Aggregate Production Plan — LP Formulation for Office Chairs: 23-Ind-A4 Production Management, December 2013, Q4

Question 4: Aggregate Production Plan — LP Formulation for Office Chairs (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Twelve months of forecast demand and the cost/capacity data below; workforce starts at 7 workers (24 chairs/worker/month regular capacity); beginning inventory $I_0=143$, ending inventory $I_{12}=0$, and backorders forced to zero at month 12.

Month $t$	1	2	3	4	5	6	7	8	9	10	11	12
Forecast demand $D_t$	151	150	163	181	112	143	152	85	147	164	211	149

Cost item	Value
Regular time	$\$115$/unit
Overtime	$\$163$/unit
Subcontract	$\$204$/unit
Inventory carrying	$\$26$/unit-month
Back-order	$\$103$/unit-month
Hiring	$\$1523$/worker
Firing	$\$2512$/worker

Find. A linear program — decision variables, objective, and constraints — that yields the minimum-cost aggregate production plan for this data (formulation only, not the numerical solution).

Approach. Model each month's production, workforce, and inventory/backorder position as linked decision variables carried forward from the previous month, then attach capacity caps and the stated boundary conditions.

Decision variables (for each month $t=1,\dots,12$). $P_t$ = regular-time production (chairs); $O_t$ = overtime production; $Sub_t$ = subcontracted units; $W_t$ = workforce size (workers) during month $t$; $H_t,F_t$ = workers hired / fired at the start of month $t$; $I_t$ = ending inventory; $B_t$ = ending backorder. All variables $\ge0$.
Workforce balance. The workforce evolves by net hiring/firing, starting from the 7 workers on hand: $$W_t=W_{t-1}+H_t-F_t\quad(t=1,\dots,12),\qquad W_0=7.$$
Production capacity. Regular output cannot exceed what the current workforce can make (24 units/worker/month); overtime and subcontracting are capped at the stated flat limits: $$P_t\le24\,W_t,\qquad O_t\le16,\qquad Sub_t\le8\qquad(t=1,\dots,12).$$
Inventory / backorder balance. Supply in month $t$ (production plus whatever was on hand or owed from month $t-1$) must cover demand, with the surplus or shortfall carried as inventory or backorder: $$I_{t-1}-B_{t-1}+P_t+O_t+Sub_t-D_t=I_t-B_t\qquad(t=1,\dots,12).$$
Boundary conditions. The stated starting and ending stock levels fix the two ends of the horizon: $$I_0=143,\quad B_0=0,\qquad I_{12}=0,\quad B_{12}=0.$$
Objective. Minimize total cost across all cost-bearing decisions over the 12 months: $$\boxed{\min Z=\sum_{t=1}^{12}\Big(115P_t+163O_t+204Sub_t+26I_t+103B_t+1523H_t+2512F_t\Big)}.$$

Element	Formulation
Variables	$P_t,O_t,Sub_t,W_t,H_t,F_t,I_t,B_t\ge0$ for $t=1,\dots,12$
Objective	$\min\sum_t(115P_t+163O_t+204Sub_t+26I_t+103B_t+1523H_t+2512F_t)$
Workforce balance	$W_t=W_{t-1}+H_t-F_t$, $W_0=7$
Capacity caps	$P_t\le24W_t$; $O_t\le16$; $Sub_t\le8$
Inventory/backorder balance	$I_{t-1}-B_{t-1}+P_t+O_t+Sub_t-D_t=I_t-B_t$
Boundary	$I_0=143$, $B_0=0$, $I_{12}=0$, $B_{12}=0$

Check

The problem statement says the manager is "preparing an aggregate production plan for the next six months" but then supplies twelve months of forecast demand and explicitly imposes the no-backorder condition "at month 12" — an internal inconsistency in the source. The formulation above uses the full $T=12$-month horizon actually given, since that is the only horizon consistent with both the demand table and the month-12 boundary condition; restricting to $t=1,\dots,6$ would leave the month-12 constraint referring to a period outside the model. Separately, note the regular-time capacity with the starting workforce ($24\times7=168$ chairs/month) is below the peak forecast month (211, month 11) — so overtime, subcontracting, and/or hiring are not merely optional refinements but are economically necessary in at least one month, which is what makes the $H_t,F_t,O_t,Sub_t$ variables load-bearing in the optimum, not just formal slack.

Concept & Theory

Topic. Linear-programming formulation of aggregate (multi-month) production planning with a mixed workforce/production/inventory decision structure.

Key relations. Workforce balance $W_t=W_{t-1}+H_t-F_t$; inventory/backorder balance $I_{t-1}-B_{t-1}+(\text{supply}_t)-D_t=I_t-B_t$; capacity constraint linking production to workforce size, $P_t\le(\text{output per worker})\times W_t$.

Why this works. Aggregate planning is fundamentally a multi-period transportation-style balance: every month's "supply" (regular + overtime + subcontract + carried-over inventory) must clear that month's demand, either immediately or via a carried backorder, and every lever that changes supply (hiring, overtime, subcontracting) carries its own unit cost that the objective must weigh against carrying cost. Writing the workforce and inventory as stage variables linked by simple balance equations, rather than trying to express $t$-month decisions independently, is what keeps the model linear and tractable.

Common pitfall. Forgetting that $P_t$ is bounded by the *workforce*, not a fixed constant — because $W_t$ is itself a decision variable (via $H_t,F_t$), the capacity constraint $P_t\le24W_t$ is what makes hiring/firing economically meaningful in the objective; omitting it would let the model produce without ever needing workers.

Source. Nahmias & Olsen, Production and Operations Analysis, Ch. 3 (aggregate planning, linear-programming formulation); Sipper & Bulfin, Production: Planning, Control, and Integration, Ch. 6.

Question 4 of 7: Aggregate Production Plan — LP Formulation for Office Chairs

Question 4: Aggregate Production Plan — LP Formulation for Office Chairs (20 marks)