COEF 15.000000 8.000000 -6.000000 -1.500000 -1.000000 -1.000000  OBJ COEFFICIENT RANGES ALLOWABLE ALLOWABLE INCREASE DECREASE .966667 .533333 .266667 .483333 2.133334 INFINITY INFINITY 4.500000 1.000000 5.333335 1.000000 7.249999  CURRENT RHS 50.000000 60.000000 160.000000 .000000 400.000000 100.000000 320.000000


DUAL PRICES .100000 .066667 3.866666 6.000000 4.500000 .000000 .000000  NO. ITERATIONS =  5  RANGES IN WHICH THE BASIS IS UNCHANGED:  VARIABLE P1 P2 OT RM A1 A2  ROW 2 3 4 5 6 7 8  CURRENT COEF 15.000000 8.000000 -6.000000 -1.500000 -1.000000 -1.000000  OBJ COEFFICIENT RANGES ALLOWABLE ALLOWABLE INCREASE DECREASE .966667 .533333 .266667 .483333 2.133334 INFINITY INFINITY 4.500000 1.000000 5.333335 1.000000 7.249999  CURRENT RHS 50.000000 60.000000 160.000000 .000000 400.000000 100.000000 320.000000  RIGHT-HAND SIDE RANGES ALLOWABLE ALLOWABLE INCREASE DECREASE 110.000000 876.666600 20.000000 1315.000000 27.500000 2.500000 6.666667 55.000000 6.666667 55.000000 INFINITY 87.666660 INFINITY 16.000000  (a) If overtime 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 proï¬t per week beâor 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 remained unchanged), what would the companyâs proï¬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 produceâor 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 two-tier pricing scheme such that the ï¬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 ï¬rst purchase 200 bf of the $2.00 raw material.) To modify the LP to compute an âoptimalâ production/advertising policy, we deï¬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 ï¬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 reï¬ect the new pricing scheme? ii. If p < 2, what other changes in the previous LP make it properly reï¬ect the new pricing scheme? 8. Consider a production line with four workstations, labeled j = 1, 2, 3, and 4, in tandem (all products ï¬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 proï¬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


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 (dÌit 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  dÌ At d At dÌ Bt d Bt dÌCt 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 proï¬t minus holding and backorder costs 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 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 hour. Unit proï¬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 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 proï¬t (i.e., net sales 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 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 proï¬t, and compare it to the proï¬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-ordering the products according to proï¬tability by a cost accounting scheme? What does this say about the effectiveness of using accounting methods to plan production schedules?